DRAFT VERSION FEBRUARY 5, 2018 Preprint 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. S TEPHAN1,WILL M. FARR3 1Physics and Astronomy Department, University of California, Los Angeles, CA 90024 2Mani 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, UK Draft version February 5, 2018

ABSTRACT Recent 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 other stellar objects. These companions might be inclined with respect to the inner planets, and could potentially excite the eccentricities of the inner planets through the Eccentric Kozai-Lidov mechanism. These eccentricity excitations could perhaps cause the inner orbits to cross, disrupting the inner system. We study the stability of two-planet systems in the presence of a distant, inclined, giant planet. Specifically, we derive a stability criterion, which depends on the companion’s separation and eccentricity. We show that our analytic criterion agrees with the results obtained from numerically integrating an ensemble of systems. Finally, as a potential proof-of-concept, we provide a set of predictions for the parameter space that allows the existence of planetary companions for the Kepler-56, Kepler-448, Kepler-88, Kepler-109, and Kepler-36 systems.

1. INTRODUCTION may be the cause of water being delivered to earth (Scholl & Recent ground and space-based observations have shown Froeschle 1986; Morbidelli 1994), the potential instability of that multi-planet systems are abundant around main-sequence Mercury’s orbit (Neron de Surgy & Laskar 1997; Fienga et al. stars (e.g., Howard et al. 2010, 2012; Borucki et al. 2011; Lis- 2009; Batygin & Laughlin 2008; Lithwick & Wu 2011), and sauer et al. 2011; Mayor et al. 2011; Youdin 2011; Batalha the obliquity of the exoplanets’ in general (e.g., Naoz et al. et al. 2013; Dressing & Charbonneau 2013; Petigura et al. 2011; Li et al. 2014b). 2013; Christiansen et al. 2015). These studies reveal that the Recently, several studies showed that the presence of a gi- architecture of planetary systems can drastically vary from ant planet can affect the ability to detect an inner system (e.g., our solar system. For example, systems consisting of mul- Hansen 2017; Becker & Adams 2017; Mustill et al. 2017; tiple low mass (sub-Jovian) planets with relatively compact Jontof-Hutter et al. 2017; Huang et al. 2017). Specifically, dy- orbits usually have periods that are shorter than Mercury’s. namical interactions from a giant planet, having a semi-major NASA’s Kepler mission found an abundance of compact axis much greater than the planets in the inner system, can multi-planet super-Earths or sub-Neptune systems (e.g., Mul- excite eccentricities and inclinations of the inner planets. A lally et al. 2015; Burke et al. 2015; Morton et al. 2016; Hansen possible effect is that the inner system becomes incapable of 2017). These systems seemed to have low eccentricities (e.g., having multiple transits or completely unstable. Volk & Glad- Lithwick et al. 2012; Van Eylen & Albrecht 2015). In addi- man (2015) showed that observed multi-planet systems may tion, it was suggested that these many body systems are close actually be the remnants of a compact system that was tighter to the dynamically stable limit (e.g., Fang & Margot 2013; in the past but lost planets through dynamical instabilities and Pu & Wu 2015; Volk & Gladman 2015). It was also proposed collisions (see also Petrovich et al. 2014). Interestingly, ver- that single planet systems might be the product of systems ini- ifying these problems could reconcile the Kepler dichotomy tially consisting of multiple planets that underwent a period of (e.g., Johansen et al. 2012; Ballard & Johnson 2016; Hansen disruption, i.e, collisions, resulting in reducing the number of 2017). planets (see Johansen et al. (2012) Becker & Adams (2016)). In this work, we investigate the stability of compact sub- Giant planets may play a key role in forming inner plan- Jovian inner planetary systems in the presence of a distant arXiv:1802.00447v1 [astro-ph.EP] 1 Feb 2018 etary systems. Radial velocity surveys, along with the exis- giant planet (see Figure 2 for an illustration of the system). tence of Jupiter, have shown giants usually reside at larger Over long time scales, a distant giant planets’ gravitational distances from their host star than other planets in the sys- perturbations can excite the eccentricities of the inner plan- tem ( 1 AU)(e.g., Knutson et al. 2014; Bryan et al. 2016). ets’ to high values, destabilizing the inner system. (e.g., Naoz For example, there were some suggestions that the near- et al. 2011). However, if