Planet Population Synthesis
2017 Sagan Exoplanet Summer Workshop
Microlensing in the Era of WFIRST
Yann ALIBERT
The National Centres of Competence in Research (NCCR) are a research instrument of the Swiss National Science Foundation Outlines
1- Exoplanets and population synthesis
2- From disks to planets: integrated models
3- Population synthesis: comparison and results
4- Pebbles versus planetesimals Planet formation models
protoplanetary disks planets
⇒
What is the origin of the diversity of planetary systems? Importance of stellar evolution models
Kepler - bias corrected Fulton et al. 2017
The majority of planet properties are based on stellar evolution models Stellar evolution
the full numerical approach the stellar evolution models Baraffe et al. 1998, Alibert Baraffe et al. 1999 Integrated models and HR diagram
initial condition + stellar evolution = HR diagram
-> age, other properties Padova Outlines
1- Exoplanets and population synthesis
2- From disks to planets: integrated models
3- Population synthesis: comparison and results
4- Pebbles versus planetesimals From GMC to present day planets Giant Molecular Cloud Protoplanetary disk
99% gas 1% solids AU ~100s 7 T1000disk < AU10 yr ⇒
2.5 Light years ~160000 AU ~160000 2.5Lightyears ⇒ Top down or bottom-up?
Protoplanetary disk
GI/DI model Core-accretion model
pre-solar grain Top-down models: the disk instability model
Clump formation depends critically on disk cooling
⇒ formation of massive planets ⇒ formation in outer parts of the disk Origin of enrichment in heavy elements/formation of low mass (Earth,Neptune) planets? Bottom up model: the core accretion model
Solar CoreCore formationformation System Core formation Gas accretion byby planetesimalsplanetesimals SlowSlow corecore by solid beyond critical (“solids”)(“solids”) accretionaccretion accretionaccretion massmass accretion Protoplanetary disk
Extrasolar planets 99% gas 1% solids 7 Tdisk < 10 yr
TheThe core-accretioncore-accretion modelmodel gas giant ice giant terrestrial planet What are these solids?
Planetesimals-based model Pebbles-based model
dust ~ microns ⇒ ⇒
pebbles ~ cm ⇒ ⇒ high efficiency low efficiency
planetesimals ~ km Planet formation: the players
⇒ Planet formation: the interactions
• solid-solid interactions • gas disk structure and evolution • solid accretion • solid-gas disk interactions • planet-disk interactions • planet internal structure and gas capture • planet-planet interactions Integrated models
protoplanetary disks planets
⇒
What is the effect of the combined processes in shaping planetary systems? Integrated models Outward Inward
Outward migration Bitsch et al. 2012 Integrated models Outward Inward
Outward migration Bitsch et al. 2012 Integrated models Outward Inward
Outward migration
Neptune mass planets should be found at 5 AU and 16 AU !! Bitsch et al. 2012 Integrated models Outward
The observed mass/period distribution of planets is the result of combined processes Inward Migration / disk evolution / planetary growth
Outward migration
Neptune mass planets should be found at 5 AU and 16 AU !! Bitsch et al. 2012 determination of the migration rate requires the computa- gration velocity as well as its direction depend sensitively determination of the migration rate requires the computa- gration velocity as well as its direction depend sensitively tion of the 2D or 3D structure (including the thermodynam- upon the detailed dynamical and thermal structure of discs, tion of the 2D or 3D structure (including the thermodynam- upon the detailed dynamical and thermal structure of discs, ics) of the proto-planetary disc. The computer time required leading to a number of sub-regimes of type I migration (lo- ics) of the proto-planetary disc. The computer time required leading to a number of sub-regimes of type I migration (lo- for carrying out such detailed multi-dimensional simula- cally isothermal, adiabatic, (un-)saturated). Recently, a new for carryingtions of out these such processes detailed over multi-dimensional a timescale covering simula- planetcallysemi-analytic isothermal, descriptionadiabatic, (un-)saturated). of type I migration, Recently, which a repro- new tions offormation these processes is prohibitively over a high. timescale Hence, covering in the population planet semi-analyticduces the results description of Paardekooper of type I migration, et al. (2011), which has repro- been formationsynthesis is prohibitively approach, planetary high. Hence, migration in the is computed population usingducesderived the results (Mordasini of Paardekooper et al., 2011a; etKretke al. (2011), and Lin has, 2012). been synthesisfits toapproach, migration planetary rates resulting migration from is hydrodynamical computed using cal-derivedIt includes (Mordasini the effect et al. of, 2011a; co-rotationKretke torques and that Lin, can 2012). lead fits toculations migration (see rates Chapter resulting by Baruteau from hydrodynamical et al. and references cal- It includesto outward the migration effect of inco-rotation non-isothermal torques discs. that can This lead new culationstherein). (see Chapter by Baruteau et al. and references to outwardformalism migration has been in implemented non-isothermal in recent discs. simulations This new by therein). Planetary migration occurs in different regimes depend-formalismAMB who has have been shown implemented that the inscaling recent factor simulations determining by Planetarying upon migration the mass occurs of the in planet. different For regimes low mass depend- planets,AMBthe who migration have shown speed introduced that the scaling in earlier factor models determining (Eq. 47) ing uponi.e. planets the mass not of massive the planet. enough For to open low a mass gap in planets, the proto-thebecomes migration much speed less introduced important (inAlibert earlier et models al., 2013). (Eq. 47) i.e. planetsplanetary not massive disc, migration enough occurs to open due a gap to the in imbalance the proto- be-becomesThe much transition less important mass between (Alibert type etI and al., type 2013). II migration planetarytween disc, the migration Lindblad and occurs corotation due to torques the imbalance excerted be- on the Theis M transitiong,vis (Eq. mass 52) in between IL’s prescription type I and and typeM IIg,th migration(Eq. 53) tweenplanet the Lindblad by the andinner corotation and outer torques regions excerted of the disc. on the Thisis M(more exactly,(Eq. 52) the in condition IL’s prescription derived by andCridaM et al.