Constraining protoplanetary disks with exoplanetary dynamics: Kepler-419 as an example
MMNRAS , 1–9 (2020) Preprint 15 September 2020 Compiled using MNRAS L A TEX style file v3.0
Constraining protoplanetary disks with exoplanetarydynamics: Kepler-419 as an example
Mohamad Ali-Dib , (cid:63) & Cristobal Petrovich , Institut de recherche sur les exoplan`etes, Universit´e de Montr´eal, 2900 boul. ´Edouard-Montpetit, Montr´eal, H3T 1J4, Canada Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA Centre for Planetary Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada Canadian Institute for Theoretical Astrophysics, 60 St. George St., Toronto, ON M5S 3H8, Canada
Accepted XXX. Received July 2020.
ABSTRACT
We investigate the origins of Kepler-419, a peculiar system hosting two nearly copla-nar and highly eccentric gas giants with apsidal orientations librating around anti-alignment, and use this system to place constraints on the properties of their birthprotoplanetary disk. We follow the proposal by Petrovich, Wu, & Ali-Dib (2019) thatthese planets have been placed on these orbits as a natural result of the precessionaleffects of a dissipating massive disk and extend it by using direct N-body simulationsand models for the evolution of the gas disks, including photo-evaporation. Based on aparameter space exploration, we find that in order to reproduce the system the initialdisk mass had to be at least 95 M
Jup and dissipate on a timescale of at least 10 yr.This mass is consistent with the upper end of the observed disk masses distribution,and the dissipation timescale is consistent with photoevaporation models. We studythe properties of such disks using simplified 1D thin disk models and show that theyare gravitationally stable, indicating that the two planets must have formed via coreaccretion and thus prone to disk migration. We hence finally investigate the sensitivityof this mechanism to the outer planet’s semi major axis, and find that the nearby 7:1,8:1, and 9:1 mean-motion resonances can completely quench this mechanism, whileeven higher order resonances can also significantly affect the system. Assuming thetwo planets avoid these high order resonances and/or close encounters, the dynamicsseems to be rather insensitive to planet c semi major axis, and thus orbital migrationdriven by the disk. Key words: planets and satellites: formation – planets and satellites: gaseous planets– planet-disc interactions
Kepler-419 is a two gas giants system with well character-ized architecture. The two planets b and c are respectively2.77 and 7.65 Jupiter masses, orbiting at 0.374 and 1.697AU, with eccentricities of 0.81 and 0.18, in apsidally anti-aligned orbits ( (cid:36) b - (cid:36) c ∼
180 deg) (Ford et al. 2012; Daw-son et al. 2014; Almenara et al. 2018). The origins of theseunique orbits merit an explanation. Petrovich, Wu, & Ali-Dib (2019) (PWA19) showed using secular theory (approx-imate orbit-averaged equations of motion) that the systemcould have originated in the inner gap of a slowly dissipatingmassive disk that forced the apses to anti-align through itsprecessional effects. The general dynamics and eccentricityevolution of exoplanets due to disk dispersal were initially (cid:63)
E-mail: [email protected] explored by Nagasawa, Lin, & Ida (2003). In this mecha-nism, the system needs to start with an angular momentumdeficit (AMD) for planet c , that is transferred in the processto planet b . PWA19 proposed that this initial AMD mightbe due to either planetˆa ˘A¸Sdisk interactions where the outerLindblad resonances can increase the planet’s eccentricity(Bitsch, et al. 2013), or from planetˆa ˘A¸Splanet scattering(Lega, Morbidelli, & Nesvorn´y 2013). It is unlikely howeverthat scattering alone can lead to an anti-aligned and nearlycoplanar system (Barnes & Greenberg 2006; Chatterjee etal. 2008), hence the need for the disk dispersal mechanism.We refer the reader to PWA19 for further background dis-cussions.An alternative explanation was proposed by Jackson,Dawson, & Zalesky (2019), who showed using N-body in-tegrations that for a small region of parameter space, thepresence of an undetected third planet could excite planet c (cid:13) a r X i v : . [ a s t r o - ph . E P ] S e p Ali-Dib & Petrovich b ’s eccentricity periodically without distabilizing the system.In this paper we build on and extend the work of PWA19.We first verify the accuracy of their results using N-bodyintegrations and do a parameter study over disk mass anddispersal timescale to constrain the values allowing the for-mation of Kepler-419 (sections 3 and 4). We then study theproperties of these disks and compare them against obser-vations to verify the realism of this idea (section 4.2). Wefinally study the effects of changing planet c ’s semi-majoraxis to understand the sensitivity of the system (and forma-tion mechanism) to this parameter (section 4.3). All simulations in the work were done using the
