DRAFT VERSION FEBRUARY 5, 2018Preprint typeset using LATEX style emulateapj v. 12/16/11

HIDDEN PLANETARY FRIENDS: ON THE STABILITY OF 2-PLANET SYSTEMS IN THE PRESENCE OF A DISTANT,INCLINED COMPANION

PAUL DENHAM1, SMADAR NAOZ1,2, BAO-MINH HOANG1, ALEXANDER P. STEPHAN1, WILL M. FARR3

1Physics and Astronomy Department, University of California, Los Angeles, CA 900242Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA and

3School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UKDraft version February 5, 2018

ABSTRACTRecent observational campaigns have shown that multi-planet systems seem to be abundant in our Galaxy.

Moreover, it seems that these systems might have distant companions, either planets, brown-dwarfs or otherstellar objects. These companions might be inclined with respect to the inner planets, and could potentiallyexcite the eccentricities of the inner planets through the Eccentric Kozai-Lidov mechanism. These eccentricityexcitations could perhaps cause the inner orbits to cross, disrupting the inner system. We study the stabilityof two-planet systems in the presence of a distant, inclined, giant planet. Specifically, we derive a stabilitycriterion, which depends on the companion’s separation and eccentricity. We show that our analytic criterionagrees with the results obtained from numerically integrating an ensemble of systems. Finally, as a potentialproof-of-concept, we provide a set of predictions for the parameter space that allows the existence of planetarycompanions for the Kepler-56, Kepler-448, Kepler-88, Kepler-109, and Kepler-36 systems.

1. INTRODUCTION

Recent ground and space-based observations have shownthat multi-planet systems are abundant around main-sequencestars (e.g., Howard et al. 2010, 2012; Borucki et al. 2011; Lis-sauer et al. 2011; Mayor et al. 2011; Youdin 2011; Batalhaet al. 2013; Dressing & Charbonneau 2013; Petigura et al.2013; Christiansen et al. 2015). These studies reveal that thearchitecture of planetary systems can drastically vary fromour solar system. For example, systems consisting of mul-tiple low mass (sub-Jovian) planets with relatively compactorbits usually have periods that are shorter than Mercury’s.

NASA’s Kepler mission found an abundance of compactmulti-planet super-Earths or sub-Neptune systems (e.g., Mul-lally et al. 2015; Burke et al. 2015; Morton et al. 2016; Hansen2017). These systems seemed to have low eccentricities (e.g.,Lithwick et al. 2012; Van Eylen & Albrecht 2015). In addi-tion, it was suggested that these many body systems are closeto the dynamically stable limit (e.g., Fang & Margot 2013;Pu & Wu 2015; Volk & Gladman 2015). It was also proposedthat single planet systems might be the product of systems ini-tially consisting of multiple planets that underwent a period ofdisruption, i.e, collisions, resulting in reducing the number ofplanets (see Johansen et al. (2012) Becker & Adams (2016)).

Giant planets may play a key role in forming inner plan-etary systems. Radial velocity surveys, along with the exis-tence of Jupiter, have shown giants usually reside at largerdistances from their host star than other planets in the sys-tem (� 1 AU)(e.g., Knutson et al. 2014; Bryan et al. 2016).For example, there were some suggestions that the near-resonant gravitational interactions between giant outer plan-ets and smaller inner planets can shape the configuration of aninner system’s asteroid belt (e.g., Williams & Faulkner 1981;Minton & Malhotra 2011). Hence, it was suggested that oursolar system’s inner planets are of a second generation, whichcame after Jupiter migrated inward to its current orbit (e.g.,Batygin & Laughlin 2015). Furthermore, secular resonance

pdenham629@gmail.comsnaoz@astro.ucla.edu

may be the cause of water being delivered to earth (Scholl &Froeschle 1986; Morbidelli 1994), the potential instability ofMercury’s orbit (Neron de Surgy & Laskar 1997; Fienga et al.2009; Batygin & Laughlin 2008; Lithwick & Wu 2011), andthe obliquity of the exoplanets’ in general (e.g., Naoz et al.2011; Li et al. 2014b).

