On the Correlation between Hot Jupiters and Stellar Clustering: High-eccentricity Migration Induced by Stellar Flybys
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On the Correlation between Hot Jupiters and Stellar Clustering: High-Eccentricity MigrationInduced by Stellar Flybys
Laetitia Rodet, Yubo Su, and Dong Lai Cornell Center for Astrophysics and Planetary Science, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
ABSTRACTA recent observational study suggests that the occurrence of hot Jupiters (HJs) around solar typestars is correlated with stellar clustering. We study a new scenario for HJ formation, called “FlybyInduced High-e Migration”, that may help explain this correlation. In this scenario, stellar flybysexcite the eccentricity and inclination of an outer planet (or brown dwarf) at large distance (10-300au), which then triggers high-e migration of an inner cold Jupiter (at a few au) through the combinedeffects of von Zeipel-Lidov-Kozai (ZLK) eccentricity oscillation and tidal dissipation. Using semi-analytical calculations of the effective ZLK inclination window, together with numerical simulations ofstellar flybys, we obtain the analytic estimate for the HJ occurrence rate in this formation scenario. Wefind that this “flyby induced high-e migration” could account for a significant fraction of the observedHJ population, although the result depends on several uncertain parameters, including the density andlifetime of birth stellar clusters, and the occurrence rate of the “cold Jupiter + outer planet” systems.
Keywords:
Hot Jupiters — Close Encounters — Exoplanet dynamics INTRODUCTIONHot Jupiters (HJs), giant planets orbiting with very-short period ( ∼ Corresponding author: Laetitia [email protected] a r X i v : . [ a s t r o - ph . E P ] F e b initial planetary systems, and the numerical approachlimits the generality of the results.In this paper, we examine the likelihood of HJs form-ing through high-e migration triggered by flybys in astellar environment, using a combination of analyticalcalculations (for high-e migration) and numerical simu-lations (for stellar flybys). In Section 2 we outline theproposed scenario and its key ingredients. In Section 3,we review the conditions for the ZLK mechanism to pro-duce a HJ, and extend previous works to examine thecase of a planetary-mass perturber. In Section 4, weevaluate the extent to which a stellar encounter can raisethe eccentricity and inclination of an outer planet. InSection 5, we derive the occurrence rate of HJs as afunction of the properties of the system and its stellarneighborhood. Finally, in Section 6, we conclude anddiscuss alternative explanations for the observed corre-lations between HJs and stellar overdensities. FLYBY INDUCED HIGH-E MIGRATIONSCENARIOAccording to the current understanding of planetaryformation, giant planets form preferentially beyond thesnow line at a few astronomical units. Their migrationto the close neighborhoods of their host stars can betriggered by a combination of eccentricity excitation andtidal dissipation—the so-called high-e migration mech-anism (see references in Section 1). This requires thepresence of a misaligned and eccentric outer perturber(see Section 3). To evaluate the probability to form HJsthrough high-e migration, we thus need to estimate theoccurrence rate of such perturbers.On one hand, misaligned perturbers could be natu-rally associated with binary companions. The orienta-tion of stellar companions with respect to the protoplan-etary disk plane of the primary is expected to be randomat large separations. This is also likely the case for atleast some fraction of the substellar companions (browndwarfs). This possibility is examined in Section 6.On the other hand, a scattering encounter betweena planetary system and a passing star could raise theeccentricity and inclination of an initially coplanar andcircular outer body, which then drives the inner coldJupiter into a high-e orbit, leading to HJ formation. Theprobability of this HJ formation channel is strongly de-pendent on the stellar density. The combination of flybyand high-e migration could account for the observed cor-relation between HJ occurrence and stellar overdensity.High-e migration can be triggered by the ZLK mech-anism or by planet-planet scattering. This paper willfocus on the former, and the latter is discussed in Sec-tion 6. For the basic setup, we consider two giant planetsaround a solar-type star (with mass M ), an inner coldJupiter ( m J ) at a J ∼ m p ) at larger separations ( a p ∈ M )with velocity at infinity v ∞ and periastron q . In thefollowing sections, we will examine the requirements forthe passing star to trigger the high-e migration of thecold Jupiter via excitation of the outer planet’s orbit. EFFECTIVE RANGE OF ZLK ECCENTRICITYEXCITATION AND HIGH-E MIGRATION3.1.
