Institute for Advanced Study
Formation of resonant multi- planetary systems
Hanno Rein @ NAOJ, Tokyo, March 2012 Statistics of multiple planets (using iPhone App)
Available for free on the Apple AppStore. Radial velocity planets
80 RV planets
70 ] 60 10 jup [M 2 50 + m 1
40
number of pairs 30
1 20 total planet mass m
10
0 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 300 0.1 KOI planets
250 ] jup
Rein, Payne, Veras & Ford [M (2012 in prep) 200 2 + m 1
150 number of pairs 100 total planet mass m 50
0 0.01 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 80 RV planets
70 ] 60 10 jup [M 2 50 + m 1
40
number of pairs 30
1 20 total planet mass m
Kepler's 10transiting planet candidates
0 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio 300 0.1 KOI planets
250 ] jup [M
200 2 + m 1
150 number of pairs 100 total planet mass m 50
0 0.01 1 1.33 1.5 2 3 4 5 6 7 8 9 period ratio
Rein, Payne, Veras & Ford (2012 in prep) Recipe
Migration
Resonances Migration in a non-turbulent disc Planet formation
Image credit: NASA/JPL-Caltech Migration - Type I
• Low mass planets • No gap opening in disc • Migration rate is fast • Depends strongly on thermodynamics of the disc Migration - Type II
• Massive planets (typically bigger than Saturn) • Opens a (clear) gap • Migration rate is slow • Follows viscous evolution of the disc Gap opening criteria
Disc scale height Stellar mass
3 H 50M + ⇤ 1 4 R M Hill pR Viscosity -1 Planet mass
Crida et al 2006 Migration - Type III
• Massive disc • Intermediate planet mass • Tries to open gap • Very fast, few orbital timescales Resonance capture 2:1 Mean Motion Resonance
Star Planet 1 Planet 2 2:1 Mean Motion Resonance 2:1 Mean Motion Resonance
2 1
2
1 Resonant angles
• Fast varying angles ⇥ ⇥ 1 1 2 2 • Slowly varying combinations ⇥ = 2 + ⇤ 1 2 1 2 ⇥ = 2 + ⇤ 2 2 1 1 ⇤ = ⇤ ⇤ 1 2 • Two are linear independent Non-turbulent resonance capture: two planets
= 2 + $ 1 2 1 2
parameters of GJ 876 GJ 876
Lee & Peale 2002 Take home message I
planet + disc = migration
2 planets + migration = resonance HD 45364 HD45364
40 Observations Correia et al
30
20
10
0
-10 radial velocity [m/s]
-20
-30
-40 53000 53200 53400 53600 53800 54000 54200 54400 54600 JD-2400000 [days]
Correia et al 2009, Visual Exoplanet Catalogue Formation scenario for HD45364 • Two migrating planets • Migration speed is crucial • Infinite number of • Resonance width and resonances 3:2 1:3 libration period define critical migration rate 7:82Rein,PapaloizouandKley:TheDynamicalOriginofHD453643:4 1:2 Jupiter and Saturn in the early solar system. They also 2.8 1400 found that the 2:1 resonance forms at early stages. However, 2.6 in their case the inner planet had the larger mass whereas 1200 2.4 the planetary system that we are considering has the heav- 1000 2.2 ier planet outside. In this situation the 2:1 resonance can 800 be unstable, enabling the formation of a 3:2 resonance 600 2 period ratio time (years) later on and the migration rate may stall or even reverse 1.8 400 (Masset & Snellgrove 2001). 200 1.6 1.4 400 450 500 550 600 650 700 2.2. The 2:1 mean motion resonance migration timescale τa,1 (years) We found that if two planets with masses of the observed Rein, Papaloizou & Kleysystem 2010 are in a 2:1 mean motion resonance, which has been Fig. 1. Period ratio P2/P1 as a function of time (y-axis) and form via convergent migration, this resonance is very sta- migration timescale of the outer planet τa,2 (y-axis). The ble. An important constraint arises, because as indicated migration timescale of the inner planet is τa,1 =2000yrs. above, provided the planets start migrating outside any low The inner planet is initially placed at r1 =1AU.Theec- order commensurability, at the slowest migration rates a 2:1 centricity damping is given through K ≡ τa/τe =10. resonance is expected to form rather than the 3:2 commen- surability