Recently, several studies showed that the presence of a gi-ant planet can affect the ability to detect an inner system (e.g.,Hansen 2017; Becker & Adams 2017; Mustill et al. 2017;Jontof-Hutter et al. 2017; Huang et al. 2017). Specifically, dy-namical interactions from a giant planet, having a semi-majoraxis much greater than the planets in the inner system, canexcite eccentricities and inclinations of the inner planets. Apossible effect is that the inner system becomes incapable ofhaving multiple transits or completely unstable. Volk & Glad-man (2015) showed that observed multi-planet systems mayactually be the remnants of a compact system that was tighterin the past but lost planets through dynamical instabilities andcollisions (see also Petrovich et al. 2014). Interestingly, ver-ifying these problems could reconcile the Kepler dichotomy(e.g., Johansen et al. 2012; Ballard & Johnson 2016; Hansen2017).

In this work, we investigate the stability of compact sub-Jovian inner planetary systems in the presence of a distantgiant planet (see Figure 2 for an illustration of the system).Over long time scales, a distant giant planets’ gravitationalperturbations can excite the eccentricities of the inner plan-ets’ to high values, destabilizing the inner system. (e.g., Naozet al. 2011). However, if the frequency of angular momentumexchange between the inner planets is sufficiently high, thenthe inner system can stabilize. Below we derive an analyticstability criterion (Section 2). Then we analyze our nominalsystem in Section 3 and provide specific predictions for Ke-pler systems in Section 3.3. Finally, we offer our conclusionin Section 4.

Near the completion of this work, we became aware of (Pu& Lai 2018) a complementary study of stability in similar sys-tems. Here we provide a comprehensive stability criterion asa function of the companion’s parameters. Furthermore, we

arX

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0044

7v1

[astr

o-ph

.EP]

1 F

eb 2

018

2

FIG. 1.— Here is a schematic of the systems being analyzed. The inclina-tion of the third planet, i3, is shown and measured relative to the z-axis, alongthe spin axis of the host star. M,m1,m2, and m3 are the masses of the hoststar, and the first, second and third planets respectively.

provide a set of predictions for possible hidden companionorbits in several observed systems.

2. ANALYTICAL STABILITY CRITERION

Here we develop a generic treatment of the stability of two-body systems in the presence of an inclined outer planet. Con-sider an inner system consisting of two planets (m1 and m2)orbiting around a host star M with a relatively tight config-uration (with semi-major axis a1 and a2, respectively). Weintroduce an inclined and eccentric companion that is muchfarther from the host star than the planets in the inner system(m3 and a3, see Figure 1 for an illustration of the system).We initialize this system to have orbits far from mean-motionresonance, and have a1,a2 << a3. The three planets’ orbitshave corresponding eccentricities e1, e2 and e3. We set an ar-bitrary z-axis to be perpendicular to the initial orbital plane ofthe inner two planets, thus, the inclinations of m1, m2 and m3are defined as i1, i2 and i3. Accordingly, for planets one, two,and three we denote the longitude of ascending nodes and theargument of periapse ⌦1, ⌦2, ⌦3, !1,!2, and !3.

The outer orbit can excite the eccentricities, via the Ec-centric Kozai-Lidov mechanism (EKL) (Kozai (1962), Lidov(1962), see Naoz (2016) for review) on each planet in theinner system. The EKL resonance causes angular momen-tum exchange between the outer and inner planets, which inturn causes precession of the periapse of each of the inner or-bits. However, angular momentum exchange between the in-ner two planets also induces precession of the periapse ("theso-called Laplace-Lagrange interactions, e.g., Murray & Der-mott 2000). If the inner orbits’ angular momentum exchangetakes place at a faster rate than that induced by the outer com-panion, then the system will not be disrupted by perturbationsfrom the tertiary planet. The quadrupole approximation to thetimescale of the EKL between the third planet m3 and the sec-ond one, m2 is given by:

⌧k(2,3) ⇠

1615

a

33

a

3/22

pM + m2 + m1

m3p

G

(1 - e

23)3/2 , (1)

(e.g., Antognini 2015) where G is the gravitational constant.This is roughly the timescale at which the second planet’s ar-gument of periapse precesses. Note that here we consideredthe timescale of EKL excitations from the tertiary on m2 sincethis timescale is shorter than the timescale of EKL excitations

between m3 and m1. Planet m2’s argument of periapse alsoprecesses due to gravitational perturbations from the inner-most planet, m1. The associated timescale is (e.g., Murray &Dermott 2000)