Semi-major axis window
Not every semi-major axis ratio a J /a p can lead to theformation of HJs. High-e migration requires the innerplanet ( m J ) to attain a sufficiently large eccentricity.The maximum eccentricity depends on i p , the mutualinclination between the planet and the perturber ( m p ),and the competition between eccentricity driving by theZLK mechanism and its suppression due to short-rangeforces (e.g., Fabrycky & Tremaine 2007; Liu et al. 2015,hereafter LML15). In our case, the most important ofthese forces is the tidal force from the star on the innerplanet. Over all possible values of i p , the inner planetcannot become more eccentric than the limiting eccen-tricity e lim , as determined by LML15:˙ ω tides ˙ ω ZLK (cid:12)(cid:12) e J = e lim = 818 , (1)where ˙ ω tides and ˙ ω ZLK are the precession frequenciesdue to the tidal force and ZLK mechanism, respectively.They are given by:˙ ω tides = 152 k , J M m J (cid:18) R J a J (cid:19) e + e (1 − e ) n J , (2)˙ ω ZLK = m p M (cid:18) a J a p , eff (cid:19) n J (cid:112) − e , (3)where a p , eff = a p (cid:113) − e , k , J is the tidal Love numberof the planet, R J its radius and n J ≡ (cid:112) GM /a itsmean motion. We find1 − e lim ≈ − (cid:18) k , J . (cid:19) (cid:18) R J Jup (cid:19) (cid:16) a p , eff
40 au (cid:17) × (cid:18) m p Jup (cid:19) − (cid:18) m J Jup (cid:19) − × (cid:18) M (cid:12) (cid:19) (cid:16) a J (cid:17) − . (4) n the Correlation between Hot Jupiters and Stellar Clustering H i gh e cc en t r i c i t y e xc i t a t i on r eg i on High e p , moderate i p Most likely outcome of the fly-byHigh i p Unlikely outcome of the fly-by
Figure 1.
High eccentricity excitation region for the innerplanet in the ZLK problem at the octupole order, in the ε oct - i p phase space (where i p is the inclination of the externalperturber, and ε oct is given by Eq. 6). The boundary ofthe light red zone is given by the fitting formula of Mu˜nozet al. (2016), and systems in this zone can attain extremeeccentricity e lim [Eq. (4)], assuming the inner planet has anegligible angular momentum compared to that of the outerperturber. High inclinations are harder to produce with aflyby than high eccentricities (corresponding to high valuesof ε oct , see Section 4). To produce a HJ with a (circular) semi-major axis a HJ ( ∼ .
04 au, corresponding to a 3-day orbit), we requirethe pericenter distance a J (1 − e J ) to reach below a HJ / a J a p , eff (cid:38) (cid:18) k , J . (cid:19) (cid:18) R J Jup (cid:19) (cid:18) m J Jup (cid:19) − × (cid:18) M (cid:12) (cid:19) (cid:18) m p Jup (cid:19) − (cid:16) a p , eff
40 au (cid:17) − (cid:16) a HJ .
04 au (cid:17) − . (5)The possibility of creating a HJ thus depends mostlyon the semi-major axis ratio a J /a p , and weakly on a p alone. For giant planets initially at 5 au and the fiducialparameters, Eq. (5) amounts to a p , eff (cid:46)
300 au. As mostplanets are expected to lie inside this limit, it followsthat Eq. (5) does not provide a strong constraint on thesemi-major axis of the outer planet/perturber.3.2.
Eccentricity and Inclination window
In the ZLK mechanism, at the quadrupole order, aninitially circular planet can only reach extreme eccentric-ities if the outer perturber’s inclination is close to 90 ◦ .This picture is valid under two assumptions: the innerplanet is effectively a “test particle” (i.e., its orbital an-gular momentum is negligible compared to that of theouter perturber) and the octupole-order corrections aresufficiently weak. In the test-particle limit, it has been shown that the octupole-order effect can induce extremeeccentricities ( e J (cid:39) e lim ) in the inner orbit if the outerorbit’s inclination belongs to a finite window around 90 ◦ (LML15; Mu˜noz et al. 2016, hereafter MLL16) whosesize depends on the octupole parameter : ε oct = a J a p e p − e . (6)As ε oct increases, the inclination window grows untilit saturates at [48 ◦ , ◦ ] for ε oct (cid:38) .