REBOUND
N-body integrator (Rein & Liu 2012; Rein & Tamayo 2015;Rein & Spiegel 2015), along with its
REBOUNDx add-ons pack-age (Tamayo, et al. 2020). Simulations were mostly done us-ing the symplectic Wisdom-Holman integrator
WHFAST , un-less otherwise is explicitly stated. This is because we areonly interested in the evolution of stable systems that donot undergo close encounters. The system is integrated witha timestep equal to 0.025 × the smallest orbital period in thesystem. All simulations are run for 6 × yr. The disk po-tential is implemented following Binney & Tremaine (2008)eq. 2.156, leading to the radial acceleration: F D ( r ) m = 4 Gr (cid:90) r da a √ r − a dda (cid:90) ∞ a dr (cid:48) r (cid:48) Σ ( r (cid:48) ) √ r (cid:48) − a (1)where we are setting to 0 the z component of the potential,and hence are not considering its effects on the inclinationsof the system. In fact we treat the system purely as 2 di-mensional, in contrast with the 3D treatment of PWA19.The disk force is implemented using the add_custom_force method of REBOUNDx .For all cases, we initiate the system with a 1.39 M (cid:12) star, and two planets b and c of 2.77 and 7.65 M J . Their re-spective initial eccentricities are set to 0.05 and 0.4 (follow-ing PWA19), while inclinations and Longitudes of ascendingnodes are set to 0. We again follow PWA19 by initially set-ting ω b − ω c = 60 ◦ For the system’s secular dynamics to evolve as suggestedby Petrovich, Wu, & Ali-Dib (2019), a main requirementis for the two planets to be located inside a common gap,with a massive disk beyond their orbits. One possible phys-ical mechanism for this scenario is a photoevaporating disk,where the planets are initially fully embedded in a disk withno cavity, then photoevaporation slowly carves up a gap inthe inner disk.Here we test this hypothesis numerically using oursetup, and assuming a disk surface density profile thatevolves as a function of time as:Σ( r, t ) = Σ ( r ) − ˙Σ w × ∆ t (2) where Σ ( r ) is defined through the following functional form:Σ( R ) = Σ (cid:18) rr in (cid:19) − γ (3)where Σ is one of the main parameters we vary, r in is setto 0.05 AU, and γ is fixed at 1.5. The outer edge of thedisk is always kept at 50 AU. The photoevaporation rate˙Σ w is defined following the functional fit to hydrodynamicsimulations of Owen, Clarke and Ercolano (2012) :˙Σ w ( y ) = (cid:20) a b exp ( b y ) R + c d exp ( d y ) R + e f exp ( f y ) R (cid:21) × exp (cid:20) − (cid:16) y (cid:17) (cid:21) (4)and ˙Σ w ( y <
0) = 0, where y = 0 .
95 ( R − R hole )1AU (cid:18) M ∗ (cid:12) (cid:19) − (5)and the dimensionless constants are a =-0.438226, b =-0.10658387, c =0.5699464 d =0.010732277, e =-0.131809597, f =-1.32285709.We use R hole =0.05 AU. We normalize this photoevapo-ration rate by 10 , to get a total mass loss of 6 × − M (cid:12) /yr .The system’s evolution is shown in Fig. 1. The diskstarts with a mass of ∼
10 M J . Photoevaporation then dis-perses the disk on a timescale of ∼ yr, opening a gapspanning the region between the two planets after ∼ × yr. At this point ∼ J of gas is remaining outside the or-bit of planet c . Before the gap opening, planet c eccentricityremains constant while planet b ’s oscillate with moderateamplitude, consistent with the Laplace-Lagrange solutionfor two planets orbiting a star, with the axi-symmetric diskhaving no effect as the two planets are still fully embedded.When the gap is opened at ∼ t=8 × yr however, angularmomentum exchange proceeds with planet b ’s eccentricityincreasing quickly to 0.8, and planet c ’s decreasing to 0.2,values consistent with current observations of Kepler-419.Finally, the gap opening is followed by the anti-alignment ofthe planets apses, the second major dynamical characteristicof Kepler-419.In reality however, this approach is not truly self-consistent since, as shown in section 4.2, planets this mas-sive will open deep gaps in the disk that are wide enoughto merge, quickly clearing up the disk regions interior tothe outer planet. In section 4 we consider a more realistic,although simpler, setup where the planets start embeddedin a common gap. The dynamics considered in the currentsection however might still be relevant for sub-Neptunes notcapable of fully carving gaps. For more sophisticated modelsof disk evolution with planet-carved gaps we refer the readerto the recent work by Toliou et al. (2019). In this section we investigate the disk mass and disper-sal timescales needed for a 2 planets system to evolve intoKepler-419 like configuration. We hence generate and evolvea large number of systems while varying two parameters: 1-
MNRAS000
MNRAS000 , 1–9 (2020) onstraining disks Time [yr] E cc e n t r i c i t y Planet bPlanet c Time [yr] c o s ( b − c ) -1 r [AU] Σ g [ g / c m ] Figure 1.