⌧LL

⇠

A22 + A21

✓e1

e2

◆cos(v2 - v1)

- B22 - B21

✓i1

i2

◆cos(⌦2 -⌦1)

�-1

, (2)

where vj

= !j

+⌦j

for j = 1,2. The A

i, j, and B

i, j are laplacecoefficients, which are determined by:

A22 = n21

4⇡m1

M + m2

✓a1

a2

◆2

f , (3)

A21 = -n21

4⇡m1

M + m2

✓a1

a2

◆f2 , (4)

B22 = -n21

4⇡m1

M + m2

✓a1

a2

◆2

f , (5)

B21 = n21

4⇡m1

M + m2

✓a1

a2

◆f . (6)

and

f =Z 2⇡

0

cos ✓

1 - 2⇣

a1a2

⌘cos +

⇣a1a2

⌘2◆ 3

2d , (7)

f2 =Z 2⇡

0

cos2 ✓

1 - 2⇣

a1a2

⌘cos +

⇣a1a2

⌘2◆ 3

2d . (8)

The system will remain stable if angular momentum ex-change between the two inner planets takes place faster thanthe precession induced by m3. Accordingly, there exists tworegions of parameter space: one contains systems where per-turbations on m2 are dominantly from m1, and the other con-tains systems where m2 is dominated by m3. The former iscalled the Laplace-Lagrange region, and the ladder is calledEKL region. The transition between the Laplace-Lagrangeregion and EKL region is determined by comparing the twotimescales relating the frequencies of precession from eachmechanism, i.e.,

⌧k

⇠ ⌧LL

. (9)

We have related the timescale for Kozai oscillations inducedby m3 on m2 to the Laplace-Lagrange timescale between m1and m2. Equating these timescales yields a simple expressionfor the critical eccentricity of the third planet, e3,c, as a func-tion of a2, i.e.,

e3,c ⇠

0

@1 -

"1516

m3p

GpM� + m1 + m2

a

3/22

a

33

1f

LL

# 231

A

12

, (10)

where

f

LL

= A22 + A21e1,ic

e2,iccos(v1 - v2) - B21

i1,ic

i2,iccos(⌦1 -⌦2) - B22 .

(11)e

j,ic and i

j,ic, are the initial eccentricity and inclination for thetwo inner planets, i.e., j = 1,2. As we show below, during the

3

evolution of a stable system ⌦1 ⇠ ⌦2, v2 ⇠ v1, e1 ⇠ e2, andi1 ⇠ i2. Thus, we define a minimum stable configuration off

LL,min

f

LL,min = A22 + e1,ic

e2,icA21 - i1,ic

i2,icB21 - B22 . (12)

Numerically we find that during the evolution of an unstablesystem, e1/e2, i1/i2, ⌦1/⌦2, and v2/v1 largely deviate fromunity, where cos(v1 - v2) can be negative. Thus, we also de-fine the maximum stability as:

f

LL,max = A22 - e1,ic

e2,icA21 - i1,ic

i2,icB21 - B22 , (13)

where the difference between Eq. (12) and (13) is the signof A21. The stability of systems transitions from the mini-mum to the maximum f

LL

, as a function of e3. In other wordswe find a band of parameter space, between e3,c( f

LL,min) ande3,c( f

LL,max), where systems are nearly unstable or completelyunstable. If the third planet’s eccentricity is larger than theright hand side of Equation (10), then the inner system is morelikely to become unstable. In the next section we test this sta-bility criterion and show that it agrees with secular numericalintegration.

We note that this stability criterion is based on the Laplace-Lagrange approximation, which assumes small eccentricitiesand inclinations for orbits in the inner system. Thus, it mightbreak down for initial large eccentricities or mutual inclina-tions of the inner two planets. Furthermore, this stability cri-terion assumes that the inner system is compact and that theeccentricity excitations from the second planet on the first aresuppressed. On the other hand, in the presence of these condi-tions the stability criterion depends only on the shortest EKLtimescale induced by the far away companion and the corre-sponding Laplace-Lagrange timescale. Thus, it is straightfor-ward to generalize it to more than two inner planets.

