Planet formation and disk mass dependence in a pebble-driven scenario for low mass stars
MMNRAS , 1–13 (2019) Preprint 1 October 2020 Compiled using MNRAS L A TEX style file v3.0
Planet formation and disk mass dependence in apebble-driven scenario for low mass stars
Spandan Dash (cid:63) and Yamila Miguel , Leiden Observatory, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Measured disk masses seem to be too low to form the observed population of plane-tary systems. In this context, we develop a population synthesis code in the pebbleaccretion scenario, to analyse the disk mass dependence on planet formation aroundlow mass stars. We base our model on the analytical sequential model presented inOrmel et al. (2017) and analyse the populations resulting from varying initial disk massdistributions. Starting out with seeds the mass of Ceres near the ice-line formed bystreaming instability, we grow the planets using the Pebble Accretion process and mi-grate them inwards using Type-I migration. The next planets are formed sequentiallyafter the previous planet crosses the ice-line. We explore different initial distributionsof disk masses to show the dependence of this parameter with the final planetarypopulation. Our results show that compact close-in resonant systems can be prettycommon around M-dwarfs between 0.09-0.2 M (cid:12) only when the disks considered aremore massive than what is being observed by sub-mm disk surveys. The minimumdisk mass to form a Mars-like planet is found to be about × − M (cid:12) . Small variationin the disk mass distribution also manifest in the simulated planet distribution. Theparadox of disk masses might be caused by an underestimation of the disk masses inobservations, by a rapid depletion of mass in disks by planets growing within a millionyears or by deficiencies in our current planet formation picture. Key words: planets and satellites: formation – planets and satellites: general
One fascinating example of a multiple planet system isthe TRAPPIST-1 system which is a compact resonantlocked system of 7 exoplanets around a low mass star(Gillon et al. 2017). Apart from the number and natureof exoplanets discovered in this system, the fact that it isa M dwarf system is even more significant. M dwarfs arethe most common and longest lived of all low mass starsin the galaxy. The vast number of such stars makes themattractive options for exoplanet surveys.Since the discovery of TRAPPIST-1, exoplanet surveyshave uncovered some more of these planetary systems withEarth mass planets e.g. Proxima Cen b (Anglada-Escud´eet al. 2016) and the recently discovered Teegarden’s starsystem (Zechmeister et al. 2019) and the task of actuallyexplaining the formation of such systems is now gainingsteam. Important constraints for any planet formation (cid:63)
E-mail: [email protected] hypotheses around low mass stars are: (1) Explaining theformation and ubiquity of Earth mass exoplanets aroundM-dwarfs, as well as (2) The locked in resonant architectureof all these planetary systems.The properties of the protoplanetary disk (like its mass)and their correlation with the host star mass are importantparameters that influence the planet formation process(Alibert & Benz 2017). With sub-mm surveys it has becomepossible to probe the disk mid-plane and estimate the dustmasses in disks. However, total disk masses still remainuncertain with the scaling factor between dust mass anddisk mass (the gas to dust mass ratio) still not being ableto be determined precisely. Direct estimation of gas massesfrom molecular lines provide evidence that this scalingfactor may not be universal and can vary between a factorof 10 to 1000 (Anglada-Escud´e et al. 2016). Moreover,disk surveys around low mass stars have only been ableto characterize disks as young as 1 million years (Manaraet al. 2018). © a r X i v : . [ a s t r o - ph . E P ] S e p S. Dash et al. Mass of star in M M a ss o f p l a n e t s o r d i s k s i n M Planet mass-Disk mass comparison
Upper limit for most observed Gas disk massesSimulated Gas disk massesSimulated Dust disk masses Mass of star in M M a ss o f p l a n e t s o r d i s k s i n M Sum of planet massMaximum observed dust disk massesTRAPPIST-1 systemYZ Cet systemGJ3323 systemLHS 1140 systemProxima Cen bGJ 1214bGJ 1132 systemRoss 128bTeegarden star system
Figure 1.
Comparison between (top) observed gas disk massesand 1000 simulated dust and gas disk masses around low massM dwarf stars and (bottom) observed dust disk masses, planetmasses and the upper limit of masses in these planetary systems.The dust and gas disk masses are simulated using the dust massto star mass relation for
Lupus disks with a variable gas to dustratio (See Section 2.3). Planet parameters are same as in Figure5. Data for observed dust and disks is from Ansdell et al. (2016).