The evolution of a two planets system into Kepler-419 in a photoevaporating disk. Left: Time evolution of the gas surfacedensity profile of a disk governed by eq. 2. Center: The apses of the two planets are forced into anti-alignment by the disk’s precession,and then remain in this state due to its adiabatic dispersal. Right: Starting with an AMD, the eccentricity of planet b increases from anear zero to ∼ c decreases from the initial 0.4 to ∼ the disk mass through Σ in eq. 3, and 2- the disk’s dispersaltimescale τ d . Therefore, instead of using eq. 2 as above, wesimply the scheme by assuming the disk mass to decrease as M t disk exp ( − t/τ d ). This implies that the disk is dissipatingsimultaneously at all radii, rather than inside-out as sug-gested by photoevaporation models.This is however justifiable, since, within the simplifi-cations of the model (axisymmetric disk, no feedback fromthe planet onto the disk), the dynamics we are interestedin has a stronger dependence on the disk’s total mass, thanon its distribution. This is shown in PWA19’s eqs. 9 and11 where for a minimal mass solar nebula density profile:˙ (cid:36) p ∝ M disk / √ r out for a given r in .We set r in in eq. 3 to 1.5 × a c AU with Σ( r < r in ) = 0to emulate a gap, and the semi-major axis of planets b and c to respectively 0.374 and 1.697.We try disk masses of 1, 20, 40, 50, 75, 100, and 200 M Jup . The higher values are probably unrealistic as suchmassive disks can be prone to gravitational instabilities, butwe include them as limiting cases. We explore disk dispersaltimescales τ d of 10 , 10 , and 10 yr. In Fig. 2 we show the time evolution of the planets’ eccen-tricities and apses for three representative cases, with thefull results for all cases shown in appendix Figures 6 and 7.In this plot we identify multiples distinct regimes. For M d = 1 and 20 M Jup , the disk mass is small enough for thedisk-planet potential to be weak compared to that planet-planet interaction potential. Hence this case is equivalentto a three-body problem with a central star and two sig-nificantly less massive planets where the eccentricities aresecularly forced and the the apses circulate. This problemis well described by the classical Laplace-Lagrange seculartheory (Murray & Dermott 2000).On the other extreme end, for the massive disks with M d ≥ M Jup , we recover the results of PWA19 remarkably.In all of these cases the apsidal precession of the planets isdominated by the disk, rather than the planet-planet secularinteractions. The apses therefore always evolve into liberat-ing around anti-alignment, while the AMD is transferred from planet c to planet b , giving eccentricities consistentwith Kepler-419 as seen again in Fig. 2. Finally, for the in-termediate cases of M d ∼ − M Jup , we do see possibleevolution towards (sometimes transient) anti-aligned apses,however the amplitude of the oscillations are very large andthus we consider this case incompatible with Kepler-419.The same can be said for the eccentricities.We identify the minimal disk mass necessary for aKepler-419 like system to be around 75 M J , a value fewtimes higher than that used in section 3 where the systemdid evolve into K419. The two setups however are not fullyequivalent. In section 3 the planets start fully embeddedin the disk, and hence the precession rate of the planets isretrograde (e.g., Rafikov & Silsbee 2015) and dominated bythe inner planet ( | ˙ (cid:36) in | ∝ M disk n in ). In both cases, we have∆ ˙ (cid:36) ≡ ˙ (cid:36) out − ˙ (cid:36) in > (cid:36) = 0) in a similar fashion. However, the mass ofthe disk required to cross the resonance is lower in the em-bedded case roughly by a factor of n in /n out = P out /P in ∼ ∼ M J / M J = 7 .