3. LONG TERM STABILITY OF MULTI-PLANET SYSTEM

3.1. The Gaussian averaging method

To test our analytic stability criterion we utilize Gauss’sMethod. This prescription allows us to integrate the systemover a long timescale (set to be 10 Gyr, see below), in atime-efficient way. In this mathematical framework, the non-resonant planets’ orbits are phase-averaged and are treated asmassive, pliable wires interacting with each other. The line-density of each wire is inversely proportional to the orbitalvelocity of each planet. The secular, orbit-averaged methodrequires that the semi-major axes of the wires are constants ofmotion. We calculate the forces that different wires exert oneach other over time.

This method was recently extended to include softenedgravitational interactions by introducing a small parameter inthe interaction potentials to avoid possible divergences whileintegrating (Touma et al. 2009). Furthermore, the method hasbeen proven to be very powerful in addressing different prob-lems in astrophysics, ranging from understanding the dynam-ics in galactic nuclei to describing how debris disks evolve.(e.g., Touma et al. 2009; Sridhar & Touma 2016; Nesvoldet al. 2016; Saillenfest et al. 2017).

3.2. Stability test on a nominal system

We tested our stability criterion by numerically integratingan ensemble of systems with identical initial conditions ex-cept for the semi-major axis and eccentricity of the second

and third planets respectively. In this ensemble, we initializeall systems with a primary star M = 1 M� orbited by threeplanets where the two inner planets, m1, and m2, each havemasses of 1 M�. The innermost planet, m1, was placed inan orbit with a semi-major axis of a1 = 1 au, and both innerplanets were set initially on circular orbits in the same orbitalplane. We set the third planet with a mass of m3 = 1 M

J

, ata3 = 100 au. Throughout the ensemble, a2 ranges from 1.4 auto 5.9 au, in steps of 0.5 au, and e3 ranges from zero to 0.95 insteps of 0.05. In all such systems, the third planet’s inclinationwas set to be 45� relative to the inner system. We integratedeach system to 10 Gyr, or until orbits crossed. From the re-sults we made a grid labeling the stable/unstable systems ofthe ensemble. The grid revealed that stable orbits are boundedby the stability criterion. The same process was done on anensemble which had the third planet’s inclination set to 65�

instead; the result was the same.

FIG. 2.— Here is the parameter space relating the third planets’ eccentric-ity with the second planet’ separation. The distant companion’s inclinationwas set to 45�. The color code shows log(1 - �), Eq. (14). This parameterestimates the closest the two inner orbits came to each other during the evo-lution. A darker color characterizes far away separation while a lighter colorcharacterizes orbit crossing. The stability criterion, Eq. (10), is plotted overthe grid to show that stable orbits are bounded. The bottom green line repre-sents e3,c( f

LL,min) and the top green line represents e3,c( f

LL,max), and thus theshaded band is the transition zone (see text). Systems above the zone undergoan instability episode while systems below the zone are stable.

In Figure 2 we show the grid of systems we integrated inthe parameter space relating a2 and e3. The color code is de-termined by a proxy that characterizes the stability of the sys-tem. The proxy is defined as log(1-�), where � is a parameterwhich describes how close the two inner orbits came duringthe evolution:

� = min

a2(1 + e2) - a1(1 - e1)a2 - a1

�. (14)

In Figure 2, a lighter color means a smaller value of �, whichwhen zero yields orbit crossing, while a darker color repre-sents a stable configuration. The criterion for stability is plot-ted over the grid to show that it is in agreement with the prob-ability for orbits to cross. Above the stability curve, systemsare more likely to destabilize during evolution.