that is actually observed. We can estimate the critical relative migration timescale gration theory only. The parameter space of orbital config- τa,crit above which a 2:1 commensurability forms from the urations produced by planet disc interactions (low eccen- condition that the planets spend at least one libration pe- tricities, relatively small libration amplitudes) is very small. riod while migrating through the resonance. The resonance As in case of the GJ876 system, this can provide strong ev- semi-major axis width ∆a associated with the 2:1 resonance idence on how the system formed. Finally, we summarise can be estimated from the condition that two thirds of the our results in section 6. mean motion difference across ∆a be equal in magnitude to 2π over the libration period. This gives ω a 2. Formation of HD45364 ∆a = lf 2 (1) n2 2.1. Convergent migration and resonance capture where a2 and n2 are the semi major axis and the mean mo- In the core accretion model (for a review see e.g. Lissauer tion of the outer planet, respectively. The libration period 1993) a solid core is firstly formed by dust aggregation. 2π/wlf can be expressed in terms of the orbital parameters This process is much more efficient if water exists in solid (see e.g. Goldreich 1965; Rein & Papaloizou 2009) but is, form. In the proto-stellar nebula this happens beyond the for convenience, here measured numerically. If we assume ice line where the temperature is below 150 K at distances the semi-major axes of the two planets evolve on constant larger than a few AU. Subsequently, after a critical mass is (but different) timescales |a1/a˙ 1| = τa,1 and |a2/a˙ 2| = τa,2, attained (Mizuno 1980), the core accretes a gaseous enve- the condition that the resonance width is not crossed within lope from the nebula (Bodenheimer & Pollack 1986). Both alibrationperiodgives planets in the HD45364 system are interior to the ice line, implying that they should have migrated inwards. 1 a2 n2 τa,crit ≡ ! ! ! 2π =2π 2 (2) The migration rate depends on many parameters of the !1/τ 1 − 1/τ 2 ! ω ∆a ω ! a, a, ! lf lf disc such as surface density and viscosity as well as the mass ! ! of the planets. The planets are therefore in general expected to pass through the 2:1 MMR. to have different migration rates which leads to the possi- If the planets of the HD45364 system are placed in a bility of convergent migration. In this process the planets 2:1 resonance with the inner planet being located at 1 AU, approach orbital commensurabilities. If they do this slowly the libration period 2π/ωlf is found to be approximately enough, resonance capture may occur (Goldreich 1965) af- 75 yrs. Thus, a relative migration timescale shorter than ter which they migrate together maintaining a constant pe- τa,crit ≈ 810 yrs is needed in order to pass through the 2:1 riod ratio thereafter. resonance. For example, if we assume that the inner planet Studies made by a number of authors have shown that migrates on a timescale of 2000 years, the outer planet has when two planets, of either equal mass or with the outer to migrate with a timescale one being the more massive, undergo differential convergent τ 2 " 576 yrs. (3) migration, capture into a mean motion commensurability is a, ,crit expected to occur provided that the convergent migration We have run several N-body simulations to explore the rate is not too fast (Snellgrove et al. 2001). The observed large parameter space and confirm the above estimate. The inner and outer planet masses are such that, if (as is com- code used is similar to that presented in Rein & Papaloizou monly assumed for multiplanetary systems of this kind) the (2009) and uses a fifth order Runge-Kutta as well as a planets are initially widely enough separated so that their Burlish Stoer integrator, both with adaptive time-stepping. period ratio exceeds 2, at low migration rates a 2:1 commen- Different modules deal with migration and stochastic forc- surability is expected to form (e.g. Nelson & Papaloizou ing. Non conservative forces are calculated according to 2002; Kley et al. 2004). the procedure presented in Lee & Peale (2002) where the Pierens & Nelson (2008) studied a similar scenario migration and eccentricity damping timescales τa = |a/a˙| where the goal was to resemble the 3:2 resonance between and τe = |e/e˙| are imposed for each planet individually. Formation scenario for HD45364