5) between the fully em-bedded simulations and this case.It is interesting that Kepler-419 is recovered even fordisks dispersing on a 10 yr timescale, an order of magnitudelower than the value assumed by PWA19. This is reassuringsince in photoevaporation models, once an inner cavity hasbeen carved (which is our starting assumption), disks usu-ally proceed to disperse inside-out very quickly due to directstellar irradiation. This puts a constraint on the amount of“adiabacity” necessary for this mechanism to operate. No-tice that, as one would expect, for a fixed disk mass, longerdisk dispersal timescales lead to the same end results, butover longer time. On the other hand for fixed τ d but increas-ing the disk mass, the amplitude of the secular oscillationsaround the equilibrium values of the eccentricity and apsesdecreases. In this section we analyze the properties of our lowest massdisk compatible with Kepler-419, that is 75 M J ( ∼ (cid:12) ). First we compare this disk to observations. In a MNRAS , 1–9 (2020)
Ali-Dib & Petrovich
Figure 2.
Examples for typical behaviors observed in our Kepler-419 like setups, for three disk mass values, and τ d = 10 yr. Top: thedisk mass is too low to have any effects on the system evolution. Middle: Transition case where the disk mass forces the planets apses intoanti alignment but with noticeable oscillation amplitudes. Bottom: Massive disk leading to Kepler-419 like system with low oscillationamplitudes. recent ALMA survey of the Lupus complex, Ansdell et al.(2016) found dust masses ranging from ∼ . ⊕ ,while their gas masses where mostly below Jupiter mass,and almost all of them below the MMSN (10 − M (cid:12) , 10M J ). Pascucci et al. (2016) and Long et al. (2017) alsoused ALMA to measure the dust mass in the ChamaeleonI star-forming region, and found a mass range consistentwith Ansdell et al. (2016). Gas masses as measured inthe infrared (with HD lines) seem to be higher, whereMcClure et al. (2016) for example constrained the massesof GM Aur and DM Tau to respectively 2.5-20.4 × − and 1.0-4.7 × − M (cid:12) . Bergin, et al. (2013) on the otherhand found a lower mass limit of TW Hya around 0.05M (cid:12) (50 M J ). Our lowest mass disk hence lies towards theupper end of these distributions. Note that estimating disk gas mass from CO lines is problematic, as it depends onthe assumed CO/H ratio, that is affected by the complexphysical-chemistry of CO. Yu et al. (2017) for examplefound that CO observations underestimate gas mass byan order of magnitude. The values reported above arehence probably lower limits in most cases. Note that sincedust mass is measured using the sub-millimeter continuumflux that is insensitive to solids larger than ∼ cm, dustmasses are also lower limits that does not account for thetotal solids budget. Dust mass is hence a bad tracer for thetotal disk’s mass, even when the stellar metallicity is known.To fully understand the Kepler-419 birth protoplane-tary disk, we need to account for the planets mass as well.This increases our minimal mass disk to 95.07 M J (0.094 MNRAS000
Examples for typical behaviors observed in our Kepler-419 like setups, for three disk mass values, and τ d = 10 yr. Top: thedisk mass is too low to have any effects on the system evolution. Middle: Transition case where the disk mass forces the planets apses intoanti alignment but with noticeable oscillation amplitudes. Bottom: Massive disk leading to Kepler-419 like system with low oscillationamplitudes. recent ALMA survey of the Lupus complex, Ansdell et al.(2016) found dust masses ranging from ∼ . ⊕ ,while their gas masses where mostly below Jupiter mass,and almost all of them below the MMSN (10 − M (cid:12) , 10M J ). Pascucci et al. (2016) and Long et al. (2017) alsoused ALMA to measure the dust mass in the ChamaeleonI star-forming region, and found a mass range consistentwith Ansdell et al. (2016). Gas masses as measured inthe infrared (with HD lines) seem to be higher, whereMcClure et al. (2016) for example constrained the massesof GM Aur and DM Tau to respectively 2.5-20.4 × − and 1.0-4.7 × − M (cid:12) . Bergin, et al. (2013) on the otherhand found a lower mass limit of TW Hya around 0.05M (cid:12) (50 M J ). Our lowest mass disk hence lies towards theupper end of these distributions. Note that estimating disk gas mass from CO lines is problematic, as it depends onthe assumed CO/H ratio, that is affected by the complexphysical-chemistry of CO. Yu et al. (2017) for examplefound that CO observations underestimate gas mass byan order of magnitude. The values reported above arehence probably lower limits in most cases. Note that sincedust mass is measured using the sub-millimeter continuumflux that is insensitive to solids larger than ∼ cm, dustmasses are also lower limits that does not account for thetotal solids budget. Dust mass is hence a bad tracer for thetotal disk’s mass, even when the stellar metallicity is known.To fully understand the Kepler-419 birth protoplane-tary disk, we need to account for the planets mass as well.This increases our minimal mass disk to 95.07 M J (0.094 MNRAS000 , 1–9 (2020) onstraining disks M (cid:12) ), still around the uppermost limits of the observed pop-ulation. Assuming ISM dust/gas ratio of 0.01, the minimalmass dust disk is ∼
