# Demographics of Exoplanets in Binaries (DEB). I. Architecture of S-Type
Planetary Systems Revealed by the RV Sample.

Xiang-Ning Su  Ji-Wei Xie  Ji-Lin Zhou  School of Astronomy and Space Science,
Nanjing University, Nanjing 210023, People’s Republic of China  Key Laboratory
of Modern Astronomy and Astrophysics, Ministry of Education, Nanjing 210023,
People’s Republic of China  Philippe Thebault  LESIA, Observatoire de Paris,
Meudon Principal Cedex, France

(Received April 23, 2021; Revised August 27, 2021; Accepted September 28,
2021)

###### Abstract

Although the sample of exoplanets in binaries has been greatly expanded, the
sample heterogeneity and observational bias are obstacles toward a clear
figure of exoplanet demographics in the binary environment. To overcome the
obstacles, we conduct a statistical study that focuses on S-type
(circumstellar) planetary systems detected by the Radial Velocity (RV) method.
We try to account for observational biases by estimating, from available RV
data, planet detection efficiencies for each individual system. Our main
results are as follows: (1) Single (resp. multiple) planetary systems are
mostly found in close (wide) binaries with separation $a_{B}<(>)\sim$ 100-300
AU. (2) In binaries, single and multiple-planet systems are similar in 1-D
distributions of mass and period as well as eccentricity (in contrast to the
“eccentricity dichotomy” found in single star systems) but different in the
2-D period-mass diagram. Specifically, there is a rectangular-shaped gap in
the period-mass diagram of single-planet systems but not for multiples. This
gap also depends on binary separation and is more prominent in close binaries.
(3) There is a rising upper envelope in the period-mass diagram for planets in
wide binaries as well as in single stars but not in close binaries. More
specifically, there is a population of massive short period planets in close
binaries but almost absent in wide binaries or single stars. We suggest that
enhanced planetary migration, collision and/or ejection in close binaries
could be the potential underlying explanation for these three features.

Exoplanets — Binary Stars — Statistics

††journal: AJ

##  1 Introduction

Stars are thought to be commonly born and found in binary/multiple systems
(Duquennoy & Mayor, 1991; Raghavan et al., 2010). Therefore, the demographics
of exoplanets in binaries play a crucial role in statistically studying the
whole exoplanet population in our Galaxy. Furthermore, the diverse orbital
configurations and rich dynamics in planet-bearing binary systems provide
valuable conditions to test various theories and models of planet formation
and evolution (Thebault & Haghighipour, 2015; Marzari & Thebault, 2019).

There are generally two orbital configurations of planets in binaries (Dvorak,
1982). One is called P-type, where planets orbit both the binary stars, i.e.,
circumbinary planets, e.g., Kepler 413 ABb (Kostov et al., 2013). The other is
called S-type, where planets orbit one of the binary stars, i.e.,
circumstellar planets, e.g., $\gamma$ Cephei Ab (Campbell et al., 1988; Hatzes
et al., 2003). Most planets in binaries are currently found in S-type. At the
time of writing this paper (04-March-2021), by combining the two catalogues
(see section 2.1 for details) retrieved from the Extrasolar Planets
Encyclopaedia and the Catalogue of Exoplanets in Binary Star Systems (Schwarz
et al., 2016), there are 211 confirmed S-type planets. Among them, 116 are
from radial velocity (RV) surveys (e.g., $\gamma$ Cephei Ab (Campbell et al.,
1988; Hatzes et al., 2003), Tau Bootis b (Butler et al., 1997)), 89 from
transit surveys (e.g., Kepler-68 b (Gilliland et al., 2013), HD 202772 b (Wang
et al., 2019)) and 6 from others, e.g., microlensing surveys (Bennett et al.,
2016; Gould et al., 2014), and direct imaging surveys (e.g. 51 Eri (Montet et
al., 2015)). There are only 28 confirmed P-type planets, and a large fraction
of them are from transit surveys, e.g., Kepler-16AB b (Doyle et al., 2011) and
Kepler-47 b,c and d (Orosz et al., 2012, 2019) etc. Note, the above numbers
only reflect a small but relatively well characterized portion of planet-
bearing binary systems. Many candidate planet-bearing binary systems, e.g.,
those from imaging followup observations of Kepler planet host stars (Wang et
al., 2014a, b; Kraus et al., 2016; Ziegler et al., 2017; Furlan et al., 2017;
Ziegler et al., 2018) are currently not included in the above two catalogues
because either the planetary nature or stellar binarity remains to be
confirmed.

When mining the observational data, there are two commonly used approaches to
study the effect of stellar binarity on planetary systems. On the one hand, by
calculating the binary fraction of planet hosts and comparing it to those of
non-planet hosts or field stars, one can learn whether/how planet formation is
suppressed by binary stars (Bonavita & Desidera, 2007; 2020Galax...8...16B;
Eggenberger et al., 2011; Wang et al., 2014a, b; Kraus et al., 2016; Ziegler
et al., 2017, 2018; Matson et al., 2018; Moe & Kratter, 2019; Fontanive et
al., 2019; Fontanive & Bardalez Gagluiffi, 2021). One difficulty with this
approach is to remove or at least quantify the effects of observational bias
against finding planets in binaries. A major bias often comes from the target
selection process itself, which is severely biased against binaries in most RV
surveys, but also significant for transit surveys, e.g, Kepler (Wolniewicz et
al., 2021). On the other hand, by investigating the distributions of planet
properties (e.g., mass, period, and etc.) in known planet-bearing binary
systems, one can learn how planetary architecture is sculpted by binary stars,
e.g., (Roell et al., 2012). It is the latter approach that we adopt in this
paper. Furthermore, we only focus on S-type planet systems from RV surveys to
reduce the sample complexity and thus better take into potential observational
bias. Since we just care about ‘relative’ properties (e.g., period-mass
distribution) rather than ‘absolute’ properties (e.g, absolute occurrence
rate), our analysis is not affected by binary-hostile target selection biases
in RV surveys. The main potential bias we need to correct for is the varying
planet detectability performances depending on planetary mass and period,
which could over-represent (or under-represent) some types of planets in our
sample. We could also have, for the same location in the planetary mass-period
diagram, different detectabilty thresholds depending on the considered system.
For example, some systems may have poorer RV measurements due to large stellar
activity or light contamination of close binary stars, which would thus bias
against finding smaller planets on longer periods in these systems.

Some previous statistical studies have already explored the properties of
S-type planets in RV surveys, but for more limited samples. With a small
sample of 9 S-type planets, Zucker & Mazeh (2002) found that planets in
binaries seem to follow a different period-mass distribution than planets in
single star systems. Increasing the sample size to 19, Eggenberger et al.
(2004) confirmed this trend by pointing out that the few most massive
($Msini>2M_{\rm Jup}$) short-period ($P<40$ days) planets are exclusively in
binary systems. Doubling the sample size to $\sim$40, Desidera & Barbieri
(2007) found that the above mass difference is significant only in tight
binaries (binary separation, $a_{B}<100-300$ AU). Duchêne (2010) then found
that planets in tight binaries ($a_{B}\lesssim 100$ AU) are significantly more
massive than those in wide binaries ($a_{B}\gtrsim 100$ AU). Roell et al.
(2012) compiled a larger sample of planets in binaries and found that (1) no
planets are in binaries with a projected separation of less than 10 AU (2)
multi-planets systems are exclusively in wide binaries ($a_{B}>100$ AU) and
(3) planetary mass decreases with an increasing binary separation.

In this paper, which is the first (Paper I) of a series of studies on the
Demographics of Exoplanets in Binaries (DEB for short), we focus on the
architecture of S-type planetary systems detected by the RV method. With the
significantly expanded sample that we compile (110 S-type RV planets), we aim
to (1) finding new patterns and (2) revisiting the above reported patterns,
e.g., the period-mass distribution, by taking into account the RV detection
efficiency in detail.

This paper is organized as the follows. In section 2, we describe our planet
sample and the method used to consider the effect of the RV detection
efficiency. In section 3, we present our results. Finally, we discuss and
summarize the paper in sections 4 and 5.

Figure 1: The mass (or Msini) histogram of all companion objects in our
sample. Apparently, there is gap (grey shaded) between 20 and 50 Jupiter
masses, which separates planetary objects from stellar objects.  Figure 2:
Overview of our sample: Planet mass as a function of binary separation. All
the single-planet systems are represented by green circles, while all planets
in multiple-systems are represented by blue squares. The light yellow dash
line indicates the Jupiter mass.  Figure 3: Two examples of RV signal
injection-recovery test. 30 Ari is a close binary system with one planet (red
star in the left panel), and HD 125612 is wide binary with three planets (red
stars in the right panel). The filled dark grey circles (light squares) show
where the hypothetical planets can (can not) be recovered from the simulated
RVs. The green shaded regions mark where $RV_{signal}=(1-2)\times rms$. The
purple shaded regions mark where $period=(1-2)\times the\ time\ baseline$ of
the simulated RV data.

