Giant planet formation

Transcription

1 Lecture 12 Giant planet formation Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini Based partially on script of Prof. W. Benz National Geographics Mentor Prof. T. Henning

2 Lecture 12 overview 1. Core accretion paradigm 1.1 Structure equations 1.2 Boundary conditions 1.3 Planetesimal-envelope interaction 1.4 Gas accretion in runaway 2. Early models of giant planet formation 3. Classical models of giant planet formation 4. Extended and improved models of giant planet formation

3 1. Core accretion paradigm

4 Core accretion paradigm Giant planets (such as Jupiter or Saturn) being mostly gas planets, it is quite obvious that their formation must be completed prior to the disappearance of the gas in the disk. In a previous chapter, we have seen that disks have typical lifetime of 3-10 Myr, hence giant planet formation must be completed within this time period. Two competing giant planet models are presently being discussed. Both have advantages and disadvantages depending which aspect is weighted more. We have already studied the so called gravitational instability or direct collapse model. Core accretion or nucleated instability Perri & Cameron 1974; Mizuno et al 1978; Mizuno 1980; Bodenheimer & Pollack 1986; Pollack et al 1996; Alibert et al 2005 According to the core accretion paradigm, the formation of giant planets proceeds in a two step way: A solid core accretes first. Once this core reaches a critical mass (of order ~10 Mearth) the gaseous envelope is accreted in a runaway process. Basic requirement: A critical core must form before the gas disappears. This is difficult to achieve especially at large distances, as we have seen.

5 Constraints from Jupiter and Saturn The internal structure of the planets is obtained through modeling. A core-envelope structure is adopted a priori and then adjusted so as to meet observations (mass, radius, gravitational moments, surface abundance, etc.) An important observation is that both Jupiter and Saturn are enriched in heavy elements compared to solar: Jupiter: enriched times solar. Saturn: enriched 6-14 times solar). There are however large uncertainties, partially coming from the EOS. Guillot 1999 Saumon & Guillot 2004 This is regarded as a indication that core accretion lead to the formation of Jupiter and Saturn. Recently it was however found that direct collapse can also lead (under certain circumstances) to enriched planets.

6 Core accretion models In numerical models of core accretion, one follows the concurrent growth both in core and gas mass of an initially small solid core (ices, rocks) surrounded by a gaseous envelope (H2 & He) in an (evolving) protoplanetary disk. One divides the problem in three modules: Solid accretion rate/growth of the core. This is usually calculated with methods like we have seen in earlier lectures. Gas accretion rate/growth of the envelope Planetesimal-envelope interaction. Links the two modules, gives the captures radius, and the energy and mass deposition profile in the envelope. Y. Alibert

7 1.1 Structure equations

8 Structure equations The internal structure of the envelope of the planet is computed using the traditional equation of stellar structure in 1D (spherically symmetric). This gives the pressure, temperature, mass and luminosity as a function radius, and, as we shall see below, also the gas accretion rate. There is one notable difference to stars: in planets there is no internal nuclear energy generation. However, the infall of planetesimals and the corresponding dissipation of their kinetic energy in the interior of the growing planet is actually an energy source which needs to be taken into account, as it is dominant in the early stage. The equation correspond to the usual conservation of mass, momentum and energy, and energy transfer: (1) dm dr =4πr2 dp ρ dr = Gm ρ r 2 (2) ( ) (3) dl dr =4πr2 ρ ɛ T S dt t dr = T dp P dr (4) e.g. Bodenheimer & Pollack ) Mass conservation The equation of mass conservation for a spherical shell of mass dm and thickness dr is simply dm = ρ dv = ρ 4πr 2 dr dm dr = 4πr2 ρ.

9 Structure equations II 2) Equation of motion and hydrostatic equilibrium m+dm Considering an cylindrical mass element of mass P(r+dr) dm m dm = ρ dr ds ds P(r) F g one finds for the net acceleration under the influences of gravity and pressure r = 2 r/ t 2 r rdm= gdm+ P(r) ds P(r + dr) ds dr With Taylor, we can write P(r + dr) = P(r) + (dp/dr) dr, This gives the equation of motion for gas inside the planet r = Gm r 2 1 ρ dp dr It is usually assumed that the evolution of a planet always occurs in quasi hydrostatic equilibrium (although opposite views exist, Wuchterl 1991). Then we simply have dp dr = Gm r 2 ρ,

10 Structure equations III 3) Energy conservation A mass shell can have its energy changed by external energy injection (nuclear fusion, planetesimal deceleration), expansion or contraction (volume work) and change of the internal energy. This change of energy in a layer must lead to a change of the luminosity l: With the laws of thermodynamics, we can write this in terms of entropy: dl dr =4πr2 ρ ( ɛ T S t The energy source originating from the infall of planetesimals is given by ϵ. Note that this term is not evident to compute and will be described in the next section. The term with the entropy establishes an explicit temporal dependence between two structures at t and t+dt. )

