Orbital migration. Lecture 14. Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini & PD Dr. H. Klahr. Based partially on script of Prof. W.

Transcription

1 Lecture 14 Orbital migration Klahr & Kley Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini & PD Dr. H. Klahr Based partially on script of Prof. W. Benz Mentor Prof. T. Henning

2 Lecture 14 overview 1. Basic considerations 2. Impulse approximation 3. Gap formation and type I migration 4. Resonant torques 5. Type I migration 6. Recent results on type I migration

3 1. Basic considerations

4 Orbital migration We have learned in the last lecture why one could expect from planet formation theory (necessity of a 10 Mearth core formed during the disk lifetime, i.e. rapidly) that giant planets should form in a region outside the iceline, i.e. at ~3-5 AU. The fact that the giant planets are found in our Solar System at such a distance and further out was regarded as a good confirmation of this theory. The detection of the first extrasolar planet by Mayor and Queloz in 1995, which was a giant planet at an orbital distance of just 0.05 AU was therefore for many a major surprise. ApJ, 241, 425 (October 1, 1980) It let to the revision of the standard picture of planet formation (~in situ formation), and the insight that the orbital migration of planets represents a key aspect of the theory which must be included. Ironically, migration was discovered 15 years before the first exoplanet by theoretical considerations.

5 Basic mechanism and types The presence of a planet orbiting the star creates a non-axisymmetric time varying gravitational potential. The gas reacts to this perturbation in the potential by the formation of density waves. These density waves create additional perturbations in the potential which are seen by the planet as well. The torques originating from these perturbations change the planet s angular momentum and give rise to migration. We here consider migration due to the gravitational interaction with the gaseous disk only. Migration can also occur due to the interaction with the planetesimals disk. Orbital decay due to direct gas drag is negligible at planetary masses. Note: - for small mass planets the density waves propagate through the disk - for larger mass planets, a gap opens in the disk Type I migration migration mode of small mass planets, no gap surface density Type II migration migration mode of large mass planets, with gap Simulations by P. Armitage The movie shows the transition by ramping up the planet mass.

6 Inertial and rotating frame These hydrodynamic simulations show the development of density waves once in the inertial and one in the co-rotating (with the planet) frame. The basic mechanism of angular momentum exchange is that the heading density enhancement pulls the planet forward (leading to outward migration, while the trailing density enhancement pulls the planet backwards (leading to inward migration). forward pull: Outwards migration inertial frame rotating frame backward pull: Inwards migration Simulations by C. Baruteau

7 Net torque The net torque is the sum of all the torques. For most of the disk structures, this net torque is such that it induces inward migration. W. Kley

8 2. Impulse approximation

9 Impulse approximation A simple approach to compute the torque exerted on a planet is the so called impulse approximation (Lin & Papaloizou 1979). We consider the gravitational interaction between the planet and gas flowing past. We neglect the effect that we are in a corrotating frame (around the sun), and treat the interaction in the two body problem. Here we follow the derivation in Papaloizou & Terquem 1999 and Alexander v ϕ v ϕ We start by deriving the general expression for the gravitational deflection angle for the case of a body of mass m, initial relative velocity v and an impact parameter b encountering a big body with mass M. The force perpendicular to the initial velocity means for v s F = m dv dt = v = dv = 1 m F dt r " From the geometry of the encounter ( ) ( b GMm F = F sin θ = F = r r 2 )( ) b r

10 Impulse approximation II For small angles, we can use the Born approximation, where for the total velocity vinit vfinal v v dt = vdt = ds = dt = ds v )( )( v = 1 F dt = 2 ( )( )( ) GMm b 1 Thus m m 0 r 2 ds r v Since Defining r =(s 2 + b 2 ) 1/2 v = 2GM v x = s/b 0 b 2GM ds = (s 2 + b 2 3/2 ) v 0 ds/b (1 + (s/b) 2 ) 3/2 allows us to evaluate easily the integral to find the traversal velocity v = 2GM dx 2GM x = vb 0 (1 + x 2 3/2 ) vb (1 + x 2 ) 1/2 = 2GM vb For the (small) angle we have ϕ = v /v ϕ = 2GM v 2 b 0 and thus we finally find for the angle

11 Impulse approximation III We can now use our results form the previous page to calculate the momentum exchange. For this, we now associate the velocity v of the body with mass m with the velocity difference between a gas parcel and the planet and define for the changes in velocity: v ( v = v gas v p ) The change in the perpendicular component of the velocity is thus given as before by: δv = 2GM p b v Note that since this velocity change occurs radially, it does not correspond to any change in angular momentum. Since in a two body problem an encounter conserves energy, a change in the perpendicular component also implies a change in the parallel component. From the conservation of energy (and geometry) we have v 2 = δv 2 +( v δv ) 2 Evaluating this, and neglecting the quadratic term in (small deflection) δv 1 2 v 2GMp b v 2

12 Impulse approximation IV The change of angular momentum of the gas parcel associated with must be balanced by the change of angular momentum of the planet. For a planet with a semimajor axis a, this implies a change in specific angular momentum: j = a.δv = 2G2 M 2 p a b 2 v 3 Note that gas exterior to the planet is overtaken by the planet and therefore interactions lead to a net loss of angular momentum for the planet (a gain for the gas) while the gas interior is overtaken by the planet and therefore interactions lead to a net gain in angular momentum (a loss for the gas). The interaction is frictional. The net direction of migration thus depends on the difference between the interior and exterior torque. Note also that the trajectory prior to the local scattering is assumed to be linear (corresponding to a circular orbit). Subsequent returns of disc matter on circular orbit are necessary to make the frictional interaction persistent. The scattering itself, of course, disturbs this geometry, so that there is an implicit assumption that dissipative or other processes are at work to restore circular orbits for returning trajectories. The hypothesis is that disc viscosity is able to provide such an effect.

13 Impulse approximation V The net torque will be the sum of all the torques (inside and outside) and will depend on the exact structure of the disk. To compute this net torque, let us integrate the single particle torque over all the gas in the disk. Let us consider a small annulus outside the orbit of the planet at distance a. The mass in the interval (b;b+db) is given by dm 2πaΣdb If the planet has an orbital frequency and the gas has, the gas parcel suffers impulses separated by 2π t = Ω Ω p For small displacements b<<a, a first order expansion of the angular frequencies yields: Ω Ω p dω p da b 3Ω p 2a b The total temporal change of the angular momentum of the planet must be the integral over the angular momentum transfer of all interacting gas parcels per unit time: dj dt = j.dm t We can eliminate v by assuming quasi Keplerian orbits and again a first order expansion v Ω p ab =(3/2)Ω p b.

14 Impulse approximation VI Substituting yields dj dt = 0 8G 2 MpΣa 2 db 9Ω 2 p b4 This integral diverges at the inner boundary, but if we specify some minimum impact parameter bmin>0, we easily find (for a constant surface density) dj dt = 8G2 M 2 p Σa 27Ω 2 p b3 min Hence, this formalism explicitly neglects the presence of the so-called co-rotation torques. Therefore, the above analysis is applicable only in a nonlinear regime when there is a significant gap, corresponding to type II migration. Values of bmin are between the Hill radius (for low-mass planets) and the disc scale-height H (for massive planets). Then, one finds a torque which agrees approximately with that obtained from more detailed analyses. While there exist more rigourous methods to derive the net torque, this expression yields the proper scaling. In particular, note that: - the torque scales with the surface density of the disk - the torque scales with the square of the planet mass - the migration time scale varies with the planet mass as τ mig = J dj/dt For fixed disk conditions, more massive planets migrate faster. 1 M p

