Strong Scatterings of Cold Jupiters and Their Influence on Inner

MNRAS 000, 000–000 (2020) Preprint 3 September 2021 Compiled using MNRAS LATEX style ﬁle v3.0

Strong Scatterings of Cold Jupiters and their Inﬂuence on Inner Low-mass Planet Systems: Theory and Simulations

Bonan Pu1 ★, Dong Lai1,2 † 1. Department of Astronomy, Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853, USA 2. Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China

3 September 2021

ABSTRACT Recent observations have indicated a strong connection between compact (푎 . 0.5 au) super- Earth and mini-Neptune systems and their outer (푎 & a few au) giant planet companions. We study the dynamical evolution of such inner systems subject to the gravitational effect of an unstable system of outer giant planets, focussing on systems whose end configurations feature only a single remaining outer giant. In contrast to similar studies which used on N- body simulations with specific (and limited) parameters or scenarios, we implement a novel hybrid algorithm which combines N-body simulations with secular dynamics with aims of obtaining analytical understanding and scaling relations. We find that the dynamical evolution of the inner planet system depends crucially on 푁ej, the number of mutual close encounters between the outer planets prior to eventual ejection/merger. When 푁ej is small, the eventual evolution of the inner planets can be well described by secular dynamics. For larger values of 푁ej, the inner planets gain orbital inclination and eccentricity in a stochastic fashion analogous to Brownian motion. We develop a theoretical model, and compute scaling laws for the final orbital parameters of the inner system. We show that our model can account for the observed eccentric super-Earths/mini-Neptunes with inclined cold Jupiter companions, such as HAT-P- 11, Gliese 777 and 휋 Men. Key words: Dynamics - Extra-solar planets - Planetary Systems

1 INTRODUCTION ties have smaller multiplicity of inner Super-Earths, suggesting that cold Jupiters have influenced the inner planetary system. Masuda Exoplanets with masses and radii between that of the Earth and Nep- et al.(2020) found that these CJ companions are typically mildly tune, commonly referred to as “super-Earths” or “mini-Neptunes”, misaligned with their inner systems with a mutual of Δ휃 ∼ 12 deg. have been discovered in large quantities in recent years. Indeed, These mild inner-outer misalignments could potentially explain the such planets appear to be ubiquitous in the Galaxy: about 30% of apparent excess of Kepler single-transit Super-Earth systems (Lai Sun-like stars host super-Earth planets, with each system containing & Pu 2017). an average of 3 planets (Zhu et al. 2018a). The observed super-Earth systems have compact orbits, with periods typically less than 200 The question of how low-mass inner planet systems may be days. In recent years, an increasing number of such systems have influenced by the presence of one or more external giant planets has been found to host long-period giant planet companions (i.e. “Cold attracted recent attention (e.g. Carrera et al. 2016; Gratia & Fabrycky arXiv:2008.05698v3 [astro-ph.EP] 2 Sep 2021 Jupiters” or CJs). Zhu et al.(2018b) analysed a sample of ground- 2017; Lai & Pu 2017; Huang et al. 2017; Mustill et al. 2017; Hansen based radial velocity (RV) observations of super-Earth systems and 2017; Becker & Adams 2017; Read et al. 2017; Jontof-Hutter et al. an independent sample of Kepler transiting Super-Earths with RV 2017; Pu & Lai 2018; Denham et al. 2019). This paper is the third follow-up, and found that cold Jupiters are three times more com- in a series where we systematically investigate the effect of outer mon around hosts of super-Earths than around field stars: about companions on the architecture of inner super-Earth systems. In Lai 30% the inner super-Earth systems have cold Jupiter companions, & Pu(2017) and Pu & Lai(2018) we study the secular evolution and the fraction increases to 60% for metal-rich stars. Bryan et al. of an inner multi-planet system perturbed by an inclined and/or (2019) found a similar result, and gave the estimated occurrence rate eccentric external companion. Combining analytical calculations of 39 ± 7% for companions between 0.5-20푀퐽 and 1 − 20 au. There and numerical simulations (based on secular and N-body codes), is evidence that that stars with cold Jupiters or with high metallici- we quantify to what extent eccentricities and mutual inclinations can be excited in the inner system for different masses and orbital parameters of super-Earths and cold Jupiter. When the perturber is ★ E-mail: [email protected] sufficiently strong compared to the mutual gravitational coupling † E- mail: [email protected] between the inner planets, the inner system becomes dynamically

© 2020 The Authors 2 Bonan Pu hot and may be unstable. Even for milder perturbers that do not 2017; Pu & Lai 2018). As mentioned above, because of the hier- disrupt integrity of the inner system, the small/modest excitation archy of dynamical timescales, it is difficult to study the systems of mutual inclinations can nevertheless disrupt the co-transiting where the inner super-Earths and outer giants are well separated geometry of the inner planets and thereby reduce the number of using brute-force 푁-body simulations, especially when the ejection transiting planets (e.g. Brakensiek & Ragozzine 2016). Other related time of giant planet is large – and yet such systems are most rele- works can be found in Boué & Fabrycky(2014a); Hansen(2017); vant to the observed super-Earths with cold Jupiter companions. To Becker & Adams(2017); Read et al.(2017); Jontof-Hutter et al. this end, we developed a hybrid algorithm, combining 푁-body sim- (2017); Denham et al.(2019) (see also Boué & Fabrycky 2014b; ulations of outer giant planets undergoing strong scatterings with Lai et al. 2018; Anderson & Lai 2018, for the effect of external secular forcing on the inner planets, to compute the evolution of the companion on the stellar obliquity relative to the inner planets). inner planets throughout the “violent” phase. In this paper we study the dynamical evolution of inner planet A major part of this paper is devoted to the dynamics of strong systems under the influence of a pair of external giant planets with scatterings between two giant planets (Sec.2). Although there have initially unstable orbits. A number of previous works (based on been many previous studies on giant planet scatterings (e.g. Rasio N-body simulations) have already investigated this problem, illus- & Ford 1996; Weidenschilling & Marzari 1996; Lin & Ida 1997; trating that the strong scatterings of unstable giant planets can affect Ford et al. 2000; Ford & Rasio 2008; Chatterjee et al. 2008; Jurić the orbits of the inner planets in different ways (e.g. Matsumura et al. & Tremaine 2008; Matsumura et al. 2013; Petrovich et al. 2014; 2013; Carrera et al. 2016; Gratia & Fabrycky 2017; Huang et al. Frelikh et al. 2019; Anderson et al. 2020; Li et al. 2021), they all 2017; Mustill et al. 2017). For example, the outer scatterings can focused on the final outcomes of the unstable giant planets (e.g., the send a giant planet inward, sweeping up all the inner planets along eccentricity distribution of the remaining planets), and did not in- its wake and totally destroying the inner system. Also, the scattering vestigate the timescale (“ejection time”) of violent phase. As noted events can excite the eccentricities and mutual inclinations of the in- above, this “ejection time” directly influences the perturbations the ner planets beyond the threshold of their stability, causing the inner inner planets receive from the “outer violence”. In addition to ob- system to also undergo scattering events of their own, resulting in a taining the “ejection time” distribution, we also obtain a number of pared down inner system. In this paper we attack this problem more new analytic and scaling results for strong scatterings between two systematically, going beyond previous works in several ways. Our giant planets. rationales are: (i) Previous works were restricted to small number We then develop a theoretical model for the “violent” phase of of numerical examples, often considering specific orbital parame- the scattering process, and model the inner planet’s secular evolu- ters. As such, it is difficult to obtain a quantitative understanding tion as a linear stochastic differential equation. We obtain analytic or scaling relations (even approximate) in order to know “what sys- estimates for both the expectation values and the distributions of the tems lead to what outcomes”. (ii) Previous works often considered final orbital parameters of the inner planets, and test these results systems where the inner planets are not too detached from the outer against direct numerical integrations. A major achievement of this planets. This was adopted for numerical reason: If the inner planets work is the derivation for the marginalized “violent-phase” boost have too small a semi-major axis compared to the outer planets, factor 훾, which summarizes the entire dynamics of the “1+2” scat- their dynamical times would be much shorter than the outer planets, tering process in a single, dimensionless parameter. We derive an and it would be difficult to simulate the whole system over a long analytical expression for the distribution of 훾, which agrees robustly time or simulate a large number of systems. As a result, previous with numerical simulations over a wide range of initial system pa- works tended to over-emphasize the more “disruptive” events. In rameters. reality, for sufficiently hierarchical systems, the scattering events This paper is structured as follows. In Sec.2, we study the may only mildly excite the eccentricities and mutual inclinations of scattering process between two unstable giant planets using N-body the inner planets; in this case, the super-Earths themselves are pre- simulations, focusing in particular on the planet ejection timescale. served, but their mutual inclinations may be large enough to “hide” through N-body simulations. In Sec.3, we outline our hybrid 푁- the inner planets from simultaneously transiting their host stars – body and secular algorithm to study the effect of giant planet scat- such “mild” systems or events may be most relevant to the currently terings on the inner super-Earth system. In Sec.4, we present the observed super-Earths with cold Jupiter companions. (iii) Most im- results of these simulations, as well as theoretical scaling results for portantly, there is a wide range of “ejection times” associated with the final outcome of these systems. These results are extended to the evolution of the unstable giant planets (e.g., for some systems, systems with more than one inner planets in Sec.5. We provide a the lighter cold Jupiter may be ejected very quickly, while for others summary of our results, and apply them to several “Super-Earth + the ejection may take place over much longer time). As we show CJ” systems of interest in Sec.6, as well as providing suggestions in this paper, the degree of influences on the inner system from for further studies. the outer planets is directly correlated with the ejection time of the unstable giant planets. Thus, numerical studies that only consider restricted examples would not capture the whole range of dynamical 2 GRAVITATIONAL SCATTERINGS OF TWO GIANT behaviors of the “inner planets + outer giants” system. PLANETS Thus, the goal of this paper is to systematically examine how strong scatterings of outer giant planets influence the inner super- The topic of gravitational scatterings between two or more giant Earth system. We aim at obtaining an understanding of the whole planets on unstable orbits is a classic one and has been the subject range of different outcomes and deriving relevant scaling relations of numerous previous studies (e.g. Rasio & Ford 1996; Weiden- for different systems (with various planet masses and orbital pa- schilling & Marzari 1996; Lin & Ida 1997; Ford et al. 2000; Jurić rameters) and different ejection times. Of particular interest are the & Tremaine 2008; Ford & Rasio 2008; Chatterjee et al. 2008; Ida “mild” systems where the inner planets survive the “outer violence”. et al. 2013; Matsumura et al. 2013; Petrovich et al. 2014; Frelikh We elucidate the connections between the “violent” phase and the et al. 2019; Anderson et al. 2020; Li et al. 2021). These studies ensuing “secular” phase studied in our previous papers (Lai & Pu focused on the final states of unstable systems, such as the eccen-

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tricity distribution of the remaining planets. We return to this topic an ejection has occurred (i.e. the orbit of one of the planets be- to re-focus our attention on the scattering/ejection timescale 푡ej, a comes unbounded). For each combination of 푘0, 푚1 and 푚2/푚1 we quantity that plays a key role in the interaction between the scatter- performed computations until 200 systems that resulted in ejected ing CJs and the inner super-Earth system, but hitherto ignored by systems were obtained. The reason we perform such large numbers previous studies (but see Fig. 1 of Anderson et al. 2020 and Fig. 7 of of simulations is to have sufficient data to test various statistical Li et al. 2021). In particular, we seek to understand the distribution hypotheses that will arise later in the paper. The results of these of 푡ej and how the ejection outcome may scale with various system simulations are summarized in the following sections. parameters, such as the planet masses and spacing. In this section, we present our numerical results (based on 푁-body simulations) – 2.2 Final Outcomes of Scatterings: Orbital Parameters these empirical findings serve as the basis for our theoretical model and analytical understanding discussed in Section 3. After the scattering process has completed, we are interested in the Consider a pair of planets with masses 푚1 and 푚2, radii 푅1 final semi-major axis, eccentricity and inclination (relative to either and 푅2 and semi-major axes 푎1 and 푎2 orbiting a star with mass the initial plane or the ejected planet) of remaining planet, which 푀★. We assume the planets are initially on circular orbits and have we denote as 푎1,ej, 푒1,ej and 휃1,ej respectively, with the subscripts a mutual inclination 0 < 휃12 1 radians. The planets are stable “0” and “ej” denoting the quantity being at time zero and at the final against close encounters for all time if the condition time immediately after the ejection of the final planet. Although √ these results have been known and presented previously in various |푎 − 푎 | > 2 3푟 (1) 2 1 퐻 contexts (see references at the beginning of section 2), we explore is satisfied (Gladman 1993), where the mutual Hill radius 푟퐻 is a broader range of planets masses and mass ratios and test the given by: analytical scalings against simulations.