##  2 Sample and Method

###  2.1 Sample

We build our RV planet sample by retrieving data (on 04-March-2021) from two
online catalogues. i.e., (1) the catalog of exoplanet-hosting binaries with
separations up to 500 AU 111http://exoplanet.eu/planets_binary/, (2) the
catalogue of exoplanets in binary star systems (Schwarz et al.,
2016)222https://www.univie.ac.at/adg/schwarz/multiple.html. Specifically, we
first retrieved all RV planet systems in the first catalog, which is almost
exhaustive for all binaries of separation less than $\sim 500$ AU. Then, we
turned to the second catalog, which provided most planets in binaries with
separation from $\sim$500 up to $\sim$15000 AU in our sample. We note that HD
7449 c is reported as planet with minimum mass of 19 $m_{J}$ in the second
catalog but not recognized in the first catalog. In fact, HD 7449 c is only
inferred from the long term RV trend which could be due to the stellar
companion (Rodigas et al., 2016). Therefore, we removed this unconfirmed
planet and thus HD 7449 is included as a single planet system in our sample.
Furthermore, in order to have a unified criteria to define our sample, we
adopted an upper mass limit (20 $m_{J}$) for planetary objects and a lower
mass limit ($50m_{J}$) for stellar objects. This criterion is motivated by the
mass (or msini) distribution of the companion objects in Figure 1, which shows
a gap between 20 and 50 Jupiter masses. Under this definition, HD 87646 c and
HD 4113 c were treated as stellar objects instead in our sample.  In addition,
systems HD 41004 AB, HD 20782/HD 20781 and HD 133131 AB have RV planets
orbiting both the primary and companion stars, and they are treated as two
single S-type planet systems in our analyses. Systems HD 80606, HD 20781, HD
103774 and HD 93385, whose RV data are not publicly available, are thus
excluded from our analyses.

With these criteria, we obtain 110 planets in 80 systems, 61 (19) of which are
single (multiple) planet systems. Table 1 lists the properties of these
planets in our sample. Figure 2 is an overview of our sample in the plane of
Msini-$a_{B}$, where Msini is the minimum mass of planet and $a_{B}$ is the
semi-major axis of binary orbit. Our samples also include 23 planets in 16
triple star systems and quadruple star systems. In these cases, $a_{B}$
reported in Table 1 should be treated as the semi-major axis of the closest
planet-host companion.

Figure 4:  Period-mass diagrams for single-planet systems (green circles in
the left panel) and multiple-planet systems (blue squares in the right panel).
The green and blue curves represent the medians of the detection limits (grey
curves) of singles and multiples, respectively. Systems with RV rms larger
than $100$ ms-1 or planets below the single median detection limit (green
curves) are marked with open symbols. The shaded rectangle marked the gap, a
planet desert in the singles (left panel, see more in the text in section
3.1.3).

###  2.2 Quantifying the RV detection constraints

Different systems could differ significantly in the precision of their RV
measurements, causing an observational bias that systems with poorer RVs bias
against finding smaller planets with longer period. In order to quantify this
bias, we perform the following RV signal injection-recovery test in each
system of our sample.

First, we collect the RV data of each system and extract the RV residual,
$RV_{residual}$, by subtracting the observed planet signals using the RV
fitting tool Systemic2 (http://www.stefanom.org/console-2/, (Meschiari et al.,
2009)). During the above RV fitting process, we also removed the long term RV
trend from binary orbits in some systems (e.g., Gliese 15, gamma Cephei, tau
Boot A, HD41004 A/B, HD196885). For a few systems (HD7449, HD 8673, HD59686,
HD87646, HD2638, HD 30856, HD 42936, HD126614 and 30 Ari), the $RV_{residual}$
were extracted directly from the figures (converting graph into numbers) in
the corresponding planet discovery papers because the RV data were not readily
available.

Next, we inject a test RV signal, i.e., $RV_{signal}$ of a hypothetical planet
with a given minimum mass ($Msini$) and orbital period (P). Ignoring the
effect of orbital eccentricity, we have

$RV_{signal}=\sqrt[3]{\frac{2\pi
G}{P}\frac{(Msini)^{3}}{(M_{*}+M)^{2}}}\cos(\omega+\frac{2\pi t}{P}),$ (1)

where $M_{*}$ is the mass of host star, $M$ is the mass of hypothetical
planet, $G$ is gravitational constant, $i$ is the inclination between the
normal of the planet’s orbit and the line of sight, $\omega$ is the argument
of periastron (drawn randomly from 0 to $2\pi$) of the planet’s orbit, and $t$
is observing time.

Then, we generate a set of simulated RV data,

$RV_{sim}=RV_{signal}+RV_{residual}+RV_{error},$ (2)

by combining the test planet signal with the RV residual and the RV
measurement error. Here, $RV_{error}$ is a random number that follows the
normal distribution, i.e., $N(0,\sigma_{RV})$, where $\sigma_{RV}$ is the
reported RV uncertainty.

Finally, we evaluate whether the injected planet signal can be recovered from
the simulated RV data. The criteria are (1) the peak location of the
periodogram (Lomb, 1976; Scargle, 1982) should be within $25\%$ away from the
orbital period of the injected planet, i.e.,
$|P_{peak}-P_{inject}|<0.25*P_{inject}$ and (2) the False Alarm Probability
(FAP) of the periodogram peak should be less than $1\%$. The FAP is calculated
via a bootstrap method (Townsend, 2010). Specifically, we randomly shuffled
the simulated RV data from Equation 2, and record the power of the periodogram
peak (Lomb, 1976; Scargle, 1982). We repeat such a random calculation 1000
times, and the FAP is set as the fraction of times when the record power is
larger than the original one.

Figure 3 shows the results of the RV signal injection-recovery tests for two
individual systems (30 Ari and HD 125612, see also in Figures 12-15 in the
Appendix for the results of all systems of our sample). For each system, the
RV signals are generated by hypothetical planets which are uniformly
distributed in the log(P)-log(Msini) plane. As can be seen, the filled dark
grey circles show where the hypothetical planets can be recovered from the
simulated RV data. The bottom envelopes of the recovered region generally
follow the line where $RV_{signal}=1-2\times rms$, and there is a long period
cut-off due to the time baseline of the RV data. On the left is 30 Ari, a
close binary ($aB=27.6$ AU) hosting a single massive planet candidates
($Msini=13.82M_{J}$), while on the right is HD 125612, a wide binary
($aB=5990$ AU) hosting three planets with masses of
$Msini=7.2,\,3.0,\,0.058M_{J}$. Figure 3 illustrates a potential detection
bias of the RV method. Indeed, among the three planets of HD 125612, only the
most massive one is close to the recovery region of 30 Ari, while the other
two less massive planets would not have been discovered even if they were
truly existing in the 30 Ari system. This could be due to the fact that,
because RV precision is generally poorer in close binaries (due to light
contamination or inappropriate stellar properties e.g., large $vsini$), only
the signal of the largest planets of these systems can be retrieved, leaving
potential additional smaller planets undetected (whereas such planets would
have been detected in wider binaries or single stars). As a consequence, the
fraction of single-planet systems could be artificially overestimated in close
binaries. To address this potentially crucial issue, we will have to take into
account the statistical average of detection limits for our respective samples
of single-planet and multiple-planet systems.

Figure 5: Cumulative Distribution Functions (CDF) of the semi-major axis of
binary orbit ($a_{B}$, first column), and the critical distance for long-term
orbital stability ($a_{c}$, second column). _Top row_ : “raw” sample defined
without taking into account potential observational biases. _Middle row_ :
removing planets below the median detection limit of the singles “cut-1”).
_Bottom row_ with both cut-1 and cut-2, i.e., removing systems with RV
rms$>$100 ms-1. Note that, in the middle and bottom rows, the single- and
multiple-planet samples are defined according to the number of planets left
_after_ applying the different detection limit cuts (so that, for example,
some “multiple systems” of the top row are categorized as “single-planet”
systems in the middle row). In each panel, we perform the two sample KS test
and print the p value of the test between singles and multiples, i.e., p(s-m).
Figure 6: Cumulative Distribution Functions (CDF) of planetary orbital periods
( first column), mimumum masses (Msini, second column) and orbital
eccentricities (third column) for single (green) and multiple (blue) planet
systems. _Top row_ : “raw” sample defined without taking into account
potential observational biases. _Middle row_ : removing planets below the
median detection limit of the singles “cut-1”). _Bottom row_ with both cut-1
and cut-2, i.e., removing systems with RV rms$>$100 ms-1. Note that, in the
middle and bottom rows, the single- and multiple-planet samples are defined
according to the number of planets left _after_ applying the different
detection limit cuts (so that, for example, some “multiple systems” of the top
row are categorized as “single-planet” systems in the middle row). In each
panel, we perform the two sample KS test and print the p value of the test
between singles and multiples, i.e., p(s-m). For comparison, in the three
right columns, we also plot the results of RV planets in single star systems
(light curves) from the NASA Exoplanet Archive and print the corresponding p
values (grey numbers).