15 Impulse approximation VII The torque changes the angular momentum, the migration rate of a planet is thus Using our earlier results for dj/dt, we can rewrite the angular momentum exchange in terms of the angular momentum injection rate for a given impact parameter where the sign accounts for the fact that material outside injects angular momentum while material inside takes away, f is an order unity constant (replacing the 8/9 in our earlier calculation) to take into account the fact that we are in a rotating frame, and where Here, the second term means that we exclude material closer than one disk scale height from the torque. Clearly, R-a is the same as our impact parameter b. We now calculate the total torque and thus the migration rate by integrating as before over the disk, using as integral variable R instead of b, and taking into account that the surface density varies as function of R:

16 Impulse approximation VIII We now calculate the total torque and thus the migration rate by integrating as before over the disk, using as integral variable R instead of b, and taking into account that the surface density varies as function of R: For conservation of angular momentum, we must take into account that the angular momentum taken away from the planet is added to the disk, and vice versa. If we describe the evolution of the disk as the one of a rotating viscous fluid, this leads to an additional term in the master equation for the evolution of the surface density which we have seen in earlier lectures: viscous evolution planet migration When we have some mean to specify the viscosity, we now can solve a closed system of equations coupling planet migration and disk evolution.

17 Impulse approximation IX Lin and Papaloizou 1986 were the first to do such calculations. Planets are inserted at different positions into a disk that is allowed to evolve viscously (outward spreading and inward accretion). The upper plot shows a planet starting inside the radius of maximum viscous couple (or velocity reversal). It is migrating inward. The lower planet is starting outside RMVC. The plot on the right shows the corresponding semimajor axis of the planets (solid lines) and of fluid elements starting at the same position (dotted) as a function of time. Note -for the masses considered here, a gap is formed quickly. -planets follow the viscous evolution of the disk (outward, inward migration). The planet f migrates first outwards and then inwards, as it is overtaken by RMVC.

18 Impulse approximation X The plot shows the semimajor axis of planets (solid lines) as a function of time for three different starting location. The left plot is for planets with a mass such that is equal to 5 (higher planet mass), while on the right it is equal to 80 (lower planet mass). If B>>1, then the protoplanet s moment of inertia is small, so that it is essentially carried along by the disk. For relatively small B, the protoplanet provides a large reservoir of angular momentum, and the evolution of the disk has little effect on the protoplanet s orbit. This means that when the planet becomes more massive than the local disk, it partially decouples. This is particularly important at small R (see also a vs b, and c vs d on in the last figure).

19 3. Gap formation

20 Gap opening We have seen previously that angular momentum is transported from the inner part of the disk to the planet and from the planet to the outer part of the disk. Hence, gas inside the planet looses angular momentum and moves inwards while gas outside gains angular momentum and moves outwards. For this mechanism to result in the opening of a gap, two conditions have to be met. Condition I (thermal condition): Hills sphere of a planet must be comparable to the disk scale height. If this is not true the disc will be able to accrete past the planet away from the disc midplane. This can be expressed by the condition: 1/3 Mp r H = r p H 3M Which implies a mass ratio planet/star of: q = M p H 3 M r 3 p =3h 3 p Typically the disk aspect ratio is h 0.05 and q corresponding to M > 0.13 MJupiter.

21 Gap opening II Condition II (viscous condition): Viscous effect must not be able to close the gap. This can be expressed by the condition: In terms of torque, this condition is written Or recalling previous expressions: 8 27 dj dt With ν = αc s H, and b min = R H Typically h 0.05, α = 10 2 τ close τ open LR G 2 M 2 p r p Σ 9Ω 2 pb 3 min we get: dj dt q 243π 8 αh2 visc 3πνΣr 2 pω p so that q corresponding to M > 2.5 MJupiter In usual conditions, it is the viscosity criterion that determines the opening of a gap.