05 (see Fig. 1).This “symmetric” inclination window is valid in the test-particle limit, when the angular momentum ratio l be-tween the inner planet and the outer perturber is small,i.e., l ≡ m J m p (cid:115) a J (1 − e ) a p (1 − e ) (cid:28) . (7)However, when both l and ε oct are not negligible, theinclination window obtained by MLL16 cannot predictwhether the inner orbit reaches very high eccentricities.To understand the critical value of l below which the pre-scription of MLL16 is accurate, we have carried out newsimulations for the evolution of the inner and outer plan-etary orbits to octupole-order, using the secular equa-tions given in LML15. We also include general relativis-tic periastron advance and tidal distortion of the giantplanet following LML15. We ignore the orbital decay ofthe inner planet due to tidal dissipation (which is ex-pected to occur over long timescales). To isolate theimpact of different values of l , we vary m p and a p suchthat the quadrupole order ZLK timescale, given by t − ≡ m p M (cid:18) a J a p , eff (cid:19) n J , (8)is constant. In particular, we fix e p = 0 . e J = 10 − and consider six values of m p = { , , , , } × M Jup and m p = M (cid:12) , while adjust-ing a p accordingly. For each value of m p , we fur-ther consider 2000 uniformly spaced initial inclinations i p ∈ [40 ◦ , ◦ ]. Then, for each inclination, we run threesimulations while randomly choosing Ω , ω ∈ [0 , π ) forboth the inner and outer orbits, the longitude of theascending node and argument of periapsis respectively,totaling 6000 simulations per combination of m p and a p . We run each simulation for 500 t ZLK and measure themaximum eccentricity attained by the inner planet. Fig-ure 2 depicts our numerical results. We see that the in-clination window predicted by MLL16 is accurate when l (cid:46) . . (9)For l (cid:38) .
1, extreme eccentricity can be achieved onlyfor inclinations larger than the test-particle result. As
Figure 2.
Maximum eccentricity of the inner planet (as 1 − e J , max ) versus initial inclination i p for different values of perturbermass m p (as labelled) and semi-major axis a p such that t ZLK [Eq. (8)] is held constant ( a p = 50, 63, 72, 85, 108, and 500au respectively). In all cases, we take e p = 0 . M = M (cid:12) , m J = M Jup , a J = 5 AU. The blue dots denote the maximumeccentricities attained by m J when the system is integrated for 500 t ZLK . The green line illustrates the analytical e J , max ( i p )curve when the octupole effect is neglected (see Eq. 50 of LML15). The purple vertical lines denote the inclination windowfor extreme eccentricity excitation as given by the fitting formula of MLL16 (see Fig. 1 or Eq. 7 of MLL16); for the simulatedsystems in the six panels, ε oct = 0 .
09, 0 .
07, 0 .
06, 0 .
05, 0 .
04, and 0 .
009 respectively. The horizontal dashed line denotes e lim asgiven by Eq. (4). When m p (cid:46) M J , corresponding to l (cid:38) .
