Planet Formation lecture by Roy van Boekel (MPIA) suggested reading: “LECTURE NOTES ON THE FORMATION AND EARLY EVOLUTION OF PLANETARY SYSTEMS” by Philip J. Armitage http://arxiv.org/abs/astro-ph/0701485 with material from Kees Dullemond, Christoph Mordasini, Til Birnstiel 1 the scales from ~13,000 km to ~140,000 km ~1 µm in or ~30,000,000,000 km (~100 AU) 2 star formation vs. planet formation • ~spherical collapse, “own • stellar gravity usually gravity” dominant dominant • rotation, shed very much • Keplerian shear, much less angular momentum net angular momentum loss • cooling • cooling! • (elemental) composition • composition can (locally) be highly enriched in heavy ~interstellar/solar elements/“dust” • scales: ~0.1 pc —> ~1 RSun • scales: <1 to ~several AU —> ~1 REarth to ~1 RJupiter 3 2 was relatively meagre, and limited to the Solar System. 2. Mass and angular momentum The last twenty years have seen a wealth of new observa- 33 tions, including direct images of protoplanetary disks, the The mass of the Sun is M⊙ =1.989 10 g, made up discovery of the Solar System’s Kuiper Belt, and the de- of hydrogen (fraction by mass X =0.×73), helium (Y = tection and characterization of extrasolar planetary sys- 0.25) and “metals” (which includes everything else, Z = tems. Although these observations have confirmed some 0.02). One observes immediately that, existing predictions, they have also revealed gaps in our ZM M , (1) theoretical knowledge. As a consequence, the big picture ⊙ ≫ p questions remain to be fully answered. We would like to i.e. most of the heavy elements! in the Solar System are know, found in the Sun rather than in the planets. If most of How terrestrial and giant planets form. the mass in the Sun passed through a disk during star • formation, the implication is that the planet formation What processes determine the final architecture of process need not be very efficient. • planetary systems. The angular momentum budget for the Solar System is dominated by the orbital angular momentum of the The frequency of habitable planets, and the abun- planets. The angular momentum in the Solar rotation is, • dance and nature of life on them. 2 2 L⊙ k M⊙R Ω, (2) The goal of these notes is to introduce the concepts neces- ≃ ⊙ sary to understand planet formation, and to provide (ob- assuming for simplicity solid body rotation. Taking viously incomplete) entry points to the literature. First Ω =2.9 10−6 s−1 and adopting k2 =0.1 (roughly × though, we briefly review observational properties of the appropriate for a star with a radiative core), L⊙ 48 2 −1 ≃ Solar System and extrasolar planetary systems that we 3 10 gcm s . By comparison, the orbital angular × might hope a theory of planet formation would explain. momentum of Jupiter is, L = M GM a =2 1050 gcm2 s−1. (3) J J ⊙ × A. Critical Solar System observations The significance of" this result is that it implies that sub- stantial segregation of mass and angular momentum must 1. Architecture have taken place during (and subsequent to) the star for- mation process. We will look into how such segregation The orbital properties, masses and radii of the plan- arises during disk accretion later. ets in the Solar System are listed in Table I. The domi- nant planets in the Solar System are our two gas giants, Jupiter and Saturn. These planets are composed pri- 3. Minimum mass Solar Nebula marily of hydrogen and helium – like the Sun – though they have a higher abundance of heavier elements as com- We can use the observed masses and compositions of pared to Solar composition. Saturn is known to have a the planets to derive a lower limit to the amount of gas substantial core. Descending in mass there are two ice that must have been present when the planets formed. giants (Uranus and Neptune) composed of water, ammo- This is called the Minimum Mass Solar Nebula (Weiden- nia, methane, silicates and metals, plus low mass hydro- schilling, 1977). The procedure is: gen / helium atmospheres; two large terrestrial planets (Earth and Venus) plus two smaller terrestrial planets 1. Start from the known mass of heavy elements (say (Mercury and Mars). Apart from Mercury, all of the iron) in each planet, and augment this mass with planets have low eccentricities and orbital inclinations. enough hydrogen and helium to bring the mixture They orbit in a plane that is approximately,Minimum but not ex- toMass Solar composition. Solar This is Nebula a mild augmentation actly, perpendicular to the Solar rotation axis (the mis- for Jupiter, but a lot more for the Earth. ◦ alignment angle is about 7 ). “What is the minimum2. Divide amount the of Solardisk material System to into make annuli, the solar with one In the Solar System the giant and terrestrialsystem planets planets?” Basicplanet idea: per annulus. Distribute the augmented mass are