The proxy also reveals a transitional zone of the parameter

4

FIG. 3.— Here are the three prominent types of dynamics from each region. From left to right, stable, intermediate, and unstable, for our nominal system.Shown in the top panels are the apocenters and the pericenters of the innermost planets (m1 in blue and m2) in red. The middle panels show the inclinations ofthe inner planets (i1, in blue and i2, in red). Note that this inclination is not with respect to the total angular momentum, but rather with respect to the initial

angular momentum of the two inner planets. The bottom panels show, colored in black, the difference between the longitude of ascending nodes of the two innerplanets as cos(⌦1 -⌦2), and, in yellow, the difference between the longitude of the periapsis as cos(v1 - v2). These two parameters are present in Equation(11). Recall that the nominal system has the following parameters: M = 1 M�, m1 = m2 = 1 M�, m3 = 1 M

J

, a1 = 1 au, a3 = 100 au, and we set initially!1 = !2 = !3 = ⌦1 = ⌦2 = ⌦3 = 0, e1 = e2 = 0.001, i1 = i2 = 0.001 and i3 = 45�. The only difference between each integrated system is the second planet’sseparation and the third planet’s eccentricity. For the stable system we chose a2 = 3 au and e3 = 0.8. For the intermediate system we had a2 = 3 au and e3 = 0.9(which placed it on the stability curve), and finally for the unstable system we had a2 = 5.9 au and e3 = 0.551.

5

space. This zone agrees with our analytically determined zone(i.e., between e3,c( f

LL,min) and e3,c( f

LL,max)), where, given theinitial conditions in the numerical runs, we set e1,ic/e2,ic ⇠ 1and i1,ic/i2,ic ⇠ 1. Closer to the stability curve, the orbits mightget their eccentricities excited and periodically move closer toone another, but their orbits may never cross. Moreover, farinto the top right of Figure 2, the instability will take placesooner in the evolution. In this region, the two inner orbitswill undergo EKL evolution independently of each other.

The third planet can excite the inner planets’ eccentric-ities on the EKL timescale. However, eccentricity excita-tions of each planet will not necessarily cause orbit cross-ing, as depicted in Figure 2. In the parameter space for ourensemble, we have identified a region containing stable sys-tems that seems to smoothly transition into the instability re-gion. Hence, we identify three regimes: The stable regime,the intermediate regime, and the unstable regime. In the sta-ble regime, the Laplace-Lagrange rapid angular momentumexchange between the two inner planets dominates over thegravitational perturbations of the outer orbit. The left columnof Figure 3 depicts this evolution; the system in this columnresides in the bottom left of the parameter space in Figure 2.The system remains stable for 10 Gyrs of evolution and the or-bits never come close to one another. The two inner planets’inclinations oscillate together around the initial z-axis, due tothe precession of the nodes (see below). In the intermedi-ate region, systems might appear to be stable for a long timebut the outer orbit perturbations become too dominant, caus-ing instability of the inner system. We show the dynamics ofthis type of evolution in the middle panel of Figure 3. Thissystem resides very close to the stability criterion plotted inFigure 2, inside the transition zone between e3,c( f

LL,min) ande3,c( f

LL,max). In the unstable regime, gravitational perturba-tions from the third planet cause high amplitude eccentricityexcitation in the inner planets, causing orbits to cross. Weshow this behavior in the right column 3. This system lies inthe top right of the parameter space depicted in Figure 2.

In the system’s reference frame (see Figure 1), the innerplanets’ inclinations are the angles between their respectiveangular momenta and the normal to the initial orbits. Thus,the inclination modulation shown in the stable system (the leftcolumn in Figure 3, is due to precession of the nodes, whichresults in maximum inclination, which is ⇠ 2⇥ i3, since wehave started with i3,0 = 45� (e.g., Innanen et al. 1997). In con-trast, the unstable regime results in misalignment between theinner two planets.

3.3. Applications: predictions for observed systems

The stability criterion derived here can be used to predictthe parameter space in which a hidden companion can ex-ist within a system without destabilizing it. As a proof-of-concept, we discuss the stability of a few non-resonant ob-served exoplanetary systems in the presence of inclined plan-ets. Specifically, for some observed systems, we provide a setof predictions that characterize the ranges of parameter spaceavailable for hidden companions to exist in without disruptinginner orbits. We focus on the following systems: Kepler-419,Kepler-56, Kepler-448, and Kepler-36. These represent thefew systems that characterize the extreme limits of two planetsystems, from tightly packed super-earths such as Kepler-36to hierarchical eccentric warm Jupiter systems such as Kepler-448. In Figure 4 we show the relevant parameter space inwhich the systems can have a hidden inclined companion andremain stable. We chose the more conservative stability cri-

name SMA [au] mass [MJ

] eccentricityKepler-36b 0.12 0.015 < 0.04Kepler-36c 0.13 0.027 < 0.04Kepler-56b 0.103 0.07 -Kepler-56c 0.17 0.57 -Kepler-88b 0.097 0.027 0.06Kepler-88c 0.15 0.62 0.056Kepler-109b 0.07 0.023 0.21Kepler-109c 0.15 0.07 0.03Kepler 419b 0.37 2.5 0.83Kepler 419c 1.68 7.3 0.18Kepler-448b 0.15 10 0.34Kepler-448c 4.2 22 0.65