22 Gap opening III Numerical simulations (Crida et al 2006) have led to the following approximate combined gap opening criterion: P = 3 H R H qr 1 where R is the Reynolds number R = Ω pa 2 ν Criterion for migration type: Kley & Dirksen 2006 type I migration: P>1 1 Σ type II migration: P <1.5 t = M_jup 2 M_jup 3 M_jup 4 M_jup 5 M_jup r The more massive the planet, the wider the gap.

23 Type II migration Once the planet is massive enough to open gap, the gas is being pushed away from the planet and hence the torques diminish. The planet is kept in the middle of the gap, as if it were to be closer to the inner edge, it would gain angular momentum, and it would migrate back outwards, while the opposite effect would happen close to the outer edge. In a static disk, the planet would also be static. As the disk is itself evolving on the viscous timescale, also the planet s orbit is evolving on this timescale: τ II = r2 p ν = r2 p αc s H = 1 α rp where we have used the fact that the viscosity is given by speed is approximated by c s = HΩ p H 2 Ω 1 p ν = αc s H and the sound Note that in the case of type II migration, the migration timescale is independent of the mass of the planet and only depends upon the mass of the star and the characteristics of the disk. We have however seen that this simple picture is valid only if the planet is not too massive (B>>1). One therefore distinguishes two regimes: Disk dominated type II (B>>1): Planet dominated type II (B<<1): Clearly, in the planet dominated regime, migration is slower.

24 4. Resonant torques

25 Resonances Imagine a planet in orbit about a star. The rotation frequency of the planet is given by its Keplerian frequency, The, there are special resonant places in the disk. Two types must be distinguished: 1) Corotation resonance located where Ω = Ω p If the disc is Keplerian (i.e., if we neglect gas pressure and self-gravity), then the corotation resonance is found at the planet s orbital radius. 2) Lindblad resonances located where m (Ω Ω p )=±κ where m is an integer, and the epicyclic frequency is In the case of a Keplerian potential, it is easy to see that: a) for the + sign (rotation faster than planet): The inner Lindblad resonances b) the - sign (rotation slower than planet): The outer Lindblad resonances The actual position of these resonances can be readily obtained: r ILR = r p m m 1 κ 2 = R dω2 dr +4Ω2 κ = Ω 2/3 m and r OLR = r p m +1 2/3

26 Resonances II A circular Keplerian disc therefore has a single corotation resonance, and a comb of Lindblad resonances that pile up close to the planet. The innermost Lindblad resonance is found where while the outermost is at. Material at these locations are particularly susceptible to excitation by the planet because of their match in frequency. 3) Mean motion resonances Imagine a second planet orbiting at Ω q the planets are said to be in resonance (n:m) if 2π Ω p 2π Ω q = P p P q = n m where n,m are small integers

27 Summing resonant torques We have studied above the impulse approximation to have a rough estimate of planetary migration rates. A full analysis instead considers the evolution of linear perturbations in a fluid disc, and was first applied to planet-disc interactions by Goldreich & Tremaine (1979). The first step is to linearize the hydrodynamic equations, and to decompose the perturbation to the Keplerian potential due to the planet into Fourier modes. For example, the potential of the planet is expressed in terms of azimuthally periodic components characterized by m, the inverse wave number. Φ m (r, ϕ,t)=φ m (r) cos[m(φ Ω p t)] The next step is to compute the response of the disc to the perturbations, and from this it is possible to calculate the total torque on the planet by summing up the torques originating at each resonance. Tanaka et al 2002

28 Summing resonant torques II The important result of such complex calculations is that the total angular momentum exchange between disc and planet can be expressed as the sum of the torques exerted at discrete resonances in the disc. These resonances correspond to the points in the disc where the planet excites waves, which take the form of spiral density waves. The importance of the resonances can be understood by the following argument: the torques at non-resonant locations in the disc do not interfere constructively, and consequently cancel out when averaged over the orbit. Using such an approach, Goldreich & Tremaine (1979) derived an expression for the torque at the Lindblad resonance: Γ m = mπ 2 Σm rdd/dr r dφ m dr + 2Ω Φ m Ω Ω p r=r L with D(r) =κ(r) 2 m(ω p Ω(r))