##  3 Results

###  3.1 Single Planets VS. Multiple Planets

We first compare the properties of single and multiple planet systems. Figure
4 shows their period-mass diagrams. In order to visualize the aforementioned
potential RV detection bias, we also overplot the detection limits of each
individual system, i.e., the boundaries between the recovery and no-recovery
regions in the RV signal injection-recovery test (e.g., Figure 12). We then
estimate the medians of the detection limits for each subsample of single-
planet and multiple-planet systems (green and blue curves in Figure 4), which
are defined by the 50% completeness contours, meaning that planets below them
in the period-mass diagram could not be detected in half systems of the
samples. To reduce the effects of such observational biases, we remove planets
in the regions with low completeness. Specifically, we implement a cut
(hereafter labelled “cut-1”) for which we remove planets below the median
detection limit of single-planet systems (the green curve) both in the single
_a_ nd the multiple subsamples. Note that the resulting planetary parameters
actually depend on the number of planets considered when fitting the RV data
(e.g., Wittenmyer et al. (2013) ). For simplicity, we ignored this effect in
this paper and adopted the same planetary parameters before and after “cut-1”.
In addition, we also consider another cut (hereafter “cut-2”) for which we
remove systems with very poor RV data (i.e., rms $>100$ ms-1), since the
detection limits of such poor RV systems deviate too much from the median, and
the detectable areas are too small in the period-mass diagram (e.g. system HD
87646, see Fig. 12). Figure 4 shows that there are 61 single-planet systems,
of which 53 are above the detection limit of cut-1 (the green curve), while
there are 53 planets in multiple systems, of which only 26 lie above cut-1.
Note that singles and multiples in Figure 4 are initially defined according to
the number of planets actually observed in the systems. However, because of
the significant fraction of planets that do not pass the cut-1 test, when
analysing statistical properties, we also consider another classification, for
which we define single- and multiple-planet systems according to the number of
“remaining” planets after applying this cut-1 criterion (see section 3.1.2).
With this alternative classification, some initial multiple-planet systems
could be “converted” into single-planet ones (see Fig.5 and Fig.6).

####  3.1.1 multiple-planet systems prefer wide binaries

Previous studies have hinted at that multiple planet systems are
preferentially found in wide binaries (Roell et al., 2012). Here, we revisit
this trend with our larger and updated sample. The three panels in the first
column of Figure 5 compare the cumulative distribution functions (CDF) of
binary separations ($a_{B}$) for single- and multiple-planet systems. Already
in the raw un-debiased sample there is clear tendency for multiple-planet
systems to have larger $a_{B}$ values as compared to that of single-planet
systems. This trend is pronounced with a KS test p value of $=0.00054$, and it
is still significant (p value of $=0.00062$) when attempting to debias the
sample by applying cut-1, or with both cut-1 and cut-2 (p value of $=0.001$).
Our results thus clearly confirm that multiple-planet systems are more
prevalent in wide binaries whereas single-planet ones are, on average, found
in tighter binaries.

In addition, we also consider the dynamical stability limit, $a_{c}$, defined
as the critical distance to the primary star beyond which orbits become
unstable due to the perturbations of the companion star. We estimate $a_{c}$
using the empirical formula given by Holman & Wiegert (1999), which is a
function of binary mass ratio, orbital semi-major axis ($a_{B}$) and
eccentricity ($e_{B}$). We adopted a typical $e_{B}=0.3$ (Duquennoy & Mayor,
1991) if the binary eccentricity is not available from the source catalog. The
right panels of Figure 5 compare the $a_{c}$ distributions between single and
multiples. We see that multiple planetary systems not only prefer ”physically
wide” (large $a_{B}$) binaries but also ”dynamically wide” (large $a_{c}$)
binaries. This trend is significant with a KS test p value of less than
$0.001$ regardless of whether cut-1 and cut-2 are applied or not.

####  3.1.2 period, mass and eccentricity distributions

The three columns of Figure 6 compare the distributions of planet properties,
i.e., orbital period, minimum mass and eccentricity for single- and multiple-
planet systems. As can be seen, singles and multiples show similar period
distributions regardless of whether the cuts correcting for observational bias
are applied or not. For mass and eccentricity, singles, as compared to
multiples, seem to be more massive (p value $=0.01$) with larger
eccentricities (p value $=0.02$) with a statistical confidence level about
2-sigma if using the whole “raw” sample. However, this trend disappears when
correcting for observational bias by applying cut-1 and cut-2, for which the
mass and eccentricity distributions become statistically indistinguishable (KS
test p values larger than p value of 0.4) between singles and multiples. As a
consequence, the initial trend in the raw data is probably only due to the
fact that an important fraction of the smaller planets with lower
eccentricities observed in multiple-systems could not have been detected in
the singles (see Figure 4).

Note that, as mentioned earlier, in the second and third rows of Figure 6,
systems are classified as singles/multiples according to the number of planets
remaining after correcting for detection limits (cut-1 and cut-2). More
specifically, 11 multiple-planet systems from the raw sample are “converted”
into singles after applying cut-1 (61 singles and 15 multiples in the second
row) and two more if further applying cut-2 (58 singles and 15 multiples in
the third row). Nevertheless, we find similar results regarding the statistics
of planetary period, mass and eccentricity distributions (no difference
between single and multiple planet systems) even when keeping the initial
criterion for defining the two subsamples (i.e., a “multiple” system is still
labelled as such even it only has one planet surviving cuts 1 and 2), which
confirms the robustness of this result. In summary, for giant planets in
binary star systems (all planets that lie above the median detection limit
have Msini $>60m_{\oplus}$), we find that their 1-D distributions in period,
mass and eccentricity is similar in single and multiple-planet systems. As for
smaller planets (Msini $<60m_{\oplus}$), the conclusion cannot be drawn with
current RV data.

####  3.1.3 a gap in the period-mass diagram of single-planet systems?

Although singles and multiples do not show significant differences in the 1-D
distribution of either period or mass (Fig.6), they seem to differ
significantly in the 2-D distribution of the period-mass diagram. As can be
seen in Figure 4, there are $N_{s}=53$ singles above the median detection
limit (green curve) but none is located in the middle shaded rectangle region
(period = 10-300 days and Msini = 160-640 $m_{\oplus}\sim$ 0.5-2.0 $m_{\rm
Jupiter}$). For comparison, there are $N_{m}=23$ multiples above the green
detection limit curve and 5, a fraction of $f=0.22$ are located in the middle
shaded rectangle. If we adopt a null hypothesis by assuming that singles and
multiples follow the same 2-D distribution in the period-mass diagram, then we
can use binominal distribution to access the significance of the observed gap.
The probability that no more than $m$ points are found in the gap follows

$P_{gap}=\sum_{k=0}^{m}C_{n}^{k}(1-p)^{(n-k)}p^{k},$ (3)

where $n$ is the total points, $m$ is the number of points observed in the
gap, $C_{n}^{k}$ the combination coefficient and $p$ the nominal probability
each point would be located in the gap under the null hypothesis. Through out
this paper, we adopt $p=f=0.22$. Here, there are $n=53$ singles being
considered, then the probability that none ($m=0$) is found in the gap by
chance is $P_{gap}=(1-p)^{53}=2.28\times 10^{-6}$.

Nevertheless, above simple calculation implicitly assumes that planets in the
rectangle are fully detectable (100% detection completeness) for all single
systems, which is not the case in reality. By taking into account the effects
of detection completeness of individual systems, the probability that no more
than $m$ out of $n$ singles are found in the gap by chance is modified as
(marked with a superscript ‘*’)

$P_{gap}^{*}=\sum_{k=0}^{m}\sum_{l=1}^{C_{n}^{k}}\left(\prod_{i=1}^{n-k}(1-p\times
D_{l,i})\prod_{j=1}^{k}p\times D_{l,j}\right),$ (4)

where $D_{l,i}$ and $D_{l,j}$ are the coefficients to correct the
detectability of individual systems in the $l$-th case in $C_{n}^{k}$
combinations.

If we assume that planets, if any, are evenly distributed in the rectangle,
then in practice, $D_{l,i}=A_{i}/A_{tot}$, where $A_{tot}$ is the total area
of the rectangle and $A_{i}$ the area of the part of rectangle that is above
the detection limit of individual systems. For example (see Fig. 12), HD 41004
B has a poor RV precision (rms=589.2 ms-1), and thus zero detectabilitiy in
the rectangle, i.e., $D_{l,i}=A_{i}/A_{tot}=0$; in contrast, HD 41004 A has a
good RV precision (rms=9.69 ms-1) and 100% detectabilitiy in the rectangle,
i.e., $D_{l,i}=A_{i}/A_{tot}=1$. Using the above Equation 3, the probability
that none of singles is found in the gap by chance is $P_{gap}^{*}=1.15\times
10^{-5}$, which is over five times of the p value ($P_{gap}=2.28\times
10^{-6}$) if observational biases of individual systems were ignored as
mentioned before.

Note, as mentioned before, that there are two criteria for classifying systems
as singles and multiples. One is based on the number of planets _actually
observed_ in the system. The other one is based on the number of remaining
planets after applying detection limit cuts. Contrary to the analysis of the
distributions of binary separation or planetary orbital parameters (Figure 5
and 6), we here chose the first criteria for defining the two subsamples of
single- and multiple-planet systems. The main reason is that the second
criterion is statistically too drastic, converting too many multiples (8 out
of 23) into singles, resulting in (1) possible contamination of any potential
feature (e.g., the gap) that is intrinsic to singles, and (2) too few
remaining multiples for statistical study in the 2-D period-mass diagram.