the frequency of angular momentum resonant gravitational interactions between giant outer plan- exchange between the inner planets is sufficiently high, then ets and smaller inner planets can shape the configuration of an the inner system can stabilize. Below we derive an analytic inner system’s asteroid belt (e.g., Williams & Faulkner 1981; stability criterion (Section 2). Then we analyze our nominal Minton & Malhotra 2011). Hence, it was suggested that our system in Section 3 and provide specific predictions for Ke- solar system’s inner planets are of a second generation, which pler systems in Section 3.3. Finally, we offer our conclusion came after Jupiter migrated inward to its current orbit (e.g., in Section 4. Batygin & Laughlin 2015). Furthermore, secular resonance Near the completion of this work, we became aware of (Pu & Lai 2018) a complementary study of stability in similar sys- [email protected] tems. Here we provide a comprehensive stability criterion as [email protected] a function of the companion’s parameters. Furthermore, we 2

between m3 and m1. Planet m2’s argument of periapse also precesses due to gravitational perturbations from the inner- most planet, m1. The associated timescale is (e.g., Murray & Dermott 2000) e ⌧ A + A 1 cos(v -v ) LL ⇠ 22 21 e 2 1  ✓ 2 ◆ i -1 - B - B 1 cos(⌦ - ⌦ ) , (2) 22 21 i 2 1 ✓ 2 ◆ where v j = ! j + ⌦ j for j =1,2. The Ai, j, and Bi, j are laplace coefficients, which are determined by: 2 1 m1 a1 A = n f , (3) 22 2 4⇡ M + m a 2 ✓ 2 ◆ 1 m a FIG. 1.— Here is a schematic of the systems being analyzed. The inclina- 1 1 , tion of the third planet, i , is shown and measured relative to the z-axis, along A21 = -n2 f2 (4) 3 4⇡ M + m2 a2 the spin axis of the host star. M,m1,m2, and m3 are the masses of the host ✓ ◆ star, and the first, second and third planets respectively. 2 1 m1 a1 B22 = -n2 f , (5) provide a set of predictions for possible hidden companion 4⇡ M + m2 a2 orbits in several observed systems. ✓ ◆ 1 m1 a1 B = n f . (6) 21 2 4⇡ M + m a 2. ANALYTICAL STABILITY CRITERION 2 ✓ 2 ◆ Here we develop a generic treatment of the stability of two- and body systems in the presence of an inclined outer planet. Con- 2⇡ cos sider an inner system consisting of two planets (m1 and m2) f = 3 d , (7) orbiting around a host star M with a relatively tight config- 0 2 2 Z 1 - 2 a1 cos + a1 uration (with semi-major axis a1 and a2, respectively). We a2 a2 introduce an inclined and eccentric companion that is much ✓ ⇣ ⌘ ⇣ ⌘ ◆ farther from the host star than the planets in the inner system 2⇡ cos2 f = d . (8) (m3 and a3, see Figure 1 for an illustration of the system). 2 3 0 2 2 We initialize this system to have orbits far from mean-motion Z 1 - 2 a1 cos + a1 resonance, and have a ,a << a . The three planets’ orbits a2 a2 1 2 3 ✓ ◆ have corresponding eccentricities e1, e2 and e3. We set an ar- ⇣ ⌘ ⇣ ⌘ bitrary z-axis to be perpendicular to the initial orbital plane of The system will remain stable if angular momentum ex- change between the two inner planets takes place faster than the inner two planets, thus, the inclinations of m1, m2 and m3 the precession induced by m3. Accordingly, there exists two are defined as i1, i2 and i3. Accordingly, for planets one, two, and three we denote the longitude of ascending nodes and the regions of parameter space: one contains systems where per- turbations on m2 are dominantly from m1, and the other con- argument of periapse ⌦1, ⌦2, ⌦3, !1,!2, and !3. The outer orbit can excite the eccentricities, via the Ec- tains systems where m2 is dominated by m3. The former is centric Kozai-Lidov mechanism (EKL) (Kozai (1962), Lidov called the Laplace-Lagrange region, and the ladder is called (1962), see Naoz (2016) for review) on each planet in the EKL region. The transition between the Laplace-Lagrange inner system. The EKL resonance causes angular momen- region and EKL region is determined by comparing the two tum exchange between the outer and inner planets, which in timescales relating the frequencies of precession from each turn causes precession of the periapse of each of the inner or- mechanism, i.e., ⌧ ⌧ . (9) bits. However, angular momentum exchange between the in- k ⇠ LL ner two planets also induces precession of the periapse ("the We have related the timescale for Kozai oscillations induced so-called Laplace-Lagrange interactions, e.g., Murray & Der- by m3 on m2 to the Laplace-Lagrange timescale between m1 mott 2000). If the inner orbits’ angular momentum exchange and m2. Equating these timescales yields a simple expression takes place at a faster rate than that induced by the outer