1/3 (i) Final semi-major axis 푎1,ej: The final semi-major axis is 푎1 + 푎2 푚1 + 푚2 푟퐻 ≡ . (2) determined by the conservation of energy, 2 3푀∗ 퐺푀★푚1 퐺푀★푚2 퐺푀★푚1 If this condition is not satisfied, the resulting system is gravitation- 퐸tot = − − ' − , (3) 2푎1,0 2푎2,0 2푎1,ej ally unstable and will inevitably undergo mutual close encounters. Generally, such an unstable system will result in either the merger which gives a final semi-major axis of −1 of two planets or the ejection of one of the planets. The exact 푎1,0푚2 prevalence depends on the initial system parameters, and planetary 푎1,ej = 푎1,0 1 + (4) 푎2,0푚1 systems with smaller semi-major axes and/or larger planetary radii for the remaining, non-ejected planet. are more likely to result in collisions/mergers rather than planet In our simulations, we find that given the same set of initial planet ejections. For gas giant planets with semi-major axes beyond a few masses and semi-major axes, the final distribution of the semi-major au’s, the most likely outcome appears to be eventual ejection of the axis is determined by Eq. (4) to within 1%. This is a consequence of least massive planet from the system. We focus on such ejection the diffusive nature of the ejection process, which proceed over many events in this section. orbits through a series of energy exchanges, each exchange shifting the ejected planet’s orbital energy by an amount 훿퐸12 퐸2,0. At ejection, the ejected planet deposits all its initial energy into planet 2.1 Numerical Set-Up 1, and the scatter in its final (positive) orbital energy is of order We perform N-body simulations of the orbital evolution of giant 훿퐸12 and is negligible compared to the total energy lost 퐸2,0. planets orbiting a solar mass star, using the IAS15 integrator in- (ii) Final eccentricity 푒1,ej: Our simulations show that the final cluded as part of the REBOUND N-body software (Rein & Liu eccentricity of the remaining planet depends strongly on the mass 2012; Rein & Spiegel 2015). IAS15 is a 15th-order integrator based ratio 푚2/푚1, and weakly on the initial separation of the two planets. on Gauss-Radau quadruature with automatic time-stepping that is Figure1 shows a plot of the distribution density of 푒1 as a function of capable of achieving machine precision; it is well suited for prob- the mass ratio 푚2/푚1 for a system with 푚1 = 1푀퐽 . For 푚2 푚1 lems involving close encounters and high-eccentricity orbits. with initial separation of order 푟퐻 , a good empirical scaling for the We performed an array of N-body simulations involving the typical value of 푒1,ej is scattering of hypothetical unstable 2-planet systems. Each system h푒1,eji ≈ 0.7푚2/푚1. (5) had an inner planet with semi-major axis 푎1 = 5 au, with the outer 푒 planet’s semi-major axis given by 푎2 = 푎1 + 푘0푅퐻 , with 푘0 ∈ The spread in the value of 1,ej increases with the mass ratio of the 푚 /푚 푚 [1.5, 2.0, 2.5]. The inner planet had mass 푚1 ∈ [10.0, 3.0, 1.0, 0.3] planet: for the case where 2 1 1 (i.e. 2 being a test particle), 휎(푒 ) ∼ . h푒 i 푀퐽 while the outer planet’s mass is 푚2, with the mass ratio 푚2/푚1 the standard deviation 1,ej is of order 0 25 1,ej , while for chosen from [1, 2/3, 1/2, 1/3, 1/5, 1/10]. Note that in our simula- the case of 푚2/푚1 ∼ 0.5 the standard deviation is ∼ 0.5h푒1,eji. tions, the outer planet is less massive than the inner planet, although The scaling of eccentricity can be understood as a consequence our analytic results apply to cases with the inner planet being more of the conservation of angular momentum: massive as well. The planets were treated as point particles (their √︃ √︃ 푚 퐺푀 푎 (1 − 푒2) + 푚 퐺푀 푎 (1 − 푒2) = const. (6) radius were set to zero), and the possibility for collisions between 1 ★ 1 1 2 ★ 2 2 planets were not considered. Both planets were started on initially We make the approximation that the apsis of the outer planet and circular orbits, and their initial orbital mutual inclination is set to be the periapsis of the inner planet change much more slowly than their ◦ 휃12,0 = 3 . The initial mean anomaly 푓 , longitude of the ascend- eccentricities and semi-major axes during close encounters, i.e. ing node Ω and longitude of pericenter 휛 were each drawn from 푝 ≡ 푎 (1 + 푒 ) = 푎 ' const. (7) uniform distributions on [0, 2휋]. We computed each system for up 1 1 1 1,0 to 3 × 107 orbits of the inner planet, terminating simulations once 푞2 ≡ 푎2 (1 − 푒2) = 푎2,0 ' const. (8)

MNRAS 000, 000–000 (2020) 4 Bonan Pu

1.4 m /m 1/2 1.2 2 1 = m2/m1 = 1/5 1.0 m2/m1 = 1/10 0.8

0.6 PDF

0.4

0.2

0.0 0 2 4 6 8 10 θ1, ej [deg.]

Figure 2. A histogram of the final inclination of the remaining planet (rela- Figure 1. A histogram of the final eccentricity of the remaining planet, for tive to the initial plane), for a system of two initial planets that have under- a system of two initial planets that have undergone an ejection event. The gone an ejection event. The initial mutual inclination of the two planets is 3◦. different colors correspond to various values of the mass ratio 푚1/푚2. Each The different colors correspond to various values of the mass ratio 푚2/푚1. histogram represents 600 simulations, with 푘0 ∈ [1.5, 2.0, 2.5] (where Each histogram represents 600 simulations, with 푘0 ∈ [1.5, 2.0, 2.5] and 푘0 ≡ (푎2 − 푎1)/푟퐻 ) and 푚1 ∈ [3.0, 1.0] 푀퐽 . Runs with different 푚1 were 푚1 = 푀퐽 . Simulations with different 푘0 were binned together as their binned together as their distributions were indistinguishable statistically. distributions were approximately identical statistically.

Combining Eqs. (7)-(8) with Eq. (6) and substituting a ﬁnal value of 푒2 = 1, we have √︃ √ − ' + ( − )( / ) −1/2 1 푒1, 푓 1 1 2 푚2 푚1 훼0 , (9) where 훼0 is the initial value of the semi-major axis ratio 푎1/푎2. In the limit that (푚2/푚1) 1, Eq. (9) reduces to

푒1,ej ≈ 0.8(푚2/푚1). (10)

(iii) Final inclination 휃1,ej: We find 휃1,ej to be determined most strongly by the mass ratio 푚2/푚1, and somewhat independent of the other parameters. Fig.2 shows our empirical results for the distribution of the inclination as a function of 푚2/푚1. We find that 휃1,ej is well-fit by a Rayleigh distribution with scale parameter 휎 ∼ 0.7휃12,0. This can be understood as a consequence of angular momentum conservation. Since the ejected planet picks up a change in its angular momentum about the z-axis of order sin 휃12,0퐿2,0, Figure 3. Probability density distribution of 푁ej from our two planet scat- angular momentum conservation requires the remaining planet to tering simulations. The different colors represent different values of 푚1, gain angular momentum in equal and opposite direction. As a result, with red, green and blue corresponding to 푚1 = 3, 1, 0.3푀J respectively. planet 1 will pick up an inclination relative to its original plane of For each histogram, we fix 푚2/푚1 = 1/10 and 푘0 = 2.0. The histograms order are empirical results from our N-body simulations, while the solid curves are obtained using the theoretical model in Eq. (19), with 푏 empirically 휃1,ej ∼ (퐿2,0/퐿1,0) sin 휃12,0 ∼ (푚2/푚1) sin 휃12,0. (11) determined using Eq. (22).

the parameters of the system. The scaling dependence of 푡 has 2.3 Timescale to Ejection inst been the subject of many studies (e.g. Chambers et al. 1996; Zhou An important quantity in the dynamical evolution of inner planet et al. 2007; Smith & Lissauer 2009; Pu & Wu 2015; Obertas et al. systems with scattering CJs is the timescale required to finally eject 2017; Wu et al. 2019), the results of which show that generally the one of the planets. We present our empirical results on the scaling instability timescale scales exponentially with the planet spacing, and dependence of the ejection timescale with system parameters. i.e. ln 푡inst ∝ Δ푎. In this study we are interested in the timescale However, before proceeding, there are some caveats with regards to required for an initially unstable system to finally eject one of the the correct metric to use for the ejection timescale. planets, a process which only occurs after 푡inst has already been Firstly, an unstable pair of planets on initially circular orbits reached (see also Rice et al. 2018, for a study on the timescale to the will first pass through a meta-stable phase where the eccentricities first planet-planet collision). Therefore, it is convenient to separate of both planets ramp up gradually, without the planets under-going the ramp-up phase from the ejection timescale by counting time violent close encounters. This ramp-up phase is called the ‘insta- only after the first close encounter. We do so by starting our count bility timescale’ 푡inst in other contexts and its length depends on of the passage of time for planet ejections only after the planets 1

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Figure 6. The maximum likelihood estimate (MLE) estimate of 푏, as a Figure 4. Same as Fig.3, except we ﬁx 푚 = 푀 , while 푚 /푚 varies as 1 퐽 2 1 function of 푚1, for various combinations of the planet mass ratio 푚2/푚1. indicated in the legend. The ﬁlled circles are the results of numerical N-body simulations, while the solid lines are given by Eq. (24). The errorbars are computed using the asymptotic variance of the MLE (Eq. 23).

simulations of 푁ej on the mass of the more massive planet 푚1. The histograms in Fig.3 show the different probability density distributions of 푁ej for systems with various 푚1 ranging from 3푀퐽 to 0.3푀퐽 . In our simulations, while systems with 푚1 = 3푀퐽 have 3 푁ej ∼ 10 , the same system with a 푚1 = 0.3푀퐽 had a typical ejection timescale that is nearly a hundred times greater. We find ∝ −2 that the scaling is very close to 푁ej 푚1 . (ii) Dependence on 푚2/푚1: For a given 푚1, 푁ej generally depends on 푚2/푚1. When 푚2/푚1 1, there is little dependence on 푚2. On the other hand, as 푚2 increases to be of similar order as 푚1, the ejection timescale starts to increase significantly. Fig.4 shows the density distribution of 푁ej for a system with all other parameters fixed, except the ratio 푚 /푚 , which is varied from 1/5 − 1/2. We Figure 5. Same as Fig.3, except we fix 푚1 = 푀퐽 , while the initial separation 2 1 푚 /푚 ) parameter 푘0 = Δ푎/푟퐻 varies as indicated in the legend. find that in comparison to the test-particle limit ( 2 1 1 , a mass ratio of 1/2 results in an ejection timescale that is ∼ 10 times 4.0 larger. We find a scaling of 푁ej ∝ (1 + 푚2/푚1) , the functional and 2 have orbits that are separated by a Hill radius or less, i.e. when form being somewhat arbitrary. 푎 (1 − 푒 ) − 푎 (1 + 푒 ) ≤ 푟 is satisfied. 2 2 1 1 퐻 (iii) Variance of 푁ej: in our simulations, we find significant We define 푡 and 푁 respectively as the time and the number ej ej variance in the distribution of 푁ej for systems that have different of pericenter passages the ejected planet (planet 2) takes between initial orbital phases but otherwise identical orbital parameters. This the first Hill-sphere crossing event and the final ejection event. Note can be seen clearly in Figs.3 and4, where similar systems can have that we use the number of orbits of the ejected planet as opposed ejection timescales that range 4-5 orders of magnitude. We find to the number of synodic periods, because at higher eccentricities that the standard deviation of log10 푁ej is approximately 0.9; this the energy exchange mainly occurs at pericenter passages and not variance is empirically independent of the other system parameters orbital conjunctions. 푁ej and 푡ej can be converted from each other such as planet masses. using the transformations (iv) Dependence on 푘0: We found that the initial planet spac- 푡 1/2 ing Δ푎 = 푘 푟퐻 plays little role in determining the final ejection 1 ∫ 퐺푀∗ 0 푁 (푡)' 푑푡 (12) timescale, as long as the initial ramp-up period of meta-stability is 2휋 3 ( ) 0 푎 푡 accounted for. Fig.5 shows a comparison in the density distribution 푁 −1/2 ∫ 퐺푀∗ of 푁 for systems with otherwise identical parameters, except with 푡(푁)' 2휋 푑푁. (13) ej 0 푎3 (푁) 푘0 varying from 1 to 2.5. (v) Relation between 푁 and ejection time 푡 : Since the semi- We focus on 푁 below, as it is the more physically relevant quantity ej ej ej major axis of the planet increases as it is being ejected, the ejection in the scattering and ejection process. The results of of our numerical time 푡 is usually significantly larger than the naive estimate 푡 ∼ simulations are shown in Figs.3-6. We summarize the key results 푒 푗 푒 푗 푁 푃 where 푃 is the initial orbital period. The discrepancy below: 푒 푗 2,0 2,0 grows larger when 푚1 is smaller, due to the fact that the to-be- (i) Dependence on 푚1: We find a strong dependence in our ejected planet can maintain larger semi-major axes before finally

