The Formation of Habitable Planets in the Four-Planet System HD 141399
TThe Formation of Habitable Planets in theFour-Planet System HD 141399
R. Dvorak , B. Loibnegger , L.Y. Zhou , and L. Zhou Department of Astrophysics, University of Vienna,A-1180 Vienna, Austria School of Astronomy & Space Science, Nanjing University,163 Xianlin Avenue, Nanjing 210046, ChinaMay 1, 2019
Abstract
The presented work investigates the possible formation of terrestrialplanets in the habitable zone (HZ) of the exoplanetary system HD 141399.In this system the HZ is located approximately between the planets c (a= 0.7 au) and d (a = 2.1 au). Extensive numerical integrations of theequations of motion in the pure Newtonian framework of small bodieswith different initial conditions in the HZ are performed. Our investiga-tions included several steps starting with 500 massless bodies distributedbetween planets c and d in order to model the development of the disk ofsmall bodies. It turns out that after some 10 years a belt-like structureanalogue to the main belt inside Jupiter in our Solar System appears. Wethen proceed with giving the small bodies masses ( ∼ Moon-mass) and takeinto account the gravitational interaction between these planetesimal-likeobjects. The growing of the objects – with certain percentage of water –due to collisions is computed in order to look for the formation of terres-trial planets. We observe that planets form in regions connected to meanmotion resonances (MMR). So far there is no observational evidence ofterrestrial planets in the system of HD 141399 but from our results we canconclude that the formation of terrestrial planets – even with an appro-priate amount of water necessary for being habitable – in the HZ wouldhave been possible.
The ’race’ to find a second Earth led to the discovery of many such candidateswith different techniques from space and from ground. Nevertheless, althoughwe have knowledge of systems hosting terrestrial planets without gas giants using the definition of a terrestrial planet meaning a planet smaller than 5 Earth-masseswith a rocky surface a r X i v : . [ a s t r o - ph . E P ] A p r ystem m star nr of nr with m > Jup (m Sun ) Planets m > Jup a > > forupdates and more detailed information). The system of 47 UMa is topic ofanother paper within our group (Cuntz et al., 2018).– for example Trappist 1 (e.g., Gillon et al., 2016) – our Solar System is anexample of hosting big planets outside of the habitable zone (HZ). This zonewas proposed by Kasting et al. (1993) and later revised by e.g., Kopparapuet al. (2013) and describes the zone where water can exist in liquid form on aterrestrial planet orbiting the star.Since the first discovery of a big planet in 1995 (Mayor & Queloz, 1995) – aso called hot Jupiter – many gas giants have been found around stars of differentspectral type.The discoveries of many hot Jupiters over the past 20 years and the impli-cations of their existence for the process of planet formation make it difficult tobelieve that terrestrial planets may co-exist in the HZ of those systems (Dawson& Johnson, 2018).Systems with only one gas giant are well represented in the long list ofextrasolar planets , but in our work we concentrate on systems with more thanone gas giant of which several ones are known (see Table 1). Of special interestto us seems the recently discovered four-planet system HD 141399 (Vogt et al.,2014).Vogt et al. (2014) carefully describe the method of determining the orbitalelements of the planets and check their stability which seems to be guaranteedbecause of the almost circular orbits of the four planets.The aim of our research is to find whether terrestrial planets could be havebeen formed in the HZ around the late K-type star.The structure of this research paper is as follows: In Section 2 we describethe system of HD 141399 in detail. The method of investigation is presented inSection 3. The results of the integration of massless bodies in the HZ between http://exoplanet.eu/ see http://exoplanet.eu/ c and d are shown in Section 4, where we present a thorough analysisof the outcomes and show examples of places where terrestrial planets can formon stable orbits. The results were gained using extensive numerical simulationswith integration times of up to 10 Myr. In Sub - section 4.2 we take into accountthat the observations were made using radial velocity measurements which yieldonly a minimum mass for the planets and concetrate on the formation of bodiesin the system with assumed larger masses for the giant planets.Section 5 shows the results of computations with massive bodies whichformed terrestrial planets through collisions. In Section 6 we describe the growthprocess from planetesimals to protoplanets in more detail. Finally in Sections 7and 8 we show the main results and draw the conclusions with