1, the angular momentum of the inner planet is nonnegligible, and itis much more difficult for prograde outer planets ( i p < ◦ ) to excite e J to e lim than it is for retrograde outer planets ( i p > ◦ ). we shall see in Section 4, for initially coplanar systems,very high inclinations are an unlikely outcome of fly-bys. Thus, in the following, we will restrict our atten-tion to systems satisfying Eq. (9). Assuming e p = 0 . m p > Jup for a p /a J = 10; and m p > Jup for a p /a J = 50. Note that the limitingeccentricity e lim depends on l (Anderson et al. 2017,Eq. 26), and is almost unchanged from Eq. (4) in theregimes we consider, as can be seen in Fig. 2. EFFECT OF FLYBY ON THE OUTER PLANETThe goal of this section is to estimate the percentageof close encounters that can drive the outer planet intothe ZLK window—which we will refer to as “success-ful flybys”. Recall that in our scenario the outer planetserves as a perturber that drives the inner planet into ahigh-e orbit. A successful flyby should raise significantlyboth the octupole parameter ε oct (or eccentricity) andthe inclination of the outer planet (Fig. 1). In the fol-lowing, we will show that this is roughly equivalent toraising the inclination i p to at least 48 ◦ . The impact of a flyby on the outer planet depends onthe dimensionless distance at closest approach ˜ q ≡ q/a p ,where q is the flyby periastron. For large ˜ q , we can aver-age over time the orbit of the planet and the trajectoryof the passing star and analytically compute the final in-clination i p and eccentricity e p of the orbit of the outerplanet (Heggie & Rasio 1996; Rodet et al. 2019). Theseanalytical expressions hold only for q (cid:38) a p , and themaximum inclination increase is about 6 ◦ , not enoughfor the system to enter the ZLK extreme eccentricity ex-citation window (which requires at least a misalignmentof 48 ◦ ). The impact of a flyby with a smaller perias-tron is chaotic and can only be studied numerically. Wethus conduct N-body simulations with the ias15 inte-grator of the rebound package (Rein & Liu 2012; Rein& Spiegel 2015). In order to limit the number of pa-rameters, the simulations include two equal-mass stars M = M and a test particle (representing planet m p ).The integration time is chosen so that the distance be-tween M and M is equal to 100 q at the beginning andend of the simulations, and the time-step is adaptive. n the Correlation between Hot Jupiters and Stellar Clustering e p P D F i p (deg) q = 0.5 q = 1.0 q = 2.0 a p / a p Figure 3.
Probability distribution of the orbital elements (eccentricity, inclination and fraction change of semi-major axis) ofthe outer planet ( m p ) after stellar flybys. Numerical results for three values of the pericenter distance q of the passing starare shown, with ˜ q ≡ q/a p = 0 . , For star M and M approaching each other with rel-ative velocity v ∞ ( ∼ q is given by v = v ∞ + 2 GM tot q = v ∞ + (10 km / s) (cid:18)
50 au q (cid:19) (cid:18) M tot M (cid:12) (cid:19) , (10)where M tot = M + M . For the regime we are con-sidering, the gravitational focusing term dominates and v ∞ is negligible. This translates to a stellar eccentric-ity close to 1. In our simulations, we adopt e = 1 . e as long as itremains close to 1.In addition to the periastron distance q , a close en-counter is specified by the inclination i between the flybyorbital plane and the initial orbital plane of the planet,and ω , the argument of periastron. Note that since theinitial planetary orbit is circular, the outcome of an en-counter does not depend on Ω, the longitude of node.Finally, the outcome of a flyby also depends on the phaseof the planet λ p at closest stellar approach.In our simulations, we sample the periastron q over auniform grid from 0 . a p to 2 a p , and choose • ω with a flat prior between 0 and 2 π , • i , with a sin( i ) probability distribution, between 0and π , • λ p with a flat prior between 0 and 2 π .For each value of q , we sample 20 × ×
20 angles (for ω , i ,and λ p ) to determine the distributions of post-flyby orbital parameters of the planet. Several examples ofthe post-encounter outcomes for three different ˜ q valuesare shown on Fig. 3. It is clear that flybys with ˜ q = 2(and larger) produce a negligible number of systems with i p (cid:38) ◦ . Thus, for our purpose, there is no need toconsider encounters with ˜ q > q , the percentage of systems that experience a suc-cessful flyby—i.e., the outer planet m p remains boundand is in the eccentricity and inclination window (seeFig. 1) for inducing extreme eccentricity excitation ofthe inner planet, assuming the orbit of the inner planet( m J ) unaffected by the flyby. The resulting probabilityof successful flybys as a function of q is shown in Fig.4 for a J /a p = 0 . ε oct > .
05 is equivalent to e p > .
