# Follow-up photometry in another band benefits reducing _Kepler_ ’s false
positive rates

Mu-Tian Wang School of Astronomy and Space Science, Nanjing University,
Nanjing 210023, China.  Key Laboratory of Modern Astronomy and Astrophysics,
Ministry of Education, Nanjing, 210023, People’s Republic of China  Hui-Gen
Liu School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China.  Key Laboratory of Modern Astronomy and Astrophysics, Ministry of
Education, Nanjing, 210023, People’s Republic of China  Jiapeng Zhu School of
Astronomy and Space Science, Nanjing University, Nanjing 210023, China.  Key
Laboratory of Modern Astronomy and Astrophysics, Ministry of Education,
Nanjing, 210023, People’s Republic of China  Ji-Lin Zhou School of Astronomy
and Space Science, Nanjing University, Nanjing 210023, China.  Key Laboratory
of Modern Astronomy and Astrophysics, Ministry of Education, Nanjing, 210023,
People’s Republic of China

(Received XXX; Revised XXX; Accepted XXX)

###### Abstract

_Kepler_ Mission’s single-band photometry suffers from astrophysical false
positives, the most common of background eclipsing binaries (BEBs) and
companion transiting planets (CTPs). Multi-color photometry can reveal the
color-dependent depth feature of false positives and thus exclude them. In
this work, we aim to estimate the fraction of false positives that are unable
to be classified by _Kepler_ alone but can be identified with their color-
dependent depth feature if a reference band ($z$, $K_{s}$ and TESS) were
adopted in follow-up observation. We build up physics-based blend models to
simulate multi-band signals of false positives. Nearly 65-95% of the BEBs and
more than 80% of the CTPs that host a Jupiter-size planet will show detectable
depth variations if the reference band can achieve a _Kepler_ -like precision.
$K_{s}$ band is most effective in eliminating BEBs exhibiting any depth sizes,
while $z$ and TESS band prefer to identify giant candidates and their
identification rates are more sensitive to photometric precision. Provided the
radius distribution of planets transiting the secondary star in binary
systems, we derive formalism to calculate the overall identification rate for
CTPs. By comparing the likelihood distribution of the double-band depth ratio
for BEB and planet models, we calculate the false positive probability (FPP)
for typical _Kepler_ candidates. Additionally, we show that the FPP
calculation helps distinguish the planet candidate’s host star in an
unresolved binary system. The analysis framework of this paper can be easily
adapted to predict the multi-color photometry yield for other transit surveys,
especially for TESS.

Eclipsing binary stars (444), Multi-color photometry (1077), Bayesian
statistics (1900), Transits (1711)

††journal: AJ††software: Astropy (Astropy Collaboration et al., 2018),
Matplotlib (Hunter, 2007), Pandas (pandas development team, 2020), isochrones,
(Morton, 2015), scipy (McKinney, 2010)

##  1 Introduction

_Kepler_ (Borucki et al., 2010) is a space mission dedicated to search for
periodic dimming in a star’s brightness as evidence of planetary transits.
However, non-planetary astrophysical objects can also produce transit-like
light curves (Brown, 2003). These objects are called false positives (FPs).
One common source of _Kepler_ FPs is a pair of background eclipsing binary
(BEB) co-aligning with the _Kepler_ target star, in which the eclipse is
diluted by the foreground star’s flux and thus appear as the transit of a
planet-class object. BEBs are a major source of FPs at early _Kepler_ data
release, nearly accounting for 40% of the Kepler Objects of Interests (KOIs)
discovered in lower galactic latitude region (Batalha et al., 2010, 2013;
Borucki et al., 2011; Bryson et al., 2010). Besides, BEBs can easily mimic the
transits of earth-sized candidates due to the large brightness difference
between foreground and background components (Fressin et al., 2013). Another
frequent source of FPs is planets transiting in binary systems. The planet
radius inferred from the contaminated light curves will be miscalculated in
absent knowledge about the stellar companion. Nearly 30% of planet candidates’
host stars have nearby stellar companions within 4 arcseconds (Law et al.,
2014; Ziegler et al., 2016; Furlan et al., 2017; Ziegler et al., 2018), which
is the one pixel-scale of _Kepler_. If the planet is transiting the brighter
target star, the flux contamination by the companion is at most 50%. However,
suppose the planet is transiting the much fainter companion star, termed
companion transiting planets (CTPs). In that case, the dilution could be much
higher than 50%, and the planet’s radius interpreted from diluted transit
depths could be severely underestimated, leading to wrong planet
classifications and affect occurrence rate study (Borucki et al., 2011;
Ziegler et al., 2016; Wang et al., 2018; Teske et al., 2018). For example, the
occurrence rate of Earth-size planets could be overestimated by tens of
percents if not accounting for the binarity of KOI host star (Bouma et al.,
2018). Given the ubiquity of these two kinds of FPs, separating FPs from
genuine (and correctly classified) planets is crucial for transit survey like
_Kepler_ , whose main objects are Earth-size planets in habitable zones.

One way to separate these FPs from genuine planets is to measure the mass of
transiting body through radial velocity (RV) measurements. However, RV
observations are primarily based on several premium ground-based facilities
that are of high demand and are best targeted for planet candidates with high
mass from bright host stars, which are not typical for the ensemble of
_Kepler_ candidates (Borucki et al., 2011; Batalha et al., 2010). Moreover,
directly conducting RV observations to identify BEBs and CTPs could be
problematic in that (1) the spectrum of background eclipsing binary can be
concealed by the contamination of foreground star and (2) the RV residuals of
stellar companions cannot be easily corrected without knowing the binarity of
system. Thus, the conventional RV measurements are not the optimal way to
comprehensively vet _Kepler_ candidates.

Instead of directly measuring the planet’s mass, _statistical validation_ is
proposed to validate the transit candidates. _Statistical validation_ aims to
argue that the transit signal is more likely to be created by planet transits
than any other foreseeable FP scenarios. The prototype of this procedure,
BLENDER, was first embodied and employed by _Kepler_ team (Torres et al.,
2005, 2011). More recently, Morton (2012), and Díaz et al. (2014) develop more
efficient tools to evaluate the candidates’ false positive probability (FPP)
under Bayesian framework. Most notably, the VESPA has successfully validated
over a thousand of _Kepler_ candidates (Morton et al., 2016). A commonality of
these methods is that any available ancillary information of the candidate
(e.g. contrast curve, RVs, multi-color transit photometry, etc.) can be
utilized to further constrain and clear the stellar parameter space of FPs
that fail to reproduce the observed light curves. When the FPP is below the
defined threshold, the candidate can be safely validated.

Using multi-color photometry to vet transit candidates is based on the fact
that blends containing stars with different colors will project color-
dependent depth signature on different observing bandpasses (Rosenblatt, 1971;
Tingley, 2004). As the color-difference between different components
increases, the difference between mutil-band transit depths will increase,
too. While the apparent depths of the non-emitting planet transiting single
star will remain constant in different bandpasses (neglecting the depth-
changing effect caused by stellar limb darkening). Based on the chromatic
depth of blending objects, O’Donovan et al. (2006) first utilized the multi-
color photometry to identify and rule out blending eclipsing binaries in the
wide-field transit survey. Later, several _Kepler_ follow-up observations
eliminated KOIs that displayed chromatic transit depths by ground-based high-
precision photometry (Colón & Ford, 2011; Colón et al., 2012; Tingley et al.,
2014; Colón et al., 2015) and by parallel _Spitzer_ observations (Torres et
al., 2011; Fressin et al., 2012; Désert et al., 2015). All these successful
practices have demonstrated that multi-color photometry is effective in
revealing the blends, thereby sparing resources of follow-up observation for
other candidates with more scientific interest.

In this paper, we study how many false positives can be identified by
combining an additional reference photometry to _Kepler_ bandpass in a
_Kepler_ -like survey. This double-band photometry observation strategy
applies to both simultaneous observation and a follow-up photometry campaign,
assuming the reference photometry will accumulate time baseline comparable to
_Kepler_ ’s ($\sim$ 4 year).

We consider three reference photometric bands: SDSS $z$
111http://classic.sdss.org/dr7/instruments/imager/index.html, 2MASS $K_{s}$
(Cohen et al., 2003), and TESS (Ricker et al., 2014). As the color-dependent
apparent depth is a strong indicator of stellar blends, we use it as the
discriminator to identify and rule out the transit-mimicking blending signals,
and estimate the overall rule-out fraction of false positives achieved by
_Kepler_ and chosen reference photometry combinations. In addition, we also
evaluate how an extra photometry observation would improve the FPPs of
candidates compared to those derived from _Kepler_ photometry alone, providing
an alternative metric for false positive identification.

The false positives considered in this paper are background eclipsing binaries
and companion transiting planets, which are most pervasive in transit surveys
and may remain undetermined in more sophisticated vetting procedures.

The paper is structured as follows. In Section 2 we define the FPs in
consideration and introduce the simulation prescription. In Section 3 we
compute the fraction of FPs that can be identified with variable transit depth
and study how different stellar properties affect the detectability of depth
variations. For those signals which cannot be instantly identified due to
insignificant depth variations, we adopt a Bayesian FPP computation framework
present the double-band FPP results for typical cases in Section 4. We compare
these two approaches and discuss some issues in Section 5, and summarize our
work in Section 6.