(Eq.2006) 53) in A. Fortier et al.: Planetary formation g,vis g,th Planet formation: full numericalA. Fortier or et planet al.:integrated Planetaryregime by the formation is inner called andmodels? ”type outer I migration” regions of and the the disc. correspond- This (moreAMB’s exactly, prescription. the condition The comparisonderived by Crida between et al. gap2006) opening in regimeing is migration called ”type rate I has migration” been derived and by the linear correspond- and numer-AMB’scriteria prescription. is treated in The more comparison details in section between 2.7. gap opening ing migrationical calculations rate has (e.g. beenWard derived, 1997; by linearTanaka and et numer- al., 2002;criteriaInitially, is treated population in more details synthesis in section models 2.7. have assumed an the full numerical approach the integrated models 2 1 ical calculationsPaardekooper (e.g. andWard Papaloizou, 1997;, 2009;TanakaPaardekooper et al., 2002; et al., Initially,isothermal population migration synthesis rate reduced models by C1 have10 assumed− 10− anin 3 ∼2 − 10 AU2011). For higher mass planets, i.e. planets massive IL’s simulations and C 10− 10− in2 AMB’s1 sim- Paardekooper and Papaloizou, 2009; Paardekooper et al., isothermal migration rate reduced1 ∼ by−C1 10− 10− in enough to open a gap in the proto-planetary disc, the planet ulations. The values of C1 less3 than∼2 unity were− needed 2011). For higher mass planets, i.e. planets massive IL’s simulations and C1 10− 10− in AMB’s sim- is confined in the gap by Lindblad torques and thus follows to prevent cores of growing∼ giant− planets to fall into the enough to open a gap in the proto-planetary disc, the planet ulations. The values of C1 less than unity were needed is confinedthe global in the disc gap accretion. by Lindblad This torques regime and is thus called follows ”type IIto preventhost star. cores These of findings growing by giant population planets syntheses to fall into were the an the globalmigration.” disc accretion. Type II migration This regime is itself is called sub-divided ”type in II twohostimportant star. These motivation findings to by develop population physically syntheses more were realistic an 1 AUmodes: disc-dominated type II migration, in the case the non-isothermal migration models. This is one out of sev- migration.” Type II migration is itself sub-divided in two important motivation to develop physically more realistic local disc mass exceeds the planetary mass, and planet- eral examples in which population synthesis can be used to modes: disc-dominated type II migration, in the case the non-isothermal migration models. This is one out of sev- dominated type II migration in the opposite case (see also test in a statistical sense detailed modelling of individual local disc mass exceeds the planetary mass, and planet- eral examples in which population synthesis can be used to Lin and Papaloizou, 1986; Ida and Lin, 2004a; Mordasini processes. Population synthesis models did not provide a dominatedet al., type 2009a). II migration In the former in the opposite case, the case migration (see also rate istestbetter in a statistical understanding sense of detailed the migration modelling itself but of individual pointed out 0.1Lin AU andsimply Papaloizou given by, 1986; the localIda and viscous Lin, evolution 2004a; Mordasini of the proto-processes.that the current Population prescription synthesis did models not result did in not planet provide popula- a et al.,planetary 2009a). disc, In the while former the migration case, the is migration decelerated rate by the is in-bettertions understanding with the observed of the characteristics. migration itself It is but worth pointed pointing out simplyertia given of the by planet the local in the viscous latter case. evolution of the proto- thatout the that current with prescription this new formalism did not for result type in I migration, planet popula- which planetaryInitial disc, whilepopulation the migration synthesis ismodels decelerated by both by IL the and in- AMBtionshas with been the implemented observed characteristics. in recent simulations It is worth by pointing AMB, an ertia of thede planetde in the latterde case. de out that with this new formalism for type I migration, which made use= of the conventional+ formula+ of type I migration arbitrary scaling factor (Eq. 47) slowing down migration Fig. 13. Example of the formation of a planetary system. Left panel showsInitialderived the population evolutiondt fordt locally of the synthesis semi-major isothermaldt models axis discs of by thedt both ( protoplanetsTanaka IL and et versus al. AMB, 2002). time. hasis been no longer implemented an absolute in recent necessity simulations (Alibert by et AMB, al., 2013). an Accreted planets are indicated with a big dot, ejected planetsFig. with 13. a cross.Example The of final the formation masse ofGD the of a surviving planetary VS,E planets system. is Left written panelVS,p on shows the right the evolution of the of the semi-major axis of the protoplanets versus time. figure. Right panel depicts the eccentricity of planetesimals inAccreted the discmade planets as aIn use function areorder of indicated the of to time conventionalinvestigate with (x-axis) a big and dot, how semi-major formula ejected sensitive planets axis of the type (y-axis). with results a I cross. migration The are Thecolour on final bar the massearbitraryWhile of the surviving scaling this represents planets factor is (Eq. written a definitive 47) on the slowing right progress, of the down difficulties migration re- figure. Right panel depicts the eccentricity of planetesimals in the disc as a function of time (x-axis) and semi-major axis (y-axis). The colour bar indicates the eccentricity values. derivedmagnitude for locally of this isothermal migration, discs a scaling (Tanaka factor et al.C,1 2002).was intro-is nomain. longer They an are absolute linked necessity to the sensitivity (Alibert of et the al. migration, 2013). indicatesIn the order eccentricityduced: to investigate values. how sensitive the results are on the Whilerate this to the represents saturation a definitive of the corotation progress, torque, difficulties to a partial re- their final masses are determined by the coupled evolution ofmagnitude the Giant of this planet migration, formationa aby scaling accretion factor of 100C km1 was planetesimals intro- main.gap They formed are by linked relatively to the large sensitivity migrating of planet, the migration and to or- τ = disc and the planet. Considering both migration and oligarchictheir finalduced: massesresults are quitemig1 determined unlikely,a˙ by if the not coupled impossible. evolution In our of simulations, the Giant to ac- planetratebital formation to the eccentricity. saturation by accretion A of further of the 100 corotation km difficulty planetesimals istorque, due to to the a fact partial that growth for the solid component of the planet has consequencesdisc and thetually planet. form Considering giant planets both we migration had to reduce2 and oligarchic the planetesimalresults size, quitegap unlikely,the formed onset if not byof impossible. efficient relatively gas In large our accretion simulations, migrating onto to the planet, ac- core, and and the to or- sat- a 1 1 cs M M 1 in the mass interval of a few tens Earth masses. Thesegrowth planets for theatτmig1 least solid= down component= to 0.1 of km. the However, planet has assuming consequences∗ a uniform∗ tuallyΩ− popula- form gianturation planets of we the had corotation to reduce the torque planetesimal occur at size, a similar mass (of find it to hard to survive, as their migration rate is larger than tion of smalla˙ planetesimalsC 3.81 whichaΩ sizeM remainsa unchanged2Σ K dur- bital eccentricity. A further difficulty is due to the fact that in the mass interval of a few tens1 Earth! masses.K " Theseplanet planets g at least down toorder 0.1 km. 10 However,M ), so assumingthat a self-consistent a uniform popula- coupled approach of their accretion rate (dominated by the accretion of solids), and ing the whole formation of the2 planet is also1 hard to explain. It the onset of efficient gas accretion onto the core, and the sat- find it to hard to survive,1 as1 their migrationcs 1 rateMM is larger−M thana 1tion of small planetesimals which⊕ size remains unchanged dur- = 5 ∗c ∗ Ω− the two processes is necessary. they are often lost in the central star. Planets that are betweentheir accretion 10 is not rate yet (dominated clear1 how.5 by planetesimal the10 accretion formation of solids),2 proceeds. and K ing Models the wholeuration formation of the of the corotation planet is torquealso hard occur to explain. at a similar It mass (of and 100 Earth masses are usually undergoing inward type I mi- that explainC1 the≃3.81 formation×aΩKC of1f planetesimalsgMplanetM a Σ byg1AU direct collapse For type II migration, as long as the mass of the planet they are often lost in the central star.! Planets" that! are⊕ between" 10 is not yet clearorder how 10 planetesimalM ), so that formation a self-consistent proceeds. Models coupled approach of gration. Only in the cases where the gas component of the disc in vortices in turbulent regions3/2 (Johansen1 et# al. 2007)$ predict remains⊕ smaller than the local disc mass (of the order and 100 Earth masses are usually5 M undergoing1 M inwardc − type Ia mi- that explainthe the two formation processes of planetesimals is necessary. by direct collapse dissipates during this process allows to end up with planets in a fast formation1.5 of10 very∗ big planetesimals (r > 100 km). On 2 gration. Only in the cases where the gas componentyrs. of them disc in vortices(47) in turbulentof πΣr regions), the (Johansen migration et timescale al. 2007) predict (τmig2(a)=a/ vr ) this mass range. Indeed, if we calculate in situ models, planets in the other≃ hand,× coagulation× MC1fg modelsM can not1AU be ruled out. A re- For type II migration, as long as the mass of the planet| | ⊙ ! ⊕ " dissipates during this process allows! 3/2 to" end up with planets# in$ a fast formationis of given very big by planetesimals the local viscous (rm > 100 diffusion km). On time, tvis(a) this mass range are produced in the same fraction as giant plan- cent study of Windmark et al. (2012) shows that direct growth of remains smaller2 than the local disc mass (of the order∼ 3 4 this mass range.The Indeed, expression if weM calculateof Tanaka in et situ al. models,(2002) planets corresponds in the to otherC = hand,(2 coagulation/3)(a /ν models). For acan steady not be accretion ruled out. disc A re- with F 3πΣν ets in the mass bin M6 (10 to 10 M ). This is shown in Fig.14, planetesimals via dust∗ collisionsyrs. can lead to the growth of(47) 0.11 km 2 of πΣr ), the migration timescalea (τmig2(a)=2∼a/ vr ) where we have used the same mass bins are in the previousthis mass fig- rangeplanetesimals.1, while are produced×C1 Indeed,M< in1 theimplies initially same fraction slower small as planetesimals migration giant plan- rates. showcent ILstudy better as- of Windmarkand Σ et al.1/a (2012), Mdisc shows(a)= that direct2π growthaΣda of=2πΣa .| Then,| !3 ⊙ 4" ets in the mass bin M6 (10 to 10 M ). This is shown in Fig.14, planetesimalsis via given dust collisions by∝ the local can lead viscous to the growth diffusion of 0.1 km time, tvis(a) ures, but now the colour bars of the histograms represent their matchessume type to the I migration observed ceases size distribution inside the of inner objects boundary in the as- of τmig22(a) tvis(a) Mdisc(a)/F (Hasegawa and Ida∼ , fixed location and not the change in semi major axis. Planetswhere thatThe we have expressionteroid used belt the and of sameTanaka among mass the et bins al. TNOs. are(2002) in Weidenschilling the corresponds previous fig- (2011) toplanetesimals.C1 shows= (2/ Indeed,3)(a / initiallyν).∼ For small a steady planetesimals∼ accretion% show disc better with F 3πΣν the disc. 2013). For a planet morea massive than the inner2∼ disc are in the mass range M4 are also planets that did notures, have1 but the, while nowthat theC the colour1 size< 1 distribution barsimplies of the slowerhistograms currently migration observed represent in rates. their the asteroidmatches IL as- belt toand the observedΣ 1/a size, M distributiondisc(a)= of objects2πaΣ inda the=2 as- πΣa . Then, fixed location andSince not the the change publication in semi ofmajorTanaka axis. Planets et al. (2002), that teroid radiative belt and amongmass∝ theMdisc TNOs.