focus on possibleterrestrial planets in the HZ.In Appendix A.1 we investigate the dynamics around the Lagrangian Point L of planet d in detail and describe the acting secular resonances in the systemin Appendix A.2. We note that the Trojan region of planet d is just outsideof the HZ. In the presented article we included the computations for possiblyhabitable planets formed in this region taking into account the eccentricity ofplanet d . On its orbit it will sometimes be inside and sometimes just outsidethe defined HZ. It is important to have an overview of possibly stable terrestrial planets in knownmultiple planetary systems with gas giants (Agnew et al., 2018). Nevertheless,we think that each system should be treated separately because of the differentarchitecture. The dynamical stability of additional – possibly terrestrial – plan-ets in a planetary systems with gas giants is only one side of the coin, the otherside is to know how these planets could have formed. The present knowledge isthat in our Solar System the gas giants formed prior to the terrestrial planetsin the Solar environment of gas and dust. Different scenarios are possible (e.g.,Grand Tack as described by Morbidelli et al., 2011). We know that in our SolarSystem no stable terrestrial planet could exist between Jupiter and Saturn –they all formed inside the Jovian zone.We looked into the literature to find an interesting system quite similar toour own and found that HD 141399 would be a good example (Vogt et al., 2014).The HZ in HD 141399 is located between two gas giants (see Table 2), namelybetween planet c (a = 0.7 au) and planet d (a = 2.1 au). Calculation of theboundary values for the habitable zone shows that it lies between 0.8 au and2.0 au from the star (Selsis et al., 2007).HD 141399 is a K0V star with m = 1.07 m Sun , r = 1.46 r
Sun , and T eff = 5600 K.The system was presented by Vogt et al. (2014) who used radial velocity datasets from Keck-HIRES and the Lick Observatory’s Automated Planet FinderTelescope and Levy Spectrometer on Mt. Hamilton. 91 observations from over10.5 yrs were analyzed and the parameters of four planets shown in Table 2have been found. Interestingly the planets HD 141399 b and HD 141399 c are3lanets Period (d) e a (au) mass (m Jupiter )HD 141399 b 94 0.04 0.4225 0.46HD 141399 c 202 0.05 0.7023 1 363:2 MMR 732 1.63HD 141399 d 1070 0.06 2.1348 1.22HD 141399 e 3717 0.00 4.8968 0.69Table 2: Properties of the planets in the system HD 141399. The 3:2 MMR iscalled ”Hilda region” and resides in the HZ of HD 141399.close to the 2:1 mean motion resonance (MMR) which might give a clue on thehistory of the system. Vogt et al. (2014) argue, that a possible inward migrationwould point towards a capture in resonance, which cannot be the case, giventhe values derived from the observations. Conversely, Batygin & Morbidelli(2013) show, that a divergence away from exact resonance behavior is a naturaloutcome of dissipative evolution of resonant planetary pairs. Vogt et al. (2014)carefully describe the method of determining the orbital elements of the planetsand checked their stability, which seems to be guaranteed because of the almostcircular orbits of the four planets.
The method used for the investigation of the formation of possible terrestrialplanets between HD 141399 c and HD 141399 d was the Lie-ingration methodwhich has already been used extensively by our group (e.g., Dvorak , 1986;Delva, 1984; Hanslmeier & Dvorak, 1984; Lichtenegger, 1984). It is well adaptedfor the integration of the equations of motion for planetary systems hostingmassless and massive bodies and treats close encounters and collisions with ahigh precision due to its automatically chosen step size. The method has beencompared to other methods in detail by Eggl & Dvorak (2010).We ask the question where and how these terrestrial planets could haveformed in HD 141399. For being habitable the planets need to have waterand therefore we included water content of the planetesimals in our formationcomputations.Making use of the Lie integration method four main runs were undertaken: • First we distributed 500 massless bodies in the region between 0.7 auand 2.1 au (the location of planet c and planet d ) with initial condi-tions randomly distributed for the orbital elements: semi-major axis a(0 . au < a < . au ), eccentricity e < .
13, inclination i (0 . ◦ < i < . ◦ ), Ω = 154 ◦ , and ω = 0 . ◦ . The mean anomaly was randomly dis-tributed. The whole system was integrated for 1 Myr for 8 differentlychosen initial conditions. See Figure 1.4igure 1: Example of the initial inclinations and eccentricities versus semi-majoraxis. The orbital elements were chosen randomly for each of the different runs.The vertical line marks the position of planet c . • In the next step we replaced the massless bodies with massive ones (0 .