##  2 Simulating the false positive signal on different photometric band

As the objective of this paper, we investigate chromatic transit depth
originating from two typical false positive scenarios: background eclipsing
binary (BEB) and companion transiting planet (CTP). For clarity, we define
their configuration as follows:

  * •

The background eclipsing binary consists of a luminous foreground star
(_target star_) and a pair of faint eclipsing binary in the background
(_blended companion_).

  * •

The companion transiting planet consists of a binary system, with a planet
transiting the lower-mass stellar object (_blended companion_), and the signal
is largely diluted by the brighter primary star (_target star_).

In both cases, the objects that produce the signal are not the _target stars_
themselves, but rather are the unidentifiable _blended companion_. We do not
consider higher multiplicity systems in our simulation since their occurrence
rates are lower than the binary star systems (Raghavan et al., 2010).

###  2.1 Transit depth calculation

The observed depth of BEB and CTP on certain wavelength $\lambda$,
$\delta_{o,\lambda}$, is only a fraction of the true depth
$\delta_{t,\lambda}$,

$\delta_{o,\lambda}=\frac{F_{b,\lambda}}{F_{b,\lambda}+F_{tar,\lambda}}~{}\delta_{t}=d_{\lambda}\cdot\delta_{t,\lambda}$
(1)

where $F_{b,\lambda}$ and $F_{tar,\lambda}$ is the fluxes from blending
objects and target stars measured on observed wavelength. The ratio between
the flux of the blending object and the total flux is denoted as the dilution
factor $d_{\lambda}$. If the observed signal comes from a single and isolated
system, the dilution factor will be unity.

The difference between the observed depth in the _Kepler_ (KP) and the
reference band (RF) can be expressed as

$\Delta\delta_{\mathrm{RF-KP}}=\delta_{\mathrm{t,KP}}\cdot
d_{\mathrm{KP}}-\delta_{\mathrm{t,RF}}\cdot d_{\mathrm{RF}}$ (2)

For a planet transit, the true depth is fixed at the planet-star surface
ratio. Therefore the observed depth-difference from CTPs is merely the
difference of dilution factors on different photometry. For eclipsing binary,
the undiluted primary/secondary eclipse depths will also vary across
wavelengths when the binary components have internal temperature difference,
and so does the dilution factor, which will further change the depth of the
signal.

###  2.2 False positive simulation

To produce the representative population for the _target stars_ and the
_blended components_ defined at the beginning of this section, we generate
synthetic stellar populations that are not only physics-based but also
consistent with the observational constraints provided by _Kepler_ and other
surveys. Due to the main purpose of this paper, we need accurate stellar
photometry to calculate the apparent depths, which is mainly associated with
the stellar masses. We generate these stellar properties with the python
package isochrones (Morton, 2015), which perform trilinear interpolation in
mass-metallicity-age parameter space for Mesa Isochrones and Stellar Tracks
(MIST; Choi et al. (2016);Dotter (2016)). isochrones also predicts the stellar
photometry using the bolometric correction grid
222http://waps.cfa.harvard.edu/MIST/model_grids.html##bolometric provided by
MIST.

The reference catalog of our _target star_ population is Kepler Input Catalog
(KIC, Brown et al. (2011)), and we use this star population to assign line-of-
sight background eclipsing binaries or bounded stellar companions, which will
be described in following sections. We list the simulation parameters of
_target star_ population in Table 1. We compare the distributions of mass,
apparent _Kepler_ magnitude, and effective temperature of the _target star_
population, along with those of KIC stars, in Figure 1. The agreement between
KIC and our _target star_ population is satisfying, though our simulation
contains more hotter stars. We will adress this difference and its impact in
Section 5.3.

Table 1: Input parameters of _target star_ population

Initial mass function (IMF) | Chabrier IMF
---|---
[Fe/H] | Double-Gaussian in (Casagrande et al., 2011)
Age | Uniform(0,10) Gyr
Ava | 0.1-1.6 mag $~{}\cdot\mathrm{kpc}^{-1}$
Distance |  $dN\sim d^{2},~{}d\in[0,1800]$ pc
Magnitude limit (_Kepler_) | 16

$a$. Drawn from https://ned.ipac.caltech.edu/extinction_calculator, depending
on the observing location.

Figure 1: The cumulative density function of target stars’ mass, _Kepler_
magnitude and effective temperature, compared to the stars in Kepler Input
Catalog.

To infer the stellar properties of background stars and their spatial
gradients in background fields, we use TRILEGAL to simulate background stars
(Girardi et al., 2005). TRILEGAL is a web-based Milky Way star population
simulator that considers different galactic components. The request can be
called by setting the coordinate of desired fields and galactic structures (we
use defaults of structure parameters). We use the stellar mass and distance
provided by TRILEGAL to generate eclipsing binaries for BEB simulation.

We provide detailed descriptions about how we construct the appropriate BEBs
and CTPs in following Section 2.2.1 and Section 2.2.2.

####  2.2.1 Simulating background eclipsing binaries

We downloaded one-square-degree background stars in the centers at each of the
21 modules in _Kepler_ field of view via TRILEGAL. For the sake of consistency
in deriving stellar properties as we generate _target stats_ populations, we
extract the returned parameters, e.g., initial mass, metallicity, age,
distance, and extinction factor, from TRILEGAL and pass them to ischrones to
obtain the multi-band photometry and radius of the background primaries. Next,
according to the flat mass-ratio and the binary fraction as the function of
primary’s spectral type reported in Raghavan et al. (2010), we generate
secondary stars to selected primaries.

The eclipse of a binary system only occurs when the impact parameter meets the
criterion (Winn, 2010):

$b=\frac{a~{}\cos i}{R_{1}}\frac{1-e^{2}\cos\omega}{1\pm
e\cos\omega}<1+\frac{R_{2}}{R_{1}}$ (3)

where “$+$” stands for the presence of primary eclipse and “$-$” for secondary
eclipse, $b$ for the impact parameter of the eclipse, $a$ for the semi-major
axis, $R_{1},~{}R_{2}$ for the radius of primary and secondary star, $e$ for
eccentricity, $i$ for inclination and $\omega$ for argument of periastron. We
assume the orbital elements of each binary pair follow such distributions:

  * •

Period and eccentricity distribution from Multiple Star Catalog (Tokovinin,
2018), and truncate the period up to 300 days.

  * •

Uniform distribution between $0\sim 360^{\circ}$ for $\omega$ and a cosine
distribution for orbital inclination.

  * •

Reject the systems that are in each other’s Roche Lobe (Equation 2 in Eggleton
(1983)).

For each background eclipsing binary, we assign a foreground star from the
_target star_ population, and then we calculate the diluted primary/secondary
eclipse according to Equation 1. Next, we calculate the diluted eclipse’s
signal-to-noise ratio (SNR) according to

$SNR_{\mathrm{pri/sec,KP}}=\frac{\delta_{\mathrm{pri/sec,KP}}}{\sigma_{\mathrm{eff,KP}}},~{}\sigma_{\mathrm{eff,KP}}=\sigma_{\mathrm{6hr-
CDPP}}~{}\sqrt{\frac{T_{\mathrm{pri/sec}}}{6~{}hr}~{}\frac{P}{4~{}yr}}$ (4)

where $\delta_{\mathrm{pri/sec,KP}}$ is the observed primary/secondary depth
on _Kepler_ band, $\sigma_{\mathrm{eff,KP}}$ is the effective photometric
precision, which is based on the target star’s 6-hour Combined Differential
Photometric Precision (CDPP) inferred from, eclipse duration
$T_{\mathrm{pri/sec}}$ and the orbital period $P$, and decide whether they are
detectable on _Kepler_ band. To model the signal detection condition of
_Kepler_ , we reject BEBs whose both of the eclipses’ SNR${}_{\mathrm{KP}}<7$
(Jenkins et al., 2010), which is the nominal detection threshold of _Kepler_
pipeline to avoid misidentification due to statistical fluctuation in the
ensemble of all target stars.

Central for a BEB to mimic planet-transit is it must not show distinguishable
difference between primary and secondary eclipse. Based on the calculated SNR
in _Kepler_ bandpass, we classify the BEBs that have passed the secondary
eclipse test into two categories:

  * •

Type 1: the primary and secondary eclipse are both detectable, but are
indistinguishable in depths (within $3\sigma$), which are taken as candidates
with periods of half of the actual eclipse period, and the number of detected
‘transits’ will be doubled. Thus, the detection threshold of this type of BEB
will be $\frac{7}{\sqrt{2}}\sim 5$.

  * •

Type 2: one eclipse is ‘missing’ in the observation, which may result from
high-eccentric orbits, or the primary/secondary eclipse is diluted to be
undetectable (SNR${}_{\mathrm{pri/sec}}<3$) by the foreground star while the
other eclipse is still detectable.