(a), Weidenschilling the viscous torque (2011) shows from the outer disc possibility to start the runaway gas accretion phase. Planetssume ac- in type the I range migration of 10 to ceases 100 km inside can be the better inner explained boundary by an of ini- τmig2(a) tvis(a) Mdisc(a)/F (Hasegawa and Ida, effects on the type I migration rate have been investigated pushes∼ the planet (mass∼ M%planet) rather than the inner disc. creting gas in a runaway fashion grow exponentially withare time,inthe the disc. masstial range population M4 are of also 0.1 planets km planetesimals. that did not Kenyon have the andthat Bromley the size2013). distribution For currently a planet observed more in massive the asteroid than belt the inner disc so they leave the mass range of a few tens Earth masses very (2012)(e.g. Paardekooper conclude, by combining and Mellema observations, 2006; Masset of the hot and and Casoli cold, Then, replacing Mdisc with Mplanet, τmig(a) Mplanet/F possibility to start the runaway gas accretion phase. Planets ac- in the range of 10 to 100 km can be better explained by an ini- ∼ fast and jump to several Jupiter masses. This has alsocreting an e↵ect gasSince inpopulations2010; a runaway thePaardekooper publication of fashion TNOs grow of with etTanaka al. exponentially time, 2011). constraints et Ital. was with(2002), shown on time, their radiative thattial formation the population mi-mass( ofHasegawa 0.1Mdisc km( planetesimals.a) and, the Ida viscous, 2013). Kenyon torqueIn summary, and Bromley from the the type outer II migra- disc in planets with masses between a Saturn mass and a fewso Jupiter theyeffects leaveprocess, on the the mass that type range TNOs I migration of form a few from tens rate a Earth massive have masses been disc verymainly investigated(2012) composed conclude,pushes by the combining planet observations (mass Mplanet of the) rather hot and than cold the inner disc. masses: once the runaway of gas starts, planets can easilyfast and(e.g. ac- jumpPaardekooperof to 1 km several planetesimals. Jupiter and masses. Mellema This, 2006; has alsoMasset an e↵ect and Casolipopulations, Then, of TNOs replacing with timeM constraintswith M on their, formationτ (a) M /F disc planet mig ∼ planet crete hundreds of Earth masses of gas, therefore it is morein planets di2010; - withPaardekooper masses between et a al. Saturn, 2011). mass It and was a fewshown Jupiter that theprocess, mi- that(10Hasegawa TNOs form fromand Ida a massive, 2013). disc In mainly summary, composed the type II migra- cult to form a planet in this mass range. This can be seen both in Most probably the initial population of planetesimals in pro- masses: oncetoplanetary the runaway discs of is gas not starts, uniform planets in size, can but easily follows ac- aof size 1 kmdistri- planetesimals. the histograms of in situ formation and when migrationcrete is con- hundreds of Earth masses of gas, therefore it is more di - sidered: the mass bin M (planets with masses between 100 and bution. Being the origin of planetesimals still under debate,Most we probably the initial population of planetesimals in pro- 5 cult to formhave a planet some in this freedom mass torange. make This assumptions can be seen on both the in initial size dis-10 1000 M ) is the one with less amount of planets. However, when toplanetary discs is not uniform in size, but follows a size distri- the histograms of in situ formation and when migration is con- comparing the mass fraction of each mass bin in the in situ and tribution of planetesimals. From what we have shown, without sidered: thesmall mass bin planetesimals M (planets giant with planet masses formation between 100 is di and cultbution. to explain, Being the origin of planetesimals still under debate, we migration histograms one should keep in mind that the numbers 5 have some freedom to make assumptions on the initial size dis- where calculated considering only the surviving planets.1000 While M ) isat the least one in with the less way amount we understand of planets. it However, now. However, when even with an comparing theinitial mass population fraction of of each small mass planetesimals, bin in the in thesitu collisions and tribution among of planetesimals. From what we have shown, without this is always the case for the in situ calculations, it is not for small planetesimals giant planet formation is di cult to explain, the migration case. Most of the planets lost due to migrationmigration are histogramsthemselves one are should likely keep to be in disruptive mind that as the soon numbers as their random ve- locities start to be excited by a planetary embryo. In fact,at least the time in the way we understand it now. However, even with an in the mass bin M4 (10 to 100 M ). Therefore, in the inwhere situ his- calculated considering only the surviving planets. While this is alwaysfor the fragmentation case for the of in 0.1 situ km calculations, planetesimals it is in not the for neighbourhoodinitial population of small planetesimals, the collisions among tograms, the fraction of planets in this mass bin represents all the 4 themselves are likely to be disruptive as soon as their random ve- planets formed in this mass range. On the other hand, whenthe migration mi- of case. a forming Most of planet the planets is around lost 10 dueyears, to migration much less are than the for- mation timescale of the planet. Therefore, it is also unlikelylocities that start to be excited by a planetary embryo. In fact, the time gration is considered this fraction only accounts for thein planets the mass bin M4 (10 to 100 M ). Therefore, in the in situ his- that survived. tograms, thean fraction initial of population planets in ofthis only mass small bin represents planetesimals all the canfor be usedfragmentation to of 0.1 km planetesimals in the neighbourhood explain the formation of giant planets. Moreover, we have shown 4 planets formed in this mass range. On the other hand, when mi- of a forming planet is around 10 years, much less than the for- gration is considered this fraction only accounts for the planets mation timescale of the planet. Therefore, it is also unlikely that an initial population of only small planetesimals can be used to that survived. 19 explain the formation of giant planets. Moreover, we have shown