92 m
Moon 23 m Moon ) with a total mass of approximately 5 m Earth , which wasfor example also used in Quintana & Lissauer (2014) . The orbital ele-ments were initially chosen randomly like in the first run. These compu-tations were performed for a total set of 50 different initial conditions. • Again the initial conditions were chosen randomly but we took into ac-count that the estimated masses by the observers are only spectroscop-ically determined. Thus three additional computations were performedfor massless bodies but with more massive four giant planets (the mostprobable masses given by the most probable inclinations, Voigt, 2012). • In a forth run we checked the outer region between planets d and e bypopulating it with massless bodies (see Figure 3).The water content of the planetesimals was calculated to be initially between5% to 10% according to the mass of the body and the distance to the centralstar.We note that the number of bodies diminished with integration time becauseof • collisions amongst them resulting in formation of bigger planets (for thecomputations with massive planetesimals – perfect merging was assumed ) • collisions with one of the gas giants • collisions with the host star • escape from the system after a close encounter with one of the giant planets(as escape criterion for a body we fixed: e ¿ 0.99). several authors used initially different values of up to 10 Earth-masses (e.g., Raymond etal., 2006). the parameters of each collision (impact velocities, impact angle) have been stored forpossible follow up studies with SPH (Smoothed Particle Hydrodynamics). The ’Belt’ of Massless Bodies The reason for the choice of a close investigation of the region between planets c and planet d (we call it the belt) is the following: we are interested in theformation of possible terrestrial planets in the HZ of HD 141399.In a recent work of Agnew et al. (2018) amongst other multi-planetary sys-tems with gas giants the dynamical stability of HD 141399 is proven, usingbest-fit planetary and stellar parameters from the NASA Exoplanet Archive.They show that a potential terrestrial planet (m ∼ Earth ) can inhabit a sta-ble orbit inside the HZ of this particular system for the integration period of10 yrs.In order to have a first overview of the dynamics of orbits inside this belt wecompute 8 ’packages’ of 500 massless bodies each. All the orbits were integratedfor 1 Myr. The structure visible in Figures 2 and 4 looks very similar to the Kirkwood gapsin the main belt of asteroids in our Solar System. In Figure 2 we marked theMMRs.When we compare the results of an integration of massless bodies betweenthe planets d and e in Figure 3 with the region between Jupiter and Sat-urn in our Solar System there is a big difference evident. In the Solar Sys-tem there are no stable families of asteroids with small eccentricities between5 . au < a < . au . In HD 141399 we found accumulations of bodies in certainresonances. Even in the equilateral Lagrange points of planet d and planet e many planetesimals are captured and form Trojan families.The extension of computing bodies outside of planet d has been undertakenfor the sake of completeness of the dynamical study of HD 141599. It becomesrelevant as one recognizes the Trojans around planet d (see Figure 3). This wasone of the reasons to explore the Trojan region of planet d in more detail (seeAppendix). Note that around Saturn there is no a Trojan cloud (Dvorak et al.,2008, 2014). Nevertheless, it would be worth to do a more detailed investigation. The following Section deals with a check of stability and formation of familiesof massless bodies in the system of HD 141399 assuming larger masses for thefour gas giants. We performed these computations with three different initialconditions (compare Section 3 and Figure 4).For these computations we assumed the most probable inclination of theorbital plane of the giant planets – the statistical mean value. We assume equallydistributed inclinations (sin ( i ) = 0 . 56, Voigt, 2012) which leads to an increasingfactor of 0.59 and gives m planet b = 8.035, m planet c = 2.378, m planet d = 2.127and m planet e = 1.15 (masses in M Jupiter ). Because of the small eccentricities,the gas giants are stable for Gyrs in this configuration, too. For the distribution6igure 2: The results of the integration of 8 x 500 massless bodies between theplanets c and d . The number of remaining bodies is plotted separately for eachof the 8 runs (y-axis) versus the semi-major axis (x-axis). The gaps and ’groups’are very well visible. The numbers describe the resonances with planet d . Notethat the green numbers show stabilizing resonances, while the red numbers showdestabilizing resonances.Figure 3: Results of the integration of 3 x 500 massless bodies (with slightlydifferent initial conditions between the planets d and planet e). Semi-major axis(x-axis) versus eccentricity (y-axis). The number of remaining bodies is plottedin green, the initial distribution in red and the 4 planets as blue squares. Alsovisible are gaps and groups as for the inner belt.7igure 4: Belt of massless bodies for increased masses of the giant planets c and d (as described in Section 4.2). Semi-major axis is plotted versus eccentricitywith different colors indicating different runs.of massless bodies in the HZ one can observe that gaps and groups appear asin our previous computations (see Figure 4). A comparison of Figures 2 and 4shows similarities of the gaps and groups (areas of accumulation of bodies) inthe distribution of bodies after an integration time of 1 Myr. These gaps andgroups result from the acting of resonances (MMRs) with the giant planets. As an example of the formation process taking place in the system of HD 141399we picked out one simulation for discussion. Semi-major axis versus mass of thebodies of this particular simulation are shown in Figure 5. One can observethat after 0.1 Myr most of the original Moon-sized objects are gone either dueto collisions (with one of the gas giants or the host star) or they were thrownout of the system after close encounter(s) with one of the gas giants. A closeencounter can lead to large eccentricity (e > c and planet d for 1 Myr. An extension ofthe integration time to 10 Myr in this particular case shows that they obtainstable orbits over the whole integration time.In Figure 6 mass versus semi-major axis of the remaining bodies after 0.05Myr, 0.1 Myr, and 1 Myr are shown. We remark that after 0.05 Myr (red points)there are still many – now protoplanets – present in the system. Additionallystill some of the originally Moon-sized bodies remain. We observe the formationof a planet of 20 Moon-masses at ∼ c and planet d . The smaller one at 1.5 au hasapproximately Mars-mass while the bigger one at 1.1 au has approximately 0.5Earth-masses. The bottom line shows a small body with a high eccentricity.Compare Figure 6 which shows results of the same run. ∼ ∼ d andsubsequent collision or ejection (not checked in detail here) the number of re-maining bodies after 0.1 Myr (green points) is small.Note that Figure 5 and Figure 6 show results of the same simulation whileFigure 7 depicts results from another run. There are two ways of looking at the the growth of planets:a) follow one particular body during its evolution (monitoring each collisionin order to watch the growth to the final protoplanet/ planet)b) follow the development of the whole belt of bodies (checking the numberof remaining bodies and the growth after several time steps)We follow the growth of a planetesimal to the moment of reaching the massof a terrestrial planet. A special case is shown in Figure 7 where after 9 colli-sions (perfect merging was assumed) a terrestrial planet is formed. The body9igure 6: Remaining bodies after 0.05 Myr (red points), 0.1 Myr (green points),and 1 Myr (open blue circles). Semi-major axis is plotted versus mass of thebodies in (m Moon ). The horizontal line marks the size of 10 Moon-masses forclarity. The orbits of the planets depicted by the two open blue circles are shownin Figure 5.considered has a mass of 25 m Moon (blue lines in Figure 7). The second mostmassive one has only 20 m Moon (green lines in Figure 7).The growth process can be divided into different steps: In the beginningsmall bodies collide and form larger and larger objects until they reach a massof several Moon-masses. The collision of two bigger bodies leads to the formationof a terrestrial planet like object. Yet, there are still many small bodies aroundwhich are ’added’ to the big body via collisions. We show one out of 50 integrations and follow the 500 Moon-sized bodies overtime in order to depict the evolution of the belt. We show the initial conditionsand snapshots after 1 kyr, 10 kyr, and 1 Myr in Figure 8.One can observe that the eccentricities grow quite fast when we comparein Figure 8 the initial distribution to the distribution semi-major axis versuseccentricity after 1000 years (left and right upper panels in Figure 8). Bodies ofthe size of several times the Moon are formed during such a short time. After 10kyrs we can see that already some Mars-sized protoplanets were formed; like inthe former plot one recognizes also that the majority of bodies is now confinedprimarily between 0.9 au and 1.5 au (lower left panel in Figure 8).These largerbodies disappear during the continued integration of the equations of motionup to 1 Myrs; we can explain it by encounters with the gas giants which throw10igure 7: Growth process of two terrestrial planets. The blue lines show theformation process of the most massive planet formed in this particular simu-lation. The green lines depict the growth of the second most massive planetformed during the integration. The axes show time versus mass.them far away from the central star to large semi-major