Granted, the first criterion also has its own weakness, mainly that some of
the observed singles could be misidentified because of potential undetected
planets. Nevertheless, the effect of such an observational bias can be
relatively straightforward to quantify. Assuming that $N_{S-B}$ observed
singles in the left panel of Figure 4 are in reality “hidden” multiples
because of the current detection efficiency of multiples just removes
$N_{S-B}$ planets around the gap without adding any planet to it in the left
panel of Fig. 4, the gap itself should basically remain as long as $N_{S-B}$
is not particularly large compared to the total number of singles
($N_{S}=61$).

We obtain a rough estimate of $N_{S-B}$ by injecting all the second strongest
planet signals from the multiples (right panel of Fig. 4 ) into the RV data of
each single (left panel of Fig. 4 ). We find that, on average, a fraction
$\sim 43\%$ of the injected planet signals could be recovered, indicating that
$\sim 43\%*N_{S}=26$ of the observed singles are likely to be real singles.
For the remaining $\sim 57\%*N_{S}=35$ uncertain ones, if assuming a
single/multiple ratio comparable to that for the certain ones (i.e, $\sim
43\%*N_{S}/N_{M}=26/49$), then 23 of them should be intrinsic multiples,
namely $N_{S-B}=23$. We then estimate how the significance of the gap would be
modified. Given $N_{S-B}=23$ observed singles are re-classified as multiples,
the fraction of multiples in the gap reduces to $f=5/(23+23)=0.10$, and the
number of singles reduces to 38. Putting these two reduced numbers into
Equations 3 and 4, we obtain $P_{gap}=0.017$ and $P_{gap}^{*}=0.028$ (about 2
sigma confidence). Recalling the previous $P_{gap}=2.28\times 10^{-6}$ and
$P_{gap}^{*}=1.15\times 10^{-5}$, we therefore conclude that the effect of
mixing undetected multiples in observed singles is unlikely to destroy the gap
in singles though significantly reduce the statistical confidence.

Figure 7: Cumulative distribution function (CDF) of the orbital semi-major
axis $a_{B}$ of the binaries with planets lying above the detection limit as
defined by the green curve in Figure 4. Binaries with planets located in the
period-mass gap (the shaded rectangle in Figure 4) are marked with orange
circles. For comparison, the 25, 50 and 75 percentages ($a_{B}^{25\%}=155$ AU,
$a_{B}^{50\%}=383$ AU, and $a_{B}^{75\%}=1020$ AU) are also marked.  Figure 8:
Mass (Msini)-period diagrams of RV planets in close binaries (left panel),
wide binaries (middle panel) and single star systems (right panel). Here,
close/wide binaries are defined as $a_{B}$ smaller/larger than the 50
percentage, $a_{B}^{50\%}=383$ AU, of the sample. In each panel, the green
curve is the detection limit as defined by the same green curve in Figure 4,
the blue dashed line is an empirical fit to the upper envelope of the planet
distribution in the wide sub-sample (panel), and the shaded rectangle markers
the same planet desert as identified in Figure 4. Planets above/below the
green curve are marked with filled/open circles (single planet systems) and
squares (multiple planet systems), respectively. In the top-left of each
panel, we print the number of planets above the detection limit ($N_{cut-1}$),
the number of planets in the gap ($N_{gap}$), and the number of planets on the
top-left side of the blue dashed line ($N_{top-left}$).  Figure 9:  Similar to
Figure 8 but here, close/wide binaries are defined as $a_{B}$ smaller/larger
than the 25/75 percentage, $a_{B}^{25\%}=155$ AU/$a_{B}^{75\%}=1020$AU, of the
sample.

###  3.2 Close VS. Wide Binaries

####  3.2.1 Is the gap more prominent in close binaries?

As there seems to be a link between planetary multiplicity and binary star
separation (section 3.1.1), we here investigate whether the period-mass gap
found in single-planet systems (section 3.1.3) is also dependent on binary
separation. As before, we only consider systems with “good detectability”,
i.e., systems that lie above the median detection limit of singles (the green
line in Figure 4). Even though the gap itself lies in its entirety above the
green line, this criterion will change the number of planets outside of the
gap that will serve to estimate the statistical significance of the gap.
Figure 7 shows the cumulative distribution of the binary separations ($a_{B}$)
of these systems. There are five binaries which host planets in the gap, and
their separations ($a_{B}=$ 142, 750 ,1065, 5796 and 15184 AU) are marked in
Figure 7. Clearly, these gap planets tend to reside in binaries with
relatively wide separations, meaning that the gap is preferentially found in
close binaries. To further demonstrate this point, we conducted two additional
analyses as follows.

In the first analysis, we divided the whole sample into two “close” and “wide”
binary sub-samples, taking as a boundary the median binary separation for our
whole sample $a_{B}^{50\%}$=383 AU. Figure 8 shows the period-mass diagrams of
planets in these two subsamples. As can be seen, in the wide subsample, out of
the 37 planets above the detection limit line, there are 4 lying in the gap,
corresponding to a fraction of $\sim 11^{+8}_{-5}\%$ (errorbars reflecting
Poisson uncertainties). In contrast, in the close-binaries subsample, there is
only 1 planet out of 39 ($\sim 2^{+6}_{-2}\%$) that lies in the gap, and
actually near the lower boundary of it.

In the second analysis, we consider a more stringent definition of our close
and wide subsamples: a binary is in the close subsample if its $a_{B}$ is
smaller than the 25% percentage of the sample ($a_{B}^{25\%}=155$ AU), and is
in the wide subsample if its $a_{B}$ is larger than the 75% percentage of the
sample ($a_{B}^{75\%}=1020$ AU). As can be seen in Figure 9, there are now 3
out of 19 ($\sim 15^{+15}_{-8}\%$) planets in wide binaries that reside in the
gap. In contrast, the number of gap planets is one (1 out of 20 or $\sim
5^{+11}_{-4}\%$ if considering Poisson uncertainty) for the close-binary
subsample.

Both analyses show that the period-mass gap identified for single-planet
systems is also preferentially found in close-binaries, even though there is a
relatively large Poisson uncertainty given the relatively small size of the
subsamples.

Figure 10:  Comparison of planetary orbital periods (first column), mimimum
masses (Msini, second column) and orbital eccentricities (third column)
between close (red) and wide (grey) binaries in samples without any
observational bias cut (top row), with cut-1, i.e., removing planets below the
median detection limit of the singles (middle row) and with both cut-1 and
cut-2, i.e., removing systems of RV rms$>$100 ms-1 additionally (bottom row).
In each panel, we perform the two sample KS test and print the corresponding p
value. Here, close/wide binaries are defined as in Figure 8, namely, $a_{B}$
smaller/larger than the 50 percentage, $a_{B}^{50\%}=383$ AU, of the sample.
Figure 11: Similar to Figure 10 but here, close/wide binaries are defined as
in Figure 9, namely, $a_{B}$ smaller/larger than the 25/75 percentage,
$a_{B}^{25\%}=155$ AU/$a_{B}^{75\%}=1020$ AU, of the sample.

####  3.2.2 period, mass and eccentricity distributions

As can be seen in Figures 10 and 11, there is no statistically significant
difference in the 1-D distribution of planetary period, mass and eccentricity
between systems in close and wide binaries. This result holds for the two
different definitions of the close and wide subsamples and regardless whether
observational bias cuts are taken into account or not. These results also
clearly show up when considering the p values of the KS test between close and
wide binaries, which are all greater than 0.5 in Figure 10 and greater than
0.1 in Figure 11.

There is, however, a difference between the 2 subsamples for the 2-D
distribution of the planetary period-mass diagram, which is different from the
already identified gap that prefers close binaries over wide binaries (section
3.1.3). As can be seen in Figures 8 and 9, there is a rising upper envelope
planets in the period-mass diagram for wide binaries which is not found in
close-binary subsample. This upper envelope empirically follows

$\rm log_{10}\left(\frac{Msini}{m\oplus}\right)=0.5\times
log_{10}\left(\frac{period}{day}\right)+2.3.$ (5)

Specifically, as shown in Figure 8, only 1 out of 37 ($\sim 2^{+6}_{-2}\%$)
planets (with filled symbols) is above the blue dashed line (an empirical
upper envelope) in the wide subsample, while this fraction number is 8/39
($\sim 21^{+10}_{-7}\%$) in the close subsample. Such a difference is still
significant ($0/19\sim 0^{+9}_{-0}\%$ of wide v.s. $4/20\sim 20^{+16}_{-10}\%$
of close) in Figure 9, in which close/wide binaries are defined with a more
stringent criterion. Note that this result that massive short period planets
are rare in wide binaries cannot be explained by observational bias because
these planets are the easiest ones to be detected. The top-left region of the
period-mass diagram has indeed the highest detection efficiency as shown in
Figure 4.

###  3.3 Single Stars VS. Binary Stars

For the sake of comparison, we also performed the above analyses to RV planets
in single star systems, which were retrieved from the
https://exoplanetarchive.ipac.caltech.edu archive.