MNRAS 000, 000–000 (2020) 6 Bonan Pu being ejected. We ﬁnd a best-ﬁt power-law with the form: initial separations, i.e., 1/2 푚 0.46 1 ∫ 2휋 ∫ ∞ ∼ 0.7 1 ¯ ≡ 2 ( ) ( ) 푡ej 8푃2,0푁ej . (14) 훿 퐹 푎2, 푓12 푓 푎2 푑푎2 푑푓12 , (16) 푀★ 2휋 0 0

where 푓 (푎2) is the (unknown) probability density function of 푎2 2.4 Theoretical Model for CJ Scattering over the course of the scattering event. Then we may assume that the distribution of energy exchanges over the scattering process can We present a simple theoretical model for the process of CJ scat- be approximated as a Gaussian distribution with a mean of zero tering to explain our empirical results of Section 2.3. As we shall and width of 훿¯. We do not attempt to compute 퐹(푎2, 푓12) or 푓 (푎2) demonstrate in this section, by assuming that the planet orbital en- explicitly; instead, we constrain them statistically from our N-body ergy undergo a random walk during the scattering process, this simulations by measuring the related parameter 푏, which is the ratio model can explain both the distribution and the scaling of the ejec- of the initial orbital energy and the RMS energy exchange and is tion time of CJ scatterings. given by Consider the limiting case of a pair of planets with 푚 푚 . 1 2 ¯ −1 The two planet orbits are ‘unstable’ such that their orbits come very 퐺푚1푚2훿 푏 ≡ |퐸2,0| . (17) close to each other and experience repeated crossings. At larger 푎1 orbital distances it is common for the two planets to remain orbit- In the limit of many successive passages, each giving a kick in crossing for extended periods of time without physically colliding. energy that is small relative to the initial orbital energy |퐸2,0| (i.e. Since 푚1 푚2, we assume the orbital parameters of 푚1 stay 푁 1 and 푏 1), the probability density distribution in Δ퐸2/퐸2,0 constant during the scattering process. after 푁 orbits is given by At every pericenter passage (or apocenter passage if 푎2 < 푎1), 2 ! 1 −(Δ퐸2/퐸2,0) planet 2 exchanges a certain amount of orbital energy with planet 푓 (Δ퐸 /|퐸 |) = √ exp . (18) 2 2,0 2 1. The amount of energy exchanged, 훿퐸12 depends on the orbital 2휋푁 2푁푏 properties of the two planets. We hypothesize that 훿퐸 can be 12 푁 is the lowest value of 푁 such that Δ퐸 /|퐸 | = 1; it is known as approximated as follows: ej 2 2,0 the ‘stopping time’ of the Weiner process and its probability density 퐺푚1푚2 distribution is given by the Levy distribution (see, e.g. Borodin & 훿퐸12 ∼ 퐹 (푎2, 푓12) , (15) Salminen 2002): 푎1 푏 where 퐹 is a dimensionless function, and 푓 is the difference of the ( | ) (− 2/ ) 12 푓 푁ej 푏 = √︃ exp 푏 2푁ej . (19) two planets’ true longitudes at time of pericenter passage of planet 3 2휋푁ej 2. Note that in general, 퐹 should depend on 푒2 as well. However, given some 푎 , the possible values of 푒 is narrowly constrained ( ) ∝ −3/2 2 2 The distribution in Eq. (19) is long-tailed since 푓 푁ej 푁ej for due to conservation laws (see Sec. 2.5 below), so to a first order 푁 푏2, and all of its moments including the arithmetic mean approximation, it is sufficient to know only 푎 . ej 2 diverge. The geometric mean is h푁 i = exp (2훾 )푏2 ≈ 3.17푏2 Due to symmetry, for a fixed value of 푎 the function 퐹 is odd ej GM EM 2 (where 훾 ≈ 0.57 is the Euler-Mascheroni constant) and its mode with respect to 푓 , i.e. the energy exchange is equally likely to be EM 12 is equal to 푏2/3. Another useful quantity is the harmonic mean, positive and negative, and averaging over 푓 gives h퐹(푎 , 푓 )i = 12 2 12 given by 0. As a result, even though at each close approach between planet −1 2 1 and planet 2 there is a finite amount of energy exchange, in the h푁ejiHM ≡ h1/푁eji = 푏 . (20) limit that |훿퐸12| 퐸2, the long-term energy exchange is small, The standard deviation of the quantity ln 푁ej is Var(ln 푁ej) = since 푓12 is sampled almost periodically and uniformly. On the √ | | ∼ 휋/ 2 ≈ 2.2, regardless of the value of 푏, and the 68% and 95% other hand, if 훿퐸12 퐸2, then each close encounter changes 2 2 2 2 quantile ranges are 푁ej ∈ [0.25푏 , 13푏 ] and [0.1푏 , 500푏 ], re- the period of planet 2 materially, such that the value of 푓12 on the next approach is randomized. It is this randomization of the spectively. In short, 푁ej is distributed with a long tail at larger values relative phase that causes energy exchange at iterative encounters and its distribution can easily span several orders magnitude. to behave chaotically, resulting in a drift in orbital energy of planet The next step is to empirically determine the value of 푏 from 2 (a similar phenomenon occurs when highly eccentric binaries the results of our numerical simulations, given the set of system experiences chaotic tides; see, e.g. Vick & Lai 2018). parameters (푚1, 푚2, 푎2,0, etc.). To do so, we make use of the In general, the amount of random diffusion in 퐸 scales in- maximum likelihood estimate (MLE). The likelihood function for 2 ∈ [ ] versely proportional to the timescale in which the relative orbital 퐾 observations of 푁ej,i, 푖 1, 2, ...퐾 is given by phases 푓12 at successive encounters can be randomized, so the en- 퐾 Ö 푏 2 ergy exchange is most efficient at large values of 푎2, and suppressed 퐿(푏) = exp (−푏 /2푁ej,i). (21) √︃ 3 when 푎2 is small. When eventually 퐸2 drifts to a positive value, the 푖 2휋푁ej,i planet is ejected and the process terminates. Now we study the question of for how long this process occurs, Maximizing ln 퐿 with respect to 푏, we have i.e. the mean value and distribution of 푁 . To do this, we make use 퐾 !−1/2 ej √ ∑︁ of a Brownian motion approximation in 퐸 (for a recent application −1 2 푏MLE = argmax 퐿(푏) = 퐾 푁ej,푖 . (22) of this idea in a different context, see Mushkin & Katz 2020). 푏 푖 Suppose we are able to find the RMS value of the function Its variance is given by the asymptotic variance of the MLE: 퐹(푎 , 푓 ) over the course of two-planet scattering, weighted by 2 12 −1 the likelihood of each 푎 occurring during the scattering process. 휕2퐿(푏) 2 Var(푏 ) = 퐾 = −2푏2 /퐾. (23) We call this quantity 훿¯(푚 , 푚 , 푎 , 푎 ), which depends on the MLE 2 MLE 1 2 1 2,0 휕푏 푏=푏MLE

MNRAS 000, 000–000 (2020) Scattering 1+2 7

In Fig.6 we show the empirical values of 푏MLE estimated empirical results: using Eq. (22) as functions of 푚1 and 푚2/푚1. We find that 푏 can 푎2 (1 − 푒2) 1/3 be well-approximated by . 1 + 2(푚1/3푀★) . (27) 푎1 (1 + 푒1) 푐2 푐3 푐4 푚1 푚2 푎1,0 The above constraint asserts that the maximal planet separation 푏 ≈ 푐1 1 + , (24) 푀★ 푚1 푎2,0 should not exceed 2 Hill radii at all times. with 푐1 = 0.06 ± 0.02, 푐2 = −0.98 ± 0.03, 푐3 = 2.14 ± 0.07 and The above three constraints reduce the degree of freedom to 2 푐4 = −1.4 ± 0.5; the above model has a value of 푅 = 0.99 when 1, which means that given any one variable, the other 3 variables fitted against the empirical values of 푏 (as estimated by MLE). are uniquely determined. One can then optimize for the lowest al- This empirical scaling is in fact consistent with the results lowed values of 푎2 and 푎2 (1 − 푒2). This then produces a theoretical of past studies, which showed that for comets with 푎2 푎1 and lower limit on 푎2 during the scattering process. However, it is not (1 − 푒2) 1, the RMS energy exchange per pericenter passage is a given that this minimum can always be reached, for two reasons: of order 훿퐸12 ∼ 퐺푚1푚2/푎1 (see, e.g. Wiegert & Tremaine 1999; Firstly, since Δ퐸2 undergoes an approximate Brownian motion, it Fouchard et al. 2013). This result would imply that 푐2 = 푐4 = −1, is likely to spend large fractions of time being positive, such that which is in agreement with our empirical results. 푎2 is never much below its initial value. Secondly, energy exchange Eqs. (19) and (24) provides an accurate description of the becomes less efficient as 푎2 decreases, since the timescale for the distribution for 푁ej as long as 푚2/푚1 . 1/3. However, this model randomization of the relative orbital phase becomes larger. breaks down in the comparable mass regime (푚1 ∼ 푚2), where 푁ej We show these limits for 푎2 and 푟2 ≡ 푎2 (1 − 푒2), compared is usually much larger than predicted by Eq. (24). This is because for with empirical results from our simulations, in Fig.7. We see that planets of comparable mass, as 푎2 increases 푎1 will decrease by a generally, 푎2,min/푎1,0 ∼ 1/2, and decreases with increasing 푚1. comparable value. As a result, the energy exchange becomes much The theoretical constraints agreed well with empirical results when less efficient as 푎2 increases since the planet can only come close 푚2/푚1 1, but breaks down when 푚2/푚1 & 0.2. We also find to one another when planet 1 and planet 2 are simultaneously at that 푟2,min decreases strongly with increasing 푚2/푚1, and can reach their apocenter and pericenter respectively. A theoretical model for 푟2,min/푎1,0 . 0.05 for 푚2 ∼ 푚1. this strong scattering process at comparable masses is an intriguing While it is possible for the less massive planet to be sent deep question in its own right, and necessary for further refinements on into the inner system during the scattering process when 푚2 ' 푚1, the results presented here, but beyond the scope of this paper. in practice this is an unlikely outcome. In Fig.8 we show the cumulative density distribution of realized 푎2,min/푎1,0 and 푟2,min/푎1,0 from our suite of N-body simulations. We find that 푎2,min/푎1,0 2.5 Scattering into inner system and 푟2,min/푎1,0 have a broad distribution: for 푚1 = 10푀퐽 and 푚 ' 푚 , 푎 /푎 reaches below 1/10 only ∼ 10% of the Aside from the orbital parameters and ejection timescale, another 2 1 2,min 1,0 time. When 푚 ' 푚 , the empirical distribution for 푎 /푎 quantity we are interested in is the minimum approach distance a 1 2 1,min 1,0 and 푟 /푎 are very similar to 푟 /푎 and 푎 /푎 re- planet might have with its host star. Since planet ejections occur 1,min 1,0 2,min 1,0 2,min 1,0 spectively, due to symmetry. Thus, even for initial parameters most gradually in a random walk-like manner, the ejected planet may likely to result in giant planets scattered deep into the inner sys- first meander a significant amount inwards before being eventually tem (i.e. 푚 = 10푀 and 푚 ' 푚 ), the likelihood of one of the ejected. If the to-be-ejected giant planet at some point comes too 1 퐽 2 1 planets reaching a pericenter distance less than 1/10th of the initial close to the inner system, it can undergo non-secular interactions semi-major axis is only ∼ 20%. with the inner system, causing our semi-secular approximation (see Section3) to break down. Therefore, it is important to quantify the extent to which the giant planet might first move inward. First, due to conservation laws, there is a limit to how deeply 3 SEMI-SECULAR ALGORITHM FOR “N+2” inwards a planet can meander during the scattering process. If we SCATTERINGS assume the planet orbits remain (approximately) co-planar, then We now consider how an inner low-mass planet system respond the 4 relevant variables are 푎1, 푎2, 푒1 and 푒2, which satisfy the to an outer pair of giant planets undergoing strong scatterings. We constraints label the inner planets as 푗 ∈ [푎, 푏, 푐, ...], while the outer planets • Energy conservation: are labeled 푝 ∈ [1, 2]. In Sec.4 we focus on inner systems with only one planet, and we extend our results to cases with 2 inner planets in ∑︁ ∑︁ 푚 푗 /푎 푗,0 = 푚 푗 /푎 푗 . (25) Sec.5, although our method can work for a general number of inner 푗 푗 and outer planets. We imagine the inner system to be consistent with those discovered by Kepler, i.e. the planets have semi-major • Angular momentum conservation: axes typically between 0.02 − 0.5 au and are super-Earths in mass ∑︁ √︃ ∑︁ √︃ (푚 ∼ 3 − 20푀 ). We have a system of outer planets with semi- 푚 푎 (1 − 푒2 ) = 푚 푎 (1 − 푒2). (26) 푗 ⊕ 푗 푗,0 푗,0 푗 푗 푗 major axes beyond ∼ 2 − 3 au that are gravitationally unstable 푗 푗 √ (푘0 ≤ 2 3), and at least one of the planets have a fairly large mass • Second law of thermodynamics: The system must not sponta- (≥ 100푀⊕), although 푚2 may be more comparable to super-Earths neously ‘scatter’ itself into a state that is indefinitely stable, even in size. We assume that the inner system is well-separated from the if this is permitted by the conservation laws. In general, the stabil- outer system (푎 푗 푎1, 푎2), such that the inner planets do not ity criterion for 2 planets with general masses, eccentricities and participate directly in the outer scattering process. inclinations is complicated (see, e.g. Petrovich 2015). In the limit As noted in Section1, to address the question of how the of co-planar orbits with 푚1 푚2, we find that requiring planets inner planets are affected by the outer scattering, a direct approach to follow the criterion below results in best agreement with the based on N-body simulations is inadequate. The issue lies in the