11 Structure equations IV 4) Energy transfer Finally, in the energy transport equation, the temperature gradient must be calculated taking into account whether the energy is carried by radiation or by convection. This depends upon the value of the opacity which can be obtained again from tables (for example Bell and Lin 1994). dr r dt dr = T P dp dr Which of the two energy transfer modes operates can be seen with the Schwarzschild criterion: If the radiative temperature gradient is larger than the adiabatic gradient, the atmosphere is unstable and convective energy transport sets in (without proof: entropy likes to float ). In the convective regime, to good approximations, the atmosphere is adiabatic because the eddies do not have time to exchange energy with their surrounding. The effective temperature gradient is therefore the smaller of the two (radiation, convection:adiabatic). = d ln T d ln P =min( ad, rad ) rad = 3 64πσG κlp T 4 m Here, κ is the Rosseland mean opacity which is mainly dominated by the dust. The adiabatic gradient is directly given by the EOS. ad = ln T ln P We see that convection is expected to occur in opaque parts in the envelope, or in where l/m is large, i.e. regions with a large energy flux. s

12 1.2 Boundary conditions

13 Boundary conditions To integrate the system, we need to specify boundary conditions. a) inner boundary conditions: Two conditions connected to the size and luminosity of the core: Rcore, Lcore. Rcore is obtained by solving the structure equations for the core. As a first approximation, we can assume that the core has some fixed density. Lcore is obtained from the residual kinetic energy of the accreted planetesimals which is obtained from solving the infall equations. Additional core luminosity comes from: radioactive decay on the core, core contraction and cooling of the core. b) outer boundary conditions: The atmosphere Three types of outer boundary conditions have to be considered 1) attached or nebular: The planet s envelope and the disk are continuous 2) detached or transition: The planet s envelope is no longer in contact with the disk 3) isolated or evolution: The planet is no longer in an accretion disk

14 Attached or nebular phase T R Lint RCore At low masses (Mcore < ca Mearth), the envelope of the protoplanet is attached smoothly to the background nebula, and the conditions at the surface of the planet are (approximately) simply the pressure and temperature in the surrounding disk. The radius of the planet in this case is given by the minimum of the accretion (or Bondi) and the Hill sphere radius. ere Tneb Pneb Pneb R A = GM c 2 s R H = M 3M 1/3 a. R = R A 1 + R A /(k liss R H ) τ = max (ρ neb κ neb R, 2/3) T 4 = T 4 neb + T 4 int P = P neb T 4 int = 3τL int 8πσR 2 l(r) = L int.

15 Attached or nebular phase II The gas accretion rate is given by the ability of the envelope to radiate away energy so that it can contract, so that in turn new gas can stream in. The equation for the radius on the first line is a fit which reduces to RH if RH<<RA (the normal case) and RA if RA<<RH (at low masses). The factor kliss takes into account that not all gas is bound to the planet even when it is in the Hill sphere due to the shear in the gas disk. Lissauer et al The plot shows trajectories, in a frame rotating at the angular velocity of the planet, of tracer particles close to the disk midplane and within RH/2 of a 10 MEarth planet located at the origin. Blue circles (left) and dots (right) indicate bound particles. Orange circles or dots represent particles that leave the planet s Hill sphere and return to the circumstellar disk. These simulation indicate that kliss For the temperature we must take into account that the planet must be hotter in order to radiate into the disk. The temperature is calculated approximating the optical depth around the planet as and then the same approximative solution for the temperature structure as shown for the vertical disk structure in lecture 7. Finally, we take into account the background temperature in the disk.

16 Detached or transition phase Tneb Pneb RH Lint+Lshock P T Lint RCore R vff Once no solution satisfying the attached conditions exists any more the planet enters the second phase and contracts to a radius which is much smaller than the Hill sphere radius. This is the detached case of high mass, runaway planets. The planet now adjusts its radius to the boundary conditions that are now given by an accretion shock on the surface for matter falling onto the planet from the Hill sphere radius (or by conditions appropriate for the interface to a circumplanetary disk). For spherical accretion in free fall from RH we have Ṁ XY = Ṁdisk max v ff =[2GM (1/R 1/R H )] 1/2 M P = P neb + XY v 4πR 2 ff + 2g 3κ τ = max(ρ neb κ neb R, 2/3) Tint 4 int 8πσR 2 T 4 = (1 A)Tneb 4 + T int 4

17 Detached or transition phase II The first line, left side, says that in this phase, the gas accretion rate is no more controlled by the planetary structure itself, but by how much gas is supplied by the disk and can pass the gap formed by the planet. The right side is the velocity for free fall from RH onto the planet s surface at R. For the pressure in the second line, we have three contributions: the ambient nebula pressure (becoming quickly negligible compared to the other terms), the dynamic pressure due to the accretion shock (see lecture 4), and the photospheric pressure (see next). The optical depth is the bigger of the photospheric one (2/3) and the one due to the gas around the planet. For the temperature we take into account that the planet is still embedded in radiation field from the surrounding disk. The total luminosity coming from the planet is the sum of the intrinsic luminosity coming from the planet s interior, and the accretional luminosity coming from the shock. Hydrodynamical simulations in 2 or 3D Comparing these boundary conditions with the ones mentioned in lecture 4 for a growing star with spherical accretion, we see that we treat the planet here in a very similar fashion. The real situation is more complex, as accretion occurs via streamers and a sub-disk around the planet (for massive planets). Lubow 1999

18 Detached or transition phase III Lubow 1999 A high-resolution velocity map in the corotating frame of the planet and density distribution of the Roche lobe region around a 1 MJ planet. The planet is at location x=-1, y=0, and the star is located at the origin. The left (right) plus sign marks the L 2 (L 1) Lagrangian point. The left also shows the flow evolution of gas stream material. Material flowing along the left arrow first shocks, then orbits about the star, returns as material that flows along the two right arrows, and becomes accreted by the planet.

