The theory of planet formation

Transcription

1 The theory of planet formation Willy Benz University of Bern An incomplete and biased review Collaborators: Y. Alibert, T. Schröter, L. Fouchet, University of Bern C. Mordasini, K.M. Dittkrist, MPIA

2 Observations & theory 455 planets... Exoplanets have been found exactly where one did not expect to find them... giant planets cannot form Jupiter Saturn Neptune Uranus points towards serious gaps in our understanding of planet formation as derived from the solar system alone! Venus Earth Mars there might be more gaps! more data is needed!

3 Planet Formation: Stages gas is dynamically important dust migration giant planets type I type II 10 7 years planetesimals embryos giant collisions terrestrial planets 10 8 years gas dynamically less important

4 Growth from dust to planetesimals

5 Classical collisional coagulation - solids and gas do not orbit the star at the same speed gas drag causes dust to drift towards the star gas drag & turbulence determines the relative collision velocities maximum relative velocities? surface effects μm mm m km x 1000 x 1000 x 1000 strength regime gravity regime Difficulties: - drift timescale only 100 yr for 1 m body at 1 AU - what makes meter-sized rocks stick together? - typical velocities for 1 m bodies lead to destructive collisions

6 Gravoturbulent planetesimal formation Dust is trapped locally in transient gas vortices in a turbulent disk and eventually becomes gravitationally bound " K # f = Klahr & Johansen max(! p /! g ) M Hill /M Ceres!!!> t Sedimentation Self!gravity 10 0!10.0! t/t orb Turbulence aided growth might proceed from pebbles directly to intermediate-sized ( km) objects. Are all bodies born that big?

7 From planetesimals to protoplanets

8 Semi-analytical rate equations Rate equations: simplest possible approach. One big body & many small background planetesimals (surface density) ( Gravitational focussing! vrel key parameter. Safronov 1969 Alibert et al m isolation (4πr2 Σ) 3 2 (3M ) 1 2 Without radial excursion, growth goes up until the isolation mass is reached: the protoplanet has accreted all planetesimals in its gravitational reach (in the feeding zone, width ca 5 Hills sphere radii). ( ) ( ) a 3 3/2 Σ 0.07 M 1AU 10gcm 2.

9 Growth as a function of semi-major axis Mordasini et al xMMSN (Σ=7 g/cm 2 ) 10 5xMMSN Memb (Mearth) Memb (Mearth) a (AU) Fig. 7. Snapshots of the embryo mass (solid line) as a function of semimajor axis at four moments in time for two different solid surface densities. The dashed line is the isolation mass. The dotted line is M emb,0 = 0.6 M. The initial solid surface density at 1 AU is 7 g/cm 2 (left panel) and 35 g/cm 2 (right panel). It should be kept in mind that this kind of calculation is needed to generate the start time t start when the embryo is put into - Growth proceeds from inside out - Formation of massive cores (M > 10 MEarth) necessary to build giant planets require massive disks the formation model. The real evolution of the solid core for M > 0.6 M is in general much more complex than plotted here. In this figure, we have continued the calculations up to the isolation mass to allow comparison with other models. the results are quite similar, even if core growth proceeds at large orbital distances somewhat faster in our model. Compared to Chambers (2006) one finds that core growth in our model is a (AU) that the resulting sub-population ofobservablesyntheticplanets reasonably well reproduces the observed population (Paper II).

10 Beyond the rate equation - Monte Carlo method - follow explicitely up to 100 million bodies embedded in evolving gas disk - processes: relaxation, collisioons, gas drag, and type I migration Example: Evolution of 65 million bodies 35 km radius between AU - surface density of MMSN, iceline at 2.6 AU - maximum mass: 0.15 MEarth - no massive bodies at large distances Schröter et al. in prep. How to form the > 10 MEarth cores of giant planets??

11 Late stages: Terrestrial planets S.N. Raymond et al. / Icarus 203 (2009) Raymond et al Fig. 3. Snapshots in time from a simulation with Jupiter and Saturn in 3:2 mean motion resonance (JSRES). The size of each body is proportional to its mass(1/3) (but is not to cale on the x axis). The color of each body corresponds to its water content by mass, from red (dry) to blue (5% water). Jupiter is shown as the large black dot; Saturn is not hown. Jupiter assumed to be present early on...