As can be seen in the three right columns of Figure 6 (dashed curves), the 1-D
period and mass distributions of planets become indistinguishable between
single and multiple planet systems after applying the debiasing cut-1 and
cut-2, a result similar to the one obtained for binaries. Nevertheless, we
note that, contrary to binaries, the 1-D distribution in eccentricity remains
different between singles and multiples even after debiasing, with a p value
of 0.05 ($\sim$ 2 sigma confidence level).

The fraction of planets in the period-mass gap is 41/373$\sim 11^{+2}_{-2}\%$
for single stars (Figures 8 and 9), a value close to the one found for wide
binaries ($\sim 11^{+8}_{-5}\%$), strengthening the result found in section
3.2.1 that the gap is preferentially found in close binaries.

The rising upper envelope in the period-mass diagram identified for planets in
wide binaries is also present for planets around single stars (right panels of
Figures 8 and 9). There are indeed only 25 out of 373 planets, corresponding
to $\sim 6.7^{+1.6}_{-1.3}\%$, above the blue dashed line. This fraction is
consistent (within 1-sigma) with the one found for planets in wide binaries
($1/37\sim 2^{+6}_{-2}\%$), but significantly lower than that of planets in
close binaries ($8/39\sim 21^{+10}_{-7}\%$).

As can be seen from the right panels of Figures 10 and 11, planets around
single stars tend to have lower eccentricities than planets in binaries. The
median planetary eccentricity is 0.16 for single stars, while it is 0.24 in
binary star systems, as shown in Figure 10 with a KS test p value of
$0.01-0.06$. The result holds when only considering binaries at the two ends
of the separation distribution, i.e., $a_{B}$ smaller/larger than the 25/75
percentage, $a_{B}^{25\%}=155$ AU/$a_{B}^{75\%}=1020$ AU, of the sample. The
median eccentricity of planets in binaries increases to 0.26 with KS test p
value of 0.01-0.07 as shown in Figure 11. This result indicates that stellar
binarity increases the eccentricities of exoplanets to a certain extent, and
this effect is not confined to close binaries but still significant even in
very wide binaries (e.g. $a_{B}>1020$ AU).

##  4 Discussions: Implications to Planet Formation and Evolution

Compared to the distribution of planets in wide binaries and single stars, we
find that planets in close ($a_{B}<$100-300 AU) binaries are preferentially
associated with three characteristic features: a larger fraction of single
planets, a rectangle-shaped gap in the period-mass diagram, and the presence
of massive and short period planets (Fig. 6, 7, 8 and 9). We discuss hereafter
the possibility that these 3 features could have the same underlying
explanation.

Close binaries with separation less than a few hundred AU are indeed capable
of enhancing planet migration either in the early gas-disk phase (Kley, 2000)
or in the later disk-free phase e.g., via secular planetary
interactions(Fabrycky & Tremaine, 2007; Wu & Lithwick, 2011), causing planets
to horizontally cross the upper rising envelope in the period-mass diagram
identified in Figures 8 and 9. In addition, these planets could accrete
additional material (e.g., gas, planetesimals or other smaller planets) during
their enhanced migration, and could thus grow into much more massive objects
(Kley, 2000; Zhang et al., 2018). As a consequence, these planets would
populate the initially depleted upper-left part of the period-mass diagram.
These enhanced migrations could also lead to an enhanced level of planetary
mergers and/or ejections, explaining the lower level of planet multiplicity in
close binaries, as well as their larger eccentricities compared to those in
single star systems. As for the rectangle-shaped period-mass gap, it could be
the birth place of those line-crossing planets or planets that were
accreted/ejected by them.

As for planets in wide binaries, they share a number of common properties with
those around single stars, namely a similar fraction (about $10\%$) of planets
in the period-mass gap combined to the presence of a similar upper envelope in
the same period-mass diagram (Fig. 8 and 9). The rising upper envelope is
consistent with the positive period-mass correlation reported by previous
studies (Zucker & Mazeh, 2002; Jiang et al., 2007) and with the predictions of
current synthetic models (Ida & Lin, 2004; Mordasini et al., 2009) of planet
formation around single stars. The common upper envelope between planets in
wide binaries and single stars thus may suggest that planet formation and
evolution in wide binaries ($a_{B}>$100-300 AU) is mostly similar to that in
single star systems.

Nevertheless, planets in wide binaries differ significantly from planets in
single star systems in terms of their eccentricity distribution (Fig. 6, 8 and
9). For single stars, multiple planet systems have significantly lower
eccentricities than single planet systems (the so called “eccentricity
dichotomy”), a fact that has been well established for the Kepler planet
sample (dominated by super-Earths and sub-Neptunes, (Xie et al., 2016; Van
Eylen et al., 2019; Mills et al., 2019) and is also seen in RV planets
(dominated by Jovian planets) as shown here in Figure 6. In contrast, there is
no difference (once the data is debiased) between the eccentricity
distributions of single and multiple-planet systems for both our close- and
wide binary subsamples. In addition, these eccentricities are, on average,
larger than the eccentricities of planets around single stars (Fig. 6, 10 and
11). This may imply that there are some additional mechanisms at play to pump
up planet eccentricities in wide binaries and bring them to the same level as
for those in closer binaries. Potential eccentricity drivers could be the
Kozai mechanism (Kozai, 1962) and Galactic tides (Kaib et al., 2013). Future
studies, both with larger observational sample and with dedicated numerical
models are needed to further unveil the picture of planet formation and
evolution in binary star systems.

##  5 Summary and Conclusion

In this paper, we compile a sample (Table 1) of all S-type planetary systems
detected by the RV method in order to derive statistical characteristics as a
function of key properties of these systems, such as the number of detected
planets or binary separation. Available RV observation data allow us to
quantify the planet detection efficiencies of individual systems, enabling us
to correct for potential observational bias, notably between the single-planet
and multiple-planet sub-samples. We then perform a statistical investigation
from two different perspectives.

First, we consider the whole sample of S-type systems and look for statistical
properties that depend on planetary multiplicity. We find that, for the whole
S-type sample, single and multiple-planet systems have similar 1-D
distributions of both planetary masses and periods. However, they differ
significantly in terms of their 2-D period-mass distribution. The main
difference is an absence of planets in a rectangle-shaped gap (for planetary
periods between 10 and 300 days Msini between 160 and 640 $m_{\oplus}\sim$
0.5-2.0 $m_{\rm Jupiter}$) that is present in single-planet systems but not in
multiples. The gap is statistically significant with a P value of
$P_{gap}=2.28\times 10^{-6}$ or $P^{*}_{gap}=1.15\times 10^{-5}$ depending if
observational bias is ignored (Equation 3) or considered (Equation 4).

Second, we look for planet properties that depend on binary separation
($a_{B}$) and divide our sample into a close and wide binary subsamples. We
find that multiple-planet systems are preferentially found in wide binaries,
which is consistent with the previous study by (Roell et al., 2012). We also
find that the gap in the planetary period-mass diagram of single-planet
systems depends on binary separation: it is more prominent in close binaries
with $a_{B}$ less than a few hundreds AU than in wide binaries or single
stars. Furthermore, we find that, in the same planetary period-mass
distribution, there is a population of massive and short period planets in
close-binaries that is not found in wide binaries and single stars, for which
the period-mass distribution is limited by a rising upper envelope. Last but
not least, we find that the average eccentricity of planets in binaries (both
close and wide) is higher than in single stars.

We suggest that enhanced planetary migration, collision and/or ejection in
close binaries could explain most of the statistical properties we have
identified, but the detailed study of this effect exceeds the scope of the
present paper.

In summary, our study emphasizes the importance of taking into account
individual detection efficiencies when searching for statistical patterns for
planets in binaries. The results of this paper suggest that binary stars could
play a crucial role in shaping the architecture of planetary systems, i.e.,
reducing planetary multiplicity, exciting planetary eccentricity and modifying
the planetary period-mass diagram.

This work is supported by the National Key R&D Program of China (No.
2019YFA0405100) and the National Natural Science Foundation of China (NSFC;
grant No. 11933001). J.-W.X. also acknowledges the support from the National
Youth Talent Support Program and the Distinguish Youth Foundation of Jiangsu
Scientific Committee (BK20190005).