The two groups of BEBs are well separated and distinctive on the SNRKP diagram
of primary/secondary eclipse. An example of the SNRKP diagram is shown in
Figure 3. Type 1 BEBs lie in the diagonal of the diagram, with a median
orbital eccentricity of 0.01 and a mass ratio of 0.89. Type 2 BEBs cluster
along the ‘two wings’ symmetric to the diagonal, with a median orbital
eccentricity of 0.39 and mass-ratio of 0.52.

####  2.2.2 Simulating companion transiting planets

The CTP model consists of a planet transiting the secondary star of the binary
system. To construct the binary components, we select stars from the _target
population_ as the primary stars, in which we account for the reported binary
fraction’s dependence on primary mass. Then, we assign secondary companions to
the primaries with a uniform mass ratio between 0.1 and 1 (Morton & Johnson,
2011; Raghavan et al., 2010). We assume the binary members are co-located and
coeval, that is, the secondary stars have the same metallicity, age and
distance as primaries.

Next, we assign a planet to each secondary star. The planets are parameterized
by their radii and orbital periods. Orbital periods are randomly selected from
KOI lists (Batalha et al., 2013). The assignment of planet radii, however, is
not so straightforward. We will address this issue in detail in Section 3.2.

###  2.3 Reference bands and noise model

In our work, we have chosen three photometric bands to be the reference band
of _Kepler_ , and each of them has a distinctive effective wavelength and
coverage: SDSS $z$ (800-900 nm), 2MASS $K_{s}$ (1.9-2.3$\mu$m), and TESS
(600-900 nm) band. SDSS $z$ band is a narrow-band filter commonly used in
ground-based photometry. TESS band is wide-band photometry is employed by the
spacecraft TESS especially designed for transit photometry. 2MASS $K_{s}$ is a
near-infrared filter band and may represent future space-based infrared
photometry. Our preference for longer-wavelength photometry partly stems from
the assumption about the considered false positives: we assume the _Kepler_
targets are often blended with a lower-mass object, which is a lot redder in
colors (and interstellar extinction will further redden the color of
background binaries in BEB). Hence, a combination of photometry bands with
concentration at visible light (such as _Kepler_ band) and infrared (and even
near-infrared) would properly exploit and reveal the color-difference in BEBs
and CTPs. The aforementioned assumption is reasonable for most of the stubborn
cases of FPs that are of relatively small sizes and low brightness. Target
stars blended with higher mass (and thus brighter) objects may have already
been compiled in the KIC catalog or identified by other techniques, such as
centroid analysis (Bryson et al., 2013) or ephemeris matching (Coughlin et
al., 2014).

To quantify the uncertainties in transit depth measurements and the
significance of the depth variations in different bandpasses, it is necessary
to assign photometric precision to each simulated FP instance. The Combined
Differential Photometric Precision (CDPP) measured in _Kepler_ stars provides
the best template for our purpose. Here we assume the CDPP is only correlated
to the _Kepler_ magnitudes $m_{\mathrm{KP}}$. To derive an empirical relation,
we take median 6-hour CDPP evaluated in 1-magnitude bins between
$m_{\mathrm{KP}}=9$ and 16. We assign _Kepler_ noise to each FP instance in
terms of its apparent $m_{\mathrm{KP}}$ by linearly interpolating the
CDPP-$m_{\mathrm{KP}}$ relation explained above. For the reference bands, we
multiply _Kepler_ ’s CDPP-$m_{\mathrm{KP}}$ curve by factors of one, three,
and ten, interpolating the scaled noise-$m_{\mathrm{RF}}$ relation based on
the apparent $m_{\mathrm{RF}}$. By adopting different noise templates can we
provide a fiducial estimate for the photometry precision for the reference
band: when $\sigma_{\mathrm{RF}}=\sigma_{\mathrm{KP}}$, the reference
photometry attain the best space-based photometry precision so far, as have
been examplified by _Kepler_. If $\sigma_{\mathrm{RF}}=3\sigma_{\mathrm{KP}}$,
this is another typical precision for space-based telescope with smaller
aperture, like TESS (30cm). In the case of
$\sigma_{\mathrm{RF}}=10\sigma_{\mathrm{KP}}$, this is where the typical
ground-based photometry can achieve. Examples of _Kepler_ and reference band
noise model are presented in Figure 2.

Figure 2: The adopted photometric precision relation to evaluate the observed
depth uncertainties for our simulated false positives (red, blue and yellow
dashed curves). The photometric uncertainty on _Kepler_ band is represented by
the red curve, which is the median 6-hr CDPPs (black dots) in 9 bins among all
stars in Kepler Input Catalog with $m_{\mathrm{KP}}$ between 10 and 16. The
photometric precision on reference bands ($z,~{}K_{s},~{}\mathrm{TESS}$) is
evaluated by the apparent reference magnitude $m_{RF}$ according to the three
example precision curves, which are the scaled version of _Kepler_ precision
with factors of one, three, and ten.

##  3 Ruling out false positives with depth-difference

###  3.1 Background eclipsing binaries

####  3.1.1 Identify BEBs with two photometric band

This section demonstrates that the _Kepler_ ’s candidates mimicked by
background eclipsing binaries can be identified and excluded by joining
additional reference photometry. In the previous section, we define those BEBs
have passed the secondary eclipse test on _Kepler_ band will be
indistinguishable from genuine planet transits, i.e., on the _Kepler_ band,
the primary’s and secondary eclipse have similar depths, or only
primary/secondary eclipse is detectable. Since a joint photometry is observing
with _Kepler_ , we may also check whether these BEBs can also pass the
secondary eclipse test on the reference band. We then identify the BEBs for
which the SNR of primary-secondary eclipse difference on reference band,
SNRpri-sec,RF, is larger than 5. This quantity is expressed in Equation 5.

$SNR_{\mathrm{pri-
sec,RF}}=\frac{|\delta_{\mathrm{pri,RF}}-\delta_{\mathrm{sec,RF}}|}{\sqrt{\sigma_{\mathrm{pri,RF}}^{2}+\sigma_{\mathrm{sec,RF}}^{2}}}>5$
(5)

where $\delta_{\mathrm{pri,RF}}$ and $\delta_{\mathrm{sec,RF}}$ is the diluted
primary and secondary depth of BEB, and
$\sigma_{\mathrm{pri,RF}},~{}\sigma_{\mathrm{sec,RF}}$ is the effective noise
of primary/secondary eclipses. Some BEBs may pass the secondary eclipse test
on both photometry bands. To eliminate those stubborn cases, we proceed to
compare the apparent depth on two photometry bands. To estimate the number of
simulated BEBs that show detectable depth variations between _Kepler_ and the
reference band, we calculate the SNR of depth-difference, SNRRF-KP, and
identify those BEBs whose SNRRF-KP is larger than 5. This quantity can be
expressed as:

$SNR_{\mathrm{RF-
KP}}=\frac{|\delta_{\mathrm{RF}}-\delta_{\mathrm{KP}}|}{\sqrt{\sigma_{\mathrm{RF}}^{2}+\sigma_{\mathrm{KP}}^{2}}}>5$
(6)

where $\delta_{\mathrm{RF}},~{}\delta_{\mathrm{KP}}$ are the apparent transit
depth observed in reference band and _Kepler_.

With either of these two features observed, one can confidently identify the
blending nature of the transit signals. We present an example of the SNR of
primary-secondary difference and double-band depth variations for the BEBs
using _Kepler_ -TESS pair photometry, which is displayed in the left panel of
Figure 3, and we mark the BEBs if they show any of the two characteristics in
the right panel. The majority of Type 1 BEB can only show double-band depth
variations since most of them contain equal-mass EBs so that the primary and
secondary eclipse will remain nearly identical across all photometry. The
detectable primary-secondary difference is preferentially displayed by BEBs
with intermediate mass-ratio, primarily Type 2 BEB.

Figure 4 compares these two identification methods at different reference
bands and precision. Generally, more BEBs will show identifiable depth
variations rather than primary-secondary difference. On the $K_{s}$ band, most
of the BEBs will show both of the features, and this fraction only differs
between 80-95% under the considered reference photometry precision. In
contrast, using $z$ and TESS band can at most identify 83% and 69% of the
BEBs, respectively. And the fractional number will drop sharply to only around
12% and 6.8% when $\sigma_{\mathrm{RF}}/\sigma_{\mathrm{KP}}=10$. Moreover,
the fraction of BEBs identified via primary-secondary difference is nearly
negligible under relatively large noise
($\sigma_{\mathrm{RF}}/\sigma_{\mathrm{KP}}=3,10$) for $z$ and TESS band.