12 Giant planet formation

13 The core accretion paradigm Basic requirement: 1) Formation of a critical core a critical core must form through collisional accretion of planetesimals Mcrit 10 MEarth 2) Availability of gas gas must be available to accrete once the critical core has formed Perri & Cameron 1974, Mizuno et al. 1978, Mizuno 1980, Bodenheimer & Pollack 1986, Pollack et al A timing issue!

14 Timescales Timescale for specific processes: The heart of the problem - the growth time of a massive core: Collisional dynamics function of distance to star - the core gas accretion time scale: Radiative losses function of core size - the gas supply rate from the disk: (Magneto-)Hydrodynamics function of disk dissipation mechanism - the migration rate: Interactions function of core size and disk characteristics In some regime these timescales are similar, in others they are different need a self-consistent approach that captures this

15 Gas accretion by the core REPORTS 32. A. Morbidelli, A. Crida, Icarus 191, 158 (2007)..org on August 11, 2008 Internal structure equations Miso (MEarth) time (106 years) 16. M. J. Duncan, H. F. Levison, M. H. Lee, Astron. J. 116, 5 oximate 2067 (1998). ation of Ayliffe & Bate E. W. Thommes, L. Nilsson, N. Murray, Astrophys. J. 656, ation in L25 (2007). Figure 12. Surface density plots for locally-isothermal calculations (left), and self-gravitating radiation hydrodynamical calculations using 1% IGO ( 44 ry disk. 18. C. D.protoplanetary Murray, S. F.disc Dermott, System and standard IGO (right) calculations with our standard surfacesolar density. From Dynamics top to bottom the protoplanet masses are 22, 33, 1 final or (Cambridge Univ.use Press, Cambridge, 1999). and 333 M respectively. The radiation hydrodynamical calculations protoplanet radii of 1% of the Hill radii, while the locally-isothermal calcu 19. transfer D. J. Hollenbach, W. Yorke, Johnstone, use 5% of the Hill radii. Note that with radiative and standardh.opacities, thed.spiral shocks inprotostars the protoplanetary disc are much weaker tha s of solid a locally-isothermal equation of state, while using radiative transfer with reduced grain opacities results in2000). intermediate solutions because the discs and Planets IV (Univ. of Arizona Press, Tucson, AZ, 3 3 ts), with optically thick and are able to radiate more effectively than with standard opacities. 20. L. Hartmann, N. Calvet, E. Gullbring, P. D Alessio, n succesastrophys. J. 495, 385 (1998). minimum formation time om plan21. N.3I. Shakura, R. A. Syunyaev, Astron. Astrophys. 24, 337 c 0000 RAS, MNRAS 000, 0 dr 3 22 mulations 1) (1973). = mass conservation 22. dm D. N. C. Lin, me (thick 4πρP. Bodenheimer, D. C. Richardson, Nature 380, 606 (1996). r a core 23. dp M. J. Kuchner, M. Lecar, Astrophys. J. 574, L87 (2002). 11 G(m +M al dotted core ) core growth 24. C. Terquem, J. C. B. Papaloizou, Astrophys. J. 654, ) = hydrostatic equilibrium o corredm 4πr4 (2007). cooling time n bottom 25. G. Marcy et al., Prog. Theor. Phys. 158 (suppl.), 24 (2005). 0 me a gas 26. dt D. A. Fischer, J. A. Valenti, Scientific Frontiers in Research = ad or D. Deming, transfer 3) rad energy on Extrasolar Planets, S. Seager, Eds. (2003), al dotted 80 dp vol. 294 of Astronomical Society of the Pacific Conference s for in60 Series, pp planets). 27. N. C. Santos, G. Israelian, M. Mayor, Astron. Astrophys. e this as , 1153 (2004). time for 28. S. Udry, N. C. Santos, Annu. Rev. Astron. Astrophys. 45, core size 397 (2007). h its final 29. Scenarios have been proposed in which planets in solid 0 themselves are the source of viscosity (35, 36). Although time for this would cause the two time scales to be correlated, rgo runtenvelope less than tdisk, thus making mass be derives from the condition: giant would systematically r (AU) Thommes et al on, taken substantial migration likely in all planetary systems. Helmholtz time (39), tkh (dashed curve). As in more detailed calculations (40), we find that gas 30. M. E. Beer, A. R. King, M. Livio, J. E. Pringle, Mon. Not. R. Astron. Soc. 354, 763 (2004). commences at one particular radius, which for typical parameters lies in or near the Jupiter31. B. S. Gaudi et al., Science 319, 927 (2008). n this case at 7 AU and at time tgiant just under 2 My). Giant formation begins in a burst, with