## References

  * Bennett et al. (2016) Bennett, D. P., Rhie, S. H., Udalski, A., et al. 2016, AJ, 152, 125, doi: 10.3847/0004-6256/152/5/125
  * Bonavita & Desidera (2007) Bonavita, M., & Desidera, S. 2007, A&A, 468, 721, doi: 10.1051/0004-6361:20066671
  * Butler et al. (1997) Butler, R. P., Marcy, G. W., Williams, E., Hauser, H., & Shirts, P. 1997, ApJ, 474, L115, doi: 10.1086/310444
  * Campbell et al. (1988) Campbell, B., Walker, G. A. H., & Yang, S. 1988, ApJ, 331, 902, doi: 10.1086/166608
  * Desidera & Barbieri (2007) Desidera, S., & Barbieri, M. 2007, A&A, 462, 345, doi: 10.1051/0004-6361:20066319
  * Doyle et al. (2011) Doyle, L. R., Carter, J. A., Fabrycky, D. C., et al. 2011, Science, 333, 1602, doi: 10.1126/science.1210923
  * Duchêne (2010) Duchêne, G. 2010, ApJ, 709, L114, doi: 10.1088/2041-8205/709/2/L114
  * Duquennoy & Mayor (1991) Duquennoy, A., & Mayor, M. 1991, A&A, 500, 337
  * Dvorak (1982) Dvorak, R. 1982, Oesterreichische Akademie Wissenschaften Mathematisch naturwissenschaftliche Klasse Sitzungsberichte Abteilung, 191, 423
  * Eggenberger et al. (2011) Eggenberger, A., Udry, S., Chauvin, G., et al. 2011, in The Astrophysics of Planetary Systems: Formation, Structure, and Dynamical Evolution, ed. A. Sozzetti, M. G. Lattanzi, & A. P. Boss, Vol. 276, 409–410, doi: 10.1017/S1743921311020564
  * Eggenberger et al. (2004) Eggenberger, A., Udry, S., & Mayor, M. 2004, A&A, 417, 353, doi: 10.1051/0004-6361:20034164
  * Fabrycky & Tremaine (2007) Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298, doi: 10.1086/521702
  * Fischer & Marcy (1992) Fischer, D. A., & Marcy, G. W. 1992, ApJ, 396, 178, doi: 10.1086/171708
  * Fontanive & Bardalez Gagluiffi (2021) Fontanive, C., & Bardalez Gagluiffi, D. 2021, arXiv e-prints, arXiv:2101.12667.  https://arxiv.org/abs/2101.12667
  * Fontanive et al. (2019) Fontanive, C., Rice, K., Bonavita, M., et al. 2019, MNRAS, 485, 4967, doi: 10.1093/mnras/stz671
  * Furlan et al. (2017) Furlan, E., Ciardi, D. R., Everett, M. E., et al. 2017, AJ, 153, 71, doi: 10.3847/1538-3881/153/2/71
  * Gilliland et al. (2013) Gilliland, R. L., Marcy, G. W., Rowe, J. F., et al. 2013, ApJ, 766, 40, doi: 10.1088/0004-637X/766/1/40
  * Gould et al. (2014) Gould, A., Udalski, A., Shin, I. G., et al. 2014, Science, 345, 46, doi: 10.1126/science.1251527
  * Hatzes et al. (2003) Hatzes, A. P., Cochran, W. D., Endl, M., et al. 2003, ApJ, 599, 1383, doi: 10.1086/379281
  * Holman & Wiegert (1999) Holman, M. J., & Wiegert, P. A. 1999, AJ, 117, 621, doi: 10.1086/300695
  * Ida & Lin (2004) Ida, S., & Lin, D. N. C. 2004, ApJ, 604, 388, doi: 10.1086/381724
  * Jiang et al. (2007) Jiang, I.-G., Yeh, L.-C., Chang, Y.-C., & Hung, W.-L. 2007, AJ, 134, 2061, doi: 10.1086/522888
  * Kaib et al. (2013) Kaib, N. A., Raymond, S. N., & Duncan, M. 2013, Nature, 493, 381, doi: 10.1038/nature11780
  * Kley (2000) Kley, W. 2000, IAU Symposium, 200, 211
  * Kostov et al. (2013) Kostov, V. B., McCullough, P. R., Hinse, T. C., et al. 2013, ApJ, 770, 52, doi: 10.1088/0004-637X/770/1/52
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591, doi: 10.1086/108790
  * Kraus et al. (2016) Kraus, A. L., Ireland, M. J., Huber, D., Mann, A. W., & Dupuy, T. J. 2016, AJ, 152, 8, doi: 10.3847/0004-6256/152/1/8
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447, doi: 10.1007/BF00648343
  * Marzari & Thebault (2019) Marzari, F., & Thebault, P. 2019, Galaxies, 7, 84, doi: 10.3390/galaxies7040084
  * Matson et al. (2018) Matson, R. A., Howell, S. B., Horch, E. P., & Everett, M. E. 2018, AJ, 156, 31, doi: 10.3847/1538-3881/aac778
  * Meschiari et al. (2009) Meschiari, S., Wolf, A. S., Rivera, E., et al. 2009, PASP, 121, 1016, doi: 10.1086/605730
  * Mills et al. (2019) Mills, S. M., Howard, A. W., Petigura, E. A., et al. 2019, AJ, 157, 198, doi: 10.3847/1538-3881/ab1009
  * Moe & Kratter (2019) Moe, M., & Kratter, K. M. 2019, arXiv e-prints, arXiv:1912.01699.  https://arxiv.org/abs/1912.01699
  * Montet et al. (2015) Montet, B. T., Bowler, B. P., Shkolnik, E. L., et al. 2015, ApJ, 813, L11, doi: 10.1088/2041-8205/813/1/L11
  * Mordasini et al. (2009) Mordasini, C., Alibert, Y., Benz, W., & Naef, D. 2009, A&A, 501, 1161, doi: 10.1051/0004-6361/200810697
  * Orosz et al. (2012) Orosz, J. A., Welsh, W. F., Carter, J. A., et al. 2012, Science, 337, 1511, doi: 10.1126/science.1228380
  * Orosz et al. (2019) Orosz, J. A., Welsh, W. F., Haghighipour, N., et al. 2019, AJ, 157, 174, doi: 10.3847/1538-3881/ab0ca0
  * Raghavan et al. (2010) Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1, doi: 10.1088/0067-0049/190/1/1
  * Rodigas et al. (2016) Rodigas, T. J., Arriagada, P., Faherty, J., et al. 2016, ApJ, 818, 106, doi: 10.3847/0004-637X/818/2/106
  * Roell et al. (2012) Roell, T., Neuhäuser, R., Seifahrt, A., & Mugrauer, M. 2012, A&A, 542, A92, doi: 10.1051/0004-6361/201118051
  * Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835, doi: 10.1086/160554
  * Schwarz et al. (2016) Schwarz, R., Funk, B., Zechner, R., & Bazsó, Á. 2016, MNRAS, 460, 3598, doi: 10.1093/mnras/stw1218
  * Thebault & Haghighipour (2015) Thebault, P., & Haghighipour, N. 2015, Planet Formation in Binaries, 309–340, doi: 10.1007/978-3-662-45052-9_13
  * Townsend (2010) Townsend, R. H. D. 2010, ApJS, 191, 247, doi: 10.1088/0067-0049/191/2/247
  * Van Eylen et al. (2019) Van Eylen, V., Albrecht, S., Huang, X., et al. 2019, AJ, 157, 61, doi: 10.3847/1538-3881/aaf22f
  * Wang et al. (2014a) Wang, J., Fischer, D. A., Xie, J.-W., & Ciardi, D. R. 2014a, ApJ, 791, 111, doi: 10.1088/0004-637X/791/2/111
  * Wang et al. (2014b) Wang, J., Xie, J.-W., Barclay, T., & Fischer, D. A. 2014b, ApJ, 783, 4, doi: 10.1088/0004-637X/783/1/4
  * Wang et al. (2019) Wang, S., Jones, M., Shporer, A., et al. 2019, AJ, 157, 51, doi: 10.3847/1538-3881/aaf1b7
  * Wittenmyer et al. (2013) Wittenmyer, R. A., Wang, S., Horner, J., et al. 2013, ApJS, 208, 2, doi: 10.1088/0067-0049/208/1/2
  * Wolniewicz et al. (2021) Wolniewicz, L. M., Berger, T. A., & Huber, D. 2021, AJ, 161, 231, doi: 10.3847/1538-3881/abee1d
  * Wu & Lithwick (2011) Wu, Y., & Lithwick, Y. 2011, ApJ, 735, 109, doi: 10.1088/0004-637X/735/2/109
  * Xie et al. (2016) Xie, J.-W., Dong, S., Zhu, Z., et al. 2016, Proceedings of the National Academy of Science, 113, 11431, doi: 10.1073/pnas.1604692113
  * Zhang et al. (2018) Zhang, Y., Li, Q., Xie, J.-W., et al. 2018, ApJ, 861, 116, doi: 10.3847/1538-4357/aac6c3
  * Ziegler et al. (2017) Ziegler, C., Law, N. M., Morton, T., et al. 2017, AJ, 153, 66, doi: 10.3847/1538-3881/153/2/66
  * Ziegler et al. (2018) Ziegler, C., Law, N. M., Baranec, C., et al. 2018, AJ, 155, 161, doi: 10.3847/1538-3881/aab042
  * Zucker & Mazeh (2002) Zucker, S., & Mazeh, T. 2002, ApJ, 568, L113, doi: 10.1086/340373

##  Appendix A RV Signal Injection-Recovery Tests of the Whole Sample

In Figures 12-15, we present the results of RV signal injection-recovery tests
for all the 80 systems (110 planets) studied in this paper.