MNRAS 000, 000–000 (2020) 8 Bonan Pu

Figure 8. Top: empirical cumulative distribution of of 푎2,min as function of 푚1 and 푚2/푚1 from our sample of N-body simulations. The the red and Figure 7. Top: empirical values of 푎2,min as function of 푚2/푚1. Each blue histograms 푚1 = 10, 1푀퐽 respectively. The thick lines correspond data point represents the global minimum over all simulations. The blue, to 푚1/푚2 = 3/2 while the thin lines correspond to 푚1/푚2 = 10. Bottom green and red circles correspond to 푚1 = 10, 3, 1푀퐽 respectively. The panel: similar to the upper panel, except we plot 푟2,min = min[푎2 (1 − 푒2)] dashed lines are derived from minimizing 푎2 under the constraints given instead of 푎2,min. by Eqs. (25)-(27) in the limit of 푚2/푚1 1. We suppress error bars in the empirical results because it is unclear how to estimate the minimum of a set of observations without prior assumptions about the distribution of our data. The bottom panel is similar to the upper panel, except we plot influence of the inner planets on the outer planets is negligible in comparison with the outer planets’ own violent scatterings. Fur- 푟2,min = min[푎2 (1 − 푒2)] instead of 푎2,min. thermore, since the inner planets are sufficiently far from the outer planets as to avoid direct scattering interactions, the gravitational influence by the outer planets is well described by secular dynamics differing time-scales involved: The inner planets have short orbits (Matsumura et al. 2013). on the timescale of days, which forces the time-step of the N-body Our algorithm is as follows. First, we evolve the gravitational simulation to not more than a few hours. On the other hand, the interaction between the outer planets, in the absence of any inner outer planets have periods of ∼ 10 years and an ejection timescale planets. We then obtain a timeseries of the position-velocity vectors of potentially hundreds of Myrs. To make matters even worse, the of each of the outer planets from beginning until final ejection. In prospect of scattering events driven constantly by close encounters the case of two giant planets, we have r푝 (푡) and v푝 (푡) for 푝 = 1, 2. between planets preclude the use of fast and efficient symplectic These will be used as forcing terms to calculate the evolution of the integrators (e.g. the Wisdom-Holman mapping). inner planets, as follows. Here, we develop a hybrid method to evaluate the dynamical Define j and e as a planet’s dimensionless angular momentum evolution of an inner system perturbed by a system of unstable outer and eccentricity vectors: CJs. In this method, we decouple the timescale of the inner planets √︁ and outer planets by computing their orbital evolutions separately. j = 1 − 푒2nˆ, e = 푒 uˆ (28) This is possible because we can safely neglect the back-reaction on the outer planets by the inners: since the inner planets are much where n and u are unit vectors, n is in the direction normal to the less massive compared to their outer companions, the gravitational orbital plane and u is pointed along the pericenter. We compute the

MNRAS 000, 000–000 (2020) Scattering 1+2 9 time evolution of these vectors for the outer planet 푝 using 4 1+2 SCATTERING

1 We consider a single inner planet ("푎") with two outer CJs. j (푡) = r (푡) × v (푡) (29) 푝 1/2 푝 푝 Planet 푎 has mass 3푀⊕ and semi-major axis chosen from 푎 ∈ (퐺푀★푎 푝) 푎 {0.1, 0.15, 0.2, 0.25, 0.375, 0.5, 0.75, 1.0}, these are much smaller 1 e푝 (푡) = v푝 (푡) × (r푝 (푡) × v푝 (푡)) . (30) than the initial semi-major axes (≥ 5 au) of the outer planets so 퐺푀★ that planet 푎 typically does not participate directly in the scattering According to Laplace-Lagrange theory (e.g. Murray & Der- between planets 1 and 2. We assume all planets have initially circular and co-planar orbits, except that 휃 = 3 degrees. We integrate mott 1999), the evolution equations for the eccentricity vector e 푗 2,0 this system using the semi-secular algorithm described in Sec.3.A and unit angular momentum vector j 푗 on the planet 푗 due to the simulation is halted if any pair of planets undergo orbit crossings, action of planet 푘, in the limit that 푒 푗 , 푒푘 , 휃 푗푘 are small, are given by: or if planet 푎 attains an eccentricity greater than 0.99. We discuss the results of these simulations below. 푑e 푗 = −휔 (e × j ) + 휈 (e × j ), (31) 푑푡 푗푘 푗 푘 푗푘 푘 푗 푘 4.1 Empirical Results 푑j 푗 = 휔 푗푘 (j 푗 × j푘 ). (32) In our simulations we find a wide range of the final possible values 푑푡 푘 of the inner planet eccentricity 푒푎, inclination 휃푎 measured relative The quantities 휔 푗푘 and 휈 푗푘 are the quadrupole and octupole pre- to the original orbital plane of planet 푎 (note the orbits of planets 푎 cession frequencies of the 푗-th planet due to the action of the 푘-th and the remaining CJ are initially aligned), and mutual inclination planet, given by: 휃푎1 between the inner planet and the remaining CJ. As mentioned earlier, the evolution has two phases: the first phase is when the 퐺푚 푗 푚푘 푎< (1) system has 3 planets total, with the outer two planets (planets 1 and 휔 = 푏 (훼), (33) 푗푘 2 3/2 푎> 퐿 푗 2) under-going scattering and the inner planet (planet 푎) interacting secularly with both planets. At some point, an outer planet is ejected, 퐺푚 푗 푚푘 푎< ( ) 휈 = 푏 2 (훼). (34) 푗푘 2 3/2 and the inner planet interacts with only the remaining CJ, whose 푎> 퐿 푗 orbital properties remain a constant in time. We define the eccentricity and inclination of the inner planet at Here 푎< = min(푎 푗 , 푎푘 ), 푎> = max(푎 푗 , 푎푘 ), 훼 = 푎, 퐿 푗 ' √︁ the time of ejection as 푒푎,ej and 휃푎,ej respectively. After ejection, the 푚 푗 퐺푀∗푎 푗 is the angular momentum of the 푗-th planet, and the ( ) inner planet still undergoes secular oscillations in eccentricity and 푏 푛 (훼) 3/2 are the Laplace coefficients defined by: inclination due to interactions with the remaining CJ. We thus define the time-averaged RMS eccentricity and inclination at infinity as 휋 ( ) 1 ∫ cos (푛푡) 푏 푛 (훼) = 푑푡. (35) !1/2 3/2 / ∫ 푡 2휋 0 (훼2 + 1 − 2훼 cos 푡)3 2 1 2 푒푎,∞ ≡ lim 푒푎 (푡)푑푡 , (38) 푡→∞ 푡 − 푡ej 푡 Laplace-Lagrange theory breaks down for more general values ej 1/2 of 푒 and 휃 , and therefore, in this work we instead adopt a set ∫ 푡 ! 푗 푗푘 1 2 of modified secular equations that interpolates between Laplace- 휃푎,∞ ≡ lim 휃푎 (푡)푑푡 . (39) 푡→∞ 푡 − 푡ej 푡 Lagrange theory and secular multipole expansion. The equations ej are given in Eqs. (A2)-(A5) in (Pu & Lai 2018) and have better These quantities can be easily evaluated using secular theory (see, performance than Eqs. (31- 32) when 푒 푗 and 휃 푗푘 are large but e.g. Pu & Lai 2018). For the mutual inclination, 휃푎1 remains con- = (푎푎/푎1) 1. Thus we use these hybrid equations from (Pu & stant once ejection has occured, thus 휃푎1,∞ 휃푎1,ej. Since the final Lai 2018) in place of Eqs. (31)-(32) to compute the gravitational value of 휃1,ej is small (see Sec. 2.2), in general 휃푎1,∞ ≈ 휃푎,∞. influence of the outer planets on the inner planets. Note that the We focus on 푒푎,∞ and 휃푎,∞ as they are more representative of the adopted equations employ orbital averaging over both the inner long-term post-scattering dynamics of the inner planet. planet and outer planet orbits. Even though the outer planet orbits Fig.9 shows the values of 푒푎,∞ and 휃푎,∞ for a subset of our vary on orbital timescales due to the strong mutual scatterings, the simulations. According to Fig.9, 푒푎,∞ and 휃푎,∞ tends to increase √︁ use of secular orbital averaging is appropriate since the interactions roughly as 푁ej. We provide a theoretical model for this behavior between the outer and inner planets are secular and accumulate over in Sec. 4.2. Secondly, we find a strong dependence of the final large number of orbits, the orbit-to-orbit variations can be ignored values of 푒푎,∞ and 휃푎,∞ on the planet mass ratio 푚2/푚1, with so long as the orbital period of outer planets is much shorter than outer planet pairs having comparable masses leading to much higher the secular timescale. values of 푒푎,∞ and 휃푎,∞ compared with cases where 푚1 푚2. In summary, we compute the evolution of the inner planets The main reason is that these final values increase as the mass ratio 푗 ∈ [푎, 푏, 푐...], by the action of other inner planets 푘 ∈ [푎, 푏, 푐...] 푚2/푚1 increases, and more eccentric/inclined perturbers tend to as well as outer planets 푝 ∈ [1, 2] as follows: drive stronger perturbations on the inner planet. How to understand the diversity of final results in this pa- 푑j 푗 ∑︁ 푑j 푗 ∑︁ 푑j 푗 rameter space? The picture becomes clearer if we normalize the = + , (36) 푑푡 푑푡 푑푡 results by the “scattering-free” theoretical expectations. We intro- 푘=푎,푏... 푘 푝=1,2 푝 duce these “scattering-free” quantities as the “secular” eccentricity 푑e 푗 ∑︁ 푑e 푗 ∑︁ 푑e 푗 = + . (37) and inclination 푒푎,sec and 휃푎,sec that are the (RMS) eccentricities 푑푡 푑푡 푑푡 푘=푎,푏... 푘 푝=1,2 푝 and inclinations that would be expected on planet 푎, if the the dynamical history of the two-planet scattering were to be ignored, The results of the calculations are discussed in Sec.4. and the inner planets started their orbital evolution with 푚1 at its

MNRAS 000, 000–000 (2020) 10 Bonan Pu

Figure 9. The final values of 푒푎,∞ (top panels) and 휃푎,∞ (in radians, bottom panels) as defined by Eqs. (38)-(39), as a function of 푁ej, for a 1-planet inner system subject to the gravitational influence of two scattering giant planets. The masses of the outer planets are varied with 푚1 = 10, 3, 1 or 0.3푀퐽 (the red, green, blue and magenta points respectively), while the mass ratio 푚2/푚1 = 1/2, 1/5, 1/10 for the filled circles, triangles and stars respectively. The initial semi-major axes of the outer planets are 푎1 = 6.0 au and 푎2 = 푎1 + 푘0푟퐻 with 푟퐻 being the mutual Hill radius and 푘0 chosen randomly from [1.5, 2.0, 2.5]; the value of 푘0 matters little for the final results. The left panels show systems where the initial 푎푎/푎1 = 1/20, while the right panels have 푎푎/푎1 = 1/10.

final orbital state and 푚2 removed. In other words, 푒푎,sec and 휃푎,sec where 휔푎1,0 is the (initial) secular quadrupolar precession fre- are RMS eccentricity and inclination that planet “a” would finally quency of planet 푎 driven by planet 1 (see Eq. 33) and 푃1,0 is obtain, if it started on an initially circular, non-inclined orbit under the initial orbital period of planet 1. This boundary is consistent the influence of the perturber planet “1” with initial eccentricity and with the inner planet 푎 being driven by stochastic secular forcing inclination 푒1 = 푒1,ej, 휃푎 = 휃푎,ej. For 퐿푎 퐿1, we have (e.g. Pu from planets 1 and 2 during the ejection process: When 푁ej 푁sec, & Lai 2018): the ejection occurs much more quickly than the timescale of secular √ interactions, and the dynamical history of the ejection can be ig- 5 2푎 푒 푎 1,ej nored. On the other hand, when 푁 푁 , the stochastic ‘forcing’ 푒푎,sec = (40) ej sec ( − 2 ) 4푎1 1 푒1,ej on planet 푎 driven by the scattering perturbers will cause 푒푎 and 휃푎 √ to undergo a random walk of its own, with the value of 푒푎,∞ and 휃푎,sec = 2휃1,ej (41) √︁ 휃푎,∞ scaling proportionally to 푁ej. 휃푎1,sec = 휃1,ej (42) The final results can be summarized most succinctly if we consider the deviation of the final values of 푒 and 휃 from their (note that 휃 is the inclination of planet 1 measured relative to 푎 푎 1,ej secular predictions and define the “boost factors”: its initial orbital plane). Fig. 10 shows our numerical results of Fig. 9 for the final RMS values of 푒푎,∞ and 휃푎,∞, normalized by the secular expectations 푒 and 휃 . We find that the scaling for 2 2 푎,sec 푎,sec |푒푎,∞ − 푒푎,sec| 훾2 ≡ (44) the final values of 푒푎,∞ and 휃푎,∞ can be divided into two regimes. In 푒 2 푒푎,sec the case where 푁ej is small, 푒푎,∞ and 휃푎,∞ reduce to their “secular” 2 2 expectations. In the case that 푁 is large, the ratio 푒푎,∞/푒푎,sec and |휃푎,∞ − 휃푎,sec| ej 훾2 ≡ . (45) 휃푎,∞/휃푎,sec can be either larger or smaller than 1, and is bounded 휃 2 √ 휃푎,sec from below by 2/2; the average values scale proportionally to √︁ 푁ej, albeit with a large spread. The transition between the two Figs. 11 and 12 show the comparison of our numerical results for regimes occur approximately at 푁ej ∼ 푁sec, with 푁sec given by the values of 훾푒 and 훾휃 for a subset of our numerical integrations. −1 −1 −3/2 휔푎1,0푃1,0 1 푚1 푎푎 We find that across a wide range of parameters for 푎푎, 푎1, 푚푎, 푚1 푁sec ≡ = , (43) 2휋 2휋 푀★ 푎1 and 푚2, the quantities 훾푒, 훾휃 have a universal scaling given by

MNRAS 000, 000–000 (2020) Scattering 1+2 11

Figure 10. Same as the Fig.9, except the eccentricities and inclinations are normalized by the “secular” expectation 푒푎,sec and 휃푎,sec given by Eqs. (40)-(41).

√︁ (shown as the solid black line in Figs. 11 and 12): The simple universal scaling 푁ej can in fact be derived from √︃ the ﬁrst principles using secular Laplace-Lagrange theory, as we 훾푒 ∼ 훾휃 ∼ 푁ej/푁sec. (46) discuss below.