From Figure 1, a majority of observed dust disk masses(for disk samples greater than a million years old) aroundlow mass M dwarfs are either comparable to the sum ofall planets in a system or not massive enough. It is alsoseen that while overall simulated disk masses (comparableto gas disk masses) around low mass M-dwarfs are largerthan the current observed exoplanet masses, the simulateddust disk masses (from the dust mass to star mass relationfor
Lupus disks , see Section 2.3) are simply not enough toform the observed planets. Either way, there seems to be aproblem with explaining the presence of currently observedexoplanets around low mass M dwarfs with currently ob-served disk masses. Manara et al. (2018) had also come tothe same conclusion but for a wider range of stellar masses.This brings home the point of determining a minimumdisk mass required to allow planetary systems similar tocurrently observed exoplanetary systems around low massstars to form as well as to refine our planet formation models. One thing that could explain this underestimation ofdisk masses is dust growth or start of planet formationwithin disks within 1 million years (Manara et al. 2018).While Testi et al. (2014) had shown that the upper limit forplanet formation to start is in the 3-5 million years range,Miotello et al. (2014) and Harsono et al. (2018) showedthat dust growth to at least mm-cm levels has alreadyoccurred in the envelopes and disk like structures and inthe inner few AU of several young YSOs (Young StellarObjects). This shows that the start of planet formation iswell underway even in the embedded stages of such YSOs.The presence of mm-cm sized dust within a million yearsprovides impetus to the
Pebble Accretion mechanism whichcan quickly grow a planetesimal to a planet sized objectwithin a million years with a mm-cm pebble reservoiralready available. Keeping that in view,
Pebble Accretion has started to be increasingly used for planet formationmodels for growing low mass planets or giant planet coreswithin a million years (Coleman et al. 2019; Br¨ugger et al.2020; Ormel et al. 2017; Liu et al. 2020). In addition,pebble flux regulated planetesimal formation has also beenproposed by Lenz et al. (2019) which has been extendedby Voelkel et al. (2020) to show that a sub-million year100 km planetesimal formation by such a mechanism canform giant planets in the inner disk. A pebble-planetesimalhybrid accretion mechanism has also been used by Alibertet al. (2018) to explain formation of Jupiter. All of theseinfluence our choice to use pebble accretion as the planetformation model for our population synthesis study.Several population synthesis models have now tried tounderstand the planet formation process in disks aroundlow mass stars by trying to match the observed exoplanetdistribution with a simulated distribution of planets. Someuse the
Planetesimal Accretion model (Miguel et al. 2020;Coleman et al. 2019; Alibert & Benz 2017) and others usethe more efficient
Pebble Accretion model (Coleman et al.2019; Liu et al. 2020; Br¨ugger et al. 2020). Even the efficient
Pebble Accretion model would require dust masses largerthan what is being currently observed by at least an orderof magnitude since only some fraction of the total pebbleswill be used for planet growth (Manara et al. 2018). WhileColeman et al. (2019) showed that smaller but massive disks(10 AU with mass of 2.7-8% of star mass) around low massstars can explain observed planets around low mass stars,Miguel et al. (2020) recently concluded that a populationsynthesis model based on
Planetesimal Accretion alone isinsufficient to explain those exoplanets if extended diskswith a wide range of masses are considered (100 AU withmasses between − to − M (cid:12) ). Br¨ugger et al. (2020)showed that comparing pebble and km sized planetesimalbased accretion mechanisms with similar initial conditionsproduce different results with the pebble model favouringformation of Super-Earths while the planetesimal modelfavouring Gas Giants. Most of these models use the hoststar mass as a proxy and consider very massive disks fromthe outset. A comparative analysis of planet formationbased on the initial disk mass distribution along withcomparison to already observed exoplanets hasn’t beendone yet. Figure 1 and Manara et al. (2018)’s conclusionmotivates our study where we now use the disk mass distri- MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars bution as a proxy for the simulated exoplanet distributionand compare them with the observed exoplanet distribution.We use the model proposed by Ormel et al. (2017)which gave an outline for a sequential method of formingEarth mass planets in disks around low mass stars byutilizing Pebble Accretion . The seeds for planet growth areassumed to be generated due to
Streaming Instabilities (Youdin & Goodman 2005) close to the water ice line.These seeds subsequently grow by pebble accretion to Earthmasses while migrating inwards by
Type-I migration intotheir current orbits. This sequential formation mechanismalso allows these exoplanets to be trapped in resonanceseasily. Subsequently, a single-stage disk dispersal mechanismleaves us with a stable close-in system of exoplanets.While Ormel et al. (2017) was mostly concerned withexplaining the TRAPPIST-1 system specifically by simpleanalytical methods, we now expand it to construct apopulation synthesis model in order to form and explain theentire population of exoplanets around low mass stars. Wedescribe the construction of this model in Section 2, look atthe simulated results in Section 3 and then analyze theseresults to comment on the problem with disk masses andcompare them with results from other population synthesismodels in Section 4.
The gas surface density profile of the disk ( Σ g ) is assumedto be a simple power law disk profile with: Σ g = ( − p ) M disk π a out (cid:18) aa out (cid:19) − p . (1)Here, M disk is the total mass of the disk, a out is the diskouter radius and a is the semi-major axis. While this profilecan be used for any < p < , we adopt a value of p = .The disk temperature profile at a semi-major axis a (in AU) is: T ( a ) = K M (cid:63) . M (cid:12) (cid:18) h . (cid:19) (cid:18) a . (cid:19) − . (2)Here, M (cid:63) is the mass of the star and h is the disk aspectratio (a measure of the disk vertical profile in comparisonto its distance from the star) which is assumed to be . ×( a / AU ) / (Ida & Lin 2004). Equation 2 can be modified,by using the equation for h and the ice-point temperatureto be at 180 K, to give (in AU) the distance to the ice-lineat: a ice − line = . (cid:18) M (cid:63) . M (cid:12) (cid:19) . (3)The magnetospheric cavity radius is taken to be thepoint at which the inner disk is truncated. Using a value ofstellar radius as 0.5 R (cid:12) and stellar magnetic field of 180 G, the inner disk radius is (in AU) (Ormel et al. 2017): a inneredge , disk = . (cid:18) . M ⊕ M (cid:63) (cid:19) / (cid:18) − M (cid:12) y r − (cid:219) M g (cid:19) / . (4)Here, (cid:219) M g is the gas accretion rate from the viscous accretiondisk into the star. Since our model is a population synthesis extension to theanalytical model proposed by Ormel et al. (2017), we re-fer to that paper for the details of the parameters in themodel. However, we explain these briefly here for the sakeof completeness. The planet growth equation using pebbleaccretion can be written as: (cid:219) M pl = (cid:15) F p / g (cid:219) M g . (5)Here, (cid:219) M pl is the differential mass growth with time. (cid:15) isthe efficiency of mass growth from a radially drifting pebblefront. F p / g is the pebble to gas mass flux generated due tothe same radially drifting pebble front. This front is obtaineddue to growth from micron sized particles by collisions. (cid:219) M g is assumed to be − M (cid:12) yr − (Ormel et al. 2017). F p / g isdefined from the disk properties as: F p / g = M disk Z / (cid:219) M g a out κ / (cid:18) GM (cid:63) t (cid:19) / . (6)Here, Z is the metallicity of the disk and is assumed to be0.02 and κ is the number of e-foldings (growth by a factor ofe at each step) needed to reach pebble size and is assumedto be 10 (Ormel et al. 2017).The seeds for pebble accretion are assumed to be formedfrom streaming instability near the water ice-line where themid-plane pebble to gas density ratio can exceed 1 underspecific parameters for viscosity ( α ≥ − ) of the disk(Schoonenberg & Ormel 2017; Ormel et al. 2017). We alsoassume an ice-line width of 0.02 AU in which the seeds canbe produced which is motivated by the width over whichthe pebble to gas density ratio remains above 1 as shownby Ormel et al. (2017).As soon as seeds of about − M ⊕ (equivalent to mass ofCeres) are available, pebble accretion is assumed to start(Bitsch et al. 2015). The first phase of mass growth is withinthe ice-line width we assumed above and most pebblesaccreted onto the seed are icy pebbles. With the value of α we have ( α ≥ − ), the mass growth mechanism in thiscase is planar (2D) (Ormel et al. 2017)) and Equation 5becomes: (cid:219) M pl , D = (cid:15) D F p / g (cid:219) M g . (7)Here, the value of (cid:15) D = . × q / pl . q pl is defined as M pl / M (cid:63) where M pl is the mass of the planet. The growing core is then subject to Type I migration, firsttowards the ice-line inner edge and then further till it reaches
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S. Dash et al. the inner edge of the disk. The equation for Type-I migrationis (Lambrechts & Johansen 2014): dadt = − . q pl c mig Σ g a M (cid:63) h v K . (8)Here, v K is the Keplerian velocity at a particular a , c mig isthe reduction factor used as a measure of uncertainties andnon-linear effects affecting migration. In this paper, we usetwo values of c mig as 1 and 0.1 and we call them standardmigration and reduced rate migration models respectivelyin a similar vein to how it was done in Miguel et al. (2020)and Ida & Lin (2004).The planetary core migrates until it reaches the inneredge of the ice-line and then migrates further inwards intoa drier part of the disk. Now the pebbles being accretedare smaller silicate pebbles and the mass growth is moreinefficient. Equation 5 now becomes: (cid:219) M pl , D = (cid:15) D F s / g (cid:219) M g . (9)Here, (cid:15) D = . × ( q pl / − ) and F s / g is the silicate to gasmass flux. Schoonenberg & Ormel (2017) assume that atmost 50% by mass of the pebble beyond the ice-line is com-posed of ice. Interior of the ice-line, this ice evaporates andhence we assume a reduced metallicity of 0.5 Z in Equation6 to account for the missing mass from ice and calculate F s / g . The planet migration remains Type I till it ultimatelyreaches the inner edge of the disk. However, the mass growthstops at the
Isolation Mass at which point the planet startsforming a gap in the disk and is hence cut off from its pebblesupply. This mass is determined as (Ataiee et al. 2018): M iso , pl ≈ h √ . α + . × (cid:20) + . (cid:18) √ α h (cid:114) St + (cid:19) . (cid:21) × M (cid:63) . (10)Since streaming instabilities are only possible within theice-line for α > − , we assume the lowest value of − forthis calculation. St is the Stokes Parameter and for efficientpebble accretion has the lowest value of 0.05 (Ataiee et al.2018; Bitsch et al. 2018) which we assume here. The pebbleisolation mass ensures that not only do the planets stopaccreting pebbles but also that the inward drift of pebblesis also stopped (Bitsch et al. 2018).Planet mass growth is also stopped when the radialpebble flux stops. This happens when the outward movingpebble front reaches the outer disk edge. The time whenthis happens is (Ormel et al. 2017): t end = κ Z (cid:18) a out GM (cid:63) (cid:19) . . (11) After the planet reaches an isolation mass, the mass accre-tion moves into a period of slow gas envelope accretion and compression which is modeled as (Bitsch et al. 2015) (interms of M ⊕ masses/million years): dM g dt = . f − (cid:18) κ env cm / g (cid:19) − (cid:18) ρ c . g / cm (cid:19) − / (cid:18) M c M ⊕ (cid:19) / (cid:18) M g . M ⊕ (cid:19) − (cid:18) T K (cid:19) − . . (12)This can be further simplified a bit by assuming κ env and ρ c to be 1 cm /g and 5.5g/cm respectively, f is 0.2, M c is theisolation mass and initial M g is 0.1 times the isolation mass(Bitsch et al. 2015). Equation 12 then becomes (in terms of M ⊕ masses/year): dM g dt = . × − (cid:18) M iso , pl M ⊕ (cid:19) / (cid:18) M gas (cid:19) − (cid:18) M (cid:63) . M (cid:12) (cid:19) − / (cid:18) a AU (cid:19) / . (13)This slow accretion continues till the planet and envelopemass together reaches a critical mass of about 2 M iso , pl (Bitsch et al. 2015) after which there is rapid gas accretiononto the envelope. However, we find that none of our planetsmanage to reach this limit. The amount of gas accreted isalso very meagre and can easily be stripped off when the diskeventually dissipates. This means that we can safely neglectthis contribution for our simulations. This is similar to whatwas found in Coleman et al. (2019) who used a more preciseself-similar gas accretion model. We assume that when the first planet reaches the inneredge of the disk, it stops there and continues growing atthe same efficiency it had before. In the absence of anyconcrete hypothesis of events happening in such a case,this simplifying assumption is made just for the sake ofcalculation and can be easily modified by introducinga fudge factor for the efficiency of either pebble or gasaccretion.We assume that the second planet mass growth is triggeredas soon as the first planet crosses the ice-line. All steps asoutlined in the previous subsections are followed again untilthe planet reaches a minimum separation distance from thefirst planet and stops migrating or the gas disk is dispersedwhich also stops migration.We assume that two planet interactions here are mostlyindependent of an external influence. This is similar to theassumption made in Sasaki et al. (2010) (for a Jovian moonresonant system) and Miguel et al. (2020) (for exoplanetsaround low mass stars formed with core accretion mecha-nism). In such a case, orbits of two planets will be stableif they are separated by K × r H , m where K is a criticalparameter for planet spacing, r H , m is the mutual Hill radiusand is equal to (( a + a )/ ) × (( m + m )/ M (cid:63) ) / . Here, a i and m i correspond to semi major axis and mass of the ithplanet. For critical parameter K , we base our assumption onPu & Wu (2015) who found that compact multi-planetarysystems can be sculpted into systems with wide spacing.They found that systems where this spacing among planets MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars is characterised by K ≤ . can be weeded out in thenascent disk and hence orbits with spacing greater thanthis amount of mutual Hill Radii should be stable on abillion year timescale. On the other hand, they also foundthat Kepler planets seem to be tightly clustered around avalue of K = . Motivated by these threshold values, weassume a minimum spacing between each pair of planets atrandom in between these two limits for our simulations.An additional complication to keep in mind here isthe possibility of a planet upstream reaching pebble Isola-tion mass before the planet downstream which would thencut of pebble supply for the planet growing downstream.However, from our simulations we find that the secondplanet always has to migrate more slowly (See Section2.2.6) and has to reach a higher
Isolation mass than thefirst planet as a result. This means that our model avoidsthis complication and the interference on the growth ofthe planet downstream by the planet upstream is reducedeven though our model is a sequential formation model.Subsequently, more planets are formed and migrate to theirdesignated orbits before the disk dissipates allowing theformation of multiple planetary systems.