19 Evolutionary or isolated phase P Lint RCore T R Tequi stellar irradiation Once the nebula has disappeared, the planet evolves at constant mass. The outer boundary conditions are now (in the Eddington/gray atmosphere approximation): P = 2g 3κ Tint 4 = ) 1 2 T equi =280K ( a 1AU ( ) M M L int 4πσR 2 T 4 =(1 A)T 4 equi + T 4 int For the second line, we have assumed that the host star which irradiates the planet is a solar like star on the main sequence, because then, L goes approximately as Mstar 4. We have also assumed that the planet redistributes the stellar radiation over its entire surface. With an albedo A, this leads to a temperature T on the surface of the planet as also given on the second line. Here, we associate the surface of the planet with its photosphere, which is where the bulk of the radiation comes from and which corresponds to its visible surface. Our boundary condition approximates the photosphere with a single surface at optical depth τ =2/3. To get the pressure at this radius we use the optical depth: τ(r) = R κρdr κ ph where κph is the average value of opacity in the atmosphere which are the layers above the photosphere. R ρdr, (1)

20 Evolutionary or isolated phase II In the atmosphere we have hydrostatic equilibrium dp dr = GM R 2 ρ We can approximatively solve this in the thin, low mass atmosphere (acceleration g=cst): P(R) GM R 2 R ρdr (2) Combining eq. (1) and (2), gives for the radius where P = 2 3 GM κ ph R 2 τ(r) = 2 3 dary condi a pressure One also finds that the photosphere is the place where the mean free path of a photon and the atmospheric scale height become approximately equal. The photosphere is therefore indeed the place from where photons directly escape from the planet. Note that our treatment of the outer boundary conditions becomes inadequate for a strongly irradiated planet like a Hot Jupiter.

21 1.3 Planetesimal-envelope interaction

22 Equation of motion The third and last module of classical giant planet models describes the interaction of planetesimals and the envelope. This module links the solid accretion and the envelope structure modules. The capture radius enters the solid accretion rate, and the energy and mass deposition obtained from this module enter back into the calculation of the envelope structure, in the equation for the luminosity we have seen before. It also tells us how much mass is deposited in the envelope, and how much directly reaches to the core. The models calculating the interaction are complex models on their own, and include gravity and gas drag, thermal ablation as for shooting star, and aerodynamical disruption inspired by the modeling of the destruction of comet Shoemaker-Levy 9 in Jupiter s envelope. As mentioned above, the kinetic energy of the planetesimals represents an important energy source and hence is an important factor in determining the internal structure of the growing planet. Infalling planetesimals are slowed down by gas drag. The equation of motion for a planetesimal of mass m to be solved in order to compute the slowing down of these planetesimals is given by (planetocentric reference frame, 2 body): where CD the drag coefficient which can be written as a function of the Reynolds number and the Mach number. ρ is the density of the gas in which the planetesimal is plowing and is obtained from the calculations of the internal structure of the planet as is the mass M(r) of the planet internal to the position r of the planetesimal. S is the cross section of the planetesimal, π R 2.

23 Shock waves As the planetesimal plows through the envelope, pressure and temperature are increasing. Eventually, two effects can lead to its destruction: thermal ablation and mechanical mass loss. These effects therefore determine how deep the planetesimal is able to penetrate, thus determining where the energy and where in the planet the energy and the debris of the planetesimals are deposited. Note that in order to compute the energy input rate, the hydrodynamics of the region surrounding the planetesimals needs to be known. The most important hydrodynamic regime for massive impactors is a hypersonic, highly turbulent continuum flow. This means that strong shock waves form. Especially, the bow shock temperature needs to be computed as the radiation field generated by it can be an important heating factor of the planetesimal as temperatures in excess of K are reached. Consider only flow along the stagnation line (normal shock wave jump conditions). Known Preshock cond. (free flow conditions) c,v c,v p,,w p,,w supersonic Unknown Postshock conditions subsonic L 1 = L 2 = L 3 = Continuity Momentum Energy Here, w is the enthalpy per mass w=e+p/ρ (e=internal energy per mass). This coupled set of equation can be solved numerically. (For an ideal gas, an analytical solution exists, but not usable, due to strong non-ideal effect, like dissociation and ionization at the shock.)