Figure 3: _Left_ : The SNRKP of the primary and secondary eclipse of simulated
background eclipsing binaries (BEBs), color-coded by the mass-ratio $q$ of
background binary. Type 1 BEBs has undistinguishable primary and secondary
eclipses in _Kepler_ photometry. They lie in the diagonal and consist of
mainly equal mass ($q\sim 0.88$) and circular-orbit ($e\sim 0.01$) background
binaries . Type 2 BEBs have one undetectable eclipse. They cluster in the
upper left and bottom right corner of the plot, and they are mainly composed
of low mass-ratio ($q\sim 0.43$) and eccentric-orbit ($e\sim 0.26$) binaries;
_Right_ : the same as the left figure, but the BEBs are marked in terms of the
detectable double-band photometry feature. Light blue circles are the BEBs
that only show distinguishable depth variations (SNR${}_{\mathrm{RF-KP}}>5$),
which are mainly Type 1 BEBs and low $q$ Type 2 BEBs; deep blue circles are
those BEBs that can only be revealed by their identifiable primary-secondary
eclipse difference on the reference band, which are mainly binaries with
intermediate $q$; red filled circles are the BEBs that both of the signatures,
and the grey circles show neither depth variation nor primary-secondary
difference. The instances whose SNRKP of primary or secondary eclipse is below
1 are plotted on the edge of the figure, for visualization. The reference band
used to plot this figure is TESS, assigned with a noise estimate identical
with _Kepler_ ’s.  Figure 4: The fraction of BEBs showing detectable depth
variations or detectable primary-secondary difference or both, as a function
of the photometric precision on reference band.

####  3.1.2 Factors affecting the the amplitude of depth variations

According to Figure 4, most BEBs in our simulation show significant double-
band depth difference rather than primary/secondary difference. We may examine
how the amplitude of depth-difference is correlated with BEB system
properties. We present the SNRRF-KP of all BEBs in Figure 5, along with the
apparent depths and _Kepler_ magnitude exhibited by the systems. We find that
BEBs preferentially produce and mimick Earths and small-Neptune candidates.
This pattern agrees well with the simulation of Fressin et al. (2013).
Besides, BEBs with large apparent _Kepler_ depth tend to show large SNRRF-KP,
even though they are produced by faint foreground target stars. However, even
for small planet candidates, at least half of them can display detectable
depth variations, assuming the reference bands can achieve a _Kepler_ -like
precision.

The color-difference between different stellar components in BEB
configurations will also influence the amplitude and detectability of the
wavelength-dependent depth feature: (i). the color-difference (temperature
difference) between the foreground target star and background system, in which
foreground stars’ dilution level will differ between bandpasses and (ii). the
internal color-difference (or equivalently the effective temperature
difference) between the background binary will result in a chromatic eclipse
depth. With such a view, we may proceed to explore the color-difference
between which stellar components would impact SNRRF-KP more significantly.

To better illustrate this, we plot BEBs’ SNRRF-KP against the temperature-
ratio between background binary components and foreground-background stars
separately in Figure 6. The effect of (i) is evidenced by the positive trend
between the SNRRF-KP and background-foreground star temperature ratio (black
dots) presented in Figure 6. The effect of (ii), on the contrary, show a
negative correlation (red dots) with SNRRF-KP, as shown in Figure 6. This is
because the effective temperature not only correlates to the color of star,
but also with star’s size and luminosity. As the secondary star gets a lot
cooler than the primary, it will become smaller and less luminous, therefore
the largest possible decline in brightness during the eclipse will drop, too.
The decrease of the true eclipse depth will reduce the magnitude of apparent
diluted depth and the amplitude of depth variations. Thus, BEBs that consist
of background binary members with distinct spectral types tend to show minor
amplitude in depth variations, even if the internal color-difference is large.

Figure 5: BEBs’ SNRRF-KP as a function of the apparent _Kepler_ depth, color-
coded by the _Kepler_ magnitude of foreground star. The black dashed line
separate the detectable depth-difference (SNR${}_{\mathrm{RF-KP}}>5$) from
those undetectable. BEBs with fainter foreground stars tend to produce deeper
signals with larger SNRRF-KP. Red and blue points represent the median SNR
value of BEB subsets with hot and cold foreground star (respectively) in each
apparent depth bin. Error bars correspond to 25th and 75th percentile. Hotter
foreground stars may have larger color-difference relative to the background
binaries. Therefore they can be more easily distinguished by double-band
photometry. Figure 6: BEBs’ SNRRF-KP plotted against the effective temperature
ratio between different stellar components. Black dots are the temperature
ratio between background primaries and foreground stars; red dots are the
temperature ratio between background secondaries and primaries. The median
SNRRF-KP calculated in each $T_{\mathrm{eff}}$-ratio bin is represented by big
dots. The figure suggests that the larger color-difference between foreground
and background stars will enhance the detectability of depth variation.
However, the color-difference between background binary’s negatively impacts
the SNRRF-KP. The opposite trend results from the decreasing eclipse depth
when the secondary star is much smaller and fainter in low
$T_{\mathrm{eff}}$-ratio region.

####  3.1.3 Ruling out BEBs with different apparent depths

We learn that different types of BEBs can be identified through different
approaches, and the detectability is correlated to the stellar properties in
the system. We may predict how much BEBs in _Kepler_ can be identified
overall. The mass distribution of background star have gradient along the
galactic latitude covered by _Kepler_ Field of View (FoV); thus such mass
variance may affect the BEB’s rule-out fraction in different modules in the
entire _Kepler_ field. To estimate the overall rule-out fraction, we simulate
BEB population in the centers of 21 modules of _Kepler_ FoV via TRILEGAL,
while the pool of foreground stars are the same as introduced in Section
2.2.1. We calculate the rule-out fraction and in each field. We present the
average fraction in Figure 7, sorted by different reference bands, the size of
planet candidate they mimic (based on apparent _Kepler_ depth), and the
assumed reference photometry precision.

The _Kepler_ -$K_{s}$ photometry pair can efficiently rule out all BEBs with
any depths. Even if the depth comparison is done with the largest noise on the
reference band, the rule-out rate can still attain nearly 80% for our BEB
samples.

The rule-out fractions achieved by _Kepler_ -TESS photometry pair strongly
depend the apparent depths that the BEBs display. Although 80% of BEBs that
mimics giant planet candidates can be effectively ruled out if TESS can
achieve a _Kepler_ -like precision, only half of the BEBs with Earth-size
depth can be identified. Since most BEBs’ apparent depth is shallow, the
overall rule-out fraction is only around 69%, still below that of the $Kepler-
K_{s}$ pair with the largest noise. Moreover, the exact level of
$\sigma_{\mathrm{TESS}}$ will significantly affect the rule out rate of Earth-
like BEBs, too. When $\sigma_{\mathrm{TESS}}$ is three times of _Kepler_
precision, the rule-out rate for Earth-like BEBs drops to 12%. In the worst
precision, nearly none of Earth-like BEBs can be identified, and only those
mimicking giant planet candidates are identifiable. The performance of
_Kepler_ -$z$ photometry pair better than TESS. If $z$ band can achieve a
_Kepler_ -like precision, it can rule out 80% of the BEBs with any apparent
depths. However, shallow-depth BEBs get harder to identify as the noise on $z$
band gets larger, and thus the total rule-out rate decreases.

Figure 7: This figure sums up the BEBs’ rule-out fraction, classified by
different apparent planet radii on _Kepler_ band. The number on left column
denotes the adopted photometric precision on the reference band. The numbers
in each cell are the mean value of rule-out rate, averaged over the simulation
from 21 fields centered on _Kepler_ ’s CCD modules. The uncertainty represents
95% confidence interval. Our planet radii classification is the same as that
in Fressin et al. (2013). $K_{s}$ band can indiscriminately identify the BEBs
mimicking all sizes of planet radii. TESS’s and $z$ band’s successful
identification prefers giant planet candidates mimicked by BEBs, though $z$ is
generally better than TESS.

###  3.2 Companion transiting planets

The search of planet transit is often carried out without knowing whether the
target stars are single are not. Unrecognized binarity of the target star will
bring uncertainty to the planet radius inferred from the observed signal. If
the planet is transiting the primary star, the inferred planet radius is
underrated by $\sqrt{2}$ at most (in case of equal-mass binary) and is thus
not seriously affected. Also, there may well be a chance that this is actually
a large planet transiting an unseen stellar companion, and the transit depth
is diluted to appear much smaller due to primary’s flux contamination. In this
case, the radius correction $X_{R}$, the ratio of the planet’s true radius to
apparent radius, can be up to six (Ciardi et al., 2015). Multi-color
photometry can help identify the binarity of the system by identifying the
depth variation in transit light curves.

Part of the objective of this paper is to estimate the exclusion rate of
_Kepler_ -detected false positives via double-band photometry. However,
deriving the exclusion rate in the case of companion transiting planets is not
so straightforward as background eclipsing binaries. This is because
currently, we lack the specific radius distribution of planets that are
transiting the secondary star in a binary system, $P^{\mathrm{CTP}}(R_{p})$.
Simply extrapolating the radius distribution from other work based on
completely different contexts would undermine the reliability of our result.
Thus, we refrain from employing any functional forms of the planet radius
distributions in our simulation, nor will we derive a definitive value of the
exclusion rate. Instead, in this section, we aim to derive the specific
exclusion rate as the function of the true planet radius, i.e., the
probability that a CTP system can show detectable depth variations if the
hosting planet has a radius of $R_{p}$. These formalisms can accommodate any
forms of $P^{\mathrm{CTP}}(R_{p})$ to derive the overall exclusion rate. We
will first discuss how different spectral-type binary would influence the
specific exclusion rate in Section 3.2.1, then we present the final formalism
in Section 3.2.2.