Observations indicate that the gas disk around starsdissipates on a scale of 1-10 million years (Hartmann et al.1998; Bitsch et al. 2015). However, the photoevaporationrate after 3 million years can substantially deplete thedisk of gas (Alexander et al. 2014). Pecaut & Mamajek(2016) found that disks around K/M stars can last longerthan this lifetime. However, we find that the pebble fluxstopping time from Equation 11 takes less than 1 millionyears for the stars we consider in our simulations. Thismeans that most of the disk lifetime would then only befor the orbital evolution of already formed planets. Hence,similar to Bitsch et al. (2015), we take our disk lifetimesin the range of 1 to 3 million years. We note that taking alonger timescale would slightly affect the orbital positionsof planets not already stuck in mean motion resonances.Gaseous disk evolution will affect how our planets grow andmigrate. To model this, we use a simple exponential decayfunction (Ida & Lin 2004): Σ g = Σ g , e − t τ . (14) Σ g , is the gas surface density profile at the beginning ofthe calculation (Equation 1), τ is the disk dissipation life-time which can vary between 1 and 3 million years and t is the current disk age. The exponential contribution termis named as the dissipation factor . As the gas continuouslydepletes from the disk, the gas accretion rate into the staralso slows down as: (cid:219) M g = (cid:219) M g , e − t τ . (15)From Equations 6 and 5, it is easy to see that the product F p / g (cid:219) M g is independent of the dissipation factor. This meansthat the efficiency of accretion of planets remains indepen-dent of the disk age. Equations 8 and 14 makes it clear thatthe dissipation factor would result in slowing of the planet Mass of Disks in M F r e q u e n c y Number distribution of Disks
Figure 2.
Comparison of all simulated disk mass distributions. orbital migration rate as the disk grows older. Change inother parameters of the model is mentioned in Section 2.3.
The model we use is made from scratch in Python 2.7 andis available online . We consider star masses from 0.09 M (cid:12) to 0.2 M (cid:12) . We select a mass at random for each iterationfrom a uniform list of masses in the above mentioned range.Given the uncertainty in determining disk mass around lowmass stars, we assume two possibilities. The first is an arbi-trary disk mass to star mass relation of 0.03-0.05 (averagevalued of 0.04 used by Ormel et al. (2017) and hence thesedisks are herewith called as Ormel disks ) and the other is amore precise disk dust mass to star mass relation obtainedfrom observations in the Lupus and Taurus molecular clouds(Ansdell et al. 2016). There is evidence that the gas to dustmass value in disks is not 100 (Bohlin et al. 1978) and in-stead varies between 10 to 1000 (Ansdell et al. 2016). Hence,we use a value of gas to dust mass ratio ( g / d ) from a uni-form list between 10-1000 to scale this relation to a diskmass-star mass relation. The modified relation from Ansdellet al. (2016) becomes: lo g M disk = . lo g M (cid:63) + . + lo g ( g / d ) + lo g ( × − ) . (16)These disks will be henceforth called as Lupus disks for con-venience sake. To incorporate more variety in our disk sam-ples, we also consider disk masses between 3-8% and 3-10%star mass which are more massive than
Ormel disks . Weshow all the disk mass distributions in Figure 2. For eachiteration, we choose one disk at random from the distribu-tions we have constructed based on our runs (Run 1 through4, See Table 1).Apart from the above discussed main independent param-eters, there are also several other dependant parameterswhich we list below: • Disk dispersal lifetime: We pick a random age from auniform list of ages from 1 to 3 million years (Section 2.2.6). • Water ice-line: We use Equation 3 for this. https://github.com/dashspandan/planetMNRAS000
Ormel disks . Weshow all the disk mass distributions in Figure 2. For eachiteration, we choose one disk at random from the distribu-tions we have constructed based on our runs (Run 1 through4, See Table 1).Apart from the above discussed main independent param-eters, there are also several other dependant parameterswhich we list below: • Disk dispersal lifetime: We pick a random age from auniform list of ages from 1 to 3 million years (Section 2.2.6). • Water ice-line: We use Equation 3 for this. https://github.com/dashspandan/planetMNRAS000 , 1–13 (2019) S. Dash et al. • Disk inner radius: We use Equation 4 for this. Thisvalue will increase with time as it is inversely proportionalto (cid:219) M g (Equation 15). We assume that any inwards migrat-ing planet will be stopped when it reaches this disk edge(similar to the assumption in Coleman & Nelson (2016) fornon-gap forming planets). • Disk outer radius ( a g ): This value is assumed to be 200AU (Ormel et al. 2017), which falls in the range of observedouter radii of gas disks (Ansdell et al. 2018). • Disk metallicity ( Z ): This is taken to be 0.02 (super-solar) to ensure that sufficient rocky material is available toform multiple planets as well as to make streaming instabil-ities easy by increased clumping of dust particles (Johansenet al. 2009). • Minimum spacing between planets: The spacing be-tween each planet pair is chosen at random from a uniformlist of separations between 8 and 12 r H , m (See Section 2.2.5).For our simulations, simplifying this in terms of the Hill ra-dius of just the preceding planet would make computationeasier. We assume that m ∼ m and a ∼ a which is approx-imately valid for close in compact systems like ours whereconsecutive planets are found to be quite similar in finalmasses. • Maximum number of planets: We limit the maximumnumber of planets that can be formed in a planetary systemto 20 due to numerical constraints and keeping in view thecomplexities of large N. Since all planets are formed insidethe ice-line width and migrate inwards, the disk inner edgeand the outer edge of the ice-line width form the innermostand outermost extent of the semi-major axis distributionfor planets. We observe that all disks in our reduced ratemigration model and in the standard migration model formless planets in a system than the maximum limit.