com- for the critical eccentricity of the third planet, e3,c, as a func- panion, then the system will not be disrupted by perturbations tion of a2, i.e., from the tertiary planet. The quadrupole approximation to the 1 timescale of the EKL between the third planet m and the sec- 2 2 3 p 3/2 3 ond one, m2 is given by: 15 m3 G a2 1 e3,c 1 - 3 , (10) ⇠ 0 16 pM + m1 + m2 a fLL 1 16 a3 pM + m + m " 3 # ⌧ 3 2 1 (1 - e2)3/2 , (1) k(2,3) ⇠ 15 3/2 p 3 @ A a2 m3 G where (e.g., Antognini 2015) where G is the gravitational constant. e1,ic i1,ic fLL = A22 + A21 cos(v1 -v2) - B21 cos(⌦1 - ⌦2) - B22 . This is roughly the timescale at which the second planet’s ar- e2,ic i2,ic gument of periapse precesses. Note that here we considered (11) the timescale of EKL excitations from the tertiary on m2 since e j,ic and i j,ic, are the initial eccentricity and inclination for the this timescale is shorter than the timescale of EKL excitations two inner planets, i.e., j =1,2. As we show below, during the 3 evolution of a stable system ⌦ ⌦ , v v , e e , and and third planets respectively. In this ensemble, we initialize 1 ⇠ 2 2 ⇠ 1 1 ⇠ 2 i1 i2. Thus, we define a minimum stable configuration of all systems with a primary star M =1M orbited by three ⇠ fLL,min planets where the two inner planets, m1, and m2, each have masses of 1 M . The innermost planet, m1, was placed in e1,ic i1,ic fLL,min = A22 + A21 - B21 - B22 . (12) an orbit with a semi-major axis of a1 = 1 au, and both inner e2,ic i2,ic planets were set initially on circular orbits in the same orbital Numerically we find that during the evolution of an unstable plane. We set the third planet with a mass of m3 =1MJ, at a3 = 100 au. Throughout the ensemble, a2 ranges from 1.4 au system, e1/e2, i1/i2, ⌦1/⌦2, and v2/v1 largely deviate from to 5.9 au, in steps of 0.5 au, and e3 ranges from zero to 0.95 in unity, where cos(v1 -v2) can be negative. Thus, we also de- fine the maximum stability as: steps of 0.05. In all such systems, the third planet’s inclination was set to be 45 relative to the inner system. We integrated e1,ic i1,ic each system to 10 Gyr, or until orbits crossed. From the re- fLL,max = A22 - A21 - B21 - B22 , (13) e2,ic i2,ic sults we made a grid labeling the stable/unstable systems of the ensemble. The grid revealed that stable orbits are bounded where the difference between Eq. (12) and (13) is the sign by the stability criterion. The same process was done on an of A . The stability of systems transitions from the mini- 21 ensemble which had the third planet’s inclination set to 65 mum to the maximum fLL, as a function of e3. In other words instead; the result was the same. we find a band of parameter space, between e3,c( fLL,min) and e3,c( fLL,max), where systems are nearly unstable or completely unstable. If the third planet’s eccentricity is larger than the right hand side of Equation (10), then the inner system is more likely to become unstable. In the next section we test this sta- bility criterion and show that it agrees with secular numerical integration. We note that this stability criterion is based on the Laplace- Lagrange approximation, which assumes small eccentricities and inclinations for orbits in the inner system. Thus, it might break 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 the eccentricity excitations from the second planet on the first are suppressed. On the other hand, in the presence of these condi- tions the stability criterion depends only on the shortest EKL timescale 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 FIG. 2.— Here is the parameter space relating the third planets’ eccentric- To test our analytic stability criterion we utilize Gauss’s ity with the second planet’ separation. The distant companion’s inclination Method. This prescription allows us to integrate the system was set to 45. The color code shows log(1 - ), Eq. (14). This parameter over a long timescale (set to be 10 Gyr, see below), in a estimates the closest the two inner orbits came to each other during the evo- lution. A darker color characterizes far away separation while a lighter color time-efficient way. In this mathematical framework, the non- characterizes orbit crossing. The stability criterion, Eq. (10), is plotted over resonant planets’ orbits are phase-averaged and are treated as the grid to show that stable orbits are bounded. The bottom green line repre- massive, pliable wires interacting with each other. The line- sents e3,c( fLL,min) and the top green line represents e3,c( fLL,max), and thus the density of each wire is inversely proportional to the orbital shaded band is the