####  3.2.1 Depth variation detection probabilty for CTPs with different
spectral types

According to Equation 2, the amplitude of depth variation of CTP model comes
from the color-difference between the binary members and the radius ratio
between the transiting planet and secondary star. In this regard, we may
expect the amplitude of CTP’s depth variation is correlated to the spectral
types of binary members and the true radius of the planet. However, the
detectability of depth variations not only depends on the amplitude of
variations but also on the planet orbital periods and apparent _Kepler_
magnitude, which are needed to calculate the effective photometry precision.

To start with, we simulate a simplified population of CTPs, which all locate
at the same distance (500 pc) and uniformly have a Jupiter-size planet. We
calculate the SNRTESS-KP of one-transit signal, so that the SNRTESS-KP is
mostly determined by binary’s spectral type composition. In the left panel of
Figure 8, we present these CTPs’ SNRTESS-KP as well as the effective
temperatures of binary components. It can be seen that CTP instances with
similar SNRTESS-KP form a distinctive U-shape pattern, which suggests these
CTPs, despite that they are made up of different spectral types of stars, have
a similar amplitude of depth variation. For a given primary star, the SNRTESS-
KP increases as the $T_{\mathrm{sec}}/T_{\mathrm{pri}}$ decreases from one, in
which the binary components have more discrepant color and the amplitude of
depth variation subsequently increases, too. Then SNRTESS-KP decreases as
temperature-ratio keeps dropping after some point. When the primary star is
much more massive than the secondary star, its flux contamination is so heavy
that the apparent transit depth is attenuated on both photometry band, and the
depth-difference will be small. The primary’s dilution effect on SNRTESS-KP is
most extreme at the bottom of the figure, where the binaries are mainly an M
dwarf bound to an F/G dwarf, rendering the depth variations to be almost
undetectable.

The U-shape pattern no longer holds if we add in distance and planet orbital
distributions to CTPs (the right panel of Figure 8), which introduce
fluctuations to the effective precision of individual CTP system. However, the
absolute amplitude of CTP depth variation is determined by the system’s
spectral type composition. Therefore we may expect there are certain
populations of CTPs that are most amenable to display identifiable depth
variation.

To evaluate the probability that the depth-difference signal arising from a
binary system with certain spectral types can attain sufficient SNRRF-KP in a
_Kepler_ -like survey. We redo the simulation in the right panel of Figure 8;
in each trial we change the binaries’ distances and planets’ orbital periods.
We calculate the fraction of systems whose SNRRF-KP is larger than 5 in
gridded $T_{\mathrm{pri}}-T_{\mathrm{sec}}/T_{\mathrm{pri}}$ space. Finally we
average the value over ten trials.

We present the depth variation detection probabilities of CTP systems for each
reference bands in Figure 9. Contours are the radius correction factor $X_{R}$
for CTPs with different spectral types.

Most notably, the _Kepler_ -$K_{s}$ photometry can resolve the depth variation
of CTPs with wide ranges of spectral type combinations. Although CTPs
comprising stars with nearly identical spectral types
($T_{\mathrm{eff,sec}}/T_{\mathrm{eff,pri}}>0.9$) are hard to show detectable
depth variations, the planet radius estimate from this CTP population is close
to the true value so they do not plague the planet classifications as severely
as the other CTPs do.

Using $z$ and TESS band as the reference band is not as effective in resolving
the depth variations. The CTP system involving a K dwarf primary ($\sim
4500$K) and an M dwarf secondary ($\sim 3500$K) is most ameanble to display
resolvable depth variations for _Kepler_ -$z$ and -TESS photoemtry, even in
the case where $\sigma_{\mathrm{z,TESS}}/\sigma_{\mathrm{KP}}=10$. The
adequate binary color-difference and the small size of the secondary star
would allow larger amplitude of depth variations and render this CTP
population the optimal targets for the search of depth-difference.

When $\sigma_{\mathrm{z,TESS}}/\sigma_{\mathrm{KP}}=3$, depth variations can
be resolved for more than 40% of the CTPs with temperature-ratio of 0.6-0.9.
CTPs containing stars with very similar spectral types (i.e. two G dwarfs or
two F dwarfs, etc.) or with very different spectral types (i.e. an F/G dwarf
paired with an M dwarf) are still difficult to display detectable depth
variations, which is due to the lack of color-difference and the heavy flux
contamination of primaries (respectively), as explained above. When TESS and
$z$ band can achieve a _Kepler_ -like photometric precision, one can expect to
observe the depth variations of most CTPs, but only comparable to the most
noisy $K_{s}$ photometry ($\sigma_{\mathrm{K}}/\sigma_{\mathrm{KP}}=10$).

Figure 8: _Left_ : an example of the SNRTESS-KP distribution of equal-distance
($\sim$ 500 pc) CTP simulations plotted against the primary star effective
temperature $T_{\mathrm{eff,~{}pri}}$ versus the primary-secondary temperature
ratio $T_{\mathrm{eff,~{}sec}}/T_{\mathrm{eff,~{}pri}}$. CTPs showing
undetectable depth variations (black crosses) are mainly binary systems having
identical primary and secondary stars. Therefore there is little color-
difference in between. The other part of the diagram shows a U-shape
distribution at a fixed value of SNRTESS-KP, which is the result of primary
dilution effect and stellar color-difference; _Right_ : the same as the left
figure, but the CTPs are spatially distributed far out to 1800 pc. The spatial
distribution will randomly decrease the photometric noise of the system Figure
9: The probability that a CTP system show detectable depth variations, in
terms of the effective temperature of primaries and secondaries. The
probability is evaluated in a grid defined in
$T_{\mathrm{pri}}-T_{\mathrm{sec}}/T_{\mathrm{pri}}$ space, averaged over 10
trials for each reference band and noise. Blocked area is the region where the
mass of secondary star is below 0.1$M_{\odot}$. Contours are the planet radius
correction factor defined as the same in Ziegler et al. (2016).

####  3.2.2 Depth variation detection probability of CTPs with different
planet radii

We present a framework to calculate the overall exclusion rate of _Kepler_
-detected CTPs. We list the following key ingredients needed to compute the
overall exclusion rate of _Kepler_ -detected false positives in double-band
photometry. We start with the probability that the planet in CTP configuration
having a radius of $R_{p}$, $P^{\mathrm{CTP}}(R_{p})$. Only a fraction of CTPs
with given planet radii can be still detectable on _Kepler_ band after the
dilution. To obtain the modified planet radius distribution of _Kepler_
-detected CTPs, we derive the _Kepler_ detection probability for CTPs with
given $R_{p}$, $P_{\mathrm{KP}}(R_{p})$. This quantity is plotted as the black
thick line in Figure 10. More than 70 % of giant planets
($R_{p}>0.6~{}R_{\mathrm{jup}}$) can still be detected by _Kepler_ band.
However, the detection probability will decrease sharply for small planets as
their apparent transit depths are susceptible to get diluted to be
undetectable by the primary star. For those Earth-sized planets that occur in
CTP systems, there is nearly zero chance for their diluted transits to be
detected by _Kepler_ band.

Next we calculate the specific exclusion rate for a _Kepler_ -detected CTP
with given $R_{p}$, $f^{\mathrm{CTP}}_{\mathrm{spec,ex}}(R_{p})$. This
quantity can be obtained by integrating the detection probability shown in
Figure 9 over the whole $T_{\mathrm{pri}}-T_{\mathrm{sec}}/T_{\mathrm{pri}}$
space. The expected specific exclusion rates computed with different reference
bands and photometric precisions are displayed as the colored lines in Figure
10. The specific exclusion rate increases with the radius of the resident
planet since the amplitude of depth difference scales with planet size. Earth-
size planets’ specific exclusion rate is nearly zero for all reference bands.
$K_{s}$ band has a specific exclusion rate larger than 80% for all planets
larger than $0.5R_{\mathrm{jup}}$. The reference band photometric precision
has a small impact on the $K_{s}$ band’s specific exclusion rate while
affecting the rate of $z$ and TESS more seriously. The specific exclusion rate
for noisy ($\sigma_{K}/\sigma_{\mathrm{KP}}=10$) $K_{s}$ photometry is
comparable to that of $z$ and TESS with best photometry precision
($\sigma_{\mathrm{z,TESS}}/\sigma_{\mathrm{KP}}=1$).

Based on all the quantities, the overall exclusion rate,
$f^{\mathrm{CTP}}_{\mathrm{ex}}$, can be computed. With selected planet radius
function in CTPs, the probability density function of a _Kepler_ -detected CTP
having a planet with radius $R_{p}$, $P^{\mathrm{CTP}}_{\mathrm{KP}}(R_{p})$,
can be written as

$P^{\mathrm{CTP}}_{\mathrm{KP}}(R_{p})=N_{0}P_{\mathrm{KP}}(R_{p})\times
P^{\mathrm{CTP}}(R_{p})$ (7)

$N_{0}$ is the nomalization factor. The overall exclusion rate can be computed
by integrating the specific exclusion rate with Equation 7 over $R_{p}$

$f^{\mathrm{CTP}}_{\mathrm{ex}}=\int
f^{\mathrm{CTP}}_{\mathrm{spec,ex}}(R_{p})\times
P^{\mathrm{CTP}}_{\mathrm{KP}}(R_{p})dR_{p}$ (8)

Figure 10: This figure presents the probability of a CTP system will show
detectable double-band depth variation (colored lines) and the probability of
a CTP system can show detectable transit signals on _Kepler_ band (black thick
line), as a function of the true radius of resident planet.