300 M ⊕ , a factor of few times abovethe upper limits of the measured values. On the M ∗ -M dust diagrams of Ansdell et al. (2017), our minimal mass disk fiton the curves found for the younger disks in Taurus, Lupus,and Cham I.Our minimal mass K419 disk can moreover be comparedto transitional disks with a large inner cavity. These disksare an intermediate step between photoevaporating proto-planetary disks and debris disks. A Spitzer survey of ∼ ∼
7% of there sample has a total mass ≥
100 M J , eventhough there is a much higher fraction with masses between10-100 M J . These masses were obtained through (dust) SEDfitting by RADMC-3D radiative transfer models, assuming ISMdust to gas ratio and account for dust growth using the pre-scription of Andrews et al. (2011). A
Herschel data analysisof Chamaeleon I by Ribas et al. (2016) found that SZ Chaand CS Cha have respectively M dust = 10 − . and 10 − . M (cid:12) , implying roughly 10 − . M (cid:12) total mass. This is a fac-tor of 3 times less than our minimal disk. Finally, a recentcomplete ALMA survey of the Lupus Star-forming Regionby van der Marel et al. (2018) found that 3 out of 11 diskshave dust masses consistent with our minimal mass disk.Therefore, transition disks massive enough to form K419are consistent with observations. Note that, strictly speak-ing, transition disks masses should be compared only to theminimum disk mass excluding the actual planets (75 M J ),while protoplanetary disks masses should be compared tothe minimal disk in addition to the planets (95 M J ). Thesignificant uncertainties on the disks masses, due to both ob-servational error bars and the modeling assumptions, renderthis distinguishment unnecessary however.Now we focus on the possible formation pathways forthe Kepler-419 planets. Since a very massive disk is neededto form the system, we investigate whether such a disk isgravitationally stable. We hence construct a standard radia-tive steady-state thin disk (Pringle 1981), as described inAli-Dib, Cumming & Lin (2020). This gives: T d, rad = T / s (cid:18) κµ Ω ˙ M παk B γ (cid:19) / = 373 K r − / α − / − ˙ M / − . × (cid:18) M (cid:63) M (cid:12) (cid:19) / (cid:18) κ cm g − (cid:19) / (6)where α − = α/ .
01 and ˙ M − . = ˙ M/ − . M (cid:12) yr − . Thedensity is then ρ d, rad = 1 . × − g cm − r − / α − / − ˙ M / − . × (cid:18) M (cid:63) M (cid:12) (cid:19) / (cid:18) κ cm g − (cid:19) − / . (7)We hence control three disk parameters: turbulence vis-cosity parameter α , the disk opacity κ , and its accretion rateonto the star ˙ M . We try α = 10 − and 10 − , in addition to κ = 1 and 0.01. For each case we change the ˙ M value to getthe total disk mass we need. We finally calculate the Toomreparameter: Q ≡ c s Ω πG Σ (8) where
Q < Q inFig. 3 (left hand panel) for our three disk models: highly tur-bulent with high opacity (“hot” disk), and weakly turbulentwith both high and low opacity.We find that, in all cases, Q (cid:29) r /ν . For r = 30 AU and ν ∼ cm /s , t mig ∼ × yr, which is comparable to the disk lifetime.Assuming these calculations stands for more sophisticateddisk and migration models with proper radiative transfer,this imply that the Kepler-419 planets probably formed viacore accretion.It is also of interest to check whether the Kepler-419planets are capable of carving gaps in such massive disks.This is an important self consistency check since an innergap in the disk embedding both planets is a prerequisite forthe secular dynamics we are considering, and this can be analternative to the photoevaporation carved gap discussed insection 3. Crida, Morbidelli & Masset (2006) showed thatfor a planet embedded in a disk to open a gap, the followingcondition needs to be satisfied: P gap ≡ HR H + 50 q R (cid:46) R H is the planet’s Hill radius, q = M p /M s , and R = r p Ω p /ν is the Reynolds number. In Fig. 3 (center) we plotthis quantity for our disk models, and M p = 2 .