As mentioned in Section 2.2.2, we consider two different val-ues of c mig for our analysis i.e. 1 and 0.1 (standard modeland reduced migration model respectively). c mig = (standard model) To determine the effect of disk masses on a planetarysystem’s evolution, we run our simulation for a 0.1 M (cid:12) star and three different disk masses, evolving the system inall cases for 1 million years. Three different cases of diskmasses are used and the results are shown in Figure 3: (top) Lupus disk with g / d = , (middle) Disk mass is 3% starmass and (bottom) disk mass is 5% star mass.In Figure 3 we see that the effect of increasing diskmasses mostly affects the number of planets formed persystem (top panel has 1, middle panel has 4 and bottompanel has 7). This is because migration rate is faster aroundmore massive disks (from Equations 1 and 8) and henceplanets can cross the ice-line fast enough to trigger theformation of subsequent planets in our model. In general,Figure 3 shows that migration in planets starts beingsignificant at around the Mars mass limit. Hence, increasing disk masses also means that most planets in the systemcan cross this threshold in order to migrate away from theice-line.In the middle panel of Figure 3, the first two planetsare separated by the minimum spacing required for stableorbits for both while the other two stopped migratingbefore reaching that limit due to disk dispersal. The secondplanet is more massive than the first planet and also moremassive than the third and fourth planets as the pebblesupply ends (Equation 11) after it reaches its Isolationmass limit but before the third planet reaches this limit.Regarding migration, since each succeeding planet startsevolving a bit later than its predecessor, its migration isslower (due to gas depletion which slightly reduces the gasmass in the disk; from Equations 8 and 14) while the massaccretion rate is constant (See Section 2.2.6). This allowsit to accrete more mass at the same value of semi-major axis.In the bottom panel of Figure 3, the effect of a suc-ceeding planet accreting more mass at similar semi-majoraxis is more pronounced. Since migration rate is the fastest,more planets reach their respective
Isolation masses (withthe first planet quickly reaching a lower value of disk innerradius (See Equation 4)). As succeeding planets accretemore mass at similar semi-major axis, each succeedingplanet manages to reach its
Isolation mass at a larger valueof the semi-major axis. Hence in this system, the first 5planets form a convoy with increasing order of mass as wemove outward. The 6th and 7th planets cannot accreteenough mass as the pebble supply stops before they canreach the
Isolation mass limit (Equation 11). The 7thplanet stops migrating before it can migrate to an orbitwith the lowest limit for spacing from its preceding planet.Every other planet before it manages to reach that limitbefore disk dispersal. c mig = . (reduced rate migration model) We now reduce the migration rate by a factor of 10 for thesame set of three simulations done above to see the effect ofa longer migration timescale for same disk masses.From Figure 4 we see that the effect of delaying themigration rate by a factor of 10 leads to a reduced numberof planets for all simulation except
Lupus disks (top has1, middle has 2 and bottom has 3) for each system forsimilar disk masses. This is expected for heavier disks asslower migration means that a lesser number of planets cancross the ice-line and trigger subsequent planet formation.For
Lupus disks , the planet grows to similar mass as theaccretion rate remains unchanged due to slower migrationbut the planets can’t grow to Mars mass limit to cross theice-line in both cases. The slower migration means thatmost planets manage to reach their
Isolation masses insidethe ice-line width at larger values of semi-major axis. Thismeans that the most massive planets formed in each systemis more massive than its counterpart for faster migration.
MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 1) Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 1) Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 1) Figure 3.
The effect of increasing disk masses on planetary sys-tem evolution around a 0.1 M (cid:12) star. Disk masses are: (top) Lupus disk with g / d = , (middle) 3% star mass and (bottom) 5%star mass. c mig = (standard model) To compare our model results with a distribution ofobserved exoplanets, we utilize a population synthesisapproach and run 4 simulations (labeled in Table 1 with c mig = ) to construct 1000 planetary systems each usingthe 4 different disk mass distributions we formulated(Section 2.3). The simulations are shown in Figure 5 alongwith planets from different observed systems within thestellar mass range assumed. The different types of planetarysystems (according to number of planets) are shown inFigure 6. Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 0.1) Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 0.1) Semi-major axis in AU M a ss o f p l a n e t i n M Evolution of planets ( c mig = 0.1) Figure 4.