axes and eccentricitiesand may also lead to escapes. But even collisions with the four large planets letthem ’disappear’. The left over after 1 Myr consists now of only two Mars-sizedplanets and two Moon-like ones (lower right panel in Figure 8). The results from the integrations are summarized in the following plots whichshow (note that the water content will be discussed in Section 7.1) • Figure 9: We show the resulting planets formed out of 50 x 500 plan-etesimals distributed between 0.7 au and 2.1 au (between planet c and d ). Only two of the planets with mass comparable to the mass of Earthwere formed via collisions and consecutive merging (these planets gained0.9 Earth-masses). In addition many Mars-sized planets are formed. Onecan observe that most of the more massive planets inhabit low eccentricorbits. The biggest bodies have eccentricities e < ∼ e ∼ . Figure 10: The distribution of bodies at the end of the integration (0.5Myr) shows that most of the bodies remain between the planets c and d .An accumulation of objects can be observed at certain semi-major axes.The majority of the formed planets with up to 0.9 Earth-masses reside inthe area 0 . < a < . d . They have been captured into the 1:1 MMR as Trojan planets.Furthermore there appears to be a more or less empty region inside ofplanet d . Clearly, empty regions analogue to the ’Kirkwood gaps’ in ourSolar System are seen. The groups where many larger planets are formed,are observable at 1.15 au (by far the biggest one), 1.35 au, and 1.5 au. • Figure 11: The distribution of bodies at the end of the integration showsthat most of them stay in the plane of the gas giants. In particular themore massive ones show only small values for inclination (i < ◦ ). Insome simulations bodies were scattered to retrograde orbits during theformation process. • Figure 12: The plot shows, that some of the bodies captured as Trojansof planet d have orbits with inclinations of up to 80 ◦ . Some are even onretrograde orbits. Nevertheless, the majority of bodies stays in the planeof the gas giants with i < ◦ . • Figure 13: Statistical summary of all the remaining bodies after the endof the integration (included are all 50 runs). One can see that most ofthe planetesimals remained small. Nevertheless, some (to be exact: 2)managed to gain a mass of up to 0.9 Earth-masses. At the beginning of the integration each of the approximately Moon-sized bodiesin the belt between the planets c and d is assigned with a water mass fractionof 5 – 10 % depending on the distance from the star. We assume that aftereach collision the bodies perfectly merge. Figure 9 and 10 show the summaryof all 50 runs. Depicted are the mass and water content of the bodies afterthe end of the integration time. One can see that, obviously, the more massiveones collected more water (in units of Earth-masses). Nevertheless, the watercontent in percentages is the same for small and massive bodies.SPH (smooth particle hydrodynamics) computations of collisions betweenbodies show that they suffer from essential loss of water depending on the im-pact velocity, the impact angle etc. (Burger et al., 2018; Maindl et al., 2013,2014; Sch¨afer et al., 2016). All these parameters were not yet included in oursimulations. As we assume that no mass is lost during a collision we clearlyoverestimate the water mass fraction present in each planet at the end of thesimulation.The more realistic models of collisions with the aid of SPH-codes need timeconsuming simulations for every single collision and should be treated in com-bination with a n-body code. 13igure 9: Accumulated bodies of 50 different runs with initially 500 Moon-sizedbodies between planet c and planet d after 1 Myr. Semi-major axis is plottedversus mass with the color indicating the water content (in Earth-masses) ofeach planet.Figure 10: Same as Figure 9 with the color indicating the water content (inEarth-masses) of each planet. 14igure 11: Same as Figure 10 but for mass versus inclination.Figure 12: Same as Figure 10 but for semi-major axis versus inclination. Onecan see that some of the bodies captured as Trojans of planet d have inclinationsof up to 80 ◦ . 15igure 13: Histogram of the planets depicted in Figure 10. Number versusmass. One can see that most of the planetesimals remained small for the wholeintegration time. Nevertheless some were able to gain a mass of up to 0.9Earth-masses.Nevertheless, our results show that the planets formed in between the planets c and d could foster life as they remain on stable low eccentric orbits in the HZaround HD 141399 with possibly water on their surfaces. The goal of the presented work was to investigate the probability of the existenceand formation of terrestrial planets in an extrasolar planetary system with sev-eral gas giants. We found a very interesting example: The system HD 141399,where a K0V star hosts 4 giant planets in distances to the