clearly segregated in orbital radius, with the inner for each planet uniformly across the annuli, to zone occupied by the terrestrial planets being(1) separatedConsider the disk yieldregion a characteristicfrom which each gas surfaceplanet (given density Σ (units from the outer giant planet region by the maincurrent asteroid mass / location)gcm−2 would) at the accrete location material of each planet. belt. The orbital radii of the giant planets coincide with where we expect the protoplanetary disk to(2) haveConsider been the amountThe result of refractory is that between elements Venus in each and Neptuneplanet (and ignoring the asteroid belt) Σ r−3/2. The precise nor- cool enough for ices to have been present. This(add is a volatiles/ices sig- beyond iceline) ∝ nificant observation in the classical theory of giant planet malization is mostly a matter of convention, but if one formation, since in that theory the time scale(3) assume for giant diskneeds bulk composition a specific number is solar; the mostadd H+He common gas value used is planet formation depends upon the mass of condensibleaccordingly that due to Hayashi (1981), materials. One would therefore expect faster growth to r −3/2 Result (Hayashi 1981): Σ =1.7 103 gcm−2. (4) occur in the outer ice-rich part of the protoplanetary disk. × AU # $ Σ is gas surface density (~total dens., H+He gas ~99% of mass) This estimate relies on several assumptions and is only an approximate result 1981IAUS...93..113H Minimum Mass Solar Nebula Hayashi (1981) 5 Minimum Mass Solar Nebula more recent rendition, ~equivalent to previous plot Ruden (2000) (Giant) Planet Formation Two main theories of planet formation: (1) Core accretion: formation of solid core (terrestrial planet), if core sufficiently massive (~8 M⊕) subsequent accretion of gas (gas giant planet). (2) Gravitational Instability: direct collapse from gas phase (only gas giant planets, relatively far out in the disk) 7 Gravitation Instability (GI) Main idea: (1) instability causes initial over- density, subject to self-gravity (2) if these “clumps” can get rid of potential energy faster than pressure and differential rotation smooth them out again, they can collapse (3) fast process; planet composition ~ (local) disk bulk composition W.K.M. Rice, P.J. Armitage, M.R. Bate & I.A. Bonnell, MNRAS, 339, 1025 (2003) 8 Gravitation Instability (GI) 47 We have already derived the necessary conditions for angular momentum transport also leads to lower surface Safronov-Toomregravitational Criterion instability for disk to occur. stability We against need the isothermal Toomre density and, again, enhanced stability (Lin & Pringle, collapseQ inparameter keplerian to disk: be low enough, specifically, 1990). Given these consideration, when will a disk fragment? csΩ Gammie (2001) used both analytic arguments and local Q <Q> crit 1(213) ≡ πGΣ ≃ numerical simulations to identify the cooling time as the control parameter determining whether a gravitationally where cs is the sound speed in a gas disk of local sur- -1 unstable disk will fragment. For an annulus of the disk Q is “Toomreface density parameter”,Σ and cs theis sound disk speed mass is[cm assumed s ], Ω is to orbital be -1 3 -1 -2 we can define the equivalent of the Kelvin-Helmholtz time frequencysmall [rad enough s ], G that = gravitational the distinction constant between [cm g the s orbital], Σ is -2 scale for a star, surfaceand density epicyclic [g cm frequencies]. is of little import. If we con- sider a disk with h/r =0.05 at 10 AU around a Solar U Disk will stable against fragmentation where Q > Qcrit. Typical mass star, then the relation h/r = cs/vφ yields a sound tcool = 4 (217) value Qcrit ≈ 1. Having Q < Qcrit is necessary but not sufficient 2σTdisk speed c 0.5kms−1. To attain Q = 1, we then require condition fors fragmentation; a surface≃ density, where U is the thermal energy content of the disk per unit surface area. Then for an ideal gas equation of state 3 2 Σ 1.59 10 gcm . (214) with γ =5/3 Gammie (2001) found that the boundary ≈ × for fragmentation is: This is much larger than estimates based, for example, −1 on the minimum mass Solar Nebula, from which we con- tcool ! 3Ω — the disk fragments. • clude robustly that gravitational instability is most likely t " 3Ω−1 — disk reaches a steady state in which to occur at an early epoch when the disk mass is still high. • cool Recalling that the characteristic wavelength for gravita- heating due to dissipation of gravitational turbu- 2 lence balances cooling. tional instability is λcrit =2cs/(GΣ), we find that the mass of objects formed if such a disk fragmented would This condition is intuitively reasonable. Spiral arms re- be, sulting from disk self-gravity compress patches of gas 4 within the disk on a time scale that is to order of mag- 2 4πcs nitude Ω−1.