24 Shock waves II The plot shows the postshock temperature as a function of ambient gas density and velocity of the impactor. For reference, the escape velocity from Jupiter s surface is 60 km/s. C. Mordasini 2004 Note: The post-shock temperatures are for a SL9 type impact of order Jupiter K This is much more than the pre-shock ambient temperature assumed to be T1=81 K. For impacts into Earth (different EOS), at vesc, Tps K 3 [g/cm ] The table compares the post-shock temperature using a realistic equation of state EOS SC, and an ideal gas with various γ=cp/cv. It is clear that one cannot use ideal gas, especially as the shock radiation goes as Eshockrad Tps 4

25 Thermal ablation A part of the energy dissipated by the drag is used to heat up the impactor. This leads do the classical equation for thermal ablation/mass loss (Öpik 1958, Bronsthen 1983) The dependence on the third power C H of : Heat the velocity transfer comes coef. from the fact that power is given as. Qabl is the heat of ablation (vaporization or melting). Q abl : Heat of ablation The heat transfer coefficient CH gives the fraction of the total dissipated energy used to heat the body. In the hypersonic continuum flow, the most energy comes from the radiation of the shockwave, so that one writes for the energy input per second and unit surface: The energy input onto the body is equal to the sinks, which are re-radiation, transport into the interior, and mass loss. At high energy inputs the mass loss term dominates (ablative cooling). Here, the Ai are geometrical factors, κ the thermal conductivity and ε the emissivity. The surface temperature is linked to the mass loss rate via the Knudsen-Langmuir equation:

26 Mechanical destruction Besides thermal effects, mechanical effects can result in the more or less rapid destruction of the planetesimal. The main effect to consider is the large pressure difference between the front of the planetesimal (stagnation point) and the back where the pressure almost vanishes. This pressure difference (if large enough) leads to a lateral spreading of the body (the so called pancake model) and eventually its disruption. This is due to Rayleigh-Taylor (RT) instabilities which grow due to the deceleration on the front side of the body. Mechanical effects start if and (big bodies) Afterwards, one (approximately) assumes a fluidized impactor. The fluid reacts to the pressure with spreading: C. Mordasini 2004 This leads to a radius increase as (Zahnle et al. 1992) R w b Ṙ p stagn At the same time, RT fingers grow into the front of the impactor. v RT is a general hydrodynamic instability that develops when a dense fluid (the impactor) is pushed (here decelerated) by a less dense fluid (the gas).

27 Mechanical destruction II During the later, nonlinear phase, large perturbations ( fingers ) dominate the RT instability. Here hinst, the mixing length i.e. the depth to which RT fingers have grown into the dense fluid is: Korykansky et al The combined action of flattening and the growth of the RT finger eventually lead to the fragmentation of the body and a fragmentation cascade (Mordasini & Benz 2004). This results in a fast destruction of the impactor (terminal explosion). R w= h inst R cloud Rfrag wfrag b b h inst p stagn p stagn v Ṙ v Ṙ w cloud Mordasini & Benz 2004

28 de/dr [erg/cm] Energy and mass deposition profile The plot shows the fate of planetesimals as a function of protoplanet s core (and thus envelope) mass, and initial planetesimal size. As the core grows, it surrounds itself with an increasingly thick envelope, which is increasingly difficult to penetrate. This leads to an auto-regulation by self shielding of the maximal core mass that can be built up by planetesimals directly hitting the core. Later on, other processes as sedimentation of heavy elements or core dissolution can still modify the core mass. Mordasini et al Mcore=15 Mearth Mevn = 9 Mearth Rcore= 2x10 9 cm Shoemaker - Levy 9 peak energy deposition R [cm] In these impact, huge energies are released for 100 km or 1000 km planetesimal as we can see here, which are several orders of magnitude larger than for SL9. And the interesting point is that these impact must happen often, of order 10 times per year. An open question is if this could lead to the formation of a hot plume as seen for SL9, so that some energy could directly leave the envelope. This could in turn reduce the formation time and might be observable from Earth.

29 Energy and mass deposition profile planetesimal energy deposition convective zone The two plots show the internal structure of a low mass growing planet. The planet has a solid core of Mcore= 8 Mearth and a gaseous envelope of Menv= 1.04 Mearth. The corresponding radius is cm. Note: Sudden increase in L at the location of planetesimal destruction. Debris is assumed not to sink to the core. Entropy constant in convective regions. In the detached/runaway phase, the scale height in the planet s structure become much smaller, so that the planetesimal destruction happens in a terminal explosion. Energy depostion [erg/cm] 1e+25 1e+24 1e+23 1e+22 1e+21 1e+20 1e+19 1e+18 1e+17 1e Masse [Me] Mass depostion [g/cm] 1e+13 1e+12 1e+11 1e+10 1e+09 1e+08 1e+07 1e Masse [Me] Mcore=22 Mearth, Mtot=82 Mearth The energy and mass is initially deposited into a shell of about 3 Mearth width, far above the core. Debris is here assumed to sink to the core. C. Mordasini 2011 Lumi [LJ] sinking contribution L direct contribution L total L Masse [Me]