Figure 12:  RV signal injection-recovery tests of 21 binary systems with $a_{B}\leq 137$ AU. The red star symbols indicate the location of observed planets. The filled dark grey circles (light squares) show where the hypothetical planets can (can not) be recovered from the simulated RVs. The green shaded regions mark where $RV_{signal}=(1-2)\times rms$. The purple shaded regions mark where $period=(1-2)\times the\ time\ baseline$ of the simulated RV data. The black curve mark the detect limit of the binary system. Figure 13: Similar to Figure 12 but showing the  RV signal injection-recovery tests of 21 binary systems with 142 AU $<a_{B}\leq$ 380 AU.  Figure 14: Similar to Figure 12 but showing the  RV signal injection-recovery tests of 21 binary systems with 380 AU $<a_{B}\leq$ 1030 AU.The caption is similar to Figure 12.  Figure 15: Similar to Figure 12 but showing the  RV signal injection-recovery tests of 17 binary systems with $a_{B}>$ 1030 AU.The caption is similar to Figure 12.  Table 1: S type planet in binary stellar systems $row\\_id$ | $Planet\\_id$ |  $a_{B}$(AU) |  $a_{c}$(AU) | $N_{p}$ | $M_{p}$($M_{J}$) |  $a_{p}$(AU) | period(days) | Eccentricity | source | $N-star$
---|---|---|---|---|---|---|---|---|---|---
1 | HD 42936 b | 1.22 | 0.2 | 1 | 0.0081 | 0.0662 | 6.6732 | 0.14 | 1 | 2
2 | HD 87646 b | 1.58 | 4.2 | 1 | 12.4 | 0.117 | 13.481 | 0.05 | 1 | 2
3 | HD 59686 A b | 13.56 | 4.1 | 1 | 6.92 | 1.086 | 299.36 | 0.05 | 1 | 2
4 | HD 7449 b | 18 | 5.6 | 1 | 1.09 | 2.33 | 1270.5 | 0.92 | 1 | 2
5 | gamma Cephei b | 19 | 5.4 | 1 | 1.85 | 2.05 | 903.3 | 0.049 | 1 | 2
6 | HD 4113 Ab | 23.7 | 6.7 | 1 | 1.56 | 1.28 | 526.62 | 0.903 | 1 | 2
7 | HD 41004 Ab | 26.48 | 5.2 | 1 | 2.54 | 1.64 | 963 | 0.39 | 1 | 2
8 | HD 41004 Bb | 26.48 | 3.7 | 1 | 18.4 | 0.0177 | 1.3236 | 0.058 | 1 | 2
9 | 30 Ari b | 27.6 | 6.3 | 1 | 13.82 | 0.99 | 335.1 | 0.29 | 1 | 4
10 | GJ 86 b | 27.8 | 4.4 | 1 | 4.01 | 0.11 | 15.766 | 0.046 | 1 | 2
11 | HD 196885 b | 29 | 6.7 | 1 | 2.98 | 2 | 1326 | 0.48 | 1 | 2
12 | HD 2638 b | 32.2816 | 6.8 | 1 | 0.48 | 0.044 | 3.438 | 0.041 | 1 | 3
13 | HD 8673 b | 35 | 3.0 | 1 | 14.2 | 3.02 | 1634 | 0.723 | 1 | 2
14 | HD 126614 b | 45.6482 | 13.6 | 1 | 0.38 | 2.35 | 1244 | 0.41 | 1 | 3
15 | HD 164509 b | 46.0265 | 10.6 | 1 | 0.48 | 0.875 | 282.4 | 0.26 | 1 | 2
16 | GJ 3021 b | 85.747 | 22.4 | 1 | 3.37 | 0.49 | 133.71 | 0.511 | 1 | 2
17 | Gliese 15 b | 93 | 15.9 | 2 | 0.017 | 0.072 | 11.44 | 0.27 | 1 | 2
18 | Gliese 15 c | 93 | 15.9 | 2 | 0.114 | 5.4 | 7391 | 0.27 | 1 | 2
19 | HD 30856 b | 117.273 | 25.6 | 1 | 1.8 | 2 | 912 | 0.24 | 1 | 2
20 | HD 177830 b | 123.57 | 31.5 | 2 | 1.49 | 1.2218 | 406.6 | 0.001 | 1 | 2
21 | HD 177830 c | 123.57 | 31.5 | 2 | 0.15 | 0.5137 | 110.9 | 0.35 | 1 | 2
22 | HD 142 b | 132.405 | 37.4 | 2 | 1.25 | 1.02 | 349.7 | 0.17 | 1 | 2
23 | HD 142 c | 132.405 | 37.4 | 2 | 5.3 | 6.8 | 6005 | 0.21 | 1 | 2
24 | HD 79211 | 136.86894 | 27.4 | 1 | 0.3231 | 0.141 | 24.45 | 0.11 | 1 | 2
25 | HD 207832 b | 142.493 | 35.2 | 2 | 0.56 | 0.586 | 160.07 | 0.197 | 1 | 4
26 | HD 207832 c | 142.493 | 35.2 | 2 | 0.73 | 2.112 | 1155.7 | 0.27 | 1 | 4
27 | HD 197037A b | 150.059 | 35.4 | 1 | 0.79 | 2.07 | 1035.7 | 0.22 | 1 | 2
28 | HD 195019 b | 165.19 | 37.8 | 1 | 3.7 | 0.1388 | 18.2016 | 0.014 | 1 | 2
29 | HD 114762 b | 166.452 | 42.1 | 1 | 10.98 | 0.353 | 83.9151 | 0.3354 | 1 | 2
30 | HD 217786 b | 197.97 | 50.4 | 1 | 13 | 2.38 | 1319 | 0.4 | 1 | 2
31 | HD 98736 b | 199.23 | 39.7 | 1 | 2.33 | 1.864 | 968.8 | 0.226 | 1 | 2
32 | HD 116029 b | 215.63 | 55.4 | 2 | 2.1 | 1.73 | 670.2 | 0.21 | 1 | 2
33 | HD 116029 c | 215.63 | 55.4 | 2 | 1.27 | 2.144 | 907 | 0.038 | 1 | 2
34 | HD 19994 b | 220 | 49.4 | 1 | 1.68 | 1.42 | 535.7 | 0.3 | 1 | 2
35 | tau Bootis b | 221 | 5.3 | 1 | 4.13 | 0.046 | 3.3135 | 0.0787 | 1 | 2
36 | HD 212301 b | 238 | 69.6 | 1 | 0.4 | 0.036 | 2.2457 | 0.0147 | 1 | 2
37 | HD 142245A b | 244 | 72.0 | 1 | 1.9 | 2.77 | 1299 | 0.32 | 1 | 3
38 | HD 65216 b | 256.8 | 84.9 | 1 | 1.21 | 1.37 | 613.1 | 0.41 | 1 | 3
39 | HD 185269A b | 271.11 | 68.9 | 1 | 0.94 | 0.077 | 6.838 | 0.3 | 1 | 2
40 | HD 16141 b | 281.203 | 66.3 | 1 | 0.215 | 0.35 | 75.82 | 0.28 | 1 | 2
Table 1: continued $row\\_id$ | $Planet\\_id$ |  $a_{B}$(AU) |  $a_{c}$(AU) | $N_{p}$ | $M_{p}$($M_{J}$) |  $a_{p}$(AU) | period(days) | Eccentricity | source | $N-star$
---|---|---|---|---|---|---|---|---|---|---
41 | Gliese 667 b | 300 | 52.3 | 6 | 0.0176 | 0.0505 | 7.2 | 0.13 | 1 | 3
42 | Gliese 667 c | 300 | 52.3 | 6 | 0.0119 | 0.125 | 28.14 | 0.02 | 1 | 3
43 | Gliese 667 d | 300 | 52.3 | 6 | 0.016 | 0.276 | 91.61 | 0.03 | 1 | 3
44 | Gliese 667 e | 300 | 52.3 | 6 | 0.0085 | 0.213 | 62.24 | 0.02 | 1 | 3
45 | Gliese 667 f | 300 | 52.3 | 6 | 0.0085 | 0.156 | 39.03 | 0.03 | 1 | 3
46 | Gliese 667 g | 300 | 52.3 | 6 | 0.0145 | 0.549 | 256.2 | 0.08 | 1 | 3
47 | HD 27442 b | 302.64 | 61.2 | 1 | 1.35 | 1.16 | 415.2 | 0.058 | 1 | 2
48 | HD 28254A b | 302.64 | 59.8 | 1 | 1.16 | 2.15 | 1116 | 0.81 | 1 | 2
49 | HD 106515 b | 345 | 61.9 | 1 | 9.61 | 4.59 | 3630 | 0.572 | 1 | 2
50 | HD 86081A b | 348 | 94.9 | 1 | 1.5 | 0.039 | 1.99809 | 0.0575 | 1 | 2
51 | HD 114729 b | 355.602 | 85.5 | 1 | 0.84 | 2.08 | 1135 | 0.32 | 1 | 2
52 | HD 133131 Ab | 379.22 | 80.9 | 2 | 1.43 | 1.44 | 649 | 0.32 | 1 | 2
53 | HD 133131 Ac | 379.22 | 80.9 | 2 | 0.48 | 4.36 | 3407 | 0.47 | 1 | 2
54 | HD 133131 Bb | 379.22 | 79.9 | 1 | 2.5 | 6.415 | 6119 | 0.62 | 1 | 2
55 | HD 108341 b | 383 | 95.5 | 1 | 3.5 | 2 | 1129 | 0.85 | 1 | 2
56 | HD 102365 b | 406.042 | 99.8 | 1 | 0.05 | 0.46 | 122.1 | 0.34 | 1 | 2
57 | HD 220842A b | 428.739 | 85.4 | 1 | 3.18 | 0.74 | 218.47 | 0.404 | 1 | 2
58 | HD 132563 b | 430.75 | 86.7 | 1 | 1.49 | 2.62 | 1544 | 0.22 | 1 | 3
59 | HD 46375 b | 435 | 71.5 | 1 | 0.249 | 0.041 | 3.024 | 0.0524 | 1 | 2
60 | HD 43691 b | 452.69 | 111.9 | 1 | 2.57 | 0.238 | 36.99913 | 0.085 | 1 | 3
61 | omi UMaA b | 518.271 | 103.0 | 1 | 4.1 | 3.9 | 1630 | 0.191 | 1 | 2
62 | HD 180617 b | 560.6406 | 144.5 | 1 | 0.0384 | 0.3357 | 105.9 | 0.16 | 1 | 2
63 | HD 4732 c | 605.9105 | 113.0 | 2 | 2.37 | 4.6 | 2732 | 0.23 | 1 | 2
64 | HD 4732 b | 605.9105 | 120.6 | 2 | 2.37 | 1.19 | 360.2 | 0.13 | 1 | 2
65 | HD 109749 b | 618.5205 | 121.0 | 1 | 0.28 | 0.0635 | 5.24 | 0.01 | 2 | 2
66 | HD 196050 b | 631.761 | 156.9 | 1 | 2.83 | 2.47 | 1316.24 | 0.21 | 2 | 3
67 | HD 178911 b | 640 | 173.9 | 1 | 6.29 | 0.32 | 71.49 | 0.124 | 2 | 3
68 | HD 96167 b | 645.632 | 172.1 | 1 | 0.68 | 1.3 | 498.9 | 0.71 | 2 | 2
69 | HD 99492 b | 649.415 | 121.6 | 2 | 0.087 | 0.122 | 17.1668 | 0.13 | 2 | 2
70 | HD 99492 c | 649.415 | 121.6 | 2 | 0.359 | 5.4 | 13.1 | 0.1 | 2 | 2
71 | Ups And b | 750 | 245.1 | 4 | 0.62 | 0.059 | 4.6171 | 0.013 | 2 | 2
72 | Ups And c | 750 | 245.1 | 4 | 1.8 | 0.861 | 237.7 | 0.24 | 2 | 2
73 | Ups And d | 750 | 245.1 | 4 | 10.19 | 2.55 | 1302.61 | 0.274 | 2 | 2
74 | Ups And e | 750 | 245.1 | 4 | 1.059 | 5.2456 | 3848.86 | 0.005 | 2 | 2
75 | HD 75289 b | 781.819 | 204.7 | 1 | 0.47 | 0.046 | 3.5098 | 0.021 | 2 | 2
76 | HD 188015 b | 862 | 217.6 | 1 | 1.26 | 1.19 | 456.46 | 0.15 | 2 | 2
77 | HD 147379 b | 869.4 | 171.8 | 1 | 0.0777 | 0.3193 | 86.54 | 0.01 | 2 | 2
78 | GJ 676 b | 1010 | 247.8 | 4 | 4.95 | 1.8 | 1050.3 | 0.328 | 2 | 2
79 | GJ 676 c | 1010 | 247.8 | 4 | 3 | 5.2 | 4400 | 0.2 | 2 | 2
80 | GJ 676 d | 1010 | 247.8 | 4 | 0.014 | 0.0413 | 3.6 | 0.15 | 2 | 2
Table 1: continued