TABLE 1OBSERVABLE PARAMETERS OF THE EXAMPLE SYSTEMS. THE

OBSERVATIONS ARE ADOPTED FROM: Kepler-36: CARTER ET AL.(2012). Kepler-56: HUBER ET AL. (2013) AND OTOR ET AL. (2016).

Kepler-88 NESVORNÝ ET AL. (2013), BARROS ET AL. (2014) ANDDELISLE ET AL. (2017). Kepler-109: MARCY ET AL. (2014), VANEYLEN & ALBRECHT (2015) AND SILVA AGUIRRE ET AL. (2015).Kepler-419:DAWSON ET AL. (2012), DAWSON ET AL. (2014) AND

HUANG ET AL. (2016). Kepler-448: BOURRIER ET AL. (2015), JOHNSONET AL. (2017) AND MASUDA (2017).

terion for this exercise, i.e., e3,c( f

LL,min). Each line in thefour panels of Figure 4 shows the stability criterion for dif-ferent companion masses. In particular, we consider compan-ion masses of (from top to bottom), 0.1,0.5,1,5 and 20 M�.For each companion mass, allowed system configurations liebelow the curve, and unstable configurations are above thecurve. We caution that very close to the stability curve (evenbelow the curve) systems may still undergo eccentricity exci-tations that may affect the dynamics. The parameters of theinner, observed planets were taken from observations, see Ta-ble 1. Below we discuss the specifics of the four examplesystems.

• Kepler-36 is an, approximately, one solar mass starorbited by two few-Earth mass planets on 13.8 and16.2 days orbits (Carter et al. 2012). This compact con-figuration is expected to be stable in the presence ofperturbations as shown in Figure 4, in the top left panel.For example, as can be seen in the Figure, an eccentricinclined 20 M

J

brown dwarf can reside slightly beyond2 au.

• Kepler-56 is an evolved star (M = 1.37 M�) at the baseof the red giant branch and is orbited by two sub-Jovianplanets with low mutual inclinations (e.g., Huber et al.2013). Most notably, asteroseismology analysis placeda lower limit on the two planets’ obliquities of 37�. Liet al. (2014b) showed that a large obliquity for this sys-tem is consistent with a dynamical origin. They sug-gested that a third outer companion is expected to residefar away and used the inner two planets’ obliquities toconstrain their inclinations. Follow-up radial velocitymeasurements estimated that a third companion indeedexists with a period of 1000 days and a minimum massof 5.6 M

J

(e.g., Otor et al. 2016). Here we show that in-deed a ⇠ 5 M

J

planet can exist at ⇠ 3 au, with a rangeof possible eccentricities up to 0.9. A more massiveplanet can still exist at ⇠ 3 au with a possible range ofeccentricities up to slightly below e3 ⇠ 0.8. This ex-ample of a tightly packed system yields the expectedresult, i.e, a large part of the parameter space is allowed

6

FIG. 4.— The parameter space of hidden friends for a few observed systems. We consider the companion’s eccentricity e3 and separation a3 for five Keplersystems. For each of the systems we plot the stability criterion e3,c( f

LL,min), for the different companion masses. In particular, we consider companion mass of(from top to bottom), 0.1,0.5,1,5 and 20 M

J

. The stable region exists below each curve and the instability region resides above each curve (For Kepler-448 weshaded both the stable and unstable regions below the curve corresponding to a 20 M

J

companion and above the curve corresponding to a 0.1 MJ

companionrespectively). For each system, we used the observed parameters in Equation 12 to generate the contours. The observed parameters we used for the inner planetsare specified in Table 1. We note that a third companion was reported to Kepler-56, with a minimum mass of 5.6 M

J

, which yields a 3.1 au separation (Otor et al.2016). This constrains the eccentricity of the companion to lie on a vertical line of a constant semi-major axis in the parameter space. As such, we over-plottedKepler-56d on the top right panel (dashed line).