central star between0.4 au and 5.2 au. The small eccentricities of the planets guarantee stable orbitsup to Gyrs. This is true, even when it is taken into account that the determinedmasses via spectral analysis from observations with large telescopes are onlyminimum values.In order to answer the question about existence and formation of terrestrialplanets in the system of HD 141399 we assumed a leftover belt of hundreds ofsmall bodies between the two planets c and d orbiting at 0.7 au respectively2.1 au. Furthermore we assume that the formation of the four gas giants hasbeen finished (see Section 3). The region between the planets c and d includesthe habitable zone (HZ) where water on a rocky planet can be present in liquidform. This might give the planet the chance to develop and sustain life on itssurface.The work was done in several steps. First of all we studied the evolutionof orbits of massless bodies in the HZ which yielded a structure similar to the16Kirkwood Gaps’ found in the main belt of asteroids in our Solar System –all associated to mean motion resonances (MMRs) with the gas giants. Thesesimulations have been done with the primarily determined masses of the fourgas giants. In addition the integrations were repeated with masses of the fourgiants estimated for the statistical mean value of inclination. There is only asmall difference in the final distribution of the remaining massless bodies insidethe belt (comparison of Figure 2 and 4).As a second step we integrated Moon-sized bodies for 500 different initialconditions between the planets c and d with randomly chosen orbital elements,small initial eccentricities ( e ∼ . i ∼ ◦ ).We provide numerical integrations of the equations of motion in the Newto-nian framework up to the formation of terrestrial planets. We carefully studiedthe collisions between these bodies under the assumption of complete merg-ing. Being aware of this rough estimation we started SPH (Smooth ParticleHydrodynamics) computations which will provide more realistic results for thecollisions in a future project. We performed 50 different runs with differentinitial conditions for the 500 planetary embryos. In several plots we show therespective results – e.g after how many collisions such a terrestrial planet isformed, how fast the formation happens in different stages of the evolution,how many planets form, and in which distance to the host star most of theplanets are formed.We found that there is primarily one region around 1.15 au where manyterrestrial planets formed within approximately 1 Myr (most of the masslessbodies survived in this region in previous computations (see Figure 10). Alsoin two other regions around 1.35 au and 1.55 au planets formed. The latter onecorresponds to the so-called Hilda region (named after the Hilda asteroids inthe main belt of asteroids in our Solar System) in the 2:3 MMR with planet d .Those Hildas are less numerous and less massive.Concerning formation computations with larger masses for the gas giants wefound that it is far more difficult to form such planets, because relatively soon theobjects undergo perturbations by the giant planets leading to large eccentricitiesand subsequent close encounters and/or ejections. Our test computations witha system with larger masses of the primary bodies has shown that already aftersome thousand years only half of the original planetesimals survived. Althoughthis number is comparable to the computation with the nominal masses (seeFigure 8) there is a big difference insofar that for the latter test 95 % of theoriginal bodies left the central region because of close encounters respectivelycollisions with the gas giants and also the central star and not even one biggerbody was formed within this time (see Section 4.2).The special case of a MMR with one of the giant planets – the capture of asmall body in 1:1 resonance – a so called Trojan, was investigated separately.One can observe a few aggregated bodies up to the size of Mars in the Trojanregion around planet d . In the results of our integrations of fictitious masslessTrojans for 1 Myr we found stable regions around planet d up to an inclinationof the Trojans of about 80 ◦ (compare Figure 12).In our computations we also included the water content of the bodies and17ollowed the collisional process of accumulation of more and more mass up tothe final formation of a habitable terrestrial planet. In conclusion we can saythat terrestrial like planets may exist in the extrasolar planetary system ofHD 141399 and it might even harbor life. Acknowledgements This research is supported by the Austrian Science Fund (FWF) trough grantS11603-N16 (B.L. and R.D. and L.Z.). The computational results presentedhave been achieved in part using the Vienna Scientific Cluster (VSC). Thiswork has been supported by the National Natural Science Foundation of China(NSFC, Grants No. 11473016, and No. 11333002) Special thanks to T. I. Maindlfor critically reading the manuscript and his help with Gnuplot. 