19 cause opacity in the envelope is highly uncertain. Note that at smaller core masses and therefore speeds-up the giant Mc,hydro depends on the planetesimal-accretion rate M˙ c. formation timescale (Pollack et al. 1996; Hubickyj et al. Since Mc,hydro can be comparable to an Earth-mass M 2005). ⊕ after the core accretes most of planetesimals in its feeding In order to gain computing time and to avoid numerical zone, whether the core becomes a gas giant planet is actu- convergence difficulties, the energy equation is not solved. ally regulated by a timescale of the subsequent quasi-static Instead the procedure outlined by Mordasini et al. (2012a), envelope contraction (Kelvin-Helmholtz contraction time) based on total energy conservation, is adopted with a small rather than the value of Mc,hydro. improvement which allows to take the energy of the core Because the contraction of the gas envelope also releases into account as well as described in (Fortier et al., 2013). energy to produce pressure to support the gas envelope it- The total luminosity L = Lcont +Lacc is the sum of the en- self, the contraction is quasi-static. Its rate is still regulated ergy gained through the contraction of the envelope Lcont by the efficiency of radiative/convective transfer in the en- and by the accretion of planetesimals Lacc. The contrac- velope such that tion luminosity, assumed to be constant throughout the en- Fig. 1.— Schematic of the coupling between the different processes entering in the computationvelope, is computed of a self-consistent from the change of planet energy of formation the planet dM M planet planet , (27) between the time t and t + dt model. Quantities exchanged by the different modules are indicateddt ≃ alongτKH arrows. Etot(t + dt) Etot(t) Egas,acc where Mplanet is the planet mass including gas envelope. L = − − (32) cont − dt Based on the results by 1D calculations (Ikoma et al., 2001), Different integrated modelsextremely fast and to allow exploration ofIL a approximate large number the Kelvin-Helmholtzwhich contractionq =1 timescale.5. However,where Etot is they the total found planetary that energy this and E doesgas,acc not= τKH of the envelope with dtM˙ gasuint is the energy gained by the accretion of nebular ˙ of models. However, this approach does not provide self- considerablyk2 affect thegas with results. a specific internal energy uint at a rate Mgas. The M − τ τ planet , (28) luminosity associated with the accretion of planetesimals, Nbody (incl. planetesimals)consistently a relationhydrodynamical between the disk disc surface density3D KHinterior≃ andKH1 structureM Neglecting the detailed energy balance in the disc, IL ! ⊕ " which are assumed to deposit their energy onto the core, Fig. 1.— Schematic of the coupling between the different processes entering in the computation of a self-consistent planet formation can be written the local pressure and temperature whichwhere enterτKH1 inis the the contraction cal- timescaleadopt for theMplanet equilibrium= M . temperature distribution of optically ⊕ model. Quantities exchanged by the different modules are indicated along arrows. Since there are uncertainties associated with dust sedimen- culation of the structure of the growing planets as well as thin discs prescribed by Hayashi (1981),M˙ coreMcore tation and opacity in the envelope (Pollack et al., 1996; Lacc = G (33) R in their migration rates. An intermediate approachHelled et al., consists 2008; Hori and Ikoma, 2011), IL adopt a core 8 10 1/4 range of values τKH1 = 10 10 years and k2=3– where M˙ core is the1/ mass2 accretion rate of the planetesimals extremely fast and to allow exploration of a large number ofwhich usingq a model=1.5. of However, viscously they evolving found that discs this (Shakura does not and − 9 r − L 4 with nominal parameters of k2=3and τKH1 = 10 model not possible T = 280which results in an increase in∗ the core massK, Mcore and ra-(2) of models. However, this approach does not provide self- Sunyaevconsiderably, 1973) affect for which the results. the local verticalyears. structure Eq. (27) can shows be that dMplanet/dt rapidly increases dius1AURcore. It has to be notedL that at Lcont cannot be com- A. Fortier et al.: Planetary formation # ⊙ $ consistently a relation betweenA. Fortier the et al.: disc Planetary surface formation density and Neglecting the detailed energy balance inas M theplanet disc,grows. IL However, it is limited by the global gas puted! in a straightforward" manner since in order to compute fullnumerical computed (Papaloizou and Terquem, 1999; Alibert et al., accretion rate throughout the disc and by the process of gap Etot(t + dt) the structure of the envelope at t + dt needs to the local pressure and temperature which enter in the cal- adopt the equilibrium temperature distributionformation of optically near the proto-planets’where orbits, asL discussedand later.L are respectively stellar and solar lumi- Nbody + statistics of planetesimals2005a). Such an intermediatesemi-analytical approach disk has the advantage1D interior structure∗ ⊙ be known. This difficulty can be circumvented with the help culation of the structure of the growing planets as well as thin discs prescribed by Hayashi (1981), The AMB approach consists of solving the standard in- of an iterative scheme (Fortier et al., 2013). of providing the full structure of the disc and tocause provide opacity in the the envelopenosity. is highly uncertain. IL determine Note that at the smaller position core masses of and the therefore ice speeds-up line (a theice giant) as the ternal structure equations (Bodenheimer and Pollack, 1986)˙ The internal structure equations are solved with four in their migration rates. An intermediate approach consists Mc,hydro depends on the planetesimal-accretion rate Mc. formation timescale (Pollack et al. 1996; Hubickyj et al. location at which Tboundary= 170 conditions:K, which 1) the radius translates of the core forRcore a, 2) opti- the framework to include self-consistently additional1/4 Since M physicalc,hydro candr be comparable1 to an Earth-mass M 2005). r 1/2 L cause opacity in the envelope= is highly uncertain. Note that(29)⊕ at smaller core masses and therefore speeds-up the giant of using a model of viscously evolving discs (Shakura and − after the core accretes most of planetesimals2 in its feeding˙ totalIn radius order to of gain the computing planet RM time, 3) and the to surface avoid numerical temperature of processes suchT = as 280 photo-evaporation∗ fromMK thec,,hydro centraldepends(2) on stardM ther planetesimal-accretion4callyπρr thin disc rate M intoc. (Eq.formation [2]) timescale (Pollack et al. 1996; Hubickyj et al. zone, whether the core becomes a gas giant planet is actu- theconvergence planet T difficulties,surf , and 4) the the energy surface equation pressure is notP solved.surf . With Sunyaev, 1973) for which the local vertical structure can be 1AU L Since Mc,hydro can be comparable to an Earth-mass M 2005). ally regulated bydP a timescaleGM of ther subsequent quasi-static⊕ theseInstead boundary the procedure conditions outlined the by structureMordasini equations et al. (2012a), provide a and/or nearby stars, the existence# of⊙ dead-zones,$ after the core and accretes irra- most= of planetesimals in its feeding In order to gain computing time and to avoid numerical computed (Papaloizou and Terquem, 1999; Alibert et al., ! " envelope contraction (Kelvin-Helmholtz4 contraction(30) time) based on total energy conservation,1/2 is adopted with a small dMr − 4πr aiceunique=2. solution7(L for/L a given) planetAU mass.. (3) zone,rather whether than the the core value becomes of M a gas. giant planet is actu- convergenceimprovement difficulties, which allows the to energy take the equation energy of is notthe coresolved. 