The boost factor for the mutual inclination, defined as | 2 − 2 | 4.2 Analytic Model for “1+2” Secular Evolution: Eccentricity 휃푎1,∞ 휃푎1,sec 훾2 ≡ (47) 휃,푎1 2 휃푎1,sec We model the dynamical evolution of an inner planet 푎 subject to the gravitational influence of a pair of outer perturbers under-going also shows the same scaling, but with different normalization. We gravitational scattering as a linear stochastic differential equation find that 훾휃,푎1 ∼ 1.4훾휃 ; we provide a theoretical explanation for (SDE). We define E ≡ 푒 exp (푖휛) and I ≡ 휃 exp (푖Ω) as the com- this in Sec. 4.3. plex eccentricity and inclination respectively. Note that 푒 = |E| and To make this scaling even clearer, and to show its robustness 휃 = |I|. In the discussion below we will focus on the eccentricity over a range of system parameters, in Figs 13- 15 we show the evolution and derive the boost factor 훾푒, although the inclination is mean square values of 훾2, binned by logarithmic increments of 푒 completely analogous and will have the same scaling as 훾휃 . 푁 /푁sec for various combinations of 푎푎, 푚 and 푚 . We see that ej 1 2 First, consider an inner planet 푚푎 with initial eccentricity E푎,0 the approximate scaling given by Eq. (46) agrees very well with undergoing secular evolution with an external planet 푚1 푚푎 that the simulations for values of 푎푎/푎 ranging from 1/7 − 1/20, 푚 1 1 has a constant eccentricity E1. For simplicity, we ignore for now the from 3 − 0.3푀 , and 푚 /푚 from 1/10 to 1/2, although there is 퐽 2 1 secular interaction between planet 푎 and 2. The evolution of E푎 (푡) a trend of increasing deviation from Eq. (46) when 푁ej/푁sec 1. is governed by the ODE We explore a possible reason for this deviation, and present a more accurate analytic formula for h훾2i in Sec. 4.2. In general, the above 푑E푎 (푡) = 푖휔푎1E푎 (푡) − 푖휈푎1E1 (푡), (48) scaling is accurate for 푚2/푚1 . 1/2 and 푎푎/푎1 . 1/5. When 푑푡 푚 ∼ 푚 and/or 푎푎/푎 & 1/5, it is often the case that the ejected 1 2 1,0 where 휔푎1, 휈푎1 are given by Eqs. (33)-(34). The solution to the planet can come very close to the orbit of planet 푎, resulting in strong above equation is given by non-secular interactions that causes 훾푒, 훾휃 to be much greater than predicted by Eq. (46). E푎 (푡) = E푎,free (푡) exp (푖휔푎1푡) + E푎,forced, (49)

MNRAS 000, 000–000 (2020) 12 Bonan Pu

2 Figure 13. The average value of 훾푒, binned by log (푁ej/푁sec with 4 bins per logarithmic decade, as a function of 푁ej/푁sec. For each of the points, 푚 = 푀 and 푚 /푚 = 1/5. The red, green, blue and magenta ﬁlled Figure 11. The value of 훾2 (Eq. 44) plotted as a function of 푁 /푁 (see Eq. 1 퐽 2 1 푒 ej sec circles correspond to 푎 /푎 = 1/20, 1/13, 1/10 and 1/7 respectively. The 43) for our simulations. Here 푎 = 0.3 au (corresponding to 푎 /푎 = 1/20). 푎 1 푎 푎 1 errorbars are given by the standard error, and the solid black line is given by Red, green and blue points correspond to 푚1 = 3, 1, 0.3푀퐽 respec- 2 h훾푒 i = 푁ej/푁sec. tively. The ﬁlled circles, triangles and stars correspond to 푚2/푚1 = 2 1/2, 1/5, 1/10 respectively. The black solid line is given by 훾푒 = 푁ej/푁sec.

Figure 14. Same as Fig. 13, except that we ﬁx 푎푎/푎1 = 1/10, and 푚2/푚1 varies as indicated by the plot legend.

2 Figure 12. Same as Fig. 11, except we show 훾휃 as deﬁned by Eq. (45). where 휈푎1 E푎,forced = E1, (50) 휔푎1 and

E푎,free = E푎,0 − E푎,forced. (51) Applying Eq. (48) to the secular evolution of planet 푎 after the ejection of planet 2, we have that E푎,0 = E푎,ej (where E푎,ej = E푎 (푡ej)), and the RMS eccentricity |E푎,∞| is given by

2 2 2 |E푎,∞| = |E푎,free| + |E푎,forced| 2 2 ∗ = |E푎,ej| + 2|E푎,forced| − 2Re(E푎,ejE푎,forced). (52)

Note that |E푎,∞| is what we termed 푒푎,∞ in Sec. 4.1. If the initial Figure 15. Same as Fig. 13, except that we ﬁx 푎푎/푎1 = 1/10, and 푚1 varies eccentricity of planet 푎 is zero, then√ the free eccentricity is equal to as indicated by the plot legend. the forced eccentricity, and 푒sec = 2푒forced. Now we ask the question: What happens to E푎 (푡) if, instead

MNRAS 000, 000–000 (2020) Scattering 1+2 13 of being a constant, E1 (푡) is a stochastically varying quantity, as is The expectation of the forced eccentricity is the case during the scattering process. We study a version of Eq. 휈 2 E 2 푎1 2 (48) with 1 being given by a Brownian motion stochastic process: h|E푎,forced| i = 2 휎E1푡ej. (58) 휔푎 E1 (푡) = 푍 (푡), where 푍 (푡) is a Brownian motion in the complex plane with diffusion constant equal to 휎E1, i.e. 푍 (푡) = 푋(푡) + 푖푌 (푡) From Eqs. (52)-(54), the RMS eccentricity of planet 푎 is where 푋(푡), 푌 (푡) are each given by a Gaussian distribution with 2 2 2 휈 25푎 h푒1,eji 푁ej 2 h|E |2i = 푎1 2 ∼ 푎 mean h푋i = h푌i = 0, variance Var(푋) = Var(푌) = 휎E 푡, and 푎,∞ 4 휎E1푡ej . (59) 1 휔푎 8푎2푏2 [( ( ) ( )] = [( ( ) ( )] = 2 ( ) 1 covariance Cov 푋 푠 , 푋 푡 Cov 푌 푠 ,푌 푡 휎E1min 푠, 푡 . 2 The diffusion coefficient of the perturber eccentricity, 휎E1 is We see that h|E푎,∞| i ∝ 푁ej. However, in this unconstrained case, 2 a constant that can either be calculated analytically or numerically, it is also the case that |E푎,forced| ∝ 푁ej, so that the scaling for or derived empirically from the time series of scattering planet the boost factor is 훾푒 = const., which is contrary to our empirical systems. We make a heuristic estimate of it here. Over the ejection results. This contradiction arises because we have not taken into → timescale, the eccentricity of planet 1 changes from 푒1 = 0 account the fact that E1,ej is a known quantity and not a random 푒1,ej (where 푒1,ej is the eccentricity of planet 1 when planet 2 variable. Only when we place a constraint on the Brownian motion has been ejected; see Section2). On average, this process takes at 푡ej can the desired scaling be derived. 2 푁ej ∼ 푡ej/푃1,0 ∼ 푏 orbits (see Eq. 18). Thus, one might surmise: 2 2 2 Case 2: E is known or constrained h푒1,eji ∼ 2휎E1푏 푃1,0, (53) 1,ej When the final value of E at 푡 = 푡 is known, the evolution E (푡) is where h푒2 i ∼ (푚 /푚 ) (see Sec. 2.2). This yields 1 ej 푎 1,ej 2 1 qualitatively similar, but the statistical properties change due to the Brownian motion in E being “tied down” at the final time, giving it 휎2 ∼ h푒2 i/(2푃 푏2). (54) 1 E1 1,ej 1,0 a lower variance. To recognise that this process is different from an We would like to know what are the mean, variance and distributions unconstrained Brownian motion, we label it 퐵(푡) instead of 푍 (푡). At of E푎 (푡) given the initial conditions and parameters. Note that the 푡 = 0, we have E1 = 퐵(0) = 0, while at 푡 = 푡ej, E1 = 퐵(푡ej) = E1,ej. value of E푎 (푡) at ejection is not the ultimate quantity of interest In between this time, 퐵(푡) executes a (complex) Brownian motion here, since planet 푎 still under-goes secular coupling with planet and is normally distributed, with mean and variance (Borodin & 1 after ejection. Our final goal is to derive the expectation, and if Salminen 2002) possible the distribution of E푎,∞. 푡 To proceed, note that Eq. (48), with E (푡) = 푍 (푡), has the h퐵(푡)i = E1,ej (60) 1 푡ej solution 2 2푡(푡ej − 푡)휎 ∫ 푡 [ ( )] ≡ h 2 ( )i − h ( )i2 E1 푖휔푎푡 −푖휔푎 푠 Var 퐵 푡 퐵 푡 퐵 푡 = . (61) E푎 (푡) = −푖휈푎1푒 푒 푍 (푠)푑푠, (55) 푡ej 0 Another relevant quantity is the covariance of a Brownian bridge E ( ) where we have assumed 푎 0 = 0. The statistical property of with itself at a different time, which (without loss of generality, E ( ) 푎 푡 as determined by Eq. (55) depends on whether the final value assuming 푠 < 푡) is given by of E (푡 ) = E is known (empirically measured, or otherwise 1 ej 1,ej 2 constrained by conservation laws). If E is unconstrained, then 2푠(푡ej − 푡)휎 1,ej Cov[퐵(푠), 퐵(푡)] ≡ h퐵(푠)퐵∗ (푡)i = E1 . (62) 푍 (푠) is the classic 2-D Brownian motion. If E1,ej is known a priori, 푡ej then 푍 (푠) is not a Brownian motion but rather a Brownian bridge, We can now calculate the expectation of E푎,ej. Unlike the uncon- which is given by a different density distribution that has a reduced strained case, the mean is non-zero: variance towards the end of the stochastic process. We consider 푖휔푎1푡ej ! both cases below. In this study, since the final values of perturber 휈푎1 푒 − 푖휔푎1푡ej − 1 hE푎,eji = 푖E1,ej , (63) properties are known, case 2 is the more appropriate one. We deal 휔푎1 휔푎1푡ej with case 1 first as a stepping stone. and the square of the mean eccentricity is 2 2 휈푎1 2 Case 1: Unknown E1,ej |hE푎,eji| = |E1,ej| 휔푎1 We study the expected value and distribution of E푎 at the time    1 − cos (휔푎1푡ej) − 휔푎1푡ej sin(휔푎1푡ej)  of ejection, E푎,ej = E푎 (푡ej). First, since h푍 (푠)i = 0 for all 푠, the × 1 + 2 © ª . (64)  휔2 푡2 ® integral in Eq. (55) has expectation hE푎 (푡)i = 0 for all 푡. The  푎1 ej  variance and covariances of interest can be computed using the  « ¬ The variance of the eccentricity is given by linearity of expectation. The variance of the final eccentricity is given by (see AppendixA) 2 휈  1 − cos(휔푎1푡ej)  h|E |2i−|hE i|2 = 2 푎1  − © ª 2 푎,ej 푎,ej 2휎E1 푡ej 1 2  . 휈 sin (휔푎1푡ej) 휔푎1  휔2 푡2 ® h|E |2i = 푎1 − 2  푎 ej  푎,ej 4 1 휎E1푡ej, (56)  « ¬ 휔푎 휔푎1푡ej (65) while the covariance between the final eccentricity and its forced In order to know the final RMS eccentricity E푎,∞, we also require amount (see Eq. 50) is the covariance between E푎,ej and E푎,forced, which is given by 2 ( ) ( ) − ∗ 휈푎1 sin 휔푎1푡ej 2 ∗ 2 cos 휔푎1푡ej 1 hRe(E푎,ejE푎,forced)i = 2 1 − 휎E1푡ej. (57) hRe(E푎,ejE푎,forced)i = |E푎,forced| . (66) 휔푎1 휔푎1푡ej 휔푎1푡ej

MNRAS 000, 000–000 (2020) 14 Bonan Pu

Combining these expressions with Eq. (52), the RMS eccentricity this limit is given by at inﬁnity is given by 2 ! 훾푒 −훾푒 푓 (훾푒) = exp . (72) 2  − ( )  훾¯2 2훾 ¯2 2 휈푎1 2  1 cos 휔푎1푡ej  푒 푒 h|E푎,∞| i = 2 휎 푡ej 1 − 2 © ª 휔 E1  휔2 푡2 ® 푎1  푎1 ej  Empirically, we ﬁnd that Eq. (72) is a good approximation for the  « ¬ distribution of 훾푒 even when it is not the case that 푁 푁sec.  − ( ) − ( ) − ( )  ! ej 2  3 1 cos 휔푎1푡ej sin 휔푎1푡ej 1 cos 휔푎1푡ej  +|E1,ej|  + +  .  2 휔 푡 휔2 푡2   푎1 ej 푎1 ej    (67) 4.3 Inclination Evolution