transition zone (see text). Systems above the zone undergo velocity of each planet. The secular, orbit-averaged method an instability episode while systems below the zone are stable. requires that the semi-major axes of the wires are constants of motion. We calculate the forces that different wires exert on In Figure 2 we show the grid of systems we integrated in each other over time. the parameter space relating a2 and e3. The color code is de- This method was recently extended to include softened termined by a proxy that characterizes the stability of the sys- gravitational interactions by introducing a small parameter in tem. The proxy is defined as log(1-), where is a parameter the interaction potentials to avoid possible divergences while which describes how close the two inner orbits came during integrating (Touma et al. 2009). Furthermore, the method has the evolution: been proven to be very powerful in addressing different prob- a2(1 + e2) - a1(1 - e1) lems in astrophysics, ranging from understanding the dynam- = min . (14) a - a ics in galactic nuclei to describing how debris disks evolve.  2 1 (e.g., Touma et al. 2009; Sridhar & Touma 2016; Nesvold In Figure 2, a lighter color means a smaller value of , which et al. 2016; Saillenfest et al. 2017). when zero yields orbit crossing, while a darker color repre- sents a stable configuration. The criterion for stability is plot- 3.2. Stability test on a nominal system ted over the grid to show that it is in agreement with the prob- We tested our stability criterion by numerically integrating ability for orbits to cross. Above the stability curve, systems an ensemble of systems with identical initial conditions ex- are more likely to destabilize during evolution. cept for the semi-major axis and eccentricity of the second 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 of the 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 inner planets 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 =1M , m1 = m2 =1M , m3 =1MJ , 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’s separation 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 name SMA [au] mass [MJ ] eccentricity (i.e., between e3,c( fLL,min) and e3,c( fLL,max)), where, given the Kepler-36b 0.12 0.015 < 0.04 initial conditions in the numerical runs, we set e /e 1 Kepler-36c 0.13 0.027 < 0.04 1,ic 2,ic ⇠ and i1,ic/i2,ic 1. Closer to the stability curve, the orbits might Kepler-56b 0.103 0.07 - get their eccentricities⇠ excited and periodically move closer to Kepler-56c 0.17 0.57 - one another, but their orbits may never cross. Moreover, far Kepler-88b 0.097 0.027 0.06 into the top right of Figure 2, the instability will take place Kepler-88c 0.15 0.62 0.056 sooner in the evolution. In this region, the two inner orbits Kepler-109b 0.07 0.023 0.21 will undergo EKL evolution independently of each other. Kepler-109c 0.15 0.07 0.03 The third planet can excite the inner planets’ eccentric- Kepler 419b 0.37 2.5 0.83 ities on the EKL timescale. However, eccentricity excita- Kepler 419c 1.68 7.3 0.18 tions of each planet will not necessarily cause orbit cross- Kepler-448b 0.15 10 0.34 ing, as depicted in Figure 2. In the parameter space for our Kepler-448c 4.2 22 0.65 ensemble, 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, TABLE 1 OBSERVABLE PARAMETERS OF THE EXAMPLE SYSTEMS.THE the intermediate regime, and the unstable regime. In the sta- OBSERVATIONS ARE ADOPTED FROM: Kepler-36:CARTER ET AL. ble regime, the Laplace-Lagrange rapid angular momentum (2012). Kepler-56:HUBER ET AL.(2013)AND OTOR ET AL.(2016). exchange between the two inner planets dominates over the Kepler-88 NESVORNÝET AL.(2013),BARROS ET AL.(2014)AND gravitational perturbations of the outer orbit. The left column DELISLE ET AL.(2017).Kepler-109: MARCY ET AL.(2014),VAN EYLEN &ALBRECHT (2015) AND SILVA AGUIRRE ET AL.(2015). of Figure 3 depicts this evolution; the system in this column Kepler-419:DAWSON ET AL.(2012),DAWSON ET AL.(2014)AND resides in the bottom left of the parameter space in Figure 2. HUANG ET AL.(2016).Kepler-448:BOURRIER ET AL.(2015),JOHNSON The system remains stable for 10 Gyrs of evolution and the or- ET AL.