2005a). Such an intermediate approach has the advantage diationwhere fromL and theL centralare respectively star. Both stellar approaches and solar are lumi- brieflydT c,hydro The core radius∗ can⊙ be calculated (module 8 in Fig. 1) ∗ ⊙ ally regulatedBecause by the a timescale contraction of of the the subsequent gas envelope quasi-static also releases Insteadinto account the procedure as well as outlined described by inMordasini (Fortier et et al. al., 2013).(2012a), = ad or rad (31) for a given core mass, composition (rocky or icy) and pres- outlinednosity. below.IL determine the position of the ice lineenvelope (aice contraction) as thedP (Kelvin-Helmholtz∇ ∇ contraction time) based on total energy conservation, is adopted with a small of providing the full structure of the disc and to provide the extremely time consumingenergy to produce pressure to support the gas envelope it- The total luminosity L = Lcont +Lacc is the sum of the en- Distance[AU] star to Distance[AU] star to Due to viscous diffusionsure at its surface and (relevant photo-evaporation, for planets with a massivefg H/Hede- Bernmodel whereratherself,r, than P, the T the contractionare value the of radius,M isc quasi-static.,hydro pressure,. Its rate and is temperature stillFortier et al. 2013 regulated improvementergy gained through which the allows contraction to take of the the energy envelope ofL the core framework to include self-consistently additional physical locationIL adopt at which the minimumT = 170K, mass which solar translates nebula for (MMSN)a opti- envelope). AMB solve the internal structure equationscont for a whichBecauseby are the specified the efficiency contraction as of a radiative/convective function of the gas of envelope the mass transfer alsoM releases inwhich the en- intoand account by the accretion as well ofas planetesimals described in (LFortieracc. The et contrac- al., 2013). Time [yr] creases withr time. Fordifferentiated simplicity, core using IL a simple adopt modified polytropic equa- processes such as photo-evaporation from the central star modelcally ( thinHayashi disc into, 1981) (Eq. as [2]) a fiducial set of initialrepresentsenergyvelope to conditions the produce such mass that pressure inside a to sphere support of the radius gasr envelope(including it- Thetion total luminosity, luminosity assumedL = toLcont be constant+Lacc throughoutis the sum the of the en- en- tionvelope, of is state computed for the from density the changeρ as of a energy function of the of planet pressure P theself, mass the contractionof the core isM quasi-static.dMcore). StabilityM Its rate against is still convection regulated ergy gained through the contraction of the envelope Lcont and/or nearby stars, the existence of dead-zones, and irra- f f planet planet , (27) (betweenSeager theet al. time, 2007)t and t + dt and introduce multiplicative factors1/ (2 d andisby checkedg the) efficiency to using scale the of radiative/convective Schwarzschildthe criterion transfer (e.g. inKippen- the en- and by the accretion of planetesimals Lacc. The contrac- statistical Nbody (planets) aiceparametrized=2.7(L /L disk) AU. (3)rate equationsdt ≃ τKH fg = fg,0 exp( t/τdep), (4) Fig. 13. Example of the formation of a planetary system. LeftFig. panel 13. Example shows the of evolution the formation of the of semi-major a planetary axis system. of the Left protoplanets panel shows versus the evolution time. of the semi-major axis of the protoplanets versus time. velope such that tion luminosity, assumed to be constant throughout the en- diation from the central star. Both approaches are briefly ∗ ⊙ Σ hahn and Weigert, 1994). Depending upon if convection Etot(t−+ dt) Etot(t) nEgas,acc Accreted planets are indicated with a big dot, ejected planetsAccreted with a cross. planets The are final indicated masse with of the a big surviving dot, ejected planets planets isMMSN written with on a cross. the right disc The of final the surface masse of the surviving densities planets is written of gas on the right ( of) the and planetesimalswhere Mplanet is the planet mass including gas envelope. L = ρ(P )=− ρ0 + cP− (32) (34) velope,cont is computed− from the changedt of energy of the planet figure.outlined Right panel depicts below. the eccentricity of planetesimals infigure. the disc Right as a panel function depicts of time the eccentricity (x-axis) and of semi-major planetesimals axis in (y-axis). the disc The as a colour function bar of time (x-axis) and semi-major axis (y-axis). The colour bar is presentBased or onnot, the the resultsdM adiabaticplanet by 1D calculations gradientMplanet ( (Ikomaad) or et the al., radia- 2001), indicates the eccentricity values. (Σ ). IL set ,∇ (27) between the time t and t + dt indicates the eccentricity values. s Due to viscous diffusion and photo-evaporation,tive gradientf (g radde-) is used. Thesewhere equationsτdep are solvedis the to- discwhere lifetimeEρ0, c, and (forn are detailed material parameters. discussion,E This= EOS see IL approximate∇ thedt Kelvin-Helmholtz≃ τKH contraction timescale where tot is the total planetary energy and gas,acc IL adopt the minimum mass solar nebula (MMSN) q gether with the equation of state (EOS) by Saumon et al. neglects˙ the relatively small temperature dependency of ρ creases with time.