$row\\_id$ | $Planet\\_id$ |  $a_{B}$(AU) |  $a_{c}$(AU) | $N_{p}$ | $M_{p}$($M_{J}$) |  $a_{p}$(AU) | period(days) | Eccentricity | source | $N-star$
---|---|---|---|---|---|---|---|---|---|---
81 | GJ 676 e | 1010 | 247.8 | 4 | 0.036 | 0.187 | 35.37 | 0.24 | 2 | 2
82 | HD 142022 b | 1030 | 210.6 | 1 | 5.1 | 3.03 | 1928 | 0.53 | 2 | 2
83 | 55 Cnc b | 1065 | 350.5 | 5 | 0.8 | 0.1134 | 14.651 | 0.016 | 2 | 2
84 | 55 Cnc c | 1065 | 350.5 | 5 | 0.169 | 0.2403 | 44.3446 | 0.053 | 2 | 2
85 | 55 Cnc d | 1065 | 350.5 | 5 | 3.835 | 5.76 | 5218 | 0.025 | 2 | 2
86 | 55 Cnc f | 1065 | 350.5 | 5 | 0.144 | 0.781 | 260.7 | 0 | 2 | 2
87 | 16 Cyg b | 1080 | 189.7 | 1 | 1.68 | 1.68 | 799.5 | 0.689 | 2 | 3
88 | HD 11964 b | 1272.6 | 258.8 | 2 | 0.622 | 3.16 | 1945 | 0.041 | 2 | 2
89 | HD 11964 c | 1272.6 | 258.8 | 2 | 0.079 | 0.229 | 37.91 | 0.3 | 2 | 2
90 | HD 101930 b | 2772 | 507.2 | 1 | 0.3 | 0.302 | 70.46 | 0.11 | 2 | 2
91 | 91 Aqr b | 2835 | 554.8 | 1 | 3.2 | 0.7 | 181.4 | 0.03 | 2 | 3
92 | HD 89744 b | 3100 | 856.3 | 2 | 7.2 | 0.88 | 256 | 0.7 | 2 | 2
93 | HD 89744 c | 3100 | 856.3 | 2 | 3.2 | 0.44 | 85.2 | 0.29 | 2 | 2
94 | HD 190360 b | 3585.96 | 905.3 | 2 | 1.502 | 3.92 | 2891 | 0.36 | 2 | 2
95 | HD 190360 c | 3585.96 | 905.3 | 2 | 0.057 | 0.128 | 17.1 | 0.01 | 2 | 2
96 | HD 213240 b | 4915.378 | 1301.1 | 1 | 4.5 | 2.03 | 951 | 0.45 | 2 | 3
97 | XO-2S b | 5796 | 1027 | 2 | 0.259 | 0.1344 | 18.157 | 0.18 | 2 | 2
98 | XO-2S c | 5796 | 1027 | 2 | 1.37 | 0.4756 | 120.8 | 0.153 | 2 | 2
99 | HD 222582 b | 5990 | 1416.5 | 1 | 7.75 | 1.35 | 572.38 | 0.725 | 2 | 2
100 | HD 125612 b | 5990 | 2453.9 | 3 | 3 | 1.37 | 502 | 0.46 | 2 | 2
101 | HD 125612 c | 5990 | 2453.9 | 3 | 0.058 | 0.05 | 4.1547 | 0.27 | 2 | 2
102 | HD 125612 d | 5990 | 2453.9 | 3 | 7.2 | 4.2 | 3008 | 0.28 | 2 | 2
103 | HD 147513 b | 6760 | 1326.9 | 1 | 1.21 | 1.32 | 528.4 | 0.26 | 2 | 2
104 | HD 40979 b | 8070.4 | 1764.5 | 1 | 3.28 | 0.83 | 263.1 | 0.25 | 2 | 3
105 | Prox Cen b | 10962 | 2041.7 | 2 | 0.003 | 0.0485 | 11.186 | 0 | 2 | 3
106 | Prox Cen c | 10962 | 2041.7 | 2 | 0.0182 | 1.48 | 1894 | 0 | 2 | 3
107 | HIP 70849 b | 11340 | 3081.9 | 1 | 9 | 10 | 10000 | 0.6 | 2 | 2
108 | HD 20782 b | 11550 | 2386.2 | 1 | 1.9 | 1.381 | 591.9 | 0.97 | 2 | 2
109 | HD 38529 c | 15184.962 | 3805.3 | 2 | 17.7 | 3.695 | 2134.76 | 0.36 | 2 | 2
110 | HD 38529 b | 15184.962 | 3805.3 | 2 | 0.87 | 0.131 | 14.31 | 0.25 | 2 | 2

Note. — $a_{B}$: the semimajor axis of binary orbit, calculated via the
empirical relationship (Fischer & Marcy, 1992) $a_{B}$/r = 1.261 (r is
projected binary separation) if there is no reported $a_{B}$ in the source
catalog. $a_{c}$:the critical distance to the primary star to maintain long-
term orbital stability (Holman & Wiegert, 1999). Source: 1, the catalog of
exoplanet-hosting binaries with separations up to 500
AU333http://exoplanet.eu/planets_binary/; 2, the catalogue of exoplanets in
binary star systems (Schwarz et al.,
2016)444https://www.univie.ac.at/adg/schwarz/multiple.html. N-star:2, binary
star system; 3 or 4, triple system or quadruple system.