(2017)AND MASUDA (2017). bits never come close to one another. The two inner planets’ terion for this exercise, i.e., e3,c( fLL,min). Each line in the inclinations oscillate together around the initial z-axis, due to four panels of Figure 4 shows the stability criterion for dif- the precession of the nodes (see below). In the intermedi- ferent companion masses. In particular, we consider compan- ate region, systems might appear to be stable for a long time ion masses of (from top to bottom), 0.1,0.5,1,5 and 20 M . but the outer orbit perturbations become too dominant, caus- For each companion mass, allowed system configurations lie ing instability of the inner system. We show the dynamics of below the curve, and unstable configurations are above the this type of evolution in the middle panel of Figure 3. This curve. We caution that very close to the stability curve (even system resides very close to the stability criterion plotted in below the curve) systems may still undergo eccentricity exci- Figure 2, inside the transition zone between e3,c( fLL,min) and tations that may affect the dynamics. The parameters of the e3,c( fLL,max). In the unstable regime, gravitational perturba- inner, observed planets were taken from observations, see Ta- tions from the third planet cause high amplitude eccentricity ble 1. Below we discuss the specifics of the four example excitation in the inner planets, causing orbits to cross. We systems. show this behavior in the right column 3. This system lies in the top right of the parameter space depicted in Figure 2. Kepler-36 is an, approximately, one solar mass star In the system’s reference frame (see Figure 1), the inner • orbited by two few-Earth mass planets on 13.8 and planets’ inclinations are the angles between their respective 16.2 days orbits (Carter et al. 2012). This compact con- angular momenta and the normal to the initial orbits. Thus, figuration is expected to be stable in the presence of the inclination modulation shown in the stable system (the left perturbations as shown in Figure 4, in the top left panel. column in Figure 3, is due to precession of the nodes, which For example, as can be seen in the Figure, an eccentric results in maximum inclination, which is 2 i , since we ⇠ ⇥ 3 inclined 20 MJ brown dwarf can reside slightly beyond have started with i3,0 = 45 (e.g., Innanen et al. 1997). In con- 2 au. trast, the unstable regime results in misalignment between the inner two planets. Kepler-56 is an evolved star (M =1.37 M ) at the base • of the red giant branch and is orbited by two sub-Jovian 3.3. Applications: predictions for observed systems planets with low mutual inclinations (e.g., Huber et al. The stability criterion derived here can be used to predict 2013). Most notably, asteroseismology analysis placed the parameter space in which a hidden companion can ex- a lower limit on the two planets’ obliquities of 37. Li ist within a system without destabilizing it. As a proof-of- et al. (2014b) showed that a large obliquity for this sys- concept, we discuss the stability of a few non-resonant ob- tem is consistent with a dynamical origin. They sug- served exoplanetary systems in the presence of inclined plan- gested that a third outer companion is expected to reside ets. Specifically, for some observed systems, we provide a set far away and used the inner two planets’ obliquities to of predictions that characterize the ranges of parameter space constrain their inclinations. Follow-up radial velocity available for hidden companions to exist in without disrupting measurements estimated that a third companion indeed inner orbits. We focus on the following systems: Kepler-419, exists with a period of 1000 days and a minimum mass Kepler-56, Kepler-448, and Kepler-36. These represent the of 5.6MJ (e.g., Otor et al. 2016). Here we show that in- few systems that characterize the extreme limits of two planet deed a 5MJ planet can exist at 3 au, with a range systems, from tightly packed super-earths such as Kepler-36 of possible⇠ eccentricities up to 0.⇠9. A more massive to hierarchical eccentric warm Jupiter systems such as Kepler- planet can still exist at 3 au with a possible range of ⇠ 448. In Figure 4 we show the relevant parameter space in eccentricities up to slightly below e3 0.8. This ex- which the systems can have a hidden inclined companion and ample of a tightly packed system yields⇠ the expected remain stable. We chose the more conservative stability cri- result, 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 Kepler systems. For each of the systems we plot the stability criterion e3,c( fLL,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 MJ . The stable region exists below each curve and the instability region resides above each curve (For Kepler-448 we shaded both the stable and unstable regions below the curve corresponding to a 20 MJ companion and above the curve corresponding to a 0.1MJ companion respectively). For each system, we used the observed parameters in Equation 12 to generate the contours. The observed parameters we used for the inner planets are specified in Table 1. We note that a third companion was reported to Kepler-56, with a minimum mass of 5.6MJ , 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-plotted Kepler-56d on the top right panel (dashed line). 7