In the above equation, when 휔푎1푡ej 1, the second term of the In the above analysis we have considered the eccentricity evolution 2 RHS dominates and we have |E푎,∞| ∝ 푡ej. On the other hand, when of planet 푎 subject to a stochastic forcing by the outer perturber. The 2 evolution of the inclination can be derived in the same manner as the 휔푎1푡ej 1, the first term dominates and we also have |E푎,∞| ∝ 푡ej. eccentricity, except, whenever appropriate, replacing the complex In order words, for all 푡 we have h|E |2i ∝ 푡 , in agreement with ej 푎,∞ ej E I 2 2 eccentricities with the corresponding complex inclinations , and our numerical results. Since 푒푎,sec = 2|E푎,forced| , the ensemble replacing 휔푎1 → −휔푎1 and 휈푎1 → −휔푎1. The forced inclination RMS of the boost factor h훾2i is given by 푒 is given by Eq. (41). One will eventually find that the scaling for 훾푒 and 훾 is the same: 2 2 휃 2 h|E푎,∞| i − 2|E푎,forced| 1 − cos (푥) h훾 i = ' 퐴푥 1 − 2 2 2 푒 2 2 h훾 i = h훾 i. (73) 2|E푎,forced| 푥 푒 휃 1 − cos (푥) − sin (푥) 1 − cos (푥) 1 + + + , (68) In addition, the probability density distribution for 훾휃 is also the 2 푥 푥 2 same as 훾푒, and is given by Eq. (72) (note that 훾¯푒 = 훾¯ 휃 ). Since 훾 , 훾 have the same distribution, and 훾¯ = 훾¯ , we hereafter refer where we have defined 푥 ≡ 휔푎푡ej ∼ 2휋푁ej/푁sec, and 퐴 is the 푒 휃 푒 휃 dimensionless constant to the distribution of either quantity as 훾 (although note that 훾푒 and 훾 are uncorrelated and independently distributed). 2 휃 휎E1 1 h푁ejiHM Having computed the distribution of 휃 , we now derive the 퐴 ≡ ∼ ∼ 2휋 , (69) 푎 2 2 휔푎1|E1,ej| 휔푎1푏 푃1,0 푁sec boost factor for the mutual inclination 훾휃,푎1. Note that h i 2 2 2 2 and 푁ej HM = 푏 (Eq. 24) is the harmonic mean of 푁ej. Here 휃푎1,∞ = 휃푎1,ej = |I푎,ej − I1,ej| we have made use of the fact that the final eccentricity is well 2 2 ∗ 2 2 = |I푎,ej| + |I1,ej| − 2Re(I푎,ejI ). (74) constrained by conservation laws, so h푒1,eji ≈ |E1,ej| . 1,ej ' √︁ / Eq. (68) has two regimes:√ when 푥 1, 훾푒 푥 2, while From Eq. (52) (but replacing E → I), we thus have when 푥 1, we have 훾푒 ' 퐴푥. The transition between the two 2 2 2 regimes occurs when 푥 ∼ 휋. Using our earlier estimates for 푏 (Eq. 휃푎1,∞ = 휃푎,∞ − 휃1,ej. (75) 24), 퐴 is of order √ Recall that 휃 = 휃 / 2, thus from Eq. (45)-(47) we find −3/2 4 −2 푎1,sec 푎,sec 푚 푎 푚 푎1,0 퐴 ∼ 7 1 푎 1 + 2 . (70) √ 푀★ 푎1 푚1 푎2,0 훾휃,푎1 = 2훾휃 . (76)

For the typical range of parameters relevant to Kepler planets (푚1 ∼ The above equation assumes that 휃푎1, 휃푎 1 and ignores the −3 10 and 푎푎/푎1 ∼ 1/10) one obtains 퐴 ∼ 0.3. Given the inherent contribution from planet 2. In reality, 훾휃,푎1 will deviate from Eq. scatter in the simulation results, the diﬀerence between the two (76), although the above scaling still holds on average. Once we regimes in Eq. (68) is too subtle for us to empirically measure 퐴 know the value of 훾휃 , we can convert it to the corresponding value in this study. Thus in this paper we simply adopt the approximation of 훾 to obtain the mutual inclination boost factor, and vice √︃ 휃,푎1 훾푒 ∼ 푁ej/푁sec which agrees well with the empirical results. versa. 2 Having computed the mean value h훾푒i we now comment on its distribution. The Brownian bridge has a distribution that is normally distributed over an ensemble of simulations, and any linear 4.4 Marginal Distribution of the Boost Factor transformation of normally distributed variables is also normally The distributions we have derived so far for 훾 , 훾 are contingent distributed. From Eq. (44) and Eq. (52), the boost factor can be 푒 휃 on 푁 , which is not an observable quantity. However, since we written as ej have some understanding of the distribution of 푁 , we can now ej |E |2 − 2Re(E E∗ ) marginalize over it and only deal with observable quantities. First, 푎,ej 푎,ej 푎,forced 훾2 = . (71) combining Eq. (19) and Eq. (72) we can write the joint distribution 푒 |E |2 푎,forced for 푁ej and 훾 as The quantities E and E are normally distributed complex 푎,ej 푎,forced 푏훾 −푏2 −훾2 variables with zero mean. In the limit that 푁ej 푁sec, we have 푓 (푁 , 훾) = exp exp . (77) ej √︃ 푁 2 |E |2 (E E∗ ) 훾¯2 2휋푁3 2 ej 2훾 ¯ that 푎,ej 2Re 푎,ej 푎,forced , and 훾푒 is then the length of ej a 2-D vector whose components are normally distributed with zero 2 mean; such a quantity has approximately a Rayleigh distribution. Now, from Eq. (46) we have that 훾¯ ∼ 푁ej/푁sec. Substituting into 2 1/2 We deﬁne 훾¯푒 ≡ h훾푒i (see Eq. 68), then the distribution of 훾푒 in Eq. (77), and integrating over 푁ej we thus obtain the distribution for

MNRAS 000, 000–000 (2020) Scattering 1+2 15

훾 in terms of observable quantities only: v ∫ ∞ t 1 −푏2 − 훾2푁 푓 (훾) = 푏훾푁 exp sec 푑푁 sec 5 2푁 ej 0 2휋푁ej ej 푏훾푁 = sec . (78) 2 2 3/2 (푏 + 푁sec훾 ) Now, we deﬁne 푦 as the ‘normalized’ boost factor √︃ 푦 ≡ 훾 푁sec/h푁ejiHM, (79)

2 (recall that 푏 = h푁ejiHM), then we have the rather elegant expression for the normalized boost factor 푦: 푦 푓 (푦) = . (80) (1 + 푦2)3/2 In the distribution above, the probability that 푦 is greater than some constant 푦0 is given by 1 ( ≥ 0) 푃 푦 푦 = √︁ . (81) 1 + 푦02

Just like the distribution for 푁ej (Eq. 19), the distribution 푓 (푦) is a long-tailed one, such that all its higher moments√ (e.g. mean, variance) fail to exist. Its mode occurs at 푦 = 1/ 2, its geometric mean is h푦iGM √= 2, its harmonic mean is h푦iHM = 1 and its median is 푦 = 3. The 68% and 95% conﬁdence intervals are 푦 ∈ [0.65, 6.2] and 푦 ∈ [0.23, 40] respectively. Assuming that 푎2,0 ∼ 푎1,0, the harmonic mean of 훾 is given by the following scaling:

−1/2 3/4 2 √︃ 푚1 푎푎 푚2 h훾iHM = h푁ejiHM/푁sec ∼ 1.1 1 + . 푀★ 푎1,0 푚1 (82)

From this scaling, we see that h훾iHM = 푦/훾, thus we re-interpret √︁ Figure 16. Distribution of 푦푒 ≡ 훾푒 푁sec/h푁ej iHM (see Sec. 4.4). The 푦 = 훾/h훾iHM as the boost factor ‘normalized’ by its harmonic mean histograms are empirical distributions obtained from our simulations, while and define 푦푒 = 훾푒/h훾푒iHM, 푦 휃 = 훾휃 /h훾휃 iHM for the normalized the black line is the theoretical distribution given by Eq. (80)-(82). On eccentricity and inclination boost factors respectively. Note that the the top panel, 푚2/푚1 = 1/5 while 푚1 and 푎푎/푎1 varies as shown in the scaling relation in Eq. (82) applies equally to 푦푒 and 푦 휃 . We see that legend. On the bottom panel, 푎푎/푎1 = 1/10 and 푚1 = 1푀퐽 , while 푚2/푚1 the effect of CJ scatterings on inner planets is the greatest if the CJ varies as shown in the legend. scatters are lower in mass, have semi-major axes more comparable to the inner planets, and have comparable masses. In Fig. 16 we show a comparison between our theoretical dis- 2 as it is being ejected from the system. This can be justified in tribution given by Eq. (80) for the normalized eccentricity boost the limit that 푚2/푚1 1. However, for more comparable masses, factor 푦푒 and the empirical distribution from our suite of simula- 푚2 can have an equal or even greater effect than 푚1 on the secular evolution of the inner system. Our simplification of ignoring planet tions. We find that for 푚2/푚1 . 1/3, the theoretical distribution agrees well with the empirical one over a range of different masses 2 is the main reason why our estimate from Eq. (68) becomes less accurate when 푚 ∼ 푚 . Since at the end of the ejection process, the and 푎푎/푎1. The empirical distribution starts to deviate somewhat 2 1 from Eq. (80) for more comparable masses: in particular, the dis- secular forcing by 푚2 vanishes, one way to incorporate the influence tribution becomes even more heavy-tailed, with significant fraction of 푚2 is to absorb it into the variance of the Brownian bridge, i.e. having 푦 1, although the empirical mode and harmonic mean by replacing 휎E1 → 휎E1 (1 + 휅12), where 휅12 is a dimensionless still agreed with Eq. (82) to with-in a factor of a few. ratio that depends on 푚2/푚1 (and possibly other quantities) that accounts for the added effect of secular perturbations by 푚2. For certain initial configurations, 3-body secular interactions can also 4.5 Theoretical Model: Simplifications and Refinements give rise to secular resonances that would increase the amount of eccentricity and inclination excited in the inner planet (see Lai & In developing our stochastic model for “1+2” scattering, we have Pu 2017; Pu & Lai 2018). made several simplifying assumptions. A more careful treatment • Linearity in E, I: In our theoretical model we have assumed can yield refinements to the model and more accurate estimates for that the secular evolution in eccentricity and inclination is linear. the distribution of final parameters. We discuss the most crucial Note however that our hybrid algorithm (Sec.3) allows for the simplifications and suggest possible ideas for refinement below. possibility of larger growths in eccentricity due to non-linear Lidov- • Secular forcing by planet 2: In our theoretical model we have Kozai oscillations, and that such oscillations are indeed possible ignored the secular interaction between the inner planet and planet when 휃푎 grows to large values. Unfortunately differential equations

MNRAS 000, 000–000 (2020) 16 Bonan Pu with such stochastic terms become intractable when stochasticity is involved, and one would have to resort to numerical integrations in this regime. • Constancy of 푎1: In our theoretical model we have also assumed that 푎1 (and therefore 휔푎1, 휈푎1) is constant, which is approximately the case when 푚2 푚1 but breaks down at more comparable mass ratios. In reality, 푎1 changes randomly as 푎2 undergoes strong scatterings, and its ﬁnal value can decrease by as much as 푎1,ej/푎1,0 = 1/2 in the limit that 푚2 = 푚1. There are two ways to reﬁne our model to incorporate this: First, one can absorb the stochastic changes in 휈푎1 as additional variance in 휎E1, i.e. √︃ → 2 + 2 2 by replacing 휎E1 휎E1 휎휈1, where 휎휈1 is the RMS change in 휈푎1 per unit time. In addition, one should replace 휔푎 with its expectation, i.e. h휔푎 (푡)i = 휔푎,0 + (휔푎,ej − 휔푎,0)(푡/푡ej). (83)

The above addition still allows for an analytic estimate for the final Figure 17. Similar to Fig. 11, except with 2 inner planets. We have 푚푎 = eccentricity and inclination, while incorporating the non-constancy 푚푏 = 3푀⊕, 푎푎 = 푎1/20 and 푎푏 = 1.5푎푎, while 푚1 varies as shown on the plot legend and 푚2 = 푚1/5. The boost factor for the first inner planet of 푎1, although the resulting final expressions are much less elegant. 훾 corresponds to the filled circles, while that for the second inner planet • Flat power spectrum of 휎 : We assume that the 휎 is a 푒,푎 E1 E1 is shown as filled triangles. constant that is independent of timescale. In reality, this assump- tion could break down at timescales much shorter than the orbital √︃ timescale of the outer giant planets, and the scaling 훾 ∝ 푁ej/푁sec would break down. This would be most pertinent in cases where are given by (see Pu & Lai 2018) 휔푎 1/푃1,0, and would lead to an over-estimation of the boost √ factors. 휈푎1휔푏 + 휈푎푏휈푏1 푒푎,sec = 2 푒1,ej, (85) 휔푎휔푏 − 휈푎푏휈푏푎 √ 휈 휔 + 휈 휈 푒 = 2 푏1 푎 푏푎 푎1 푒 , (86) 푏,sec 휔 휔 − 휈 휈 1,ej 5 EXTENSION TO MORE INNER PLANETS 푎 푏 12 21 휃푎1,sec = 휃푏1,sec ≈ 휃1,ej, (87) Having understood the dynamics of “1+2” scattering we now gen- ! eralize our results to the case with more than one inner planets. The 휔푎1 − 휔푏1 휃푎푏,sec = 2 휃1,ej, (88) √︁ 2 parameter space is vast when additional planets are considered, but (휔푎 − 휔푏) + 4휔푎푏휔푏푎 as we shall demonstrate, the universal scalings given by Eqs. (46) and (80)-(82) remain valid. where 휔푎 = 휔푎푏 + 휔푎1 and 휔푏 = 휔푏푎 + 휔푏1 respectively. From these “secular” values, we compute the values of 훾푒,푎, 훾푒,푏 and 훾휃,푎푏 analogous to Sec. 4.1. We show the results of our simulations 5.1 Two inner planets in Figs. 17- 18. We see that in the “2+2” case the boost factor is still consistent with the scaling law Eq. (46), even though the values For each of our N-body simulations, we consider inner systems with of 휔푎, 휔푏 and the forced eccentricities and inclinations are given 푎푎 = 푎1/20 and 푎푏 = 1.5푎푎, and 푚푎 = 푚푏 = 3푀⊕. The initial by very different expressions. eccentricities and inclinations of the inner planets are set to zero. In our simulations, the inner planets effect each other secularly, and are influenced by the outer perturbers through secular interactions, as described by Sec.3. 5.2 3 or More Inner Planets For systems with 2 inner planets and an external perturber, the dynamics of the system depends crucially on the dimensionless Having briefly studied the “2+2” scattering we make some remarks coupling parameter 휖푎푏 (Lai & Pu 2017; Pu & Lai 2018), given by on extending our theory to systems with 3 or more inner planets. The numerical algorithm described in Sec.3 works for a general 3   3/2 number of inner (and outer) planets, so long as the inner and outer 휔푏1 − 휔푎1 푚1 푎푏  3푎푎/푎푏  (푎푏/푎푎) − 1 휖 ≡ ≈   , systems are sufficiently detached that the outer planets do not come 푎푏 휔 + 휔 푚 푎  (1)  1 + (퐿 /퐿 ) 푎푏 푏푎 푏 1  푏 (푎푎/푎푏)  푎 푏 in close contact with the inner planets. However, the theoretical  3/2  (84) model in Sec. 4.2, and in particular Eq. (68) must be modified if √ there are additional of more inner planets, due to the more complex where 퐿푖 ≡ 푚푖 퐺푀★푎푖 is the circular angular momentum of the secular coupling between the inner planets. In particular, one should ( ) 푏 1 (푎 /푎 ) deal with the amplitudes of the planet eccentricity and inclination planet, and 3/2 푎 푏 is the Laplace coefficient given by Eq. (35). secular eigenmodes, and the secular precession frequency should In the parameter regime that we study in this work, the two inner be replaced with the mode frequencies. The (complex) eigenmode amplitude of the 훼-th mode should scale as planets are invariably in the “strong coupling” regime (휖푎푏 1). In this limit, assuming initially circular and co-planar orbits for √︃ planets 푎 and 푏, the “secular” eccentricities and mutual inclinations E훼,ej ∝ I훼,ej ∝ 푁ej/푁훼,sec, (89)