The effect of increasing disk masses on planetary sys-tem evolution around a 0.1 M (cid:12) star. Disk masses are: (top) Lupus disk with g / d = , (middle) 3% star mass and (bottom) 5%star mass. The rate of migration has been reduced by a factor of10 for these simulations. There were 5846 planets simulated in total in Run 1(
Ormel disks or disk mass is 3-5% star mass) with a meanplanet mass of 0.448 M ⊕ . All planetary systems formed aremulti-planetary with the majority of the systems (66.7%)having < > MNRAS000
Ormel disks or disk mass is 3-5% star mass) with a meanplanet mass of 0.448 M ⊕ . All planetary systems formed aremulti-planetary with the majority of the systems (66.7%)having < > MNRAS000 , 1–13 (2019)
S. Dash et al. Semi-major axis in AU10 M a ss o f p l a n e t i n M TRAPPIST-1 systemYZ Cet systemGJ3323 systemLHS 1140 systemProxima Cen bGJ 1214bGJ 1132 systemRoss 128bTeegarden star system
Run 1 (Ormel) with c mig = 1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 2 (3-8% star mass) with c mig = 1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 3 (3-10% star mass) with c mig = 1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 4 (Lupus) with c mig = 1 Figure 5.
Synthetic population of planets for (top left)
Ormel
Disks (3-5% star mass), (top right) a bit more heavier disks (3-8%star mass), (bottom left) a lot more massive disks (3-10% star mass) and (bottom right)
Lupus disks. The parameters of planets ineach observed system are taken from: TRAPPIST-1 (Grimm et al. 2018), YZ Cet system (Astudillo-Defru et al. 2017b), GJ 3323 system(Astudillo-Defru et al. 2017a), LHS 1140 system (Ment et al. 2018), Proxima Cen b (Anglada-Escud´e et al. 2016), GJ 1214b (Charbonneauet al. 2009), GJ 1132 system (Bonfils et al. 2018b), Ross 128b (Bonfils et al. 2018a) and Teegraden’s star system (Zechmeister et al.2019). a result of planets having preferred positions as the inneredge of the disk doesn’t really vary a lot for the stellarmasses we consider for our simulations. The size of thiscluster indicates that a majority of the simulated planetsare able to migrate close to the disk edge and becomeclose-in systems. The simulated mass and semi-major axisrange explains the formation of 7 planets which includesTRAPPIST-1 b, c, d, e and h and the first two planets inthe YZ Cet system. The rest of the planets remain out ofbounds and could indicate requiring more massive disks or a different formation scenario compared to this model.For Run 2 (disk mass is 3-8% star mass), there were8321 planets simulated with a mean mass of 0.553 M ⊕ .The increase in mean mass is expected as the average diskmass in this run is larger. All planetary systems formed aremulti-planetary with the majority of the systems (42.2%)having 7-12 planets and a close percentage (39.9%) having < MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars Disk mass Run Number of planets Mean planet mass
Ormel ) Run 1 ( c mig = ) 5846 0.448 M ⊕ c mig = ) 8321 0.5553 M ⊕ c mig = ) 10135 0.620 M ⊕ Lupus
Run 4 ( c mig = ) 1001 0.011 M ⊕ Ormel ) Run 1 ( c mig = . ) 2742 0.897 M ⊕ c mig = . ) 3779 0.980 M ⊕ c mig = . ) 4691 1.024 M ⊕ Lupus
Run 4 ( c mig = . ) 1000 0.010 M ⊕ Table 1.
Different Simulation Runs and results for c mig = and c mig = . . Figure 6.
Planetary System architectures by percentage of totalplanetary systems formed for all runs and both migration models. wider as the number of planets in convoys has gone up and isnow continuous with the previous less clustered population.The entire TRAPPIST-1, YZ Cet and Teegarden’s starsystems can now be formed. In addition, Proxima Cen band Ross 128b can now also be formed.For Run 3 (disk mass is 3-10% star mass), 10135 planetswere simulated and the mean mass is the highest at 0.620 M ⊕ . All planetary systems formed are multi-planetary withthe majority of the systems (39.4%) having 7-12 planetsbut a much increased percentage (34.3%) of systems, incomparison to all other Runs, also having 13-19 planets.The population of massive planets close to the disk edge isthe highest and the cluster now extends the farthest amongall runs as the number of planets in convoys is the highest.Even with this increase, the number of planets that can be formed remains the same as in Run 2. All other planetsremain unexplained by our model as they are either moremassive than the maximum possible upper mass constraintor are just too far away.For Run 4 ( Lupus disks ), all simulated systems mostly haveonly 1 planet with only 1001 planets being simulated intotal (only one system has 2 planets). The mean simulatedplanet mass is 0.011 M ⊕ which is less than the Marsmass threshold. From Section 3.1.1, this means that mostsimulated planets are not able to accrete enough mass toenable them to cross the ice-line (seen from the plot by thedistribution of planets in between the ice-line width) andno more planets can be formed in a system for the vastmajority of the cases. Hence, this disk mass distribution,that is the one taken from observations, doesn’t explain theformation of any of the observed planetary systems. c mig = . (reduced rate migration model) We run all four simulations again using our delayed migra-tion model, where we delay the migration by a factor of10. The simulations are shown in Figure 7 and the differenttypes of planetary systems (according to number of planets)are shown in Figure 6.2742 planets were simulated in Run 1 (
Ormel disks ordisk mass is 3-5% star mass) with a mean planet mass of0.897 M ⊕ . This mean mass is higher because most planetsreach their Isolation masses at higher values of semi-majoraxes due to slower migration but same accretion rate ascompared to its faster migration counterpart (See Section3.1.2). The slower migration rate is also responsible forreduction in the total number of simulated planets. Theplanetary systems formed are both single and multi plane-tary with 94% of the systems having < MNRAS000
Ormel disks ordisk mass is 3-5% star mass) with a mean planet mass of0.897 M ⊕ . This mean mass is higher because most planetsreach their Isolation masses at higher values of semi-majoraxes due to slower migration but same accretion rate ascompared to its faster migration counterpart (See Section3.1.2). The slower migration rate is also responsible forreduction in the total number of simulated planets. Theplanetary systems formed are both single and multi plane-tary with 94% of the systems having < MNRAS000 , 1–13 (2019) S. Dash et al. were simulated with a mean planet mass of 0.980 M ⊕ . Plan-etary systems formed are both single an multi planetarywith 71.5% of systems having < M ⊕ . Planetary systems formed are both single an multiplanetary with 53.4% of systems having < Isolation mass constraint. In general, this reduced ratemigration model accounts for the formation of the max-imum number of observed exoplanets (in Run 2 and 3 both).Run 4 (
Lupus disks ) for the delayed migration modelhas very similar behaviour as its faster migration counter-part with the mean planet mass and the total number ofsimulated planets remaining almost the same (0.010 M ⊕ and 1000 planets respectively). As discussed in Section 3.1.2(for the case of Lupus disks), this is because mass accretionis not affected by a delayed migration model but planetsdo not grow beyond the Mars mass limit in both cases andhence there is no substantial migration inwards to affectthe mass distribution. So the growth of planets to similarmasses at similar locations remains unchanged. This meansthat even this model is unable to form exoplanets with diskmasses from observations.