30 Drag enhanced capture radius We have seen in earlier lectures that the increase of the cross section of a protoplanet over its pure geometrical one by gravitational focussing is extremely important. If a planet has additionally a sufficiently massive envelope, its capture radius is even more increased, as gas drag leads to a loss of kinetic energy of the planetesimal. The planetesimal then gets captured into the Hill sphere of the protoplanet, spirals in, and eventually hits the core (or gets destroyed in the envelope). The presence of the envelope enhances the effective capture radius of the planet over the core radius by about 1 to 2 orders of magnitude. This is important because of this quadratic dependence in the accretion rate: dm Z /dt R capt 2 The capture radius is dependent on the envelope structure, the planetesimal velocity, size and composition. One obtains the capture radius by using an energy criterion. For various initial impact parameters, it is tested whether the planetesimal looses so much energy during its passage through the envelope so that it cannot leave the Hill sphere any more and go back into a heliocentric orbit (Podolak et al. 1988): E esc = 3mΩ 2 R 2 H = mgm R H

31 Drag enhanced capture radius II The plot shows trajectories of planetesimals under the influence of drag and gravity. These calculations were made in the 2 body approximation.the critical trajectory is shown in blue. For this case, the apocenter of the planetesimal s orbit just touches the Hill sphere of the protoplanet. The capture radius is finally defined as the pericenter distance at first passage for critical impact parameter parameter/critical trajectory. Mordasini & Benz 2004 When also the solar gravity is included as here, the trajectories becomes more complex, although the final phase well within the Hill sphere of course remains the same as before. The reference system is co-rotating with the protoplanet around the sun.

32 1.4 Gas accretion beyond the critical mass

33 Disk limited gas accretion rates Beyond the critical core mass (cf. below) gravity overcomes the gas pressure and in the quasistatic picture the envelope contracts slowly (the kinetic energy is negligible). The contraction delivers additional energy, increasing the luminosity, and the force balance can be achieved again. The contraction time is given by the Kelvin-Helmholz timescale In most cases, it is sufficient to treat the growth of the planet as a quasi-static contraction, even in runaway. Beyond the critical core mass the Kelvin-Helmholtz timescale typically becomes very small. In such a situation the planet accretes all the material that it can get. The gas accretion rate is thus no more controlled by the structure of the planet, but by the disk, and thus by processes like a local reservoir, the viscous accretion in the disk, and gap formation. Just after the beginning of runaway, we can assume that the accretion onto the planet is still spherically symmetric. Assuming a spherical, homogenous, unimpeded accretion flow (strictly speaking only valid if RH<<H), and saying that gas is capture at some radius Rout, we find a Bondi like accretion rate: We can estimate the density, cross section and relative velocity like this:

34 Disk limited gas accretion rates II This lead to a gas accretion rate At later stage, the gas flow in the disk itself becomes the limiting quantity, and what fraction of it can flow through the forming gap. The accretion rate in the disk (taking into account nonequilibrium effects i.e. the flux varies with distance r from the star ) is given as 3π νσ + 6πr νσ r Of such a gas flux in the disk towards the planet, about 75 to 90% are accreted onto the planet, while the rest streams by and is accreted onto the star (inside the radius of velocity reversal). Gap formation Lubow 1999 Additional complications comes from the fact that a planet massive enough to have RH>H, the disk vertical scale height (at 5 AU this minimal mass if of order the Jovian mass) starts to open tidally a gap around its orbit. This reduces the gas accretion rate roughly as ( ) 1/3 Mp e M p 1.5M J +0.04, M J Veras & Armitage 2004 It was originally thought that gap formation leads to an auto-regulation of the maximal masses to which planets can grow. It is now understood that the gap formation limit is not the necessarily the criterion (eccentric instability).

35 2. Early models of giant planet formation

36 Early models Mizuno used the hydrostatic version (and L = const) of the planetary structure equations to show that there exists a critical core mass beyond which no solutions to the hydrostatic equations can be found. In his calculations all in-falling planetesimals reach the core, and planetesimal accretion is the only source of luminosity. This means that this is not a real evolutionary sequence, but a sequence of independent (quasi) static solutions. In this early models, the core accretion rate is regarded as a free parameter, too. Mizuno has shown that when the core keeps getting larger (more massive), it will reach a point where no solution to the hydrostatic calculations can be found anymore. The largest possible core mass sustaining a static atmosphere is called the critical core mass. He did the calculations for different amounts of dust in the atmosphere which influences the opacity and thus the radiative transport. The result of Mizuno s calculations is shown in the figure to the right which was done for the position of Jupiter. For any value of the dust opacity reduction factor f (f=1: full interstellar opacities), there is a critical core mass. For f=1, the critical mass is about 12 MEarth. Furthermore, he showed that the value of this critical core mass is only very weakly dependent on nebula parameters. Mizuno critical core mass no solutions

37 Analytical toy model Mizuno s calculations were very successful because they could explain why Jupiter and Saturn have similar core sizes: the critical core mass was similar to the observed cores. Today, however, the observations have refined the core masses of the jovian planets and Jupiter and Saturn appear to have different core sizes after all. Stevenson (1982) (cf. Armitage 2007) showed that the existence of a critical mass can be reproduced in a toy model in which energy transport is radiative diffusion only (which is not usually realistic). Consider a core of mass Mcore and radius Rcore, surrounded by a gaseous envelope of mass Menv. The envelope extends from Rcore to some outer radius. If the envelope is of low mass, then the largest contribution to the luminosity is from accretion of planetesimals onto the core. This yields a luminosity L = GM corem core (1) R core L is constant across the envelope. If we assume that radiative diffusion makes the energy transport, then the structure of the envelope is given by the equations of hydrostatic equilibrium and radiative diffusion: dp dr = GM(r) r 2 ρ (2) We can combine these equations into L 4πr 2 = 16 3 dt dp = σt 3 κ R ρ dt dr (3) 3κ R L 64πσGMT 3 (4)