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J., et al. 2014, ApJ, 787, 97Zhou, L.-Y., Dvorak, R., & Sun, Y.-S. 2009, MNRAS, 398, 1217Zhou, L.-Y., Dvorak, R., & Sun, Y.-S. 2011, MNRAS, 410, 1849Zhou, L., Xu, Y.-B., Zhou, L.-Y., Dvorak, R., & Li, J., 2018, A&A, submitted19 Trojans Around Planet d Inspired by the presented results on the accumulation of bodies around planet d (shown in Figure 3) and the importance of possible Trojan planets in extrasolarplanetary systems (Dvorak et al., 2008; Schwarz et al., 2007; Dvorak et al., 2004;Laughlin & Chambers, 2002) we checked the stability of the Lagrange point L : for a grid of initial conditions 100 by 60 (semi-major axis versus initialinclination) we integrated the respective orbits of fictitious massless Trojans for1 Myr (Figure 14). There is a big stable region around a = 2.1 au visible between1.95 au < a < ◦ . The stability criterionwas the libration in the angle λ around the equilibrium point 60 ◦ ahead in theplanets orbit on one side (not shown here) and the largest eccentricity achievedduring the integration. The almost black region shows that the libration pointitself is very stable; more to the edge the eccentricity reaches values up to 0.3before they escape (yellow region). All the structures visible are comparable tothose found in other studies of Trojans in the Solar system (e.g., Dvorak et al.,2012). A.1 Dynamical Map Beside the maximum eccentricity map, we also construct the dynamical map ofthe Trojan region using the spectral number (SN) as the indicator of regularityof orbits. Concretely, the SN is defined to be the number of peaks over a giventhreshold in the power spectrum of a specific variable that can represent somebasic properties of the orbital motion. The method of SN, which could reflectthe long term stability within a relatively short integration time, was introducedby Michtchenko & Ferraz-Mello (1995) and has been successfully applied (e.g.,Michtchenko et al. , 2002; Zhou et al. , 2009, 2011).In this study, we calculate the power spectrum of the resonant angle ofeach test particle in the Trojan region, σ = λ − λ d where λ and λ d are themean longitudes of the test particle and the planet HD141399 d. After sometest runs, we set our integration time to be 4 . × yr to cover at least twocomplete periods of all the secular variation terms in the system.With the same initial conditions as the one we set for obtaining the maximumeccentricity ( e max ) map, we perform two runs of calculations and two dynamicalmaps are obtained, as shown in Figure 15. The SN in logarithm is indicated bycolours. The red and blue imply the most irregular (chaotic) and most regular(stable) motions, respectively, while the intermediates are shown in the colourbar. The empty area (white in colour) are those orbits that cannot survive ourintegration time. In the left panel, all planets in this system are nearly coplanarwith small inclination of 0 . ◦ , and in the right panel we show the dynamical mapfor the non-coplanar system, in which the planets in the HD141399 system frominside out are assumed to have the same inclinations and ascending nodes asJovian planets in our Solar System, i.e. Jupiter, Saturn, Uranus and Neptune,respectively. We will call the latter one the Jovian-analogue system hereinafter.20igure 14: The results of the stability of orbits in the regions around L ofHD 141399 d. The initial semi-major axes for fictitious Trojans (x-axes) versusthe initial inclination (y-axes) and the maximum eccentricity (z-axes - in color)as stability measure of 6000 orbits. The yellow region outside indicates escape,the black horizontal line is the most stable region.Figure 15: The dynamical map of Trojan region around the L point ofHD 141399 d. Note the different scales on the x-axes.The SN dynamical map reveals richer details than the maximum eccentricitymap. As shown in Figure 14, the e max does not change with the inclination buthas a monotonic increase from the resonance centre ( a = 2 . 09 au) outwards toboth boundaries along the semi-major axis. Contrarily, in Figure 15 evidentarc-structures can be seen, indicating that the orbital stability depends sensi-tively both on a and i. Introducing chaos into the Trojan region, these distinctstructures shrink and shred the stable region. Particularly, the left and rightbranches of the broad orange arcs corresponding to the most irregular motionare connected to each other at ( a , i ) ∼ (2 . 