Mean-Motion Resonances of High Order in Extrasolar Planetary Systems
aa r X i v : . [ a s t r o - ph . E P ] O c t Extrasolar Planets in Multi-Body Systems, Torun 2008
EAS Publications Series, 2018
MEAN-MOTION RESONANCES OF HIGH ORDER IN EXTRASOLARPLANETARY SYSTEMS
Hanno Rein and John C. B. Papaloizou Abstract.
Many multi-planet systems have been discovered in recent years. Someof them are in mean-motion resonances (MMR). Planet formation theory was suc-cessful in explaining the formation of 2:1, 3:1 and other low resonances as a result ofconvergent migration. However, higher order resonances require high initial orbitaleccentricities in order to be formed by this process and these are in general unex-pected in a dissipative disk. We present a way of generating large initial eccentricitiesusing additional planets. This procedure allows us to form high order MMRs andpredict new planets using a genetic N -body code. Of the recently discovered 322 extrasolar planets, at least 75 are in multi-planet systems(Schneider 2008). About 10% of these are confirmed to be in or very close to a resonantconfiguration.Two planets can easily be locked into a low order resonance such as 2:1 or 3:2 by dissipativeforces. This process is well understood (see eg. Lee & Peale 2002, Papaloizou & Szuszkiewicz2005). However, period ratios of multi-planetary systems seem to be non-uniformly distributedup to large period ratios of order 5. Furthermore, there is an observational bias against thedetection of resonant systems in general (Anglada-Escude et al. Department of Applied Mathematics and Theoretical Physics; University of Cambridgee-mail: [email protected] c (cid:13) EDP Sciences 2018DOI: (will be inserted later)
Extrasolar Planets in Multi-Body Systems, Torun 2008 K which is defined as the ratio of the timescales τ a and τ e , being the migration timescale andthe eccentricity damping timescale, respectively. The color encodes the final period ratio at theend of the simulation. At the beginning the planets are put far away from each other. The outerplanet migrates inwards on a timescale of τ a = 25000 years. One can see that a small initialeccentricity always results in a 2:1 mean motion resonance. At slightly higher eccentricities( e ∼ .
05) the favored outcome is a 3:1 resonance. At large eccentricities ( e ∼ .
15) it ispossible to get high resonances like 4:1 and 5:1. Note that if K is large, the eccentricitiesget damped before the planets are captured into resonance and therefore favor lower orderresonances. Fig. 1.
This figure shows the final period ratio of two convergently migrating Jupiter mass planetsas a function of the initial eccentricity e init of the outer planet and the ratio of damping timescales K (see text). Red, as indicated on the color bar, corresponds to a 2:1, green to 3:1, blue to 4:1 andyellow to 5:1 commensurabilities respectively. Black indicates that the period ratio is not close to aninteger value or the planets scattered within a short period of time. . Rein & J.C.B. Papaloizou: Formation of High Order MMRs 3 There are different ways to create the required initial eccentricity for planets to get capturedinto high order resonances. One possibility is a scattering event in the past. The planets couldeither be scattered directly into a MMR or close to it (Raymond et al. e eq > Jup , 1.0 M
Jup and 0.7 M
Jup ,respectively. Here, the masses of the inner and middle planet are similar to those in theobserved system HD108874 that is close to a 4:1 MMR (see e.g. Vogt et al. et al. N -body code that finds the formation scenario that fits best with the observedsystem. In our code initial conditions are stored in a genom consisting of genes. The initialconditions are then integrated forward in time. After each iteration, a good genom is passedon to the next generation (iteration) of initial conditions. Each gene can also mutate with agiven probability, thus creating new initial conditions. In the case presented above, we define agenom to be good if the resulting planetary system shows large similarities with the observedsystem in the eccentricities and period ratios. The genetic code we implemented proved to beboth very efficient and stable (in terms of convergence) for the problem considered here. It is possible to model the observed orbital parameters of systems with high order commensu-rabilities in a natural way by postulating additional planets. The solution presented here doesnot rely on the random effects of planet-planet scattering. The prediction of new planets canresult in a test for planet formation theories and provides attractive targets for observers.An interesting point that hasn’t been discussed here is the question whether this scenario isstable to small perturbations. These perturbations could result from a turbulent disc or otherplanets (see e.g. Rein & Papaloizou 2009).Further work is underway, including hydrodynamical simulations confirming the solutionsfound by the N -body code. Extrasolar Planets in Multi-Body Systems, Torun 2008 a [ A U ] e e e t [years] Fig. 2.
Top figure: evolution of the semi major axes of the outer, middle and inner planet respectively.Bottom figures: evolution of the eccentricities of the outer, middle and inner planet respectively. Theplanets migrate inwards convergently on slightly different time scales. The outer planet captures themiddle one in a 2:1 resonance after ∼ years. Their eccentricities rise, making a capture intothe 8:2:1 resonance after ∼ . · years possible. Finally all planets migrate together while theeccentricities remain constant. References
Schneider J., 2008, http://exoplanet.eu
Anglada-Escude G., Lopez-Morales M. & Chambers J. E. 2008, submitted to ApJL, arXiv:0809.1275
Beaug´e C., Callegari N., Ferraz-Mello S. & Michtchenko T. A. 2005, IAU Colloquium 197Lee, M. H. & Peale, S. J. 2002, ApJ, 567, 596Nelson R. & Papaloizou J. 2002, MNRAS, 333 L26-L30 . Rein & J.C.B. Papaloizou: Formation of High Order MMRs 5
Papaloizou J. & Szuszkiewicz E. 2005, MNRAS 363, 153-176Rein, H. & Papaloizou J., 2009, A&A, 497, 595Raymond, S. N., Barnes R., Armitage P. J., Gorelick N., 2009, 696, L98Butler R. P. et al. , , ApJ 646, 505-522Vogt S. S. et al.et al.