##  4 Constraining the false positive probability with double-band photometry

In the previous section, we take a clear cut in depth variation SNR and use
this as the sole metric to identify BEBs and CTPs, and estimate their rule-out
fraction. While a depth-variation detection limit of five is acceptable, it is
desirable to evaluate the signal’s status of being a false positive
quantitatively. As the apparent depth double-band difference behaves
differently between false positive and genuine planet transit, we could
incorporate the double-band photometry into the false positive probability
(FPP) computation, yielding a more improved and reliable result.

This section aims to demonstrate the capability of multi-color photometry in
constraining the FPP of background eclipsing binaries and companion transiting
planets. The calculation of FPP makes use of the details of the candidate’s
light curve and quantify how well they agree with the expectance of each false
positive scenario. We parameterize the double-band transit signals into only
two observables: the apparent “transit” depths on _Kepler_ band and reference
band. Rather than including all conceivable false positive scenarios in the
FPP estimate, which is beyond the scope of this work, we present examples
where only a certain FP scenario is most probable. As a result, the FPP
calculation will be simplified while the results remain reasonable in showing
the FPP’s improvement realized by multi-color photometry. Adopting a Bayesian
probabilistic approach, the FPP of a transit signal can be expressed as:

$FPP=\frac{p_{FP}\mathcal{L}_{\mathrm{FP}}}{p_{\mathrm{TP}}\mathcal{L}_{\mathrm{TP}}+p_{\mathrm{FP}}\mathcal{L}_{\mathrm{FP}}}$
(9)

where $p$ is the _a priori_ of the hypothesis, and $\mathcal{L}$ is the
likelihood of the hypothesis evaluated by the observed “transit” depths in two
bands, describing how well the depths obtained from double band photometry
match those predicted from the corresponding hypothetical model. The subscript
TP denotes the true positive hypothesis, which is usually the case of a
genuine planet transit with the right radius estimate, and FP denotes the
false positive. We will point out the definition of each hypothesis in the
following case study.

###  4.1 Background eclipsing binaries mimicking Earth-sized planets

False positives involving an eclipsing binary can be categorized into three
groups: undiluted eclipsing binaries (EBs), hierarchical triples (HTs), and
blended background eclipsing binaries (BEBs), and each of them tend to produce
different orders of eclipse depths, with EBs mimicking Jupiter-size transits,
HTs mimicking Neptune-size transits and BEBs mimicking the sub-Neptune size
and smaller planet transits (Morton, 2012; Morton et al., 2016; Fressin et
al., 2013). We note that this statement is not definitive but empirical in the
sense that heavy dilution will attenuate the eclipse depth shallow.

Driven by this inference, we evaluate the impact of multi-color photometry on
the FPP by the example: an earth-sized planet transiting signal (100 ppm)
found from a bright ($m_{\mathrm{KP}}=11$) G dwarf (one solar mass) located in
low galactic latitude region in _Kepler_ FoV ($b\sim 7$ degree). The reason
that we assign shallow depth and low-latitude region to the signal is that,
under such circumstance and among all the possible false positive scenarios,
background eclipsing binary is most likely to represent the false positive
hypothesis in the FPP calculation (Fressin et al., 2013; Batalha et al.,
2010). The true positive scenario, in this case, is a genuine Earth-size
planet transiting the target star.

####  4.1.1 FPP calculation

According to equation 9, we determined the prior and likelihood of BEB and
planet hypothesis in the following manners:

_Prior of BEB_ ——The prior probability of BEB $\pi_{\mathrm{BEB}}$ should be
the product of (1) the probability of a background star being blended within
the aperture of the target and (2) the probability of the blended star being
an eclipsing binary. The first term is set to be 0.37, derived from Figure 1
of Morton & Johnson (2011); the second term is set to be 0.79%, which is the
occurrence rate of short-period eclipsing binaries discovered in _Kepler_
field (Slawson et al., 2011). We have $p_{\mathrm{BEB}}=0.003$.

_Prior of planet_ ——To obtain the prior probability of the signal being a
genuine transiting Earth-size planet, we rely on the frequency of detected
Earth-size planets in entire _Kepler_ survey. We count the number of confirmed
Earth-size planets and candidates from KOI lists, then divide the number by
the number of KIC stars, which yields
$p_{\mathrm{pl}}=\frac{607}{156453}=0.004$.

_Likelihood of BEB_ ——To create the likelihood space of BEB, we generate
background EBs from the TRILEGAL simulation following the same formula
described in 2.2.1, except that all foreground stars are subject to be solar-
like G dwarf stars (a gaussian mass distribution of $1\pm 0.05M_{\odot}$).
Then we compute the diluted primary eclipse depths in _Kepler_ band and
reference bands. The sample’s logarithmic _Kepler_ depth
$\log\delta_{\mathrm{KP}}$, and double-band depth ratio
$\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}$ will be used to define the two-
dimensional probability density function (PDF) of BEB hypothesis. The
likelihood is then

$\mathcal{L}_{\mathrm{BEB}}=\int
PDF_{\mathrm{sim}}(\bm{\delta})PDF_{\mathrm{obs}}(\bm{\delta})d\bm{\delta}$
(10)

where $\bm{\delta}$ is the vector of
$(\log\delta_{\mathrm{KP}},~{}\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}})$, and
$PDF_{\mathrm{sim}}$ is the posterior PDF mapped by BEB simulation,
$PDF_{\mathrm{obs}}$ is the posterior PDF of observed signal, which is a
gaussian distribution centered on $\bm{\delta}$, with depth uncertainty as its
standard deviation. We define the observed $\delta_{\mathrm{KP}}$ at $100\pm
20$ ppm, and leave the $\delta_{\mathrm{RF}}$ as a varying value. We set the
depth uncertainties on the reference band, $\sigma_{\mathrm{RF}}$, to be 20 or
50 ppm.

_Likelihood of planet_ ——For the planet hypothesis, we compute the planet
radius based on the observed $\delta_{\mathrm{KP}}$ (within errors) and the
simulated foreground star radius. Then we compute the apparent transit depth
on reference bands with limb-darkened transit model (Mandel & Agol, 2002),
with appropriate quadratic limb darkening coefficients taken from Claret &
Bloemen (2011) and Claret (2017). The likelihoods of planets is evaluated in
the same way as for the BEBs.

Figure 11 shows an example of the PDFs of BEB and planet hypothesis. The
expected depth ratio distribution of each hypothesis is quite different. BEBs
tend to produce deeper $\delta_{\mathrm{RF}}$ than $\delta_{\mathrm{KP}}$
because the background binaries are generally of low masses and appear
consequently red in color, while the foreground G dwarf is less luminous in
longer wavelengths. Thus the flux contamination of foreground stars will be
less significant on the redder reference bands. The depth ratio expected for
planet transits is smaller than 1, which is due to the stellar limb darkening
effect, which states that light with longer wavelengths is more evenly
distributed on stellar disk than light with shorter wavelengths. Therefore the
planet will block more blue light than red light, and the
$\delta_{\mathrm{RF}}$ will be smaller than $\delta_{\mathrm{KP}}$.

Moreover, the most probable depth ratio expected for each hypothesis,
represented by the area with the deepest color in Figure 11, is also unique
when adopting different reference bands. _Kepler_ -$K_{s}$ pair has the most
separated $\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}$ distributions for two
hypotheses, with the most probable ratio value of 1.5 for BEB hypothesis and
0.88 for planet transits. _Kepler_ -TESS photometry has the least separable
depth ratios for two hypotheses. The distinction between BEB’s and the
planet’s likelihood distributions results from the photometry pair’s
capability to reveal the color change during the eclipse/transit, which is
consistent with the pattern we demonstrate in Figure 4 and 7.

Figure 11: Likelihood space for BEB and planet scenarios described in Section
4.1. The area with deeper color means a higher probability that our hypothesis
simulations have a certain value of
$(\log\delta_{\mathrm{KP}},~{}\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}})$ in
the region. Note that the depth ratio for two hypotheses is different. BEB
tends to show a larger apparent depth in the redder reference band compared to
_Kepler_. Planet transits show a smaller depth due to stellar limb darkening.
The observed depth ratio for different photometry pair is separated at
different levels, with _Kepler_ -$K_{s}$ pair showing the most distinctive
depth ratio distribution for BEB and planet scenarios; while the _Kepler_
-TESS shows a similar depth ratio between the two hypotheses, in both cases
the depth ratios are close to 1.

####  4.1.2 Improved FPP from double photometry

We present the improved FPP results in Figure 12, and compare to the single-
band FPP. Besides selecting three different photometric bands as the reference
bands, we also provide two cases when the depth uncertainty is different
(represented by solid and dashed line). The general trend is that as the
observed depth ratio increases, the FPP goes higher. When the observed depth
ratio is close to unity, the FPP will decrease, supporting the planet
hypothesis.