H. Rein et al.: The dynamical origin of HD 45364
1.8 1.8 1.6 1.6 s s
xi 1.4 xi 1.4 a a 1.2 1.2 jor jor
a 1 a 1 0.8 0.8
emi m 0.6 emi m 0.6 s 0.4 s 0.4
2.3 2.3 2.2 2.2 2.1 2.1
tio 2 tio 2 a 1.9 a 1.9 1.8 1.8 1.7 1.7 period r 1.6 period r 1.6 1.5 1.5 1.4 1.4 0.1 0.1
0.08 0.08
0.06 0.06
0.04 0.04 eccentricity eccentricity 0.02 0.02
0 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 500 1000 1500 2000 time time
Fig. 2. The semi-major axes (top), period ratio P2/P1 (middle), and ec- Fig. 4. The semi-major axes (top), period ratio P2/P1 (middle), and ec- centricities (bottom)ofthetwoplanetsplottedasafunctionoftimein centricities (bottom)ofthetwoplanetsplottedasafunctionoftimein dimensionless units for run F5 with a disc aspect ratio of h = 0.07. In dimensionless unitsRein, for Papaloizou run F4 with a& disc Kley aspect 2010 ratio of h = 0.04. In the bottom panel,theuppercurvecorrespondstotheinnerplanet. the bottom panel,theuppercurvecorrespondstotheinnerplanet.
3 0.004 1.5 0.004
0.0035 0.0035 2 1 0.003 0.003 1 0.5 0.0025 0.0025 ity ity s s y 0 0.002 y 0 0.002 den den 0.0015 0.0015 -1 -0.5 0.001 0.001 -2 -1 0.0005 0.0005
-3 0 -1.5 0 -3 -2 -1 0 1 2 3 -1.5 -1 -0.5 0 0.5 1 1.5 x x Fig. 3. Asurface-densitycontourplotforsimulationF5 after 100 orbits Fig. 5. Asurface-densitycontourplotforsimulationF4 after 150 orbits at the end of the type III migration phase. The outer planet establishes after the planets went into divergent migration. The inner planet is em- adefinitegap,whiletheinnerplanetremainsembeddedattheedgeof bedded and interacts strongly with the inner disc. The simulation uses the outer planet’s gap. a1Dgridfor0.04 < r < 0.25. similar properties to those described above when making com- rates as a result of disc planet interactions as the planets grew parisons with observations. in mass. It is possible that the solid cores of either both planets or just An issue is whether the embedded inner planet is in a rapid the outer planet approached the inner planet more closely than accretion phase. The onset of the rapid accretion phase (also the 2:1 commensurability before entering the rapid gas accre- called phase 3) occurs when the core and envelope mass are tion phase and attaining their final masses prior to entering the about equal (Pollack et al. 1996). The total planet mass depends 3:2 commensurability. Although it is difficult to rule out such at this stage on the boundary conditions, here determined by possibilities entirely, we note that the cores would be expected the circumplanetary flow. When these allow the planet to have a to be in the super earth mass range, where in general closer significant convective envelope, the transition to rapid accretion commensurabilities than 2:1 and even 3:2 are found for typical may not occur until the planet mass exceeds 60 M (Wuchterl type I migration rates (e.g. Papaloizou & Szuszkiewicz 2005; 1993), which is the mass of the inner planet (see also⊕ model J3 Cresswell & Nelson 2008). One may also envisage the possibil- of Pollack et al. 1996;andmodelsofPapaloizou & Terquem ity that the solid cores grew in situ in a 3:2 commensurability, but 1999). Because of the above results, it is reasonable that the in- this would have to survive expected strongly varying migration ner planet is not in a rapid accretion phase.
