A&A 510, A4 (2010) Astronomy DOI: 10.1051/0004-6361/200913208 & c ESO 2010 Astrophysics The dynamical origin of the multi-planetary system HD 45364 H. Rein1,J.C.B.Papaloizou1,andW.Kley2 1 University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: [email protected] 2 University of Tübingen, Institute for Astronomy and Astrophysics, Auf der Morgenstelle 10, 72076 Tübingen, Germany Received 30 August 2009 / Accepted 26 October 2009 ABSTRACT The recently discovered planetary system HD 45364, which consists of a Jupiter and Saturn-mass planet, is very likely in a 3:2 mean motion resonance. The standard scenario for forming planetary commensurabilities is convergent migration of two planets embedded in a protoplanetary disc. When the planets are initially separated by a period ratio larger than two, convergent migration will most likely lead to a very stable 2:1 resonance. Rapid type III migration of the outer planet crossing the 2:1 resonance is one possible way around this problem. In this paper, we investigate this idea in detail. We present an estimate of the required convergent migration rate and confirm this with N-body and hydrodynamical simulations. If the dynamical history of the planetary system had a phase of rapid inward migration that forms a resonant configuration, we predict that the orbital parameters of the two planets will always be very similar and thus should show evidence of that. We use the orbital parameters from our simulation to calculate a radial velocity curve and compare it to observations. Our model provides a fit that is as good as the previously reported one. The eccentricities of both planets are considerably smaller and the libration pattern is different. Within a few years, it will be possible to observe the planet-planet interaction directly and thus distinguish between these different dynamical states. Key words. planets ans satellites: formation – celestial mechanics – protoplanetary disks – accretion, accretion disks 1. Introduction stable. One possible way around this is a very rapid convergent 1 migration phase that passes quickly through the 2:1 resonance. Over 400 extrasolar planets have already been discovered In this paper we explore this idea quantitatively. The plan of and their diversity keeps challenging planet formation the- the paper is as follows. In Sect. 2 we use N-body simulations to ory. For example, the recently discovered multi-planetary sys- show that the system always ends up in the 2:1 resonance when tem HD 45364 raises interesting questions about its formation assuming moderate migration rates. We present scenarios that history. result in formation of a 3:2 resonance after a rapid migration = . = The planets have masses of m1 0 1906 MJup and m2 phase. In Sect. 3 we perform hydrodynamic simulations with a . = 0 6891 MJup and are orbiting the star at a distance of a1 variety of disc models in order to explore the dependence on . = . 0 6813 AU and a2 0 8972 AU, respectively (Correia et al. the physical setup. In Sect. 4 we briefly discuss other formation 2009). The period ratio is close to 1.5, and a stability analysis scenarios. We go on to compare the orbital parameters of our implies that the planets are deep inside a 3:2 mean motion reso- simulations with the observed radial velocity data in Sect. 5. nance. The planets have most likely formed further out in cooler We find that the orbital parameters observed in our simula- regions of the proto-stellar disc as water ice, which is an impor- tions differ from those estimated from a statistical fit by Correia tant ingredient for dust aggregation can only exist beyond the et al. (2009). However, our models reproduce the observational snow line, which is generally assumed to be at radii larger than data very well and we argue that our fit has the same level of 2AU(Sasselov & Lecar 2000). significance as that obtained by Correia et al. (2009). It is then usually assumed that migration due to planet disc Future observations will be able to resolve this issue. This is interactions has moved the planets closer to the star. Although the first prediction of orbital parameters for a specific extraso- the details of this process are still being hotly debated, the ex- lar planetary system derived from planet migration theory alone. istence of many resonant multiplanetary systems supports this The parameter space of orbital configurations produced by idea. During migration the planets can get locked into a com- planet disc interactions (low eccentricities, relatively small libra- mensurability, after which the planets migrate together with a tion amplitudes) is very small. As in the case of the GJ876 sys- constant orbital period ratio. In such a resonance, one