38 Analytical toy model II We can separate the variables to integrate this equation inward from the outer boundary, making the approximation M(r) Mt (the total mass) and taking L and also κr to be constants (!), i.e. T T 3 dt = 3κ P RL dp. T disk 64πσGM (5) t P disk Well inside the planet, we should have T 4 Tdisk 4 and that P Pdisk, so the integral is approximately T 4 3 κ R L P. (6) 16π σgm t This is the so called radiative zero solution, linking T and P at any point inside the envelope. We replace P in equation (6) with an ideal gas equation of state P = k B µm p ρt, giving us an expression for T 3. We put this expression back into equation (3) and trivially integrate again with respect to radius to obtain the temperature as function of radius ( ) µmp GMt T ( ) k B 4r (8) and, with eq. (6) and (7), also the density as function of radius. (7) ρ 64πσ 3κ R L ( µmp GM t 4k B ) 4 1 r 3. (9)

39 Analytical toy model III With this density profile the mass of the envelope is obtained easily ) ( M env = Rout R core = 256π2 σ 3κ R L 4πr 2 ρ(r)dr ( ) 4 µmp GM t ln 4k B ( Rout R core ) (10) The right-hand-side has a strong dependence on the total planet mass Mt and a weaker one on the core mass via the expression for the luminosity (eq 1) L M 2/3 coreṁcore. In principle there are more dependencies since Rout is a function of Mt (Hill sphere radius) and Rcore is a function of Mcore, but these enter only via a logarithm and can be ignored. As M core = M t M env (11) we find with eq. (10), putting all constants and near-constants into C, but keeping the formula explicit for the opacity and the core accretion rate that ( ) C M 4 M core = M t t κ R Ṁ core Mcore 2/3 (12)

40 Analytical toy model IV Armitage 2007 Solutions to equation (12) are plotted in the figure, showing the total mass as a function of core mass.. Blue is for a higher planetesimal accretion rate than red. The critical core mass is shown as the vertical dashed line, while the black solid line would show the curve if Mtotal=Mcore. One sees that with increasing core mass, the mass of the envelope increases. One also sees that for fixed core accretion rate, there is a maximum or critical core mass Mcrit beyond which no solution exists. The physical interpretation of this finding (the reason of which remains somewhat unclear even within this toy model) is that if we try to build a planet with a core mass above the critical mass, hydrostatic equilibrium cannot be achieved in the envelope. Rather the envelope has to contract (generating luminosity in this way to counteract gravity), and further gas will fall in as fast as gravitational potential energy can be radiated. This finding illustrates why the core accretion model is also called the nucleated instability model, but this name is regarded as obsolete, as most calculation show that no hydrodynamical instability occurs even beyond the critical mass, just a rapid contraction.

41 3. Classical models

42 Classical models Compared to the early models, the classical models (in particular Pollack et al. 1996) calculate the core accretion rate self-consistently. Accretion occurs from a feeding zone with a width depending on the planet s mass (proportional to the Hill sphere radius). As the core grows, the planetesimal surface density decreases. real evolutionary sequences (i.e. they include the TdS/dt term). They however still assume that: the protoplanetary disk giving the boundary conditions is static in time. the formation occurs in situ (no migration). The classical work of Pollack et al was a detailed model that we may call the baseline formation model. It is useful to study this model because subsequent works have revisited and improved various aspects of it. Their initial conditions were -formation at 5.2. AU. -10 g/cm 2 initial planetesimal surface density. This corresponds to about 4 x MMSN km planetesimals. -constant core density 3.2 g/cm 3. -rapid planetesimal accretion rate (similar as in runaway). -full ISM grain opacities. Here we study a similar model, except for: a) the core density is calculated in a realistic way b) the opacity is only x ISM. Both these setting make the simulation more realistic. The maximal gas accretion is 0.01 Mearth/yr. Once a mass of 1 MJ is approached it is artif. stopped.