29 Summing resonant torques III Since then, several groups performed similar calculations. For example, Tanaka et al (2002) studied the three-dimensional interaction between a planet and a three dimensional isothermal disk. Assuming that the perturbation are small enough, they used linear theory to derive the relevant torques. After some significant calculations, they obtain the following expression for the torque exerted on the planet: Here α Γ = ( α) Mp M r p Ω p c s 2 Σ p r 4 pω 4 p is defined by the exponent of the assumed power law surface density of the disk Σ(r) =Σ 0 r r 0 α The corresponding inward migration timescale is given by: τ mig =( α) 1 M M p M Σ p r 2 p Ω 1 p 1 M p

30 5. Type I migration

31 Type I migration Putting the results from the studies like Goldreich & Tremaine or Tanaka et al. together, the migration is determined by the net torque exerted onto the planet. This torque can be written: Γ tot = ILR Γ LR + OLR Γ LR + Γ CR This torque can be expressed in a general way as: where the generic torque is given by: Γ tot = 1 γ (C 0 + C 1 α + C 2 β)γ 0 q 2 Γ 0 = Σ 2 h p rpω 2 2 p α, β Here C 0,C 1, and C 2 are constants and are the exponents of the assumed power-law distribution of the density and temperature respectively. These constants can be derived semi-analytically or from numerical simulations. The conservation of angular momentum implies: dj p dt The angular momentum of the planet is given by: = Γ tot J p = M p (GM r p ) 1/2

32 Type I migration cont. From which we obtain the migration velocity: dr p dt = 2r p Γ tot J p Depending upon the sign of the torque (i.e. upon the signs of the Ci and the slopes of the density and temperature distribution), the migration can proceed inwards or outward. For an isothermal disk with a MMSN profile ( α =1.5) migration is inwards and rapid.

33 Migration timescales The plot shows the migration timescale as a function of planet mass and alpha viscosity parameter, in a disk about twice as massive as the MMSN. We note that before the transition to type II sets in, the migration timescale in type I are very short, ~10 4 yrs. Afterwards, migration timescale increase by 1-2 orders of magnitude from type I to type II. For type I this means: Planets seem to migrate so fast that they should all fall into the star within the lifetime of the disk (unless they grow extremely rapid)! These very short migration timescales represent another major issue in modern planet formation theory. disk lifetimes simple linear theory for isothermal disks cannot be the final word! Ward 1997

34 6. Recent results on type I migration

35 Updated type I migration rates The puzzling results of extremely fast type I migration timescales have triggered numerous studies trying to understand which processes could slow down type I migration so that it planet growth again becomes possible. Several possible processes have been put forwards of which we only address here two: 1) Random walk migration in turbulent disks There has been strong interest (e.g. Nelson & Papaloizou 2004, Uribe et al. 2011) in studying planet-disk interactions in turbulent disks, where the turbulence is magnetically generated by the magneto-rotational instability (MRI), instead of the laminar disks which we have used in the above analysis. Such laminar disks are strictly speaking not self-consistent, as they assume an abnormal viscosity (thought to be due to turbulence) without actually taking the consequence of turbulent motions into account. More realistically, angular momentum transport itself derives from turbulence, which is accompanied by a spatially and temporally varying pattern of density fluctuations in the protoplanetary disk. These fluctuations will exert random torques on planets of any mass embedded within the disk. In such turbulent disks, it is found that for low mass planets, Type I migration is no longer effective due to large fluctuations in the torque. The fluctuations in the torque created by the perturbations in the density can be larger than the mean torque expected for standard Type I migration in a laminar disk.