5. Since a successful flyby would lead to the innerplanet to migrate inward, we shall term this probabil-ity p mig . We see from Fig. 4 that p mig is approximatelyequal to the probability of producing high inclination:raising the inclination of the outer planet by 48 ◦ is thehardest constraint to get a successful flyby. This is inline with previous studies finding that the inclination isharder to raise than the eccentricity (Li & Adams 2015;Wang et al. 2020). In fact, for the entire range of semi-major axis ratios that we study (Eq. 5), most of thecases with high inclinations have (cid:15) oct > .
05, so thatthe probability of a successful flyby is most constrainedby the requirement of significant inclination excitation( i p ∈ [48 ◦ , ◦ ]). Consequently, p mig can be consideredindependent of a J /a p . Our result for p mig (˜ q ) as depictedin Fig. 4 will enable us to derive the occurrence rate of q P r o b a b ili t y p ( e p < 1) p (0.1 < e p < 1) p (0.3 < e p < 1) p (0.5 < e p < 1) p (0.8 < e p < 1) p (0.9 < e p < 1) p (0.99 < e p < 1) 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 q P r o b a b ili t y p ( e p < 1) p (0.5 < e p < 1) p ( e p < 1 and i p [48 , 132 ]) p mig ( q ) Figure 4.
Various probabilities for the outer planet’s eccentricity e p and inclination i p to be in critical ranges after a stellarflyby, as functions of the dimensionless distance at the closest approach of the passing star, ˜ q ≡ q/a p (where a p is the initialsemi-major axis of the planet). The probabilities are computed numerically following the procedure described in Section 4. Thered dots (right panel) show p mig , which is the probability for the planet to be in the “extreme” ZLK window after a flyby, i.e. theouter planet reaches a sufficiently large inclination and eccentricity to drive the inner planet into high-e migration. This extremeZLK window is approximately equivalent to i p ∈ [48 ◦ , ◦ ] and ε oct > .
05, the latter condition corresponds to 0 . < e p < a J /a p = 1 /
10. Note that p mig is almost equal to the probability that the outer planet reaches i p ∈ [48 ◦ , ◦ ] and remainsbound ( e p < successful flybys in Section 5, and to deduce the overallprobability of forming HJs from close stellar encounters. OCCURRENCE RATEIn this section, we estimate the occurrence rate of HJsthat form in our flyby-induced high-e migration scenario(Section 2). This occurrence rate can be written as η HJ = t cluster η survival (cid:90)
300 au10 au d a p d η init d a p R mig ( a p ) , (11)where d η init / d a p is the probability distribution functionof the initial conditions (two planets, one giant planetaround the snowline and one larger planet/perturber at a p , with angular momentum ratio l less than 0.1), t cluster is the lifetime of the birth cluster, η survival is the prob-ability for the giant planet to survive tidal disruptionduring high-e migration and R mig is the rate of closeencounters that result in high-e migration. The lowerlimit of the a p integration ensures dynamical stability,while the upper limit is the consequence of Eq. (5).Tidal disruption can limit the efficiency of HJ forma-tion through high-e migration. Using population syn-thesis models and analytical calculations (e.g.; Petro-vich 2015b; Anderson et al. 2016; Mu˜noz et al. 2016;Teyssandier et al. 2019; Vick et al. 2019), it has been estimated that most of the giant planets that reach e lim will be destroyed by the tidal forces of their host star.However, there are important uncertainties regardingthe fraction of surviving planets, depending on the prop-erties of tidal dissipation. Vick et al. (2019) showed thatstrong dissipation, through a mechanism called chaotictides , can sometimes save planets otherwise fated fortidal disruption by rapidly decreasing their eccentrici-ties. They estimated that η survival ∼