MNRAS 000, 000–000 (2020) Scattering 1+2 17

orbital energy conservation. The final eccentricity and inclination of the planet is 푒1,ej ∼ 0.7푚2/푚1 and 휃1,ej ∼ 0.7휃2,0푚2/푚1 for 푚2/푚1 . 0.5, where 푚2 is the mass of the ejected planet and 휃2,0 is the initial mutual inclination of the two planets. • Ejection timescale: The timescale from the first planet-planet Hill sphere crossing to the final ejection of planet 2 can be understood as the stopping time of a Brownian motion. We empirically measure the normalized dimensionless RMS energy exchange (|훿퐸12/퐸2,0|) per pericenter passage 푏 over an ensemble of N-body simulations, and present a best-fit law for it in Eq. (24). Given 푏, the distribution of 푁ej (the number of orbits of 푚2 prior to ejection) agrees well with Eq. (19). • Minimum 푎2 of ejected planet: We find that the possible values of 푎2 during the strong scattering and ejection is constrained by energy conservation, angular momentum conservation, and the requirement that the system cannot spontaneously scatter itself into an indefinitely stable state. Fig.7 shows our empirical results for the minimum value of 푎2 and 푟2 over the course of ejection. We find that generally, 푎2,min ∼ 푎1,0/2, and for 푚2/푚1 1 we have 푟2,min ∼ 푎1,0/4, although 푟2,min decreases strongly as 푚2/푚1 increases. • “1+2” Scattering - Numerical Results: For well-separated inner super-Earth and outer CJ systems, the effect of CJ scatterings on the inner planet is secular. We develop a hybrid algorithm to simulate such systems efficiently, by computing two CJ scatterings and then simulating their effects on the inner planet via secular evolution. We have performed such numerical integrations for “1+2” systems over a wide range of parameters. We find that the eccentricity and inclination of the inner planet induced by CJ scatterings can be much larger than the secular values (Eqs. 40- 41) generated by the remaining giant planet, and the enhancement increases with 푁ej (see Figs.9- 10). Despite the diversity of initial parameters and final outcomes, the dynamics of the system can be succinctly summarized by the dimensionless “boost” factor 훾 (Eqs. 44- 45). In the range of parameters we considered we find that Eq. (46) provides a Figure 18. Similar to Fig. 15, except the simulations have two inner planets. universal scaling law for the final eccentricity and inclination of the The system parameters are the same as those for Fig. 17. The top panel inner planet, as a function of the system parameters (see Figs. 11- 2 shows the eccentricity boost factor 훾푒 while the bottom panel show the 15). 2 mutual inclination boost factor 훾휃,푎푏. • “1+2” scattering - Theoretical model: We develop a theoretical model to explain the empirical scaling law in Eq. (46), by modelling the “1+2” scattering process as a linear stochastic differ- where E훼,ej, I훼,ej are the complex amplitude of the 훼-th eccentric- ential equation. We compute analytically the expected moments and ity and inclination eigenmodes respectively, and distributions for the final inner planet eccentricity and inclination −1 in terms of the boost factors, which are given by Eqs. (68)-(72). 휔훼,0푃1,0 푁 ≡ , (90) We calculate the distribution of 훾, averaged over all possible 푁 , to 훼,sec 2휋 ej derive a universal distribution function for the boost factor in terms where 휔훼,0 is the initial eigenfrequency of the 훼-th eigenmode. An of observable quantities only (Eq. 80); this analytical distribution empirical test of the above scaling is beyond the scope of this work, agrees well with empirical results (see Fig. 16). but is promising ground for further research. • Extension to “2+2” systems: We have extended our empirical investigation to “2+2” systems. We find that analogous to “1+2” systems, Eq. (46) is still valid for describing the dynamics of the 6 SUMMARY AND DISCUSSION system, although the final values of eccentricities and inclinations are substantially different due to strong secular coupling between 6.1 Summary the inner planets. We also describe how the theoretical model in Sec. 4.2 can be extended to inner systems with 3 or more planets. In this work we have studied CJ scatterings and their effect on inner planet systems. Our main results are summarized below. • Final outcome of CJ scattering: We have re-examined final outcomes of strong scatterings between two CJs on gravitationally 6.2 Caveats unstable orbits. At the semi-major axis of a few au or larger, the most likely outcome of such scatterings is ejection of the less massive In our analysis we have considered the “clean” cases. Several im- planet (see also Li et al. 2021). The remaining planet, which we portant physical effects were neglected, and we comment on them call planet 1, has a final semi-major axis that is consistent with below.

MNRAS 000, 000–000 (2020) 18 Bonan Pu

• Direct scatterings between the inner planet and outer gi- and the mean eccentricity boost factor from (Eq. 82) becomes ants: In this model we have ignored the possibility of direct hard −1/2 3/4 2 scattering between the inner planet system and the outer giants. In 푚1 푎푎 푚2 1/2 h훾푒iHM ∼ 1.1 1 + (1 + 휖푎1,GR) . our simulations, cases where the inner planet crosses orbits with 푀★ 푎1,0 푚1 one of the outer giants is discarded from our tabulated results. As (94) we have discussed in Sec. 2.5, it is possible albeit unlikely for one of the giant planets to meander deeply inwards during the scat- Note that the above equation applies only to h훾푒iHM and not the tering process. For 푚1 & 3푀퐽 and 푚2 ' 푚1, we expect such inclination. Now the forced eccentricity on planet 푎 is proportional −1 orbit crossings to occur a small fraction (∼ 20%) of the time for to 푒푎,forced ∝ (1 + 휖푎1,GR) , at the same time we also have h훾푒i ∝ 1/2 푎푎/푎1,0 = 1/10, while inner planets with 푎푎/푎1,0 . 1/20 are gen- (1 + 휖푎1,GR) , thus the final eccentricity raised on planet 푎 after −1/2 erally protected from participating directly in scatterings with the scattering scales as 푒푎,∞ ∝ (1 + 휖푎1,GR) . giant planets. Since direct scatterings between in the inner planet In comparison, in the purely “secular” scenario without scatter- and outer giants can lead to even greater excitation in eccentricity ing events, the final eccentricity raised is proportional to 푒푎,forced ∝ and inclination, our model thus under-estimates the potential to ex- −1 (1+휖푎1,GR) . Thus we see that in the stochastic forcing case, short cite large eccentricities and inclinations in the inner planet during ranged forces such as GR apsidal precession still suppresses eccen- “1+2” scattering. tricity generation, but the suppression factor is only proportional to • Physical collisions between CJs: We have focused on scatter- the inverse square root of the strength of the short-ranged force. ings between CJs that result in ejection of the less massive planet. A small fraction of systems will under-go collisional mergers instead. If the final values of 푒1, 휃1 are known, then our theoretical model 6.3 Application to Specific Systems in Sec. 4.2 applies equally to systems that result in collisions. How- We discuss our results in the context of a few specific planet systems ever, the collisional case is less interesting in terms of its impact on of interest. These systems feature an inner planet well separated the inner planetary system, because the collisional timescale tends from an exterior CJ with high orbital eccentricities and/or mutual to be much shorter due to collisional probability being highest at inclinations. Such eccentric CJs are a natural consequence of strong the initial time when planet eccentricities are low (Nakazawa et al. scatterings between CJs. As disussed below, the observed orbital 1989; Ida & Nakazawa 1989). In addition, the final eccentricity 푒 1 properties of these inner-outer systems can be explained using our and inclination 휃 of the merger product tend to be low, due to 1 model. collisions between CJs being highly inelastic (see Li et al. 2021). Typically, one can assume that the scattering history is unimportant • HAT-P-11 is a system with a transiting inner mini-Neptune for systems that result in collisions (i.e. the boost factor 훾 1). (HAT-P-11b, 푚푎 = 23.4 ± 1.5푀⊕, 푎푎 = 0.0525 ± 0.0007 au.) • Spin-orbit coupling: We have neglected the coupling between first discovered by photometry (Bakos et al. 2010) and an outer CJ the planets and stellar spin. In reality, the stellar spin and the inner (HAT-P-11c) with 푚1 sin 퐼1 = 1.6 ± 0.1푀퐽 and 푎1 = 4.13 ± 0.3 planets can exchange angular momentum, which can change the au around a mid-K dwarf with 푀★ = 0.81푀·. RV measurements inclination of the inner planets. Incorporating such evolution into report values of 푒푎 = 0.218 ± 0.03 and 푒1 = 0.6 ± 0.03 for the our theoretical model is beyond the scope of this work. In terms of two planets. The orbit of HAT-P-11c is highly misaligned relative inclination evolution, including spin-orbit coupling is equivalent to to the stellar spin 휆푎 ∼ 100 deg (Winn et al. 2010). Yee et al.(2018) adding an extra inner planet (see Lai et al. 2018). argued that such a misalignment can be explained if the two planets • Short-ranged forces: In this study we assumed that the inner are also highly mutually inclined with 휃푎 & 50 deg. This argument planets are effected by secular forces from other planets only. In is supported by recent measurements by (Xuan & Wyatt 2020), who ◦ ◦ particular, we have ignored the effects of short-ranged forces, such found that 54 < 휃푏푐 < 126 at the 1휎 level. as general relativistic (GR) apsidal precession, tidal precession, and Due to the very tight orbit of HAT-P-11b, GR apsidal preces- tidal dissipation (a discussion for the relative importance of these sion is important, with 휖푎1,GR ≈ 133. Note that despite the large effects is given in Pu & Lai 2019). The most important such effect inclination between HAT-P-11b and HAT-P-11c, Kozai-Lidov os- is GR apsidal precession, whose angular frequency (in the limit that cillations are suppressed due to the strong GR effect, and the forced −4 푒 푗 1) eccentricity is very small (푒푎,forced ∼ 1.1 × 10 ), and the required eccentricity boost factor is 훾푒 ∼ 2000. The observed value of 푒푎 3/2 3퐺푀 푀 푎 푗 −5/2 is thus highly incompatible with pure secular interactions without 휔 = ★ 푛 ≈ 6 × 10−6 ★ yr−1. (91) 푗,GR 2 푗 scattering history. 푐 푎 푗 푀 0.1au Since 푒1 = 0.6, if the observed eccentricity is the result of strong The main effect of this additional precession is to suppress eccen- scattering between HAT-P-11c and an ejected planet, it is most likely that 푚 ∼ 푚 (see Sec.2). Thus, applying Eq. (94) we have tricity generation. We define 휖 푗1,GR as the ratio between 휔 푗,GR and 2 1 the apisdal precession frequency due to secular coupling (between h훾푒iHM ∼ 40. The observed value of 훾푒 is therefore larger than planets 푗 and 1): its typical value by a factor of 푦푒 = 훾푒/h훾푒iHM ∼ 50. According Eqs. (79)-(81), the likelihood of seeing such a boost factor is 2 3 푃(푦푒 ≥ 50) = 0.02. However, Eq. (81) underestimates 푦푒 at larger 휔 푗,GR 3퐺푀★푎1 휖 푗1,GR ≡ = . (92) values when 푚2 ∼ 푚1 (see Fig. 16); from our empirical results we 휔 푗 푎4푐2푚 1 푗 1 find that for 푚2/푚1 & 0.7, 푃(푦푒 ≥ 50) ∼ 0.09. In other words, there is a 9% chance to have 푒푎 & 0.2 as a result of “1+2” scattering In the “1+2” case, the secular frequency of planet 푎 is thus changed as given by the currently observed parameters. from 휔푎1 to Now turning to the mutual inclination, since the nodal precession is not affected by GR precession, we have h훾휃 ∼ 3.5 (Eq. 82). On 휔푎 = 휔푎1 (1 + 휖푎1,GR), (93) the other hand, the ‘forced’ mutual inclination depends on 휃12,0,