083 au , ◦ ), making the stableregion at high inclination disconnected from the one at low inclination.To our experience, the arc structures in the left panel of Figure 15 arise due tosome secondary resonances, where the frequency of σ is commensurable to somesecular frequencies of the system. In addition, two V-shaped thin stripes can21able 3: The proper frequencies of four planets.The f , g and s represent thefrequencies of mean motion, periastron precession and nodal precession, respec-tively. The values are all given in a unit of 2 π/ yr. The sign ± indicates theprecession direction, ‘+’ for prograde and ‘ − ’ for retrograde. The nodal preces-sion of Planet d is so slow that we cannot detect it.Planet b Planet c Planet d Planet e f +3 . . . × − +9 . × − g +3 . × − +3 . × − +6 . × − +2 . × − s − . × − − . × − . − . × − be clearly seen, which most probably is related to the nodal secular resonances.Since the system is nearly coplanar (planets’ inclinations are all 0 . ◦ ), thesestructures may be enhanced if the planets’ inclinations are higher.In the right panel, we show the dynamical map in the Jovian-analogue sys-tem, in which the inclinations of the planets are taken as the same as the Jovianplanets (higher than 0 . ◦ ). Apparently, the V-shaped stripes are enhanced asexpected. Compared to the left panel, we find that the irregularity has devel-oped considerably. Although the Jovian inclination is still quite small in fact,the stable region in high inclination i > ◦ almost disappears now. But in thelow-inclination region, the stability increases. As a result of the disappearance ofthose cyan-yellow arc structures corresponding to irregular motion, the blue sta-ble region extends both to lower inclination and to a wider range in semi-majoraxis. It is also worth to note that the resonance centre in this Jovian-analoguesystem shifts a little from the near-coplanar system, because here all the initialconditions are osculating orbital elements. A.2 Resonance Map Generally, a resonance happens when two (or multiple) frequencies in a dy-namical system match each other or they are commensurable by some simpleintegers. So, it is possible to find the resonances if the frequencies of the sys-tem are known. Following the similar method and procedure as in Zhou et al.(2009); Zhou et al. (2018), we numerically determine the frequencies of fourplanets in this system, which are listed in Table 3.From Table 3, we notice that the mean motions of Planet c and Planet d arevery close to a ratio of 5:1. In fact, in our numerical simulations, we find thatthis near resonance plays a significant role in the dynamics of Trojans aroundPlanet d, as we will see below. The frequency of this near resonance, definedas the frequency of the angle (5 λ d − λ c ) and denoted by f , is calculated as f = 8 . × − π/ yr.To find out the resonances that sculpt the features in the dynamical map,we calculate the proper frequencies of the test Trojans on the (a , i )-plane aswell. For each test Trojan, the frequencies of mean motion f , apsidal precession g and nodal precession s are calculated and finally the empirical expressions of22igure 16: The resonances in the Trojan region around the L point ofHD141399 d, superposed on the dynamical map, which is the same as the leftpanel in Figure 15 but with a different colour code to highlight the resonances.The symbols ν i ( ν i ), indicating the secular periastron (nodal) resonance be-tween the Trojan and the i th planet (i = 1 , , , , i are obtained. Withall the frequencies in hand, we can determine the resonances on the (a , i )-plane by carefully checking the relationships among these frequencies. Themost significant resonances are found and their locations are plotted on the(a , i )-plane.The most evident structure on the dynamical map is the family of grand‘arcs’ crossing the stable region. We find that these arcs are basically relatedto the secondary resonances between the libration of σ (critical angle of the 1:1MMR) and the near resonance angle (5 λ d − λ c ). As illustrated in Figure 16, theline 3 f σ − f = 0 just crosses the unstable arc, and some other lines labelled byC1 to C4 that stand for the following equations, emplace the similar structuresnearbyC1 :3 f σ − f + g = 0 , C2 :3 f σ − f − g = 0 , C3 :3 f σ − f + g − s = 0 , C4 :3 f σ − f − g + s = 0 . (1)The secular resonance that has the strongest dynamical effects is the ν resonance. As explained above, the ν resonance indicates the situation wherethe Trojan’s periastron processes in the same rate as Planet c’s periastron,˙ (cid:36) = ˙ (cid:36) . The ν drives the Trojan’s eccentricity to vary in a large range anddestablizes its orbit, thus carves the upper boundary of the stable region. TheKozai mechanism, occurring just above the ν curve in Figure 16, may alsocontribute to the instability in this neighbourhood.Other conspicuous resonances include the nodal secular resonances ν , ν and ν . Very clearly, the ν , ν curves just sit exactly inside the V-shapedstripes. The effects of the lowest order secular resonances, particularly the ν and ν are greatly enhanced when the inclinations of the planets increase tothe Jovian-analogue system (right panel in Figure 15), while those high-order23ecular resonances, e.g. the one related to 2 s + g − g3