20% of migratingplanets could survive as HJs.Observations provide only limited information ond η init / d a p . Radial velocity surveys suggest an occur-rence rate for giant planets between 10 and 30%, with amaximum likelihood around 3 au (e.g., Fernandes et al.2019). On the other hand, direct imaging surveys pointtowards an occurrence rate of wide planetary/browndwarf companions (10-300 au) of around 5-10% (Nielsenet al. 2019; Vigan et al. 2020). The correlation betweenthe occurrences of giant planets at a few au and com-panions at 10–300 au is not known. Thus, η init ≡ (cid:90)
300 au10 au d a p d η init d a p ∼ . − . (12)The dependency of d η init / d a p on the semi-major axis a p is unconstrained. n the Correlation between Hot Jupiters and Stellar Clustering n (cid:63) ( v ∞ ), which is a function of the veloc-ity distribution f of the stellar velocities v ∞ . We take f to be a Maxwell-Boltzmann distribution with dispersion σ (cid:63) : d n (cid:63) = n (cid:63) f ( v ∞ )d v ∞ (13) f ( v ∞ ) = (cid:114) π v ∞ σ (cid:63) exp (cid:18) − v ∞ σ (cid:63) (cid:19) . (14)The cluster density n (cid:63) can take a wide range of values,from 10 − stars / pc in the nearby OB associations to10 stars / pc in the center of globular clusters. More-over, in an unbound stellar association, which corre-sponds to most stellar birth environments, the densitydecreases with time. As a rough estimate, we supposethat our cluster of interest maintains a 10 stars / pc density for the first t cluster ∼
20 Myr of its life (Pfalzner2013). We suppose that the flyby rate at later times isnegligible due to the much smaller density. The velocitydispersion σ (cid:63) is better constrained thanks to observa-tions in nearby associations and cluster (e.g. Wright &Mamajek 2018). We adopt σ (cid:63) ≈ R mig ( a p ) = (cid:90) ∞ d n (cid:63) ( v ∞ ) v ∞ (cid:90) + ∞ π d b ( q, v ∞ ) p mig (˜ q ) , (15)where p mig is depicted in Fig. 4. The impact parame-ter b ( q, v ∞ ) associated with an hyperbolic trajectory ofperiastron q and velocity at infinity v ∞ is: b = q (cid:18) GM tot qv ∞ (cid:19) ≈ GM tot qv ∞ . (16)Using the result shown in Fig. 4, we find R mig = R close (cid:90) + ∞ d˜ q p mig (˜ q ) ≈ . R close (17)where R close ≡ πa p GM tot n (cid:63) (cid:90) ∞ dv ∞ v ∞ f ( v ∞ )= 2 √ πa p GM tot n (cid:63) σ (cid:63) ≈ (cid:18) n (cid:63) pc − (cid:19) (cid:18) M tot (cid:12) (cid:19) (cid:16) a p
50 au (cid:17) × (cid:18) σ (cid:63) / s (cid:19) − Gyr − (18)is the rate of close encounters with periastron below a p . Combining Eqs. (11), (12),(17), and (18), we have η HJ ≈ . (cid:18) M tot (cid:12) (cid:19) (cid:18) (cid:104) a p (cid:105)
50 au (cid:19) × (cid:16) η init (cid:17) (cid:18) R mig / R close (cid:19) (cid:16) η survival (cid:17) × (cid:18) n (cid:63) pc − (cid:19) (cid:18) t cluster
20 Myr (cid:19) (cid:18) σ (cid:63) / s (cid:19) − , (19)where (cid:104) a p (cid:105) ≡ η init (cid:90)
300 au10 au d a p d η init d a p a p . (20)Equation (19) gives an estimate for the occurrence rateof HJs produced in our scenario, and illustrates its de-pendence on various uncertain parameters. Major un-certainties include η init , the occurrence rate of the initialtwo-planet systems, and the density and lifetime of thebirth clusters, the latter two may vary by orders of mag-nitude.The observed occurrence rate of HJs around solar-typestars is 0.5–1% (e.g. Dawson & Johnson 2018). With theadopted fiducial parameters in Eq. (19), this formationchannel can account for 10-20% of the observed HJ pop-ulation. But with more optimistic n (cid:63) and t cluster values,our estimated η HJ can be compatible with the observedvalue. SUMMARY AND DISCUSSION6.1.