77 M J . Wefind the condition to be satisfied throughout the disk forboth α = 10 − and 10 − . This imply that the Kepler-419planets can indeed open gaps in these massive disks.We finally calculate the gaps’ widths following Kana-gawa, et al. (2016):∆ gap R p = 0 . (cid:18) M p M ∗ (cid:19) / (cid:18) h p R p (cid:19) − / α − / (10)with the results shown in Fig 3 (right). Even for α = 10 − ,the gaps are found to be wide enough to merge, allowing thetwo planets to coexist inside one common gap. This implythat a photoevaporation driven inner disk gap is not the onlypossible formation channel for Kepler-419, as the planets aremassive enough to exist within a common gap anyway. P c If the Kepler-419 planets formed via core accretion as arguedabove, then it is likely they underwent disk migration atsome point in their history. This is especially true for massivedisks, since the timescale of type I migration is ∝ Σ g . Inthis section we hence explore the dependence of Kepler-419’sarchitecture on the semi-major axis / period of planet c , thatwe now free as a parameter. In Fig. 4 we show the end-resultsof simulations where we changed the period of planet c forvalues ranging between 6 and 13 × P b . The actual value forKepler-419 is P c = 9.6 P b . We use the same disk mass range(20 to 200 M J ), and a fiducial dispersal timescale of 10 yr. MNRAS , 1–9 (2020)
Ali-Dib & Petrovich -1 r [AU]10 Q ( T oo m r e ) α =10 − , =1 α =10 − , =1 α =10 − , =0 . -1 r [AU]10 -2 -1 P ga p α =10 − , =1 α =10 − , =1 α =10 − , =0 . -1 r [AU]10 -2 -1 ∆ g a p [ A U ] α =10 − , =1 α =10 − , =1 α =10 − , =0 . Figure 3.
Left: The Toomre stability parameter Q for our minimal mass Kepler-419 disk, for different values of turbulent viscosities α and disk opacity κ . In all cases the disk is fully stable against gravitational collapse in the entire planet formation region. Center: Thevalue of the gap opening criteria function P gap as defined in eq. 9, for a 2.77 M J planet. P gap is significantly lower than 1 throughoutthe disk, indicating that this planet is capable of carving a deep cavity anywhere. Right: The width of the gap opened by a 2.77 M J , ascalculated through eq. 10. For all parameters, a gap carved by a planet at 1.7 AU is deep enough to reach the inner Kepler-419 planetno matter its mass, allowing the two planets to be embedded inside a common gap, without the need for photoevaporation. Multiple distinct dynamical regimes can be identified inthis plot.Green circles in Fig. 4 represent systems that evolvedinto a Kepler-419 like architecture, with anti-aligned apses,and the observed eccentricities of both planets. These aremainly cases with P c > × P b and M d ≥
50 M J . We moreovernotice that the minimum disk mass needed for the system tofollow this evolution channel decreases with P c : while 75 M J are needed for the P c = 9.6 P b , only 40 M J are needed for P c = 13 P b . This can be readily understood within the frame-work of PWA19’s analytic model where planet c ’ precessiontimescale due to the disk is ∝ P c /M d .Blue circles on the other hand are systems that are ei-ther completely stable (with no apsidal liberation or AMDexchange between the two planets), or with minor dynamicalevolution that does not evolve the system into Kepler-419(mostly moderate amplitude secular oscillations). This is thecase for low disk masses that are unable to significantly af-fect the system’s dynamics, reducing the setup into a classic3-body problem with an inner test particle and external per-turber. More interestingly, this is also the case for some ofthe MMR cases, even for high disk masses. Prominent ex-amples are the 6:1, 7:1, and 8:1 resonance for almost all diskmasses.Pink circles are systems that undergo significant dy-namical evolution without transforming into K419. This isan umbrella for a plethora of different