MNRAS 000, 000–000 (2020) Scattering 1+2 19 the initial misalignment angle between HAT-P-11c and√ the ejected forcing from their external perturber alone, but is consistent with = /( )− planet. The actual value√ of 푦 휃 is given by 푦 휃 휃푎 3.5 2휃12,0 1 the “1+2” scattering hypothesis (푝 > 0.10 in all cases). In addition, (recall that the factor 2 arises due to the boost factor being larger direct scatterings of the inner planet by the outer giants during “1+2” for the mutual inclination; see Sec. 4.3). If we take 휃푎 = 50 deg. scattering could under certain regimes produce additional excitation and 휃12,0 = 3 deg., then 푦 휃 ∼ 3 and 푃(푦 휃 ≥ 3) ∼ 0.4, i.e. there is a in eccentricity and inclination, which further bolsters the prospects 40% chance for the observed mutual inclination to be as large as 50 the currently observed eccentricities and mutual inclinations being degrees. The probability decreases if 휃12,0 is smaller: for 휃12,0 = 1 explained by “1+2” scattering. deg., the p-value decreases to 푃(푦 휃 ≥ 9) ∼ 0.1. Note again that the empirical value of 푃 is greater than predicted by Eq. (81) due to the fact that 푚1 ∼ 푚2. ACKNOWLEDGEMENTS We conclude that for the HAT-P-11 system, the observed eccentricity of the inner planet is marginally consistent with “1+2” We thank the anonymous referee for helpful comments that im- scattering with a p-value of 푃 ∼ 0.1 for the observed eccentricity proved the manuscript. BP is supported by the NASA Earth and boost factor, while the observed inclination is consistent with “1+2” Space Sciences Fellowship. DL thanks the Dept. of Astronomy and scattering (at 푃 = 0.1 level) for 휃12,0 & 1 degree. the Miller Institute for Basic Science at UC Berkeley for hospitality • Gliese 777 A is a two-planet system detected by RV with an while part of this work was carried out. inner planet with 푚푎 sin퐼푎 = 18 ± 2푀⊕ and 푎푎 = 0.13 ± 0.008 au., and an outer CJ with 푚1 sin 퐼1 = 1.56 ± 0.13푀퐽 and 푎1 = 4 ± 0.2 au, orbiting around a yellow subgiant with 푀★ = 0.82 ± 0.17푀· DATA AVAILABILITY (Wright et al. 2009). RV measurements report 푒푎 ≈ 0.24±0.08 and Original data produced by our simulations in this work are available 푒1 ≈ 0.31 ± 0.02. upon request. The value of 휖푎1,GR ∼ 3 which gives a forced eccentricity of −3 3.5 × 10 and boost factor 훾푒 ∼ 67, thus the value of 푒푎 cannot be explained by pure secular forcing alone. Hypothesizing that the current value of 푒1 is due to scattering with an ejected planet, REFERENCES the value of 푒 ≈ 0.3 suggests that 푚 /푚 ∼ 0.4, which gives 1 2 1 Anderson K. R., Lai D., 2018, MNRAS, 480, 1402 h훾 i ∼ 8 and 푦 ∼ 8. Evaluating Eq. (81), we find that 푃(푦 ≥ 푒 HM 푒 푒 Anderson K. R., Lai D., Pu B., 2020, Monthly Notices of the Royal Astro- 8) ≈ 0.12. Thus, even though the observed value of 푒푎 is much nomical Society, 491, 1369 greater than the amount predicted by pure secular forcing, it is still Bakos G. Á., et al., 2010, ApJ, 710, 1724 consistent with “1+2” scattering theory. Becker J. C., Adams F. C., 2017, MNRAS, 468, 549 • 휋 Men is a two-planet system with an inner transiting super- Borodin A., Salminen P., 2002, Handbook of Brownian Motion - Facts and Earth (푚푎 = 4.8푀⊕, 푎푎 = 0.0684 au) discovered by TESS (Huang Formulae et al. 2018) and an external companion discovered by RV with Boué G., Fabrycky D. C., 2014a, ApJ, 789, 110 Boué G., Fabrycky D. C., 2014b, ApJ, 789, 111 푎1 = 3.3 au and 푚1 ≈ 12.9푀퐽 . The host-star is G type with Brakensiek J., Ragozzine D., 2016, ApJ, 821, 47 푀★ = 1.11푀 . Follow-up surveys have shown a significant orbital misalignment between 푚 and 푚 , with 49 deg. < 휃 < 131 deg. Bryan M. L., Knutson H. A., Lee E. J., Fulton B. J., Batygin K., Ngo H., 1 푎 푎1 Meshkat T., 2019,AJ, 157, 52 at 1휎 level (Xuan & Wyatt 2020; see also Damasso et al. 2020; Carrera D., Davies M. B., Johansen A., 2016, MNRAS, 463, 3226 Rosa et al. 2020). The external companion has an eccentric orbit of Chambers J. E., Wetherill G. W., Boss A. P., 1996, Icarus, 119, 261 푒1 ≈ 0.642 while the inner planet has 푒푎 ≈ 0.15 (Damasso et al. Chatterjee S., Ford E. B., Matsumura S., Rasio F. A., 2008, ApJ, 686, 580 2020). Damasso M., et al., 2020, A precise architecture characterization of the 휋 For this system 휖푎1,GR = 1.21, and 푒푎,forced = 0.013, thus Men planetary system (arXiv:2007.06410) 훾푒 ≈ 11, which shows the current value of 푒푎 is inconsistent with Denham P., Naoz S., Hoang B.-M., Stephan A. P., Farr W. M., 2019, Monthly pure secular forcing from 푚1 alone. If the current value of 푒1 is Notices of the Royal Astronomical Society, 482, 4146 Ford E. B., Rasio F. A., 2008, ApJ, 686, 621 due to strong scattering, the ejected planet likely has 푚2 ∼ 푚1, Ford E. B., Kozinsky B., Rasio F. A., 2000, ApJ, 535, 385 corresponding to h훾푒iHM ∼ 3.3 when GR precession is taken into account. Thus 푦 ∼ 3, which is consistent with “1+2” scattering Fouchard M., Rickman H., Froeschlé C., Valsecchi G. B., 2013, Icarus, 222, 푒 20 with 푝(푦 ≥ 3) ∼ 0.3. Thus we conclude that the observed value 푒 Frelikh R., Jang H., Murray-Clay R. A., Petrovich C., 2019, ApJ, 884, L47 of 푒1 is highly compatible with “1+2” scattering. Gladman B., 1993, Icarus, 106, 247 Now turning to the mutual inclination, we have that h훾휃 iHM ∼ Gratia P., Fabrycky D., 2017, MNRAS, 464, 1709 2.3. Taking a√ fiducial value of 휃푎1 ≈ 90 deg., we have 푦 휃 = Hansen B. M. S., 2017, MNRAS, 467, 1531 90 deg./(2.3 2휃12,0) − 1. If 휃12,0 = 3 deg., then 푦 휃 ∼ 8 and Huang C. X., Petrovich C., Deibert E., 2017,AJ, 153, 210 푃(푦 휃 ≥ 8) ∼ 0.2. On the other hand, if 휃12,0 = 1 deg., then Huang C. X., et al., 2018, The Astrophysical Journal, 868, L39 푦 휃 ∼ 27, corresponding to 푃(푦 휃 ≥ 27) ∼ 0.12. Recall that we Ida S., Nakazawa K., 1989, A&A, 224, 303 are using empirical values for 푃(푦) derived from simulations, since Ida S., Lin D. N. C., Nagasawa M., 2013, ApJ, 775, 42 Jontof-Hutter D., Weaver B. P., Ford E. B., Lissauer J. J., Fabrycky D. C., Eq. (81) breaks down when 푚1 ∼ 푚2. To conclude, the observed mutual inclination in the system can be easily generated by “1+2” 2017,AJ, 153, 227 Jurić M., Tremaine S., 2008, ApJ, 686, 603 scattering if 휃 3 deg., and is still possible with 푃 ∼ 0.12 12,0 & Lai D., Pu B., 2017,AJ, 153, 42 ∼ probability for 휃12,0 1 degree. Lai D., Anderson K. R., Pu B., 2018, MNRAS, 475, 5231 Li J., Lai D., Anderson K. R., Pu B., 2021, MNRAS, 501, 1621 In summary, we have found that each of the systems HAT-P-11, Lin D. N. C., Ida S., 1997, ApJ, 477, 781 Gliese 777 A and 휋 Men have inner planet eccentricities and mutual Masuda K., Winn J. N., Kawahara H., 2020,AJ, 159, 38 inclinations that are inconsistent with being produced by secular Matsumura S., Ida S., Nagasawa M., 2013, ApJ, 767, 129

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Murray C. D., Dermott S. F., 1999, Solar system dynamics Similarly the covariance between E푎,ej and its forced eccentricity Mushkin J., Katz B., 2020, arXiv e-prints, p. arXiv:2005.03669 is given by Mustill A. J., Davies M. B., Johansen A., 2017, MNRAS, 468, 3000 D ∫ 푡ej 휈 E Nakazawa K., Ida S., Nakagawa Y., 1989, A&A, 221, 342 ∗ −푖휔푎1 (푠−푡ej) 푎1 ∗ hRe(E푎,ejE푎,forced)i = Re −푖휈푎1푒 푍 (푠) 푍 (푠)푑푠 Obertas A., Van Laerhoven C., Tamayo D., 2017, Icarus, 293, 52 0 휔푎1 Petrovich C., 2015, ApJ, 808, 120 ∫ 푡ej 휈 −푖휔푎1 (푠−푡ej) 푎1 ∗ Petrovich C., Tremaine S., Rafikov R., 2014, ApJ, 786, 101 = Im 휈푎1푒 h푍 (푠)푍 (푠)i 푑푠 0 휔푎1 Pu B., Lai D., 2018, MNRAS, 478, 197 ∫ 푡ej 휈 Pu B., Lai D., 2019, MNRAS, 488, 3568 −푖휔푎 (푠−푡ej) 푎1 2 = 2 Im 휈푎1푒 휎E1푠 푑푠 Pu B., Wu Y., 2015, ApJ, 807, 44 0 휔푎1 Rasio F. A., Ford E. B., 1996, Science, 274, 954 2 ( ) 휈푎1 sin 휔푎1푡ej 2 Read M. J., Wyatt M. C., Triaud A. H. M. J., 2017, MNRAS, 469, 171 = 2 1 − 휎 푡 . 휔 휔 푡 E1 ej Rein H., Liu S.-F., 2012, A&A, 537, A128 푎1 푎1 ej Rein H., Spiegel D. S., 2015, MNRAS, 446, 1424 (A3) Rice D. R., Rasio F. A., Steffen J. H., 2018, MNRAS, 481, 2205 The case of the constrained perturber (Brownian bridge) is anal- Rosa R. J. D., Dawson R., Nielsen E. L., 2020, A significant mu- ogous to the case for the unconstrained perturber, except with tual inclination between the planets within the 휋 Mensae system 푍 (푠) → 퐵(푠). The expectations of 퐵(푠) are given by Eqs. (60) (arXiv:2007.08549) -(62). Smith A. W., Lissauer J. J., 2009, Icarus, 201, 381 Vick M., Lai D., 2018, MNRAS, 476, 482 Weidenschilling S. J., Marzari F., 1996, Nature, 384, 619 This paper has been typeset from a TEX/LATEX file prepared by the author. Wiegert P., Tremaine S., 1999, Icarus, 137, 84 Winn J. N., et al., 2010, ApJ, 723, L223 Wright J. T., Upadhyay S., Marcy G. W., Fischer D. A., Ford E. B., Johnson J. A., 2009, ApJ, 693, 1084 Wu D.-H., Zhang R. C., Zhou J.-L., Steffen J. H., 2019, MNRAS, 484, 1538 Xuan J. W., Wyatt M. C., 2020, MNRAS, 497, 2096 Yee S. W., et al., 2018,AJ, 155, 255 Zhou J.-L., Lin D. N. C., Sun Y.-S., 2007, ApJ, 666, 423 Zhu W., Petrovich C., Wu Y., Dong S., Xie J., 2018a, preprint, (arXiv:1802.09526) Zhu W., Petrovich C., Wu Y., Dong S., Xie J., 2018b, ApJ, 860, 101

APPENDIX A: CALCULATION OF MOMENTS OF E퐴 We demonstrate how to calculate the various moments of an inner planet subject to a stochastic secular forcing. For case 1, the unconstrained perturber, from Eq. (55) the mean of E푎 is given by

D ∫ 푡ej E 푖휔푎1 (푠−푡ej) hE푎i = 푒 푖휈푎1푍 (푠)푑푠 0 ∫ 푡ej 푖휔푎1 (푠−푡ej) = 푒 푖휈푎1h푍 (푠)i푑푠 = 0. (A1) 0

The variance of E푎 is

2 D ∫ 푡ej E 2 −푖휔푎 (푠−푡ej) h|E푎 | i = 푒 푖휈푎1푍 (푠)푑푠 0 D ∫ 푡ej ∫ 푡ej E 2 푖푖휔푎1 (푠−푡ej) 푖휔푎1 (푟−푡ej) ∗ = 휈푎1 푒 푍 (푠)푑푠 푒 푍 (푟)푑푟 0 0 ∫ 푡ej ∫ 푡ej 2 푖휔푎1 (푟−푠) ∗ = 휈푎1 푒 h푍 (푠)푍 (푟)i 푑푠 푑푟 0 0 ∫ 푡ej ∫ 푟 ∫ 푡ej ∫ 푡ej 2 2 푖휔푎 (푟−푠) 푖휔푎1 (푟−푠) = 2휎E1휈푎1 푒 푠푑푠 푑푟 + 푒 푟푑푠 푑푟 0 0 0 푟 2 ( ) 휈푎1 sin 휔푎푡ej 2 = 4 1 − 휎E1푡ej. (A2) 휔푎 휔푎푡ej

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