Section 3.2 and Figures 5 and 7 show that only Runs 1through 3 for both the standard and reduced rate migrationmodels can form planets with masses larger than Mars andmultiple planets in planetary systems for all cases. This isbecause of the increased masses of the disk distributions. Itcan already be observed from Figure 2 that
Lupus disks areless massive than the rest three distributions. This showsthat the more observationally accurate disk masses aresimply not high enough for planets to reach the Mars massthreshold and the planet migration is consequently not fast enough for these planets to migrate out of the ice-line.This also means that multiple planet formation is hinderedsince our model is a sequential model of planetary systemformation. The lower limit of the more massive and theupper limit
Lupus disk mass distributions indicates thatthe minimum disk mass needed to grow a Mars mass planetis about × − M (cid:12) .Figures 5 and 7 also provide an interesting observa-tion: Much heavier e.g. (cid:62) Ormel disks are able to formobserved exoplanets while the more precise
Lupus andless massive disks aren’t able to explain any of them.This poses the question of whether there is a significantunderestimation of disk masses around low mass starsby sub-mm disk surveys. Manara et al. (2018) also cameto this conclusion by comparing known exoplanet massesaround low mass stars and masses of protoplanetary disksaround low mass stars of age 1-3 million years. Assumingthat sub-mm emission in disks is optically thin, theyhypothesized that either growth to planetesimal and planetsizes occurs rapidly within 1 million years in disks or thereis efficient and continuous (or periodic) accretion of materialfrom the environment to the disk which replenishes thematerial accreted into the star. Zhu et al. (2019) howeverpresented a possibility of scattering in disks being a factorin making optically thick disks appear optically thin whichwould underestimate mass of disks. Nevertheless, our modelin this work shows that with our currently observed diskmasses, none of the currently observed exoplanets can beformed. Hence, this remains an open question.Signatures of dust growth to mm-cm size within a millionyears in envelopes and inner disk in Class I YSOs have beenfound by Miotello et al. (2014) and Harsono et al. (2018).This raises the possibility of
Pebble Accretion having timeto act within this lifetime if planetary seeds to efficientlyaccrete pebbles are formed by various mechanisms. Basedon this, sub-million year planet formation models basedon pebble accretion alone or a hybrid pebble-planetesimalaccretion mechanism have been proposed for forming superearths and giant planet cores by Voelkel et al. (2020),Br¨ugger et al. (2020), Coleman et al. (2019) and Liu et al.(2020). All this further strengthens the possibility thatplanet formation in protoplanetary disks starts well withina million years in which case our model predicts thatmassive disks with mass (cid:62) × − M (cid:12) would have to bepresent within a million years and substantial amount ofplanet formation would have to occur to get the observed Lupus disk masses.Liu et al. (2020) also present the case of massive disksbeing present very early in the disk lifetime (with a linear
Ormel disk to star mass relation) by integrating observedgas accretion over the disk lifetime. A steepening of the diskmass to star mass relation has been observed in sub-mmdisk surveys of low mass disks by Ansdell et al. (2016)by comparison between younger (1-3 million years)
Lupus and
Taurus disks with the older
Scorpius disks. Bothof these things considered together might indicate thatrapid planet formation and growth within a million yearsmight also substantially deplete disk masses even before1 million years. But a similar steepening of disk-mass to
MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars Semi-major axis in AU10 M a ss o f p l a n e t i n M TRAPPIST-1 systemYZ Cet systemGJ3323 systemLHS 1140 systemProxima Cen bGJ 1214bGJ 1132 systemRoss 128bTeegarden star system
Run 1 (Ormel) with c mig = 0.1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 2 (3-8% star mass) with c mig = 0.1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 3 (3-10% star mass) with c mig = 0.1 Semi-major axis in AU10 M a ss o f p l a n e t i n M Run 4 (Lupus) with c mig = 0.1 Figure 7.