16 Evolution of the gaseous disk Example: Irradiated profile with dead-zone and Lifetime: ~ years α = Very simple criterion for dead-zone: > 100 g/cm 2 Effective viscosity is obtained by z-averaging

17 Position of the ice line (IL) The position of the IL is critical in the core acretion theory. Its position is determined from the characteristics of the disk (e.g. density, alpha, etc.) 3/2 r Σ(r) =Σ 0 r α = without dead zone no irr. r ice AU Σ 5.2AU 10g/cm Mstar M with dead zone J2 Pollack et al Notes: - position of IL is a function of the assumed surface density - for massive disks, the IL can be located at significant distances - the presence of a dead-zone leads to a closer IL Σ 0 (5.2AU)

18 - Type I (low mass planets): No gap Isothermal approximation: Migration Tanaka et al free parameter: 0 to 1 - Type II (high mass planets): Gap formation Disk dominated regime M planet << Σ P a 2 P Planet dominated regime = v mig = v visc 3ν P 2a P viscous evolution M planet >> Σ P a 2 P = v mig = v visc 2Σa2 P M planet Ida & Lin 2004

19 Migration rates Migration rates change by 1-2 orders of magnitude from type to type II disk lifetimes Type I: Planets seem to migrate so fast that they should all fall into the star within the lifetime of the disk Ward 1997 simple linear theory for iso-thermal disks cannot be the final word!

20 Type I migration: Beyond isothermal Crida et al. 2006; Baruteau & Masset 2008; Casoli & Masset 2009; Pardekooper et al. 2010; Baruteau & Lin 2010 outward inward Kley et al Thermodynamics of the disk is essential

21 Type I convergence zones Time [yrs] out out in in Exact location of convergence will depend upon the detailed structure of the disk Dittkrist et al. in prep Semimajor axis [AU] Mechanism to grow large cores?