Page 5 of 8 Formation scenario for HD45364
Massive disc (5 times MMSN)
• Short, rapid Type III migration
• Passage of 2:1 resonance
• Capture into 3:2 resonance
Large scale-height (0.07)
• Slow Type I migration once in resonance
• Resonance is stable
• Consistent with radiation hydrodynamics
Rein, Papaloizou & Kley 2010 Formation scenario leads to a better ‘fit’
40 Observations Correia et al Simulation F5
30
20
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-20
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Rein, Papaloizou & Kley 2010 Migration in a turbulent disc Turbulent disc
• Angular momentum transport • Magnetorotational instability (MRI) • Density perturbations interact gravitationally with planets • Stochastic forces lead to random walk • Large uncertainties in strength of forces
Animation from Nelson & Papaloizou 2004 Random forces measured by Laughlin et al. 2004, Nelson 2005, Oischi et al. 2007 Random walk
pericenter
eccentricity
semi-major axis
time
Rein & Papaloizou 2009 Analytic growth rates for 1 planet
0.01 Dt 2 [AU] 0.001 ( a) =4 RMS a>
2 n < 0.0001
1
2 2.5 Dt 0.1 RMS ( $) = >
2 2 2 < e n a 0.01
0.001
0.01
Dt RMS 2 0.001 e> ( e) =2.5 < n2a2 0.0001 Simulations Analytic solution Analytic solution ( =1) 10 100 1000 10000 100000 time [years]
Rein & Papaloizou 2009, Adams et al 2009, Rein 2010 Two planets: turbulent resonance capture
Rein & Papaloizou 2009 Analytic growth rates for 2 planets
2 ( ⇥1) 9 f 2 = 2 2 Dt (p + 1) a1⇤lf
2 5 s ( ( ⌅)) = 2 2 2 Dt 4a1n1e1
Rein & Papaloizou 2009 Multi-planetary systems in mean motion resonance
GJ876
Earth
• Stability of multi-planetary systems depends strongly on diffusion coefficient • Most planetary systems are stable for entire disc lifetime
Rein & Papaloizou 2009 Modification of libration patterns
• HD128311 has a very peculiar libration pattern • Can not be reproduced by convergent migration alone • Turbulence can explain it • More multi-planetary systems needed for statistical argument
Rein & Papaloizou 2009 Take home message II
Migration scenarios can explain the dynamical configuration of many systems in amazing detail HD200964 The impossible system? Radial velocity curve of HD200964
• Two massive planets HD 200964, 4:3 -v- 3:2 fits 1.8 MJup and 0.9 MJup 80 4:3 A 4:3 B 60 3:2 A 3:2 B Period ratio either 40 • 3:2 or 4:3 20
0 Another similar -20 •
RV Amplitude [m/s] system, to be -40 announced soon -60
-80 13000 13500 14000 14500 15000 15500 16000 How common is 4:3? Observation Date [J days] • • Formation?
Plot by Matthew Payne Stability of HD200964
0.8 "< cat output*.txt | pgt_pm3d 3 4 5" u 1:2:($3>0?$3:0.1-$3)
0.7 4:3
0.6 5:2 2:1 0.5 2 3:2 0.4 eccentricity e 0.3
0.2
0.1
0 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
semi-major axis a2 [AU]
Rein, Payne, Veras & Ford (2012 in prep) Standard disc migration doesn't work
observed masses reduced masses
200 2.2 200 2.2
2 2 150 150
1.8 1.8
100 100
1.6 final period ratio 1.6 final period ratio
50 50
migration timescale / eccentricity 1.4 migration timescale / eccentricity 1.4
0 1.2 0 1.2 200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400 migration timescale [yrs] migration timescale [yrs]
Rein, Payne, Veras & Ford (2012 in prep) Hydrodynamical simulations I
Rein, Payne, Veras & Ford (2012 in prep) Hydrodynamical simulations II
Rein, Payne, Veras & Ford (2012 in prep) Hydrodynamical simulations III
Rein, Payne, Veras & Ford (2012 in prep) Scattering of embryos
Rein, Payne, Veras & Ford (2012 in prep) HD200964
• In situ formation?
• Main accretion while in 4:3 resonance?
• Planet planet scattering?
• A third planet?
• Observers screwed up? Take home message III
We don't understand everything (yet). Conclusions Conclusions
Formation of multi-planetary systems Number of system increases almost every week. Kepler has large number of planets, but not very suitable for detailed analysis
Multi-planetary system provide insight in otherwise unobservable formation phase Dynamical configuration keeps a record of history
GJ876 formed in the presence of a disc and dissipative forces HD128311 formed in a turbulent disc HD45364 formed in a massive disc HD200964 did not form at all