Synthetic population of planets for (top left)
Ormel
Disks (3-5% star mass), (top right) a bit more heavier disks (3-8% starmass), (bottom left) a lot more massive disks (3-10% star mass) and (bottom right)
Lupus disks all with the migration rate reduced 10times. The parameters of planets in each observed system are taken from Figure 5. star mass relation between very young ( < We now compare our results with other population synthesiscalculations. Miguel et al. (2020) assumed a planetesimalaccretion based formation model and found a bi-modaldistribution of simulated exoplanets in compact systemssimilar to TRAPPIST-1 around low mass stars and browndwarfs. This trend is almost absent in our distributionfor heavy disks for the standard migration model with asmall presence in less massive disks but the reduced ratemigration plots have a prominent bi-modal distribution(except
Lupus disks for both cases). They also found thatplanets above 0.1 M ⊕ (Mars mass) were only possible in MNRAS000
Lupus disks for both cases). They also found thatplanets above 0.1 M ⊕ (Mars mass) were only possible in MNRAS000 , 1–13 (2019) S. Dash et al. disks with masses greater than 0.01 M (cid:12) . In comparison,our model predicts that even disks with masses as low as × − M (cid:12) can form planets greater than 0.1 M ⊕ . Thisclearly shows the higher efficiency of the Pebble Accretion mechanism. However, the highest possible planet mass intheir model is much higher than our model. This is possibledue to merger of several earth mass planetesimals. However,we do not take such mergers into account for our model.Their model as well as our model require a reductionfactor of 10 for Type I migration to explain most observedexoplanets. Both of these models also predict an underes-timation of disk masses due to an inability to explain allobserved exoplanet masses with currently observed diskmasses.Alibert & Benz (2017) also developed a core accretionmodel and found that compact planetary systems similarto the TRAPPIST-1 system were possible around low massstars. However, the number of planets in their simulatedplanetary systems were limited to 4 planets. In comparison,even our reduced rate migration model simulates around 6planets in the case of
Ormel disks and 13 planets with evenhigher massive disks within a 3 million year disk lifetime(for a standard migration model, the number for
Ormel is15 and 19 for the more massive disks). This might be due tous taking the starting time of planet formation at t=0. Witha larger starting time, we expect the number of planets inour simulated systems to fall. They also found that the diskto star mass correlation and the disk lifetime are importantconstraints for the kind of planets formed and had a hugeimpact on the water content of such planets. We also foundthat the disk to star mass correlation is important for thecharacteristics of the simulated planet distribution but thedisk lifetime doesn’t seem to make a difference even for lowmass disks as
Pebble Accretion stops long before the disklifetime in general. However, we make no comments aboutthe water content of our planet distribution but we expectsome amount of water from the predominantly accreted icypebbles during growth within the ice-line.Coleman et al. (2019) presented a Pebble accretion drivenplanet growth model to compare against their own plan-etesimal accretion based model for forming TRAPPIST-1analogs around low mass stars and found that these mostlycompare favourably with each other. Although they did notformulate a population synthesis approach, the differencesthey have with our model is the consideration of planetcollisions resulting in mergers, taking smaller disks ( a out of 10 AU), different locations over the entire inner diskfor the initial planet embryo, a much larger embryo massof − M ⊕ and a much longer disk lifetime of 10 millionyears. We have considered much larger disks ( a out of 200AU), no planet collisions or mergers, embryo growth onlyinside the ice-line width, an initial embryo mass of − M ⊕ and much shorter disk lifetimes between 1-3 million years.Consequently, our simulated planets have smaller massesdue to the Isolation mass constraint as no mergers ofplanetesimals can happen. Coleman et al. (2019) also usedmore precise N-body simulations to determine the orbitsof their non-sequential planet formation model, while weconstrain our planets to have orbits determined by the minimum spacing between planets for stable orbits (fromempirical evidence) as well as disk dispersal which stops allmigration.More recently, Liu et al. (2020) proposed a populationsynthesis model using
Pebble Accretion around very lowmass stars and brown dwarfs (0.01 to 0.1 M (cid:12) ) and foundmass range of simulated planets to be around an earthmass around 0.1 M (cid:12) stars. This is very similar to oursimulated exoplanets by a reduced rate migration modelaround a similar mass star. The planet masses around a0.1 M (cid:12) star for our standard migration model is lowerthan this value which might be a result of the different Isolation mass equations we use. Our model is also closeto the Scenario A presented in their work where embryogrowth starts at the ice-line for self-gravitating disks butwith a much higher stellar mass range. Consequently, amajority of our planets grow beyond the 0.1 M ⊕ limit andalso form a larger population of close-in planets near thedisk edge. While their work is focused on analyzing the finalmass, semi-major axis and composition of their simulatedplanets by varying the stellar host masses (and consideringonly a higher disk mass and much smaller disk radius),we examine the effect of varying disk mass distributionson the simulated final planet mass and semi-major axisdistribution and compare it with observed exoplanets withconstraints from observations of protoplanetary disks. Motivated by the apparent disparity between observed diskmasses and masses needed to form exoplanets around lowmass stars presented in Manara et al. (2018), we expandan analytical model of planet evolution using the efficient
Pebble Accretion process for planet mass growth and Type-Imigration postulated by Ormel et al. (2017) and develop itinto a population synthesis model. We find that compactresonant multi-planetary systems like TRAPPIST-1 willbe common around low mass stars within a 3 million yearold disk lifetime if the disk masses are higher than what isbeing currently observed by sub-mm disk surveys of lowmass disks. In addition, migration delayed by a factor of10 for heavier disks is able to account for most observedexoplanets around low mass stars.The type of distribution of disk masses also influencesthe simulated planet distribution with heavier disksaccounting for more observed planets in general. Theminimum mass required to grow planets beyond a Marsmass threshold and form multi-planetary systems is × − M (cid:12) which is two times higher than the gas disk mass rangefor observed protoplanetary disks from sub-mm surveysaround low mass stars (from Figure 1 top panel). Thispoints to either an underestimation of mass in disks bysub-mm disk surveys or a rapid planet formation andgrowth process that depletes the mass in disks even before1 million years which while being an integral componentof several pebble accretion and hybrid pebble-planetesimalaccretion based planet formation models, cannot still beaccounted by these disk surveys as of now. This moti-vates the observation of more embedded disks in low mass MNRAS , 1–13 (2019) lanet formation and disk mass dependence in a pebble-driven scenario for low mass stars star forming regions in systems younger than a million years.We also compare our results to other population syn-thesis models based on planetesimal accretion mechanismand find that our model can form more planets in multi-planetary systems. Comparison with other pebble accretionmodels shows that the upper mass range and close-innature of our simulated planets are in line with most ofthese models unless planet mergers or different Isolationmass equations are taken into account. In future, we expectmergers to be accounted for and more computationallyintensive N-body simulations to be used for planet orbitdistribution using similar population synthesis models.
DATA AVAILABILITY
All Python codes used to generate simulated planet dataused in this paper are freely available online in a githubrepository linked in the Methods section (See footnote forSection 2.3). Disk mass data used for generating Figure 1are available online as part of supplementary material forthe Ansdell et al. (2016) paper.
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