to have an inclined companion, as depicted in Figure 4 cently, there was a reported discovery of a massive in the top right panel. In the presence of an inclined companion ( 22 MJ) with a rather hierarchical con- companion, the inner two planets will most likely have figuration (Masuda⇠ 2017). The hierarchical nature of non-negligible obliquities, as was postulated by Li et al. the inner system yields a more limited range of separa- (2014b). tions to hide a companion, see Figure 4, in the bottom panel. On the other hand, the large masses of Kepler- Kepler-88 is a star of, approximately, one solar mass. • 448b and Kepler-448c imply that a small planetary in- It has been observed to be orbited by two planets. The clined companion can still exist with negligible impli- first planet is Earth-like (m1 0.85M ) and the sec- cations on the inner system. In fact, one would expect ond planet is comparable to⇠ Jupiter (m 0.67 M ). 2 ⇠ J that the inner system will largely affect a less massive Together, the two planets form a compact inner sys- companion (e.g., Naoz et al. 2017; Zanardi et al. 2017; tem with negligible eccentricities (e.g., Nesvornýet al. Vinson & Chiang 2018). 2013; Barros et al. 2014; Delisle et al. 2017). In Figure 4, left middle panel, we show that an inclined planet can 4. DISCUSSION exist in large parts of the parameter space. For exam- We have analyzed the stability of four-body hierarchical ple, a massive companion ( 20 MJ) can exist beyond systems, where the forth companion is set far away (a3 2au with an eccentric orbit (⇠ 0.7). ⇠ a2,a1). Specifically, we focus on three planet systems, for which the two inner planets reside in a relatively close config- Kepler-109 is a star that also has, approximately, one uration and have an inclined, far away, companion. Observa- • solar mass (1.07 M ) and two planets orbiting it in a tions have shown that multiple tightly packed planetary con- compact configuration, having a small mutual inclina- figurations are abundant in our Galaxy. These systems may tion (Marcy et al. 2014; Van Eylen & Albrecht 2015; host a far away companion, which might be inclined or even Silva Aguirre et al. 2015). However, the eccentricity eccentric. An inclined perturber can excite the eccentricity of of the innermost planet is not as negligible as the sec- planets in the inner system via the EKL mechanism, which ondary’s (e 0.21 > e 0.03), but does not yield a 1 2 can ultimately destabilize the system. violation of the⇠ approximation.⇠ We show the stability We have analytically determined a stability criterion for bounds within the (e ,a ) parameter space for this sys- 3 3 two-planet systems in the presence of an inclined companion tem in the middle right panel of Figure 4. There it can (Equation 10). This criterion depends on the initial conditions be seen that this system can exist in the presence of an of the inner planetary system and on the outer orbit’s mass, ec- eccentric companion with 20 M , so long as the com- J centricity, and separation. It is thus straightforward to gener- panion is beyond 3au. alize it to n > 2 inner planetary systems. We then numerically Kepler-419 is a 1.39 M mass star which is orbited by integrated a set of similar systems using the Gauss’s averag- • two Jupiter sized planets with non-negligible eccentric- ing method, varying only the outer companion’s eccentricity ities and low mutual inclinations (e.g., Dawson et al. e3 and the second planet’s separation a2. We have character- 2012, 2014; Huang et al. 2016). The large, observed ized each numerical integrated systems’ stability and showed eccentricities of the planets violate the assumption that that stable systems are consistent with our analytical criterion. the eccentricities are sufficiently small. Recall that this A system will remain stable if the timescale for angular mo- assumption was used to derive the stability criterion in mentum exchange between the two inner orbits takes place Eq. (10). Furthermore, the quadrupole timescale for faster than eccentricity excitations that might be induced by a precession induced by Kepler-419c on Kepler-419b is far away, inclined, companion. When the system is stable, the comparable to that of Laplace-Lagrange’s. Thus, a dis- two inner orbits have minimal eccentricity modulations and tant massive companion is expected to excite