22 ependent of the radius, the torque the viscosity, the aspect ratio of extending from a given radius r0 to et al. (2006) showed that the d 2 r0 } is T ν = 3π"νr0 #0 (it can 10 per cent of the unperturbed v Crida et al n tensor). It causes a mass flow of 2 3 H 50 Transition II: ngular momentum: T ν = Fr 0 #0 = to typep = +! 1. 4 RH qr he radial velocity of the gas.opening In this of gap function of: h can also be found from the Navier Using equation (3), we have co - disk characteristics: H, R e following equality, which we- will planet & stellar mass various values of the parameter P 3 q = 10 and H/r = 0.05, and!"#$%&'%()*+)$+&,%$-),#.&-$/ #0)$$"-"*%1+23$4-&1"%&'%+,,"-% Kley & Dirksen 2006 Then, we use these parameters *+#5 (5) Wolfbig & Klahr dots 2008 depthofis shown as in Fig Large scale signature 4 for q ranging from 5 10 to planet-disk interaction n such a disc, no gas flow is allowed as crosses in Fig. 8. Furthermo he outer disc is maintained outside 0, and find the corresponding H n equilibrium is reached so that the Gap formation: Type II migration Disk eccentricity and embedded planets 372 W. Kley and G. Dirksen: Disk eccentricity and embedded planets t = M_jup 2 M_jup.15 1 M_jup 2 M_jup 3 M_jup 4 M_jup 5 M_jup.15 Disk Eccen Disk Eccen t = M_jup 4 M_jup.1 5 M_jup r Fig. 2. Disk eccentricity as a function of radius for the several models with q = up to q = at t = 2500 orbits, for the q = model at t = For the two lower curves q = and q = 0.002, the outer edge of the computational domain lies at rmax = r Fig. 2. Disk eccentricity as a function of radius for the several models with q = up to q = at t = 2500 orbits, for the q = model at t = For the two lower curves q = and q = 0.002, the outer edge of the computational domain lies at rmax = Σ Σ 1.5 t = M_jup 2 M_jup t = M_jup 2 M_jup.5 3 M_jup observationaly testable! Consequently, the planet feels from d state ocated m: q = rcular or q = ucture lower 3 M_jup 4 M_jup 5 M_jup Fig. 1. Logarithmic plots of the surface density Σ for the relaxed state 0 orbits for two different masses of the planet which is located after at r = 1.0 in dimensionless units. Top: q = , and bottom: q = r calculated with NIRVANA. The inner disk stays circular in 3. both cases but the outer disk only in the of lower mass case. For qfor = Fig. Azimuthally averaged radial profiles the surface density 3 it becomes clearly eccentric with some visible fine structure different planet masses, for the same models and times as in Fig. 2. in the gap. For illustration, the drawn ellipse (solid line in the lower 4 M_jup 5 M_jup is torque is proportional to the visss. This is the case of standard type r Fig. 3. Azimuthally averaged radial profiles of the surface density for different planet masses, for the same models and times as in Fig f(p q = 0.00

23 Population synthesis Population synthesis is a tool to: - use all known exoplanets to constrain planet formation models - test the implications of new theoretical concepts - provide a link between theory and observations Need to compute the formation of many planets - the approach and the physics must be simplified - it must capture the key effects requires separate detailed studies of all components - several different approaches are useful One learns a lot even if a synthetic population does not match the observed one!

24 Synthetic population Nominal model, no irradiation, no dead zone alpha= 7x10-3, f1=0.001, M=1 M Diversity in initial conditions leads to planet diversity transit RV micro-lensing RV The distribution of planets in the a-m plane is not uniform Planets that reached inner boarder of computational disk transit RV micro-lensing (RV) Different techniques probe different regions of the a-m plane Fig. 14. Final mass M versus final distance a of all N synt synthetic planets of the planetary population orbiting G type stars using the parameters and distributions described in 3 (α = , f I = 0.001). The feeding limit at a touch is plotted as dashed line. Planets migrating into the feeding limit have again been put to 0.1 AU. As a touch gets very large for M 20M, also a few extremely massive planets are in the feeding Mordasini et al see also: Ida & Lin 2004,...,2010

25 Formation tracks Nominal model: Isothermal migration only Mstar=1 M Nominal model Type I migration (Analytical rate reduced by fi) Type II migration (Disk dominated: Mp<Mdisk,loc) Type II migration (Planet dominated: Mp>Mdisk,loc & disk limited gas accretion) Mordasini et al. 2009

26 Formation tracks isothermal type I adiabatic type I saturated type I type II interactions between growing planets will play a key role!

27 Comparison with observations (nominal model) full synth. population detectable sub-population RV detection bias observ. comp. sample < Mstar < 1.3 -e < 0.3 -one planet / star -single host stars -KRV>10 m/s semi-major axis m sini Fe/H KS a: 64% KS msini: 96% KS Fe/H: 22%

28 Structure of the disk: Irradiation With irradiation Without irradiation Fouchet et al transition between type I and type II migration

29 Beyond a solar mass The mass of the central star enters in: - the value of the Keplerian frequency accretion timescale of solids viscous dissipation in alpha-disk - the value of the Hills radius size of feeding zone the size of the envelope at early times - the type I & II migration rate extent of migration - the position of the iceline the location of increased surface density - characteristics of circumstellar disk mass and lifetime of disk disk structure Ω = GM a 3 planet R H = a planet Mplanet 3M star R ice 1AU Σ5.2AU 10g/cm 2 1/ M M Alibert et al