43 Mordasini & al 2011 Attached Phase Jupiter in situ formation Giant planet formation traditionally focus on Jupiter, because it is the giant planet with the best observational constraints. The same model must however also explain the other (exo)planets! Accretion of the core Fast mass growth & Isolation Gas accretion allows core growth Crossover mass Detached M=cst Evolution High core accretion rate Low accretion rate Gas faster than solids End of accretion (disk disappears) Limiting dm/dt gas (disk accr. rate) approach of runaway Detached Evolution Total Gas accretion Core Gas Core accretion

44 Jupiter in situ formation II: phases The results of Polllack showed that during the attached phase, there are three important subphases I, II and III. In phase I, a solid core is built up. The phase ends when the planet has exhausted its feeding zone of planetesimals, which means that the planet reaches the isolation mass. The planet consists essentially in a solid core surrounded by a very small envelope. In phase II, the planet must increase the feeding zone. This is achieved by the gradual accretion of an envelope: An increase in the gas mass leads to an increase of the feeding zone of solids. Therefore the core can grow a little bit. This, and the cooling of the gas, leads to a contraction of the external radius of the envelope. Gas from the disk streams in, leading to an increase of the envelope mass and so on. In that time, the accretion of planetesimals makes the dominant part of the planet s luminosity. This phase takes most of the formation time. mass of gaseous envelope: MG size of feeding zone: FZ ( M 2/3 planet ) external radius of envelope: Rext mass of solids accreted: MS

45 Jupiter in situ formation III: phases In phase III, runaway gas accretion starts. It starts after the so called crossover mass is reached, i.e. the moment when the core and the envelope mass have become equal. We associate (approximatively) the crossover mass with the critical mass seen in the static calculations. At this stage, the radiative losses from the envelope can no more be compensated for by the accretional luminosity from the impacting planetesimals alone. The envelope has to contract strongly, so that the new gas can stream in, which increases the radiative loss as the Kelvin Helmhotz timescale strongly decreases with mass, so that the process runs away, building up quickly a massive envelope. The crossover core mass is of the order of Earth masses, but can be 1 to 40 in extreme cases.

46 Jupiter in situ formation: radius Attached Phase Total Capture Core Detached The plot shows the total/outer radius, the capture radius and the core radius as a function of time. Evolution Mordasini & al 2011 Note During the attached phase, the total radius is very large, as it is approximately equal to the planet s Hill sphere radius. The capture radius is about 1 to 2 orders of magnitudes larger than the core radius in the attached phase. Later, the capture radius and the total radius are about equal: The planet develops a well defined surface (small scale height). In the detached phase, a rapid contraction occurs. Even if the contraction is strictly speaking not a dynamical instability, one still speaks of the collapse phase. The total radius decreases to initially about 2 RJ. During the detached/runaway phase, the core doesn t get bigger even though it grows by about 15 Earth masses. This is due to the enormous pressure exerted by the gaseous envelope, which increases the core s density.

47 Jupiter in situ formation: luminosity Attached Phase Total Detached Evolution The plot shows the total luminosity (intrinsic plus shock), the intrinsic (cooling plus core) and the core luminosity alone as a function of time. Mordasini & al 2011 Intrinsic Core L Jup = L Note During phase I, L is given by Lcore. During phase II, the envelope starts to contribute. In the detached phase, Lshock is dominant. The core luminosity vanishes. In the evolutionary phase, Lshock vanishes. The luminosity now comes from the cooling and contraction of the envelope. The maxima of the curve correspond to moments when much gravitational energy is liberated. Planets are much more luminous when they are young! Detectable?

48 Animation of the internal structure In this animation we see the evolution of the internal structure for a planet with a final mass similar to Jupiter. The simulation is similar as the one seen before. The four panels show the pressure, the temperature, the density and the mass inside a given radius. Also plotted are the capture radius, and the minimum of the Hill sphere radius and the accretion radius. The inner end of the lines correspond to the envelope-core boundary, and the outer ones to the surface of the planet. For the pressure, temperature and density, the central values are given, and for the mass the mass of the core, and the total mass of the planet. We start with a planetary seed of 0.6 Earth masses. As we can see, is at the beginning the total mass almost equal the core mass. The capture radius is equal the core radius. Later, during phase II, an envelope is gradually accreted. After the crossover point, the evolution speeds up. Once runaway is reached, the planet contracts. Although fast compared to the rest of the evolution is the velocity of contraction of the surface much smaller than free fall velocity: no instability occurs, but a fast contraction. We also see that mainly the outer tenuous envelope collapses onto the already quite dense inner part, which is already hard to compress. The radius, core temperature and pressure at the end of the simulation are already comparable to those found today in Jupiter, even if over the following billion years, the planet will continue to cool and contract slowly, as it shown in the calculations of Guillot et al. 1995, until it reaches the values we observe today.

49 Evolution of the internal structure Pressure Temperature R capt Min(R acc,r H ) Guillot et al Density Mordasini & al 2009 Mass inside R

50 Timescale problem Σ (5.2 AU) = 10.0 g/cm 2 (4 MMSN) With the standard model of Pollack et al. 1996, for full grain opacity, one forms Jupiter in 8 Myrs in a 4 x MMSN, with a core-envelope ratio compatible with internal structure models. The timescale for formation is essentially given by phase II. It is comparable to or longer than observed disk lifetimes. Alibert & al 2004 Σ (5.2 AU) = 7.5 g/cm 2 (3 MMSN) The formation timescale is extremely sensitive to the planetesimal surface density. In a 3 x MMSN, the formation timescale is much higher than disk lifetimes. The formation timescale can also be shortened by a higher solid surface density, but this leads to too large core masses. This result lead to the (preliminary) conclusion that maybe another formation mechanism is necessary. However, migration and low opacities (as we have seen) change the game.