Summary
In this paper, we have studied a new scenario for theformation of hot Jupiters (HJs) following a close stellarencounter—we call it “flyby Induced High-e Migration”.This could account for the recently observed correlationbetween the occurrence of HJs and stellar overdensities(Winter et al. 2020). In this scenario, we suppose thatstellar flybys could excite the eccentricity and inclina-tion of an outer planet or brown dwarf (at a p ∼ Main Uncertainties
No further analysis or numerical simulations can sig-nificantly refine our estimate [Eq. (19)] until a better un-derstanding on the properties of the typical stellar birthcluster is obtained, in particular the stellar density n (cid:63) as a function of time.Furthermore, the correlation between cold Jupitersand more distant planets or brown dwarfs is currentlyunknown. So the occurrence rate of the two-giant-planetinitial systems is not constrained. Our knowledge shouldimprove with the next generation of exoplanet imagingsurveys.6.3. Comparison with other high-e migration paths
In the following we consider several other possiblemechanisms for HJ formation. All of these are lesspromising for explaining the correlation between HJsand stellar overdensities reported by Winter et al.(2020). 6.3.1.
Planet-planet scattering
Planet-planet scattering is another possible path tocreate high-eccentricity orbits, which could then lead tohigh-e migration of giant planets. Wang et al. (2020)determined that it could account for a significant forma-tion of hot Jupiters in their simulations involving stellarflybys. This mechanism does not require raising the in-clination of the outer planet to enter the ZLK regime,but only exciting the eccentricity enough to prompt aclose encounter. How high the eccentricity needs to beraised depends on the semi-major axis ratio a J /a p . For a J /a p = 0 .
5, the required eccentricity for close planet-planet encounter is about 0.5, and we found from ourflyby simulations that the probability can be ∼
40% for˜ q (cid:46) a J /a p = 0 .
1, the required ec-centricity is 0.9, and our simulations suggest a much smaller probability ( (cid:46)
Stellar companion
The probability for a solar-type star to have a stel-lar companion at separations 100-1000 au is close to20% in the field (Raghavan et al. 2010). Assuming aflat eccentricity and an “isotropic” inclination distribu-tions, a significant fraction of binary companions couldinduce ZLK migration of giant planets. This HJ for-mation channel has been well studied in the literature(see Petrovich 2015b; Anderson et al. 2016; Mu˜noz et al.2016; Vick et al. 2019, and references therein). The pre-dicted HJ occurrence rate is of order 0.1-0.2%, with oneof the main uncertainties being tidal disruption of themigrating planet. This formation channel seems moreefficient at forming HJs because of the higher occurrencerate of stellar companions at large separation and, mostimportantly, because it does not require a flyby.However, this formation channel struggles to accountfor the observed correlation between HJs and stellaroverdensities (Winter et al. 2020). While the depen-dence of the stellar multiplicity on the stellar density isnot well constrained, several surveys in nearby clustersseems to indicate that overdensity actually decreasesstellar multiplicity (King et al. 2012; Marks & Kroupa2012). 6.3.3.
Tides from the stellar cluster
Stellar clusters create a global gravitational potential,which can trigger eccentricity and inclination variationson wide companions (Hamilton & Rafikov 2019a,b).Similar to the ZLK mechanism, here the “perturber”is the global potential of the cluster. For a planetary n the Correlation between Hot Jupiters and Stellar Clustering R c from the center of thecluster, the eccentricity excitation timescale is of order T ∼ R GM c n p ∼ n p πG ¯ ρ (cid:63) ∼ (cid:18) M (cid:12) / pc ¯ ρ (cid:63) (cid:19) (cid:18)
50 au a p (cid:19) (cid:18) M (cid:12) (cid:19) , (21)where n p , a p are the (initial) mean-motion and semi-major axis of the planet, M c is the enclosed cluster massat R c and ¯ ρ (cid:63) the mean density of the cluster. Thistimescale is much larger than the lifetime of the cluster.Moreover, at such a density, about 20 close flybys are expected to occur within 1 Gyr (Eq. 18), and their in-fluence will be dominant on the dynamics of the planet.ACKNOWLEDGEMENTSThis work has been supported in part by the NSFgrant AST-17152 and NASA grant 80NSSC19K0444.YS is supported by the NASA FINESST grant 19-ASTRO19-0041. We made use of the python libraries NumPy (Harris et al. 2020),
SciPy (Virtanen et al.2020), and
PyQt-Fit , and the figures were made with
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