behaviors. For M d = 40 M J for example, while planet b’s eccentricity does in-crease significantly while that of planet c is decreasing, theamplitudes of the eccentricities secular oscillations is verylarge. Moreover, while the planets’ apses do anti-align, thisstate in these cases is transient. An intriguing case is thatof the 9:1 MMR where no AMD exchange takes place be-tween the planets at all, but their apses do liberate stablyaround anti-alignment. For periods slightly smaller or largerthan this exact commensuration however, the system evolvescleanly into Kepler-419. The 10:1 MMR is also a unique case,where the planets’ follow a dynamically very “noisy” Kepler-419 like evolution, before exiting the anti-aligned mode. Athird category included in the pink circles is for examplethe 7:1 and 8:1 MMR for M d = 200 M J , where the systemdoes evolve into K419-like state, but remains dynamicallyvery noisy, even after the disk’s dissipation. In this case, the disk contribution to the Hamiltonian is clearly of the samemagnitude as the MMR.The analytical Hamiltonian of PWA19 did not includeany resonant terms, and these results indicates that evenvery high order MMRs can play an important role in exo-planetary dynamics.To check whether these variations are indeed causedby high order MMRs, we plot in Fig. 5 the system shortterm evolution in the pseudo coordinate-momentum pair (e b cos[ (cid:36) b - (cid:36) c ], e b sin[ (cid:36) b - (cid:36) c ]) phase space for three cases:P c = 9.0, 9.6, and 10.0 P b . We notice the appearance of newhigh frequency modes for the 9:1 and 10:1 MMR, and theircomplete absence outside of these resonances when P c =9.6P b . The amplitude of these modes moreover seems to de-crease with the increasing resonance order, implying further-more that they are indeed MMRs related. Representativecases from this section were verified using the very high ac-curacy non-symplectic integrator IAS15 .In conclusion, for high enough disk masses, while thereare large parts of P c -M d parameter space where the systemcan evolve into Kepler-419, other end-results are also pos-sible. Two planets should not get trapped in certain higherorder MMRs, or be close enough to undergo close encoun-ters. Kepler-419 like systems might hence be more commonthan currently thought, and future observations can confirmor rule this out. The detailed eccentricity and apses evolu-tion of all of these cases in shown in the appendix in Figures8 and 9. In this paper we studied the origins of the highly eccentric,apsidally anti-aligned, two giant planets Kepler-419 systemusing N-body simulations where we introduced a dissipat-ing protoplanetary disk’s potential as an extra force. Wefirst show that the analytical results of PWA19 are recov-ered accurately, indicating that secular theory is adequateto describe the system’s evolution with no significant effectsfrom the ignored parts of the Hamiltonian. By exploring alarge range of disk masses, we show that the minimum diskmass to retrieve Kepler-419 is 75 Jupiter masses, increas-ing to 95 Jupiter masses if we are to include the planetsthemselves. These values are consistent with values found
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MNRAS000 , 1–9 (2020) onstraining disks Disk mass [M
Jup ] P c / P b K419-likeStable, not K419 Other
K-419
50 75
Figure 4.
The effects of relatively small changes in the period (thus semi major axis) of planet c . Blue circles are systems that becomestable, with little to no eccentricity or apsidal evolutions, and low amplitude secular oscillations. Green circles represent cases thatevolve into Kepler-419 like systems where the apses liberate around anti-alignment. Finally, Pink circles on the other hand are systemsthat does not fit either of the previous two categories. Notice the multiple breaks in the x-axis scale. Figure 5.