It is also shown from the figure that the order of FPP increment or decrement
varies across reference bands. $K_{s}$ band shows the best performance in
discriminating the planets and FP scenarios. If
$\delta_{K_{s}}/\delta_{\mathrm{KP}}=1$, the FPP will decrease by an order.
Also, it takes a smaller depth ratio for the _Kepler_ -$K_{s}$ pair to fully
confirm the false positive origin of the signal than the other two photometry
pairs (1.3 and 1.7 for the two cases). This is because _Kepler_ -$K_{s}$ pair
has the most discrepant $\delta_{K_{s}}/\delta_{\mathrm{KP}}$ distributions
for planet and BEB hypothesis (see Figure 11), which makes it easier to
distinguish the underlying hypothesized scenario that creates the observed
depth ratio.

Secondly, the comparison between two cases with different depth uncertainties
suggests that the actual value of depth uncertainty will affect the double-
band FPP more significantly than the different choice of reference band. For
example, if a depth ratio of 1.3 is observed, the corresponding FPPs derived
from $\sigma_{\mathrm{RF}}=50$ ppm are generally ten times lower than the case
of 20 ppm, while FPPs are close to each other for all three photometry pairs
at a given $\sigma_{\mathrm{RF}}$. A result of this effect is that larger
uncertainties in depth measurement would require a larger observed depth ratio
to falsify a signal. The critical depth ratio, which makes the FPPs increase
to nearly 100%, is 1.3-1.4 and 1.7-1.8 for $\sigma_{\mathrm{RF}}=20$ ppm and
$\sigma_{\mathrm{RF}}=50$ ppm, respectively.

Finally, we may compare the FPP results with the depth-difference
identification method adopted in Section 3. If we calculate the SNRRF-KP for
these critical depth-ratios, the value is only around 1.4, which is well below
the threshold of 5 we set in Section 3. This comparison suggests that
incorporating the double-band photometry in FPP calculation can more easily
identify false positives.

Figure 12: The false positive probability that an Earth-sized candidate is
caused by a background eclipsing binary, evaluated by the observed double-band
depth-ratio. The black dotted line is the single-band FPP derived from
_Kepler_ photometry alone. The solid and dashed lines are cases when the depth
observed by reference band is of different uncertainties. See Section 4.1 for
more discussion.

###  4.2 Which is the planet-host star?

Identifying the planet-host star in a binary system is crucial in correcting
the dilution of observed transit signals. Applying the double-band photometry
FPP calculation to the model of companion transiting planets provides a
natural way to distinguish the planet-host star in a binary system. We
illustrate this by providing a specific observation event. Suppose a Super-
earth-size transit candidate ($\delta_{\mathrm{KP}}\sim 200$ ppm) is observed
from a solar-like G dwarf, but it is unknown whether the target star is single
or not. To predict the multi-band transit depths for each hypothesis, we
simulate binary populations with a primary mass of around one solar mass and a
varying secondary mass, according to the procedure described in Section 2.2.1.
We assume that, before the transit on reference band is obtained, the apparent
depth observed by _Kepler_ alone provides no clues about the host star of the
planet; that is, the planet is likely to orbit either star in the binary with
equal probability, hence $\pi_{\mathrm{TP}}=\pi_{\mathrm{FP}}=50\%$. Here we
define the true positive as the planet orbiting the brighter target star,
while false positive as the planet orbiting the fainter companion of the
target star.

To determine the likelihood of each hypothesis, we first draw the apparent
$\delta_{\mathrm{KP}}$ from a Gaussian distribution with a mean value of 200
ppm and a standard deviation of 20 ppm; the latter quantity is the depth
uncertainty $\sigma_{\mathrm{KP}}$ we assigned to the _Kepler_ photometry.
Then we recover the true planet radius from the apparent depth
$\delta_{\mathrm{KP}}$ for each instance, assuming the primary/secondary star
is the planet-host star. We reject those instances with a recovered planet
radius larger than 22 $R_{\oplus}$ to avoid overestimation from systems that
unlikely exists. Then we calculate the observed depth on reference band
$\delta_{\mathrm{RF}}$ and establish the PDFs. Finally, we evaluate the
likelihoods in the similar manner described in Section 4.1.1.

We present the likelihood distribution of the two hypotheses in Figure 13. If
the primary star is the host star, the apparent radius of the planet is close
to the true value, and the depth observed on different photometry band will
not differ a lot. This is because the primary star usually makes up most total
photometry. Therefore the variance of fainter companion’s contamination on
different photometry bands would be relatively small. The situation can be
reversed when the secondary star is the host star, which is shown in the right
panel of Figure 13 that the apparent transit depth observed in the reference
bands can be up to 5-10 times of that observed on _Kepler_ band. This is
because the true planet radius is usually much larger than the apparent radius
so that the depth variation will increase, too. The apparent depth on the
$K_{s}$ band can vary in a wide range, from 170 ppm to 2000 ppm. On the other
hand, TESS and $z$ band expect a much narrower possible
$\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}$ range.

We calculate the probability of planet host star based on a set of transit
depth ratio values and present the result in Figure 14. We also provide two
cases with different $\sigma_{\mathrm{RF}}$. As expected, the depth ratios
close to unity will decrease the FPP, which suggests that the target star is
indeed the planet host star. FPPs increase with the non-unity value of depth
ratios, which alerts that the planet may be transiting the secondary companion
of the target star. Larger depth uncertainties from the reference bands will
lead to a loosely constrained FPP value. We find the SNRRF-KP of the critical
depth ratios to be 0.7 and 1.2, for two cases, respectively.

Figure 13: The likelihood distribution of the transit depth if an apparent
super-earth-size ($\delta_{\mathrm{KP}}\sim 200$ppm) candidate is observed on
three reference bands, under the assumptions that the planet is transiting the
primary star (left) or the secondary star (right).  Figure 14: The false
positive probability that a super-earth-size planet candidate is transiting
the unseen stellar companion of the solar-like target star, evaluated by
double-band depth-ratio. The black dotted line represents the uninformative
prior probability that the planet is equally likely to orbit either of the
stars in the binary system inferred from one-band photometry (prior = 50% for
each star). The solid and dashed lines are cases when the depth observed by
the reference band is of different uncertainties. See Section 4.2 for more
discussion.

##  5 Discussion

###  5.1 Comparing two multi-color approaches to identifying false positives

In this work, we demonstrate two methods by which we can use the color-
dependence feature of transit candidates to reveal and rule out false
positives. The first method is illustrated in section 3, where we measure the
SNR of the difference between observed depths and identify those false
positives whose SNR of depth-variation is large enough to confidently confirm
that the observed transit depth in two photometry is statistically
inconsistent, thereby pointing out the possible presence of blending stars or
the chance-alignment of eclipsing binary. In section 4 we adopt the second
method. We compare the observed depth ratio with a comprehensive set of depth-
ratios simulated under different hypotheses on the observed transit signal’s
origin, and we adopt a Bayesian approach to calculate the false positive
probability of the signal based on double-band photometry.

The underlying assumptions for these two methods are a bit different. The
basic assumption for the first method is that the apparent depth of false
positives should be different. To identify this difference, we set the
detection threshold SNRRF-KP as 5 to evaluate whether the difference is real
or due to statistical fluctuation. With this said, satisfying this detection
criterion (SNR${}_{\mathrm{RF-KP}}>5$ and SNR${}_{\mathrm{KP}}>7$) will
require the signal to be observed with sufficient high SNR on reference band
(SNR${}_{\mathrm{RF}}>12$), which is not only determined by the color-
difference between the blends and targets, but also by the reference band’s
relative effective wavelength compared to the main photometry, and the
photometric precision thereof. The approach is widely adopted in the early
ground-based wide-field survey to reject transit signals (O’Donovan et al.,
2006; Colón et al., 2012), but is mainly subjected to vetting the transit
signals’ with deep depths ($\sim 1\%$) for which sufficient SNR can be
attained. Small planet transit candidates may not be the optimal targets for
multi-color vetting; as suggested by Figure 7, the detectability of small
planet candidate’s depth variation is most susceptible to the choice of
reference band and precision. Thus, although we could identify blends and
exclude them based only on one such metric, the definitive threshold will
implicitly demand high SNR on the signal detection by reference photometry.
The expected yield of false positive identification is strongly limited by the
specifics of reference band and noise systematics.

The basic assumption for the FPP calculation is that the double-band depth
ratio of planet transit is different from that of false positives. In simpler
terms, we could falsify a transit signal as long as we can tell it does not
resemble the planet transit’s light curves. This is why we find in Section 4
that the critical depth ratio’s corresponding SNRRF-KP is usually much smaller
than 5 since the observed depth ratio cannot be reproduced by most of the
hypothetical planet-transit systems. However, since we adopt a Bayesian
approach to achieve this goal, the value of double-band FPP will indeed vary
as we change each hypothesis’s _a priori_ probability and as we change the way
we simulate hypothesis likelihood PDFs. If the _a priori_ probability of the
planet hypothesis is larger, the corresponding FPP will be generally lower
than before.