the two their inclinations, with respect to their initial normal orbit, re- planets’ eccentricities and inclinations. When the mu- mained aligned to one another. An example of such as system tual inclination between the two inner planets is large, is depicted in the left panels of Figure 3. the second planet can further excite the innermost plan- Assuming the normals of the inner orbits are initially par- ets’ eccentricity, thus rendering the system unstable. allel to the stellar axis allows precession of the nodes to be We have verified, numerically, the instability of the interpreted as obliquity variations. Thus, a non-negligible system is consistent with the breakdown of our crite- obliquity for two (or more) tightly packed inner orbits may be rion. We did this for several systems with far away a signature of an inclined, distant, companion (e.g., Li et al. companions with masses varying from 1,5, and 20 MJ. 2014b). The obliquity, in this case, oscillates during the sys- Thus, we suggest that a massive inclined companion tem’s dynamical evolution, which can have large implications can probably be ruled out for this system. Since our on the habitability of the planets (e.g., Shan & Li 2017). stability criterion is violated here, it is not necessary to On the other hand, a system will destabilize if the preces- show it’s parameter space in a plot. On the other hand, sion induced by the outer companion is faster than the preces- a smaller companion is not expected to cause secular sions caused by interactions between the inner bodies. In this excitations in the eccentricities of more massive plan- case, the two inner planets will exhibit large eccentricity ex- ets, since the inner system will excite its eccentricity citations accompanied with large inclination oscillations (see, and inclination (e.g., inverse EKL Naoz et al. 2017; Za- for example, right panels in Figure 3). In this type of system, nardi et al. 2017; Vinson & Chiang 2018). Therefore, each planet undergoes nearly independent EKL oscillations, the existence of a smaller inclined companion cannot and thus extremely large eccentricity values can be expected, be ruled out. as well as chaos (e.g., Naoz et al. 2013a; Teyssandier et al. 2013; Li et al. 2014a). Kepler-448 is a 1.45 M star orbited by a 10 MJ warm We also showed (e.g., Figure 2, green band) that the sta- • Jupiter (Bourrier et al. 2015; Johnson et al. 2017). Re- bility criterion includes a transition zone, where systems are 8 likely to develop large eccentricity, leading to close orbits. system will remain stable. Systems close to the transition zone, or in the transition zone, Finally, as a proof-of-concept, we used our stability crite- can be stable for long period of time, and develop instability rion to predict the parameter space in which a hidden inclined very late in the evolution. We show an example for such as companion can reside for four Kepler systems (see Figure system in the middle column of Figure 2. In this example, the 4). The systems we consider were Kepler-419, Kepler-56, system stayed stable for slightly more than a Gyr and devel- Kepler-448, and Kepler-36. These systems represent a range oped an instability that leads to orbit crossing after 1.5 Gyr. of configurations, from tightly packed systems with small or We note that our analysis did not include General⇠ Relativis- super-Earth mass planets to potentially hierarchical systems tic or tidal effects between the planets and the star because, with Jupiter mass planets. A notable example is Kepler-56, typically, including them will further stabilize systems. Gen- where the recently detected third planet was reported to have eral relativistic precession tends to suppress eccentricity ex- a minimum mass of 5.6MJ, and a 1000 days orbit. Such citations if the precession takes place on a shorter timescale a system indeed resides in the predicted⇠ stable regime. Fur- than the induced gravitational perturbations from a compan- thermore, given a mass for the Kepler-56d, we can limit its ion (e.g., Naoz et al. 2013b). Also, tidal precession tends to possible eccentricity. suppress eccentricity excitations (Fabrycky & Tremaine 2007; Liu et al. 2015). These suppression effects become more prominent the closer the inner orbits are to the host star. Thus, PD acknowledges the partial support of the Weyl Under- if eccentricity excitations from the companion take place on a graduate Scholarship. SN acknowledges the partial support longer timescale than general relativity or tidal precession the of the Sloan fellowship.

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