The evolution the system in the pseudo-coordinate-momentum phase space, over 8 ∼ × yr, starting at t=2 × yr forthe 20 M J disk case (so the disk’s potential is practically null). Note the different x and y scales used. in the infrared with Herschel, and lie towards the upperrange of masses measured with ALMA, but CO based ob-servations are probably underestimating the gas mass signif-icantly. Furthermore, Kepler-419 is recovered even for dis-sipation timescales as low as 10 yr, consistent with pho-toevaporation models. We then used a simple 1D α − diskmodel to study the stability of a protoplanetary disk thismassive, and find the Toomre Q parameter to be signifi-cantly larger than unity for all reasonable radii. This implythat the Kepler-419 planets, massive as they are, probablyformed via core accretion.Finally we ran simulations while varying the period ofplanet c , and find that higher order MMRs such as the8:1, 9:1, and 10:1 can either completely stabilize the system against the disk’s precessional effects, or sometimes renderits evolution dynamically noisy.Our results indicate that K419-like systems with highlyeccentric apsidally anti-aligned planets might not be uncom-mon. Considering the set of all confirmed planets in the ex-oplanets.eu catalogue (as of August 2020), ∼
7% of allof the planet hosting stars have a mass ≥ (cid:12) . Sincedisk and stellar masses are strongly correlated, this can beconsidered as a very crude estimate for the prevalence ofK419-capable disks in the population that surrounded theseplanet hosting stars. Moreover, in the 150 transition diskssurvey of van der Marel et al. (2016), a similar ∼ MNRAS , 1–9 (2020)
Ali-Dib & Petrovich masses, and is certainly just a theoretical upper limit on theprevalence of K419-like systems, as only a fraction of thesedisks will stochastically follow this formation scenario. Weemphasize that the stellar mass fraction is of the currentlyobserved planet-hosting stars, and not of the overall actualstellar (or planetary) population. These numbers, in addi-tion to the results of section 4.3 showing that K419 couldhave formed from a wide range of period ratios, indicate thatother apsidally anti-aligned systems could be hiding in thecurrent confirmed planets catalogue. By searching for sys-tems with two eccentric Jovian planets with P out /P in ≥ α = 10 − the dynamics we are con-sidering here are suppressed, moderate eccentricity growthis possible for α = 10 − . They did not explore the case ofa quasi laminar “dead zone” disk with α = 10 − (Gammie1996), a possibility consistent with recent ALMA observa-tions (Flaherty, et al. 2015, 2017). Moreover, their modeland also ours did not account for planet eccentricity excita-tion due to the disk’s torques, which likely operate for suchmassive planets ( m c (cid:39) M J ) through the 3:1 Lindblad res-onance (Papaloizou, Nelson & Masset 2001; Bitsch, et al.2013). These aspects merit further investigation.Overall, our paper shows how unique exoplanetary sys-tems architectures can be used to trace back the propertiesof the disk in which the planets formed. This is parallel tosome exoplanets observed in mean motion resonances (GJ876), for which disk-driven migration captures depend ondisk properties (scale height and levels of turbulence (Lee& Peale 2002; Rein 2012; Batygin & Adams 2017). Unlikethe case of MMR captures however, the ”capture” into thehigh-eccentricity secular equilibrium of Kepler-419 dependson the integrated evolution of the disk, placing a set of com-plementary constraints. Future models taking into accountthe 3D architecture of the system (Petrovich et al. 2020) willalso provide additional constrains. ACKNOWLEDGEMENTS
We thank J. R. Touma at the American University of Beirutfor interesting discussions that helped guide this project.We thank an anonymous referee for their insightful com-ments that helped improving this manuscript. M.A.-D. issupported through a Trottier postdoctoral fellowship. Thecomputations were performed on the Sunnyvale cluster atthe Canadian Institute for Theoretical Astrophysics (CITA).
DATA AVAILABILITY
The data underlying this article (Rebound simulations bi-nary archives) will be shared on reasonable request to thecorresponding author.
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Appendices
This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS , 1–9 (2020) Ali-Dib & Petrovich
Figure 6.
Appendix: Kepler-419 (P c =9.6 P b ) planets eccentricity evolution for a wide range of disk mass and dispersal timescales. Redis for planet b, and blue is for planet c. Figure 7.
Appendix: Kepler-419 (P c =9.6 P b ) planets apses evolution for a wide range of disk mass and dispersal timescales.MNRAS000
Appendix: Kepler-419 (P c =9.6 P b ) planets apses evolution for a wide range of disk mass and dispersal timescales.MNRAS000 , 1–9 (2020) onstraining disks Figure 8.
Appendix: Same as Fig. 6, but with different semi major axis values for planet c . Red is for planet b, and blue is for planet c.MNRAS , 1–9 (2020) Ali-Dib & Petrovich
Figure 9.
Appendix: Same as Fig. 7, but with different semi major axis values for planet c .MNRAS000