30 Characteristics of circumstellar disks 1) overall cluster lifetime fractions of stars showing evidence for a disk 1206 HERNA NDEZ ET AL. Hernandez et al NGC 2024 Trapezium IC 348 NGC 2362 This could indicate that the low disk presence observed in the! Velorum cluster is abnormal for its evolutionary stage, and environmental effects, such as strong stellar winds and/or strong radiation fields fromtaurus the! Velorum system, could provide the physical mechanism for the low disk frequency. Moreover,!75% of the disk-bearing stars in the! Velorum cluster show IRAC SED slopes smaller than the median values of other 5 Myr old stellar groups plotted in Figure 10 suggesting that the disks of the I! Velorum cluster have a Cha higher degree of dust settling. In particular, the median IRAC SED slopes for the disk population of NGC 2362, the k Orionis cluster, the OB1b subassociation and the! Velorum cluster are "1.72, "1.60, "1.70, and "1.82, respectively (with a typical error of 0.06). Tr 37 IC 348 Studying the young (2 3 Myr) open cluster NGC 2244, Balog et al. (2007) showed that high-mass stars (O-type stars) can affect the primordial disks of lower mass members only OB if 1they are bc within!0.5 pc of the high-mass star. We find similar resultsob 1 a for the! Velorum cluster. Using the photometric members with NGC 7160 NGC 2362 V " J > 3:5 (!M2 or later), we find a disk frequency of 4% # Upper Sco 3% at a projected distance of pc; inside 0.25 pc the central objects of the cluster contaminate the optical photometry used to select photometric candidates (x 3). The closest primordial disk is located at projected distance of!0.5 pc. At larger projected distance (>1.0 pc), the disk frequency is larger (8% # 2%). This suggests that the relative fast dispersion of disks in the! Velorum cluster is produced by the strong radiation fields and strong stellar winds from the central objects. This result must still be considered tentative given the small number of stars with Taurus disks, which results in large errors in disk frequencies. disks disappear in about 5-7 Myr formation timescale for giant gaseous planets Fig. 11. Fraction of stars with near-infrared disk emission as a function of the age of the stellar group. Open circles represent the disk frequency for stars in the T Tauri ( TTS) mass range (!K5 or later), derived using JHKL observations: NGC 2024 and Trapezium ( Haisch et al. 2001), and Chameleon I (Go mez & Kenyon 2001). Solid symbols represent the disk frequency calculated for stars in the TTS mass range using Spitzer data (see text for references). Vol. 686 Kennedy & Kenyon 2009

31 Mdisk as a function of Mstar Muzerolle et al assume adjusted to reproduce the Ṁ disk versus relation using the alpha-disk model we find M disk M 1.2 star reproduces this observation

32 Expected planet populations M =0.5M M =1.0M M =2.0M Alibert et al αd =1.2 αd =0.0 αd =2.0 no hot jupiters (correct) Changing stellar mass results in significant changes in the planet population very massive warm planets (not observed)

33 Planet populations a [AU] Low (high) mass stars lead to the formation of lower (higher) mass planets, in more (less) compact planetary systems.

34 Micro-lensing planet searches Alibert et al synthetic planets potentially detectable planet actually detected planet Expected distribution of the mass of the stars acting as lenses detection bias not known 122 Arnaud Cassan, Takahiro Sumi, Daniel Kubas 50% 25% 5% 2% Detection efficiency of the PLANET microlensing network (Cassan et al. 2008). Probing a very different region of parameter space! Figure 2. PLANET detection efficiency from the 2004 season (preliminary diagram), as a function of planet mass and orbital separation. The crosses are the detected planets with their parameter error bars. more than ten years of observations (Cassan et al. 2008). The Fig. 2 shows a prelimi-

35 Conclusions The discovery of the whole population of exoplanets is essential to provide important constraints on formation models Different detection techniques provide different constraints A comprehensive theory of planet formation is still not available: - pieces are available but dont fit together... - some pieces are still missing... Important ingredients missing - Characteristics of proto-planetary disks as a function of host star - mass, structure, lifetime, composition we are missing some of the initial and boundary conditions - Systems in the making we only see essentially old systems (end products)!