7

to have an inclined companion, as depicted in Figure 4in the top right panel. In the presence of an inclinedcompanion, the inner two planets will most likely havenon-negligible obliquities, as was postulated by Li et al.(2014b).

• Kepler-88 is a star of, approximately, one solar mass.It has been observed to be orbited by two planets. Thefirst planet is Earth-like (m1 ⇠ 0.85M�) and the sec-ond planet is comparable to Jupiter (m2 ⇠ 0.67 M

J

).Together, the two planets form a compact inner sys-tem with negligible eccentricities (e.g., Nesvorný et al.2013; Barros et al. 2014; Delisle et al. 2017). In Figure4, left middle panel, we show that an inclined planet canexist in large parts of the parameter space. For exam-ple, a massive companion (⇠ 20 M

J

) can exist beyond2au with an eccentric orbit (⇠ 0.7).

• Kepler-109 is a star that also has, approximately, onesolar mass (1.07 M�) and two planets orbiting it in acompact configuration, having a small mutual inclina-tion (Marcy et al. 2014; Van Eylen & Albrecht 2015;Silva Aguirre et al. 2015). However, the eccentricityof the innermost planet is not as negligible as the sec-ondary’s (e1 ⇠ 0.21 > e2 ⇠ 0.03), but does not yield aviolation of the approximation. We show the stabilitybounds within the (e3,a3) parameter space for this sys-tem in the middle right panel of Figure 4. There it canbe seen that this system can exist in the presence of aneccentric companion with 20 M

J

, so long as the com-panion is beyond 3au.

• Kepler-419 is a 1.39 M� mass star which is orbited bytwo Jupiter sized planets with non-negligible eccentric-ities and low mutual inclinations (e.g., Dawson et al.2012, 2014; Huang et al. 2016). The large, observedeccentricities of the planets violate the assumption thatthe eccentricities are sufficiently small. Recall that thisassumption was used to derive the stability criterion inEq. (10). Furthermore, the quadrupole timescale forprecession induced by Kepler-419c on Kepler-419b iscomparable to that of Laplace-Lagrange’s. Thus, a dis-tant massive companion is expected to excite the twoplanets’ eccentricities and inclinations. When the mu-tual inclination between the two inner planets is large,the second planet can further excite the innermost plan-ets’ eccentricity, thus rendering the system unstable.We have verified, numerically, the instability of thesystem is consistent with the breakdown of our crite-rion. We did this for several systems with far awaycompanions with masses varying from 1,5, and 20 M

J

.Thus, we suggest that a massive inclined companioncan probably be ruled out for this system. Since ourstability criterion is violated here, it is not necessary toshow it’s parameter space in a plot. On the other hand,a smaller companion is not expected to cause secularexcitations in the eccentricities of more massive plan-ets, since the inner system will excite its eccentricityand inclination (e.g., inverse EKL Naoz et al. 2017; Za-nardi et al. 2017; Vinson & Chiang 2018). Therefore,the existence of a smaller inclined companion cannotbe ruled out.

• Kepler-448 is a 1.45 M� star orbited by a 10 MJ

warmJupiter (Bourrier et al. 2015; Johnson et al. 2017). Re-

cently, there was a reported discovery of a massivecompanion (⇠ 22 M

J

) with a rather hierarchical con-figuration (Masuda 2017). The hierarchical nature ofthe inner system yields a more limited range of separa-tions to hide a companion, see Figure 4, in the bottompanel. On the other hand, the large masses of Kepler-448b and Kepler-448c imply that a small planetary in-clined companion can still exist with negligible impli-cations on the inner system. In fact, one would expectthat the inner system will largely affect a less massivecompanion (e.g., Naoz et al. 2017; Zanardi et al. 2017;Vinson & Chiang 2018).