36 Updated type I migration rates II Under these conditions, low mass planets undergo rather a random walk like migration. Only for planets more massive than ~30M = 0.1MJup the stochastic torques are found to converge to the standard type I migration torques after long-time averaging. Uribe et al The plots show the logarithm of the disk density in the mid plane (top row) and in an azimuthal cut at the position of the planet (bottom row) for q = 10 5 (~3 Mearth), q = 10 4 (~30 Mearth) and q = 10 3 (~1 MJup). The left plot shows no significant perturbation of the density by the planet, and no spiral arms are seen, indicating that random torques are dominating.

37 2) Migration in non-isothermal disk Non-isothermal migration Crida et al. 2006; Baruteau & Masset 2008; Casoli & Masset 2009; Pardekooper et al. 2010; Baruteau & Lin 2010 outward inward Another important (and not justified) assumption made in the derivation of the classical type I torque is that the gas around the planet acts isothermally. In other words, its cooling time is vanishing. Radiation hydrodynamic simulations treating correctly the energy transport in the disk find that below a certain threshold mass, migration can be directed outwards, due to a different density distribution around the planet. Kley & Crida 2009 Thermodynamics of the disk is essential

38 Horseshoe orbits A central role for the torque in non-isothermal disk is due to gas moving along so called horseshoe orbits. 1) starting at point A, the body is moving towards the planet and is being accelerated and this put on higher and higher orbits. 2) eventually, at point C, the body is moving sufficiently slower than the planet that it starts lagging behind. 3) at point D, the planet is catching up with the body and is pulling it back therefore reducing its angular momentum. The body eventually moves onto a lower orbit. 4) since the body is on a lower orbit, it will eventually catch up with the planet and the cycle can restart. width of horseshoe orbit: q x s =1.16R p H/R p libration time: τ lib = 8πR p Mp 1/2 3Ω p x s

39 Thermodynamics matter Entropy C. Baruteau Perturbed density r/rp ϕ ϕ p -3-2 simulation for 1/2 libration period ϕ ϕ p The exchange of fluid elements lead to an overdensity at smaller radii. This translates into a increased torque pushing the planet forwards and thus outwards. For this mechanism to work, the fluid has to remain adiabatic during the exchange process. In other words: τ cool >> τ u turn τ cool Σc V T Q = Σc V T 2σT 4 eff τ u turn 1.16 h 3 γq 64 9Ω p

40 Thermodynamics matter II Unfortunately, after some time, the situation becomes much less clear and the outward directed corotation torques begins to saturate (vanish)... Perturbed entropy C. Baruteau Perturbed density r/rp ϕ ϕ p -3-2 simulation for 3 libration periods ϕ ϕ p In other words, unless the viscosity re-establishes the original entropy profile, the outward migration will not last.the condition for a sustainable outward migration is therefore given by: τ lib >> τ visc τ lib = 8πr p 3Ω p x s τ visc = (2x s) 2 ν

41 τ cool τ U turn Cooling? τ cool τ U turn τlib τvisc Subtypes of type I By comparing the relevant timescales, different subtypes of type I migration can be identified. Isotherm. Inward (outer disk) Adia. Outward (inner disk) τu-turn Star τcool Saturation? Sat. Inward (High Mplanet) Unsat. Outward (Low Mplanet) Dittkrist et al. (2011, in prep) Corotation region: horseshoe orbits w. entropy gradient: needed for outward migration

42 Type I convergence zones An important consequence of nonisothermal migration is that it leads to the existence of convergence zones. These are zero torque locations in which planets get trapped. Time [yrs] out out in in The exact location of convergence will depend upon the detailed structure of the disk (temperature and surface density). The location of the convergence zone itself moves inward on a viscous timescale.this means that despite being in the type I regime, the planets will move inwards on a much slower viscous timescale, as in type II. Dittkrist et al. in prep Semimajor axis [AU] It is tempting to think that these zones are the places to grow massive planets, as they might concentrate many growing protoplanets.

43 Questions?