51 4. Improved and extended models

52 Improved models: 3D calculations Even though the inner part of a planet is roughly spherically symmetric, planets are not really 1D in their outer parts. Recently, 3D self-gravitating radiation hydrodynamical models of gas accretion with a realistic core size became possible. (Ayliffe & Bate 2009, Paardekooper & MellemaGas 2008) accretion onto planetary cores 15 Ayliffe & Bate 2009 Ayliffe & Bate 2009 (3D) Papaloizou & Nelson 2005 (1D) 1D, 15 Mearth 3D, 10 Mearth Figure 12. Surface density plots for locally-isothermal 1D, 5calculations Mearth(left), and self-gravitating radiation hydrodynamical calculations and standard IGO (right) calculations with our standard protoplanetary disc surface density. From top to bottom the protoplanet mass and 333 M respectively. The radiation hydrodynamical calculations use protoplanet radii of 1% of the Hill radii, while the locallyuse 5% of the Hill radii. Note that with radiative transfer and standard opacities, the spiral shocks in the protoplanetary disc are m a locally-isothermal equation of state, while using radiative transfer with reduced grain opacities results in intermediate solutions be optically thick and are able to radiate more effectively than with standard opacities. Figure 15. Accretion rates versus the gas mass accreted by planetary cores for 1% (thin lines) and standard (thick lines) grain opacities. We plot the results of Papaloizou & Nelson (2005) for 5 M (dashed lines), and 15 M (dot-dashed lines) cores. The solid lines show the extrapolations of our 10 M core results. Our 10 M core accretion rates lie between the accretion rates obtained by Papaloizou & Nelson for their 5 M and 15 M cores. The effect of the reduced opacities is also consistent with Papaloizou & Nelson s results. These simulation find consistent accretion rates in 1D and 3D. c 0000 RAS,

53 Improved models: Opacity Already form the early models we know that the grain opacity plays an important role in for the formation of a giant planet. People have therefore studied the effects of a reduced grain opacity. Grains are dominant for the opacity for T<~1700 K. Simulations combining planet growth and grain evolution find that in the envelope, grains coagulate (due to the higher densities compared to the background nebula), sink down and get destroyed in the hotter parts of the envelope, reducing the opacity. Opacity from Bell and Lin as a function of temperature and gas density: 10 2, 10 4, 10 6 and 10 8 g/cm 3. Solid: full ISM. Dotted: 2% ISM Mass as a function of time for a Jupiter in situ formation simulation for grain opacities of 10 4, 10 3, 10 2, 10 1 and full ISM. Region 1: ice grains. Region 2: ice grains melt Region 3: metal grains. Region 4: metal grains melt Region 5: molecules Region 6: H- Region 7: Kramer's law (bound-free and free-free absorption). Region 8: electron scattering Mordasini & al 2011 The lower opacity leads to a weaker thermal support of the envelope. In other words, the liberated gravitational energy is more easily transported out of the envelope, so that it can contract. Thus, a reduced grain opacity greatly speeds up the gas accretion timescale (phase II).

54 Disk module: Solids + gas (α-model). In contrast to classical models, the disk model yields time varying boundary conditions for the envelope, and the gas accretion rate in runaway. It is also necessary to calculate da/dt. Extended models: Migration & disk evolution One of the strongest shortcoming of the classical models is that they neglect that besides growth, other important processes occur in the disk with similar timescales: τmigration τformation τdisk evolution Modern core accretion models are therefore extended to include in a self-consistent way also these effect. These models currently are the most comprehensive models of planet formation. Alibert, Mordasini, Benz 2004 Migration module: variation of the orbital distance due to type I and II migration. Accretion module: Core and envelope growth, as in classical models.

55 Extended models: disk evolution & migration II With a planetesimal surface density same as before, a small planetary seed is started at 8 AU and is allowed to grow and migrate in a disk which is time evolving with α= Alibert, Mordasini, Benz 2004 no migration feeding zone without migration feeding zone with migration Note that in this case the formation time has been shortened by a factor of 30 (!) and is now well within the disk lifetime (the previous case without migration and disk evolution is show for comparison as thin black line). As is obvious from this figure, the reason for the formation speed-up is simply the absence of a long protracted phase 2. Migration and disk evolution prevent the growing planet from being starved of planetesimals and allow to reach the critical mass in a much shorter time. As mentioned above, the migration prevents the feeding zone of the planet to become empty. This is best illustrated by the picture above which plots the surface density of gas (solid lines) and solids (dashed lines) as a function of distance to the star initially (blue lines) and at the end of the simulation (red lines). Note that the solid surface density has been multiplied by a factor of 70. The position of the planet is marked by the large black dot. Note the difference in size of the feeding zone in the simulation allowing for migration and the one in which migration was turned off (shadowed area).

56 Extended models: disk evolution & migration III Jupiter slow type I fast type I species measured computed Saturn species measured computed Alibert et al Note: even though these models reproduce imporant observational constraints as the mass, position, formation timescale, bulk composition and enrichment, they still must be regarded as strong idealization (disk, migration, opacity, planetesimal accretion rate, gravitational interaction..)

57 Questions?