Σ For= Σ simplicity,10fg(r/10AU) IL adopt− , τKH of the envelope(1) with Ida and Lin dtMgasuint is theτEtot energy(t + gaineddt) E bytot the(t accretion) Egas,acc of nebular their final masses are determined by the coupled evolution of the Giant planet formation by accretion of 100 km planetesimals where Mplanet is the planet mass including gas envelope.2008a). ILLcont use= dep as a− free parameter− ˙ (32) rang- model (Hayashi, 1981) as a fiducialtheir final massesset of are initial determined conditions by the coupled evolution of the Giant planet formation by accretion of 100 km planetesimals (1995). The opacity is taken from Bell andk2 Lin (1994). forgas withsolids. a specific− Therefore internal it energy is sufficientdtuint at a to rate considerMgas. The only the disc and the planet. Considering both migration and oligarchic results quite unlikely, if not impossible. In our simulations, to ac- Ikoma− et al.6 7 disc and the planet. Considering both migration and oligarchic results quite unlikely, if not impossible. In our simulations, to ac- PodolakBased on(2003) the results and byMovshovitz 1D calculations andMplanet Podolak ( (2008), 2001), have equationsluminosity of associated mass conservation with the accretion and hydrostatic of planetesimals, equilibrium growth for the solid component of the planet has consequences tually form giant planets we had to reduce the planetesimal size, 2 τKH τKH1 ing from ,10 yrs(28) to 10 yrs. The self-similar solution with and introduce multiplicative factorsmodel IL (f and f ) to scale the E E = growth for thed solid componentg of the planet has consequenceswhere atually normalization form giantextremely planetsf = wef factor had to reduceexp(Σ the 10fast planetesimalt/=τ 75g) size,, /cmarguedIL approximate thatcorresponds grain the(4) opacities Kelvin-Helmholtz≃ are significantlyM contraction reduced timescale in plan- where(Eqs.which are29tot andassumedis the 30) total to deposit calculate planetary their the energy energy core’s ontoand internal thegas,acc core, structure in the mass interval of a few tens Earth masses. Thesein planets the massat interval least down of a few to 0.1 tens km. Earth However, masses. assuming These planets a uniformat popula-least down to 0.1g km. However,g,0 assuming a uniformdep popula- ! ⊕ " 1 ˙ − τKH of the envelope with Σ r− has an asymptoticdtcanMgas be writtenuint is exponential the energy gained by cut-off the accretion at of radius nebular rm findMMSN it to hard to disc survive, surface as their migration densities rate is largerfind of itthangas to hard (tionΣ to) of survive, and small planetesimals as their migration which rate size is remains larger than unchangedtion of dur- small planetesimals which size remains unchanged dur- etary envelopeswhere τKH1 asis compared the contraction to the timescale interstellar for M medium.planet = Re-M . and radius (for details, see Mordasini et al., 2012b). Re- to 1.4 times of Σ at 10AU of the MMSN model. The in- ⊕ ˙ their accretion rate (dominated by the accretion of solids), and ing the whole formation of the planet is also hard to explain. It ∝ k2 gas with a specific internal energy uint at a rate Mgas. The their accretion rate (dominated by the accretion of solids), and ing the whole formation of the planet is also hard to explain. It ducingSince the grain there opacity are uncertainties allows runaway associated− accretion with dust to sedimen- occur garding the composition, for˙ rocky material, a silicate-ironr 7 Formation & evolution model: Bern model Formation & evolution model: Bern model Seen from the edge Disk truncation by magnetic field gas disk accretion excitation 100 Mearth mass 10 Mearth grav. interactions gas drag gas drag ang. momentum 1 Myr 2 Myr ang.3 Myr momentum exchange - ti me exchange - eccentricity damping eccentricity damping Irradiation protoplanetary disk = gas disk + solids Mass (Earth masses) seeds/system system - 10 planetary 1 planetary Mercury Semi-major Semi-major axis (AU) Time Rocky Icy fraction of icy planetesimals Movie from Frederic Carron HD 134987 a system with TWO Earths at 1 AU Plot from Frederic Carron Frederic Plot from long term stability Plot from Frederic Carron Frederic Plot from Outlines 1- Exoplanets and population synthesis 2- From disks to planets: integrated models 3- Population synthesis: comparison and results 4- Pebbles versus planetesimals Population synthesis Formation & evolution model Initial Conditions Compute synthetic planet population Apply observational detection bias Predictions (going back to the full synthetic population) Observed population Comparison: Observable sub-population Model - Distribution of semi-major axis solution No match: improve, - Distribution of masses found - Distribution of luminosities Match change Noparameters match - Distribution of radii Match Ida & Lin 2004-2013, Thommes et al. 2008, Mordasini et al. 2009-2012, Miguel et al. 2011, Hellary & Nelson 2012, Alibert et al. 2013, Pfyffer et al. 2014 Mass (Earth masses) 100 planetary systems - 10 planetary seeds/system systems - 10 planetary 100 planetary Mercury Semi-major Semi-major axis (AU) Time Rocky Icy fraction of icy planetesimals Movie from Frederic Carron Planetesimal based: gas fraction M [Mearth] M a [AU] Swoboda et al. in prep Planetesimal based: mass function M2siniObservations distribution (complete sample) Synthetic Raw number corrected Correct for obs. bias observed 200 # planets 100 0 10.0 100.0 1000. M2sini [Earth Mass] Mayor et al. 2011 Benz et al. 2014 Sudden increase Typical for core accretion. Constraint on Mcrit & gas accretion rate . 12 Fulton et al. Testing models withEffect transit of Oceanobservations mass Kepler - bias corrected Fulton et al. 2017 Fig. 10.— Top: Two-dimensional distribution of planet size and incident stellar flux. The median uncertainty is plotted in the upper left. There are at least two peaks in the distribution. One class of planets has typical radii of 1.3 R and generally orbit in environments ⇠ with Sinc >100 S , while another class of slightly larger planets with typical radii of 2.4 R orbit in less irradiated environments with ⇠ Sinc < 200 S . Bottom: Same as top panel with individual planet detection points removed, annotations added, and vertical axis scaling changed. The lack of planets to the upper left of the dashed blue line is likely due to photo-evaporation. The shaded region in the lower right indicates low completeness. Pipeline completeness in this region is less than 25%. The purple and black lines show the scaling relations for the photoevaproation valley predicted by Lopez & Rice (2016) for scenarios where these planets are the remnant cores of photoevaporated Neptune size planets (dashed purple line) or that these planets are formed at late times in a gas-poor disk (dotted black line). 12 Fulton et al. Testing models withEffect transit of Oceanobservations mass Jin & Mordasini, astroph-170600251J Rocky cores Icy cores 1.01.0 8 8 0.80.8 ] ] ⊕ 4⊕ 4 0.60.6 0.40.4 Radius [ R Radius [ R 2 2 0.20.2 Fraction of Envelope Lost Fraction of Envelope Lost 1 1 0.00.0 Fig. 10.