Nevertheless, we note that this is the common feature of Bayesian probability,
which strongly depends on the models and assumptions. One advantage of double-
band FPP, which we did not explicitly address in this work, is that it can
work in parallel with other observational constraints to explore and narrow
down the possible blending systems that can produce the double-band light
curves, e.g., high-resolution imaging can reveal both physical companions in
wider orbits and line-of-sight objects. The dilution information derived from
magnitude difference helps to locate the star that harbours the observed
signal (Furlan et al., 2017; Ziegler et al., 2016). For targets including
several stars, which are hardly resolved, joint analysis of multi-color
transit light curves and SED of the targets may provide stronger
characterization of the multiplicity and configuration of the system (Miyakawa
et al., 2021).

###  5.2 The role of limb darkening

Limb darkening will also cause the planet transit’s depth to vary on different
bands. This is because fluxes with different wavelengths distribute on stellar
disk differently, with the center of the disk appear bluer and the limb
appears redder. As a consequence, the transit depths observed on bluer
photometry will be deeper than redder photometry, which projects
$\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}<1$ since our considered reference
photometry are all redder than _KP_.

To this end, one may question that whether using SNR${}_{\mathrm{RF-KP}}>5$ as
the sole metric to reject blends would ambiguously exclude the cases of limb-
darkened planet transits signals. However, for the false positives considered
in this paper the apparent depth ratio would be
$\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}>1$. According to our definition of
BEB and CTP in 2, the _blending components_ , which generate the
eclipses/transits, are often redder in color compared to their contaminant
(stars from _target population_) and thus the apparent depth would be deeper
on red reference photometry. Therefore, we could discriminate blends from
limb-darkened transits by comparing the apparent depth ratio.

One exception would be grazing giant planets for $b>0.8$ (Tingley, 2004),
which only pass through the limb of star and will block more blue light than
red light. In this case the apparent depth ratio would be
$\delta_{\mathrm{RF}}/\delta_{\mathrm{KP}}>1$, resembling the chromaticity of
stellar blends. However, high-impact-parameter transits are rather rare cases
and we do not consider it when using color-dependent feature to reject blends.

###  5.3 Impact of different stellar properties from KIC

Though our _target star_ population mainly consists of G-type stars as KIC
included, we were unable to fully recreate the KIC’s stellar mass and
effective temperature distributions in our simulation (as shown in Figure 1).
These three parameters, as demonstrated in Section 3, are the major stellar
properties that determine the detectability of double-band depth variations.
The deviation between the simulation and KIC stars partly stems from the fact
that we employed a more physical model (see Section 2.2) to construct the
stellar population, while KIC is a specially curated catalog following some
selection rules.

As for this deviation’s impact on the final results. Qualitatively speaking,
our target star population is on average hotter than KIC stars. In the BEB
model, hotter foreground stars tend to produce earth-like transit signals,
because they are brighter so the dilution is heavy, therefore the rule-out
rate of BEB signals (summarized in Figure 7) is affected the most for earth-
size BEBs, the other sizes of BEB signals only being minorly affected. The
excess of BEBs with hotter foreground stars leads to an overestimation of
earth-like BEB rule-out rate. As for the overall rule-out rate (summarized in
Figure 4&7), the case for $\sigma_{\mathrm{RF}}=\sigma_{\mathrm{KP}}$ is
affected the most, while for other reference band noise levels, the
identifiable earth-like BEBs only make up a small fraction of the total rule-
out rate and thus are not affected much.

For CTP model, we demonstrate that CTPs with hot primaries are the hardest to
rule out at low reference band precision, espeacially for $z$ and TESS band
(see Figure 9). Thus, we expect the excess of hotter stars in our sample will
lead to an underestimation on the rule-out probabilities given planet radius
(summarized in Figure 10) for $z$ and TESS band in
$\sigma_{\mathrm{RF}}/\sigma_{\mathrm{KP}}=3,~{}10$.

However, we do manage to construct a KIC-like target star simulation to
quantitatively measure the uncertainties in final results, even though we
adopt some unphysical input distributions. The rule-out probabilities in both
BEB and CTP models vary within 10%, even for the most severely affected cases.

##  6 Summary

In this work, we study how many false positives can be identified in terms of
their chromatic depth feature if another reference band different from
_Kepler_ band were added and observing with _Kepler_. We mainly focus on two
types of false positives. One is that when the target star is blended with a
pair background eclipsing binary (BEB); another is when the planet is
transiting the unresolved stellar companion to the target star (we term this
as companion transiting planet, or as CTP). We build up physics-based models
for these two false positives. To systematically estimate what fraction of
BEBs and CTPs can be identified through their color-dependent feature, we
simulate representative BEB and CTP populations according to _Kepler_
observations and galactic star population models. We derive their multi-band
photometry based on isochrone interpolation and bolometric correction. We
calculate the signal-to-noise ratio of the difference of observed transit
depth, SNRRF-KP, observed between _Kepler_ and reference bands, which include
$z$, $K_{s}$, and TESS. We identify the subset of false positives whose SNRRF-
KP is larger than five as the fraction of false positives that can be readily
ruled out and show how this fraction varies with different levels of
photometric precision we assigned to the reference bands. Some false positives
cannot gain sufficient high SNRRF-KP. However, since the depth variation
predicted by false positive hypothesis and planet hypothesis is quantitatively
different, our multi-color model enabled us to calculate the transit signals’
false positive probability based on double-band photometry. We summarize the
conclusion of this work as follows.

  * •

Background eclipsing binaries (BEBs) can be effectively ruled out by combined
identification of double-band depth difference (Equation 2) and primary and
secondary difference (Equation 5); We list the overall rule-out fraction via
identifying the depth-difference of BEB in Figure 7. $K_{s}$, $z$, and TESS,
when adopted as the reference band to _Kepler_ , can rule out 95%, 80%, and
65% of _Kepler_ candidates mimicked by BEBs if both of the photometry can
attain a precision like _Kepler_ ’s. Assuming a larger photometry precision of
the reference band, the overall rule-out fraction of TESS and $z$ band will
drop drastically towards around 10%. The sharp decrease is mainly due to the
undetectable depth variation of shallow-depth candidates that comprise most
BEB signals, and only those signals with deep depths can show detectable depth
variations. However, _Kepler_ $-K_{s}$ dual photometry is only slightly
affected even if it is assigned with the largest noise, i.e. ten times of
Kepler-like precision, reaching 70% which is generally better than the TESS
and $z$ band.·

  * •

The amplitude of the depth variations of CTPs comes from the true planet
radius and the color-difference between the binary stars the planet resides.
We calculate the detection probability for _Kepler_ -detected CTPs with
certain spectral type binary members (Figure 9). The CTPs that consist of the
smallest stars, such as a K dwarf and an M dwarf, often show detectable depth
variations, while equal-temperature CTPs are the hardest to identify for all
the considered reference bands. Next, we present the detection probability of
CTPs with given true planet radius, averaged over all possible CTP binary
masses (Figure 10). Using the provided detection probability curve, one can
obtain the overall exclusion rate of CTPs expected in a _Kepler_ -like survey
by integrating the radius distribution of planets transiting the secondary
star in a binary system.

  * •

We adopt a Bayesian approach to calculate the false positive probability (FPP)
based on the observed depth ratio between two photometry observations. We
apply the framework to identify whether an Earth-size candidate is a diluted
signal from a background eclipsing binary. For given transit SNR on the
reference band, $K_{s}$ band can better constrain the FPP of signal better
than $z$ and TESS band. However, the photometric precision of the reference
band would influence the degree of FPP improvement more than the specific
choice of reference bands. Nevertheless, using double-band photometry to
determine the FPP and reveal blends usually takes less demand on photometry
precision compared to identifying the blends via SNR${}_{\mathrm{RF-KP}}>5$
criterion. (Figure 12).

  * •

Applying the FPP calculation to our CTP model can assist in figuring out
whether an Earth-size transit signal is a larger planet transiting the bound
companion of the target star, or equivalently identifying the planet host star
in the binary system. For this purpose, assuming that the planet is equal-
likely to orbit either of the stars inferred from initial single-band
observation, the probability of the primary host can be up to 99% if the
observed depth ratio is nearly unity. The confirmation of the secondary host
will need the observed SNRRF-KP to be at least 0.8 (Figure 14).

With a more flexible treatment of input assumptions about field stars’
properties, the simulation procedure we have developed can have broader
applications to other surveys as long as we have the priors and models
specific to the surveyed area, whether physical or empirical. One possible
application is to use in the follow-up of TESS, as the pixel scale of TESS
(21”) is much larger than that of _Kepler_ (4”); thus, a considerable amount
of blended stars will be imaged. Merited for its economical cost in a single
observation, multi-color photometry can extensively search for clues of false
positive in _Kepler_ if future missions employ it, and the valuable time of
advanced telescopes would be spared.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11973028, 11933001, 1803012), and the National Key R&D Program of
China (2019YFA0706601). We also acknowledge the science research grants from
the China Manned Space Project with NO.CMS-CSST-2021-A1.

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