4. DISCUSSION

We have analyzed the stability of four-body hierarchicalsystems, where the forth companion is set far away (a3 �a2,a1). Specifically, we focus on three planet systems, forwhich the two inner planets reside in a relatively close config-uration and have an inclined, far away, companion. Observa-tions have shown that multiple tightly packed planetary con-figurations are abundant in our Galaxy. These systems mayhost a far away companion, which might be inclined or eveneccentric. An inclined perturber can excite the eccentricity ofplanets in the inner system via the EKL mechanism, whichcan ultimately destabilize the system.

We have analytically determined a stability criterion fortwo-planet systems in the presence of an inclined companion(Equation 10). This criterion depends on the initial conditionsof the inner planetary system and on the outer orbit’s mass, ec-centricity, and separation. It is thus straightforward to gener-alize it to n > 2 inner planetary systems. We then numericallyintegrated a set of similar systems using the Gauss’s averag-ing method, varying only the outer companion’s eccentricitye3 and the second planet’s separation a2. We have character-ized each numerical integrated systems’ stability and showedthat stable systems are consistent with our analytical criterion.

A system will remain stable if the timescale for angular mo-mentum exchange between the two inner orbits takes placefaster than eccentricity excitations that might be induced by afar away, inclined, companion. When the system is stable, thetwo inner orbits have minimal eccentricity modulations andtheir inclinations, with respect to their initial normal orbit, re-mained aligned to one another. An example of such as systemis depicted in the left panels of Figure 3.

Assuming the normals of the inner orbits are initially par-allel to the stellar axis allows precession of the nodes to beinterpreted as obliquity variations. Thus, a non-negligibleobliquity for two (or more) tightly packed inner orbits may bea signature of an inclined, distant, companion (e.g., Li et al.2014b). The obliquity, in this case, oscillates during the sys-tem’s dynamical evolution, which can have large implicationson the habitability of the planets (e.g., Shan & Li 2017).

On the other hand, a system will destabilize if the preces-sion induced by the outer companion is faster than the preces-sions caused by interactions between the inner bodies. In thiscase, the two inner planets will exhibit large eccentricity ex-citations accompanied with large inclination oscillations (see,for example, right panels in Figure 3). In this type of system,each planet undergoes nearly independent EKL oscillations,and thus extremely large eccentricity values can be expected,as well as chaos (e.g., Naoz et al. 2013a; Teyssandier et al.2013; Li et al. 2014a).

We also showed (e.g., Figure 2, green band) that the sta-bility criterion includes a transition zone, where systems are

8

likely to develop large eccentricity, leading to close orbits.Systems close to the transition zone, or in the transition zone,can be stable for long period of time, and develop instabilityvery late in the evolution. We show an example for such assystem in the middle column of Figure 2. In this example, thesystem stayed stable for slightly more than a Gyr and devel-oped an instability that leads to orbit crossing after ⇠ 1.5 Gyr.

We note that our analysis did not include General Relativis-tic or tidal effects between the planets and the star because,typically, including them will further stabilize systems. Gen-eral relativistic precession tends to suppress eccentricity ex-citations if the precession takes place on a shorter timescalethan the induced gravitational perturbations from a compan-ion (e.g., Naoz et al. 2013b). Also, tidal precession tends tosuppress eccentricity excitations (Fabrycky & Tremaine 2007;Liu et al. 2015). These suppression effects become moreprominent the closer the inner orbits are to the host star. Thus,if eccentricity excitations from the companion take place on alonger timescale than general relativity or tidal precession the

system will remain stable.Finally, as a proof-of-concept, we used our stability crite-

rion to predict the parameter space in which a hidden inclinedcompanion can reside for four Kepler systems (see Figure4). The systems we consider were Kepler-419, Kepler-56,Kepler-448, and Kepler-36. These systems represent a rangeof configurations, from tightly packed systems with small orsuper-Earth mass planets to potentially hierarchical systemswith Jupiter mass planets. A notable example is Kepler-56,where the recently detected third planet was reported to havea minimum mass of 5.6 M

J

, and a ⇠ 1000 days orbit. Sucha system indeed resides in the predicted stable regime. Fur-thermore, given a mass for the Kepler-56d, we can limit itspossible eccentricity.

PD acknowledges the partial support of the Weyl Under-graduate Scholarship. SN acknowledges the partial supportof the Sloan fellowship.

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