— Top: Two-dimensional distribution 0.06 of0.06 planet size and0.1 incident0.1 stellar flux. The median uncertainty is plotted0.5 in the0.5 upper 1 1 0.06 0.06 0.10.1 0.50.5 1 1 left. There are at least two peaks in the distribution. One class of planets has typical radii of 1.3 R and generally orbit in environments ⇠ with Sinc >100 S , while another class of slightly larger planets with typical radii of 2.4 R orbit in less irradiated environments with ⇠ Sinc < 200 S . Bottom: Same as top panel with individual planet detection points removed, annotations added, and vertical axis scaling changed. The lack of planets to the upper left of the dashed blue line is likely due to photo-evaporation. The shaded region in the lower right indicates low completeness. Pipeline completeness in this region is less than 25%. TheDistance purpleDistance and black lines show [AU] the[AU] scaling relations for DistanceDistance [AU] [AU] the photoevaproation valley predicted by Lopez & Rice (2016) for scenarios where these planets are the remnant cores of photoevaporated Neptune size planets (dashed purple line) or that these planets are formed at late times in a gas-poor disk (dotted black line). Observed locus of the valley agrees with evapora on -consistent with mainly rocky composition -inconsistent with mainly icy composition Planetesimal based: sucesses and problems 1- Predicted mass function is similar to observed one 2- Disk size similar to observations / masses in the higher end of the distribution 3- Works better with low mass planetesimals (~km) 4- Formation of planets at large distance - timescale problem -> ejection? -> outwards migration? -> another formation mechanism? Testing models with microlensing observations Theoretical lens population including mass and metallicity distributions Dominik 2006 Besancon model Robin et al. 2003 Alibert et al. 2011 (unpublished) Testing models with microlensing observations 0.1 0.2 0.3 0.4 0.5 1000 100 10 0.6 0.7 0.8 0.9 1.0 1000 100 10 1.1 1.2 1.3 1.4 1.5 1000 100 10 1.6 1.7 1.8 1.9 2.0 1000 100 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 0.1 1 10 a [AU] !! one-planet models !! Testing models with microlensing observations Theoretical lens population including mass and metallicity distributions Dominik 2006 Besancon model Robin et al. 2003 Alibert et al. 2011 (unpublished) Testing models with microlensing observations Theoretical lens population including mass and metallicity distributions Theoretical observable lens population Cassan, Sumi & Kubas 2008 Alibert et al. 2011 (unpublished) Testing models with microlensing observations Theoretical observable lens population Alibert et al. 2011 (unpublished) Semi-major axis Lenses at 0.5 kpc Lenses at 6 kpc Alibert et al. 2011 (unpublished) Mass Lenses at 0.5 kpc Lenses at 6 kpc Alibert et al. 2011 (unpublished) Outlines 1- Exoplanets and population synthesis 2- From disks to planets: integrated models 3- Population synthesis: comparison and results 4- Pebbles versus planetesimals What is the mass of core’s building blocks? Planetesimals-based model Pebbles-based model dust ~ microns ⇒ ⇒ pebbles ~ cm ⇒ ⇒ high efficiency low efficiency planetesimals ~ km Planetesimal based: Integrated model 1- protoplanetary disk evolution 2- planet internal structure (core & envelope) 3- orbital migration 4- planet-planet interactions (gravity & competition) Alibert, Mordasini, Benz 2004; Alibert et al. 2005, Mordasini et al. 2012, Alibert et al. 2013, Pfyffer et al. 2014, Alibert & Benz 2016, Swoboda et al. in prep… Planetesimal based: sucesses and problems 1- Predicted mass function is similar to observed one 2- Disk size similar to observations / masses in the higher end of the distribution 3- Works better with low mass planetesimals (~km) 4- Formation of planets at large distance - timescale problem -> ejection? -> outwards migration? -> another formation mechanism? Pebble based: Integrated model 1- protoplanetary disk evolution 2- no planet internal structure 3- orbital migration 4- no planet-planet interactions (one planet per system) Pebble-based model: rapid formation of planetary cores Final location of planets Formationtime Origin of pebbles as a function of time Original location of planets Bitsch et al. 2015 Pebble-based model: few Neptune mass planets - no HN Mgas = 10% Mplanet Mgas = 50% Mplanet iceline < 1 AU in < 1 Myr Bitsch et al., 2015 Pebble based: sucesses and problems 1- No timescale problem (even inversed timescale problem) 2- All disks are observed with pebbles, but they should drift to the star and disappear very rapidly -> recycling process? 3- Formation of Neptune planets very unlikely - no Hot Neptunes -> are they all formed by collision of smaller planets? -> rotation axis of exo-Neptunes? 4- Pebbles are destroyed in the planetary envelope Maximum mass of planets formed by pebble accretion Alibert,astroph-1705.06008 accreted pebble Ormel et al. 2015 -> accreted pebbles interact with the envelope (gas drag) Maximum mass of planets formed by pebble accretion Alibert,astroph-1705.06008 accreted pebble Ormel et al. 2015 pollution (1) 1-2 Mearth -> heavy material is dispersed and vaporized in the envelope and does not reach the core Maximum mass of planets formed by pebble accretion Alibert,astroph-1705.06008 accreted pebble Ormel et al. 2015 pollution (1) 1-2 Mearth fresh gas polluted gas from the disk returning to replenishment (2) the disk Maximum mass of planets formed by pebble accretion Alibert,astroph-1705.06008 accreted pebble Ormel et al. 2015 pollution (1) 1-2 Mearth fresh gas polluted gas from the disk returning to replenishment (2) the disk gas remains unpolluted if (2) is more rapid than (1), i.e. inside ~ 10 AU Maximum mass of planets formed by pebble accretion During core growth by pebble accretion, when the core is larger than ~ 1-2 Mearth: -> heavy material is dispersed and vaporized in the envelope and does not reach the core -> the heavy elements cannot accumulate in the envelope because of mass exchange with the disk, inside ~10 AU -> the total mass of heavy elements is limited to 1-2 Mearth -> the planet cannot accrete gas because it is too small heavy material cannot accumulate neither in the core nor in the envelope, gas cannot be accreted, the planetary growth stops at ~ 1-2 Mearth -> no planet larger than ~1-2 Mearth can form directly by pebble accretion inside ~10 AU Conclusions Do not mix integrated models and population synthesis 1- integrated models = stellar evolution models 2- population synthesis = compute HR diagram Population synthesis model can be used for: 1- get global picture of planetary formation 2- understand/predict planet statistics from different observation means Population synthesis models require the knowledge of the bias - ‘blind’ observations better to avoir the ‘observer bias’ Microlensing observations constraint a part of the aM diagram that is not probed by other means ⇒ strong constraint on models