or more tem, this can provide strong evidence of how the system formed. resonant angles are librating (see e.g. Lee & Peale 2001). Finally, we summarise our results in Sect. 6. For the planetary system HD 45364, this standard picture poses a new problem. Assuming that the planets have formed far apart, the outcome for the observed masses after migration 2. Formation of HD 45364 is almost always a 2:1 mean motion resonance, not 3:2 as ob- served. The 2:1 resonance that forms is found to be extremely 2.1. Convergent migration and resonance capture In the core accretion model (for a review see e.g. Lissauer 1993), 1 Schneider, J. (2009), http://exoplanet.eu a solid core is firstly formed by dust aggregation. This process Article published by EDP Sciences Page 1 of 8 A&A 510, A4 (2010) is much more efficient if water exists in a solid form. In the 2.8 1400 protostellar nebula, this happens beyond the ice line where the 2.6 1200 temperature is below 150 K at distances greater than a few AU. ) s 2.4 r 1000 tio a Subsequently, after a critical mass is attained (Mizuno 1980), the a core accretes a gaseous envelope from the nebula (Bodenheimer 800 2.2 600 2 & Pollack 1986). Both planets in the HD 45364 system are inside period r time (ye to the ice line, implying that they should have migrated inwards. 400 1.8 The migration rate depends on many parameters of the disc 200 1.6 such as surface density, viscosity, and the mass of the planets. 1.4 The planets are therefore in general expected to have different 400 450 500 550 600 650 700 τ migration rates, which leads to the possibility of convergent mi- migration timescale a,1 (years) gration. In this process the planets approach orbital commensu- / y rabilities. If they do this slowly enough, resonance capture may Fig. 1. Period ratio P2 P1 as a function of time ( -axis) and migration timescale of the outer planet τ , (y-axis). The migration timescale of occur (Goldreich 1965), after which they migrate together main- a 2 the inner planet is τ , = 2000 yrs. The inner planet is initially placed at taining a constant period ratio thereafter. a 1 r1 = 1 AU. The eccentricity damping is given through K ≡ τa/τe = 10. Studies made by several authors have shown that when two planets, either of equal mass or with the outer one more massive, undergo differential convergent migration, capture into a mean width is not crossed within a libration period gives motion commensurability is expected provided that the conver- 1 > a2 n2 gent migration rate is not too fast (Snellgrove et al. 2001). The τa,crit ≡ ∼ 2π = 2π (2) /τ , − /τ , ω Δ ω2 observed inner and outer planet masses are such that, if (as is 1 a 1 1 a 2 lf a lf commonly assumed for multiplanetary systems of this kind) the planets are initially separated widely enough that their period to pass through the 2:1 MMR. ratio exceeds 2, a 2:1 commensurability is expected to form at If the planets of the HD45364 system are placed in a 2:1 res- low migration rates (e.g. Nelson & Papaloizou 2002; Kley et al. onance with the inner planet located at 1 AU, the libration pe- π/ω 2004). riod 2 lf is found to be approximately 75 yrs. Thus, a relative τ ≈ Pierens & Nelson (2008) have studied a similar scenario migration timescale shorter than a,crit 810 yrs is needed to where the goal was to resemble the 3:2 resonance between pass through the 2:1 resonance. For example, if we assume that Jupiter and Saturn in the early solar system. They also find that the inner planet migrates on a timescale of 2000 years, the outer the 2:1 resonance forms in early stages; however, in their case the planet has to migrate with a timescale of inner planet had the higher mass, whereas the planetary system < τa,2,crit ∼ 576 yrs. (3) that we are considering has the heavier planet outside. In their situation the 2:1 resonance can be unstable, enabling the forma- We ran several N-body simulations to explore the large parame- tion of a 3:2 resonance later on, and the migration rate may stall ter space and confirm the above estimate. The code used is sim- or even reverse (Masset & Snellgrove 2001). ilar to the one presented in Rein & Papaloizou (2009) and uses a fifth order Runge-Kutta, as well as a Burlish Stoer integrator, both with adaptive time-stepping. Different modules deal with 2.2. The 2:1 mean motion resonance migration and stochastic forcing. Non conservative forces are We found that, if two planets with masses of the observed sys- calculated according to the procedure presented in Lee & Peale tem are in a 2:1 mean motion resonance, which has been formed (2002) where the migration and eccentricity damping timescales τ = | / | τ = | / | via convergent migration, this resonance is very stable.