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Gen`eve, Gen`eve, 51 Universit´e de de Observatoire hsklshsIsiu,Uiesta en Silderstras Universit¨at Bern, Institut, Physikalisches ffi V.H 56,api fpaesi : enmto resonance motion mean 3:2 a in planets of pair a 45364, HD XVI. h AP erhfrsuhr xr-oa planets extra-solar southern for search HARPS The ut fflycaatrzn ytm ihmore with systems characterizing fully of culty [email protected] [email protected] ...Correia A.C.M. tr:idvda:H 56 tr:paeaysses–tec – systems planetary stars: – 45364 HD individual: stars: e ...Correia A.C.M. : = / cgi-bin / 0 rlwms lntdtcin n the and detection, planet low- or . 6 n 0 and 168 aucitn.0774 no. manuscript / qcat?J 1 .Udry S. , . / 9,rsetvl.Adnmclaayi ftesse furth system the of analysis dynamical A respectively. 097, / A PL nvri´ eVrale an-uni,B3 91371 BP3, Saint-Quentin, Versailles Universit´e de IPSL, .Lovis C. + A m / ???? sin 2 .Mayor M. , i 2 = .Mordasini C. , 0 . 187 e5 H31 en Switzerland Bern, CH-3012 5, se M Jup etected umber, sev- d 2 earch via r ee ai ui,9bsB rg,704Prs France Paris, 75014 Arago, Bd 98bis Curie, Marie et re n 0 and .Benz W. , ABSTRACT nets a lnt,atog hr sa nlgei u oa Syste Solar our in analogue an is there although planets, lar nue h tblt ftesse vr5Gr hsi h fir the is This Gyr. 5 over system the of stability the ensures d eipratcntanso lntr omto n migrat and formation planetary on constraints important de alets 20Suen,Switzerland Sauverny, 1290 Maillettes, orp eeltepeec ftopaesobtn h sola the orbiting planets two of presence the reveal rograph ng In a- is atao 8013Aer,Portugal Aveiro, 3810-193 Santiago, 3 . 658 .Pepe F. , os hspormrvasisl svr e very as itself compan- reveals mass lower program for This searching alre ions. with while stars planets, and high-precision giant 2000) CORALIE known al. very the et to (Udry from program dedicated selected planet-search half stars is non-active About time of GTO 2003). measurements HARPS al. et the the (Mayor of th in consortium to (Chile) granted building Silla Observations La HARPS Time at Guaranteed telescope the 3.6-m of framework ESO the spe HARPS on the with trograph conducted neighborhood solar the of stars radial- co planet and the solar The detection of their characterization. helps plete precision greatly the measurements velocity improving However, programs. content. orbital of de terms in for “structure” second system and first the history useful, mining evolution very system then the is systems constraining such of evolutio analysis and dynamical formation planetary constrain of processes to the ability derstand potential our increases dramatically System. 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Table 1. Observed and inferred stellar parameters of HD45364. Table 2. Orbital parameters for the two bodies orbiting HD45364, obtained with a 3-body Newtonian fit to observa- tional data (Fig. 1). Parameter HD 40307 Sp K0V V [mag] 8.08 Param. [unit] HD 45364 b HD 45364 c B − V [mag] 0.72 Date [JD-2400000] 53500.00 (fixed) π [mas] 30.69 ± 0.81 V [km/s] 16.4665 ± 0.0002 MV [mag] 5.51 P [day] 226.93 ± 0.37 342.85 ± 0.28 Teff [K] 5434 ± 20 λ [deg] 105.76 ± 1.41 269.52 ± 0.58 log g [cgs] 4.38 ± 0.03 e 0.1684 ± 0.0190 0.0974 ± 0.012 [Fe/H] [dex] −0.17 ± 0.01 ω [deg] 162.58 ± 6.34 7.41 ± 4.30 L [L⊙] 0.57 K [m/s] 7.22 ± 0.14 21.92 ± 0.43 M∗ [M⊙] 0.82 ± 0.05 i [deg] 90 (fixed) 90 (fixed) −1 v sin i [km s ] 1 a sin i [10−3 AU] 0.1485 0.6874 ′ 1 log R −4.94 −9 HK f (M) [10 M⊙] 0.0085 0.3687 P (log R′ ) [day] 32 rot HK M sin i [MJup] 0.1872 0.6579 a [AU] 0.6813 0.8972 Photometric and astrometric data are from the Hipparcos catalogue (ESA 1997) and the stellar physical quantities from Sousa et al. (2008). Nmeas 58 Span [day] 1583 rms [m/s] 1.417 From the HARPS high-precision survey, we present an in- pχ2 2.789 teresting system of two planets in a 3:2 mean motion reso- nance around HD45364, a configuration not observed previ- Errors are given by the standard deviation σ and λ is the mean longitude ously among extra-solar planet, but similar to the Neptune-Pluto of the date (λ = ω + M). The orbits are assumed co-planar. pair in the Solar System. Section 2 gathers useful stellar infor- mation about HD45364, its derived orbital solution is described HD 45364 HARPS 16.50 in Sect.3, and the dynamical analysis of the system is discussed in Sect. 4. Finally, conclusions are drawn in Sect. 5. 16.49

16.48 2. Stellar characteristics of HD 45364 The basic photometric (K0V, V = 8.08, B − V = 0.72) and as- 16.47 trometric (π = 30.69mas) properties of HD45364 were taken 16.46 from the Hipparcos catalogue (ESA 1997). They are recalled in Table1 with inferred quantities such as the [km/s] 16.45 (MV = 5.51) and the stellar physical characteristics derived from the HARPS spectra by Sousa et al. (2008). For the complete 16.44 high-precision HARPS sample (including HD45364), these au- thors provided homogeneous estimates of effective temper- 10 5 ature (Teff = 5434 ± 20 K), ([Fe/H] = −0.17 ± 0.01), and (log g = 4.38 ± 0.03) of the stars. A projected 0 −

1 O-C [m/s] low rotational velocity of the , v sin i = 1kms was derived -5 from a calibration of the width of the cross-correlation function -10 used in the radial-velocity estimate (Santos et al. 2002). 53000 53200 53400 53600 53800 54000 54200 54400 54600 HD45364 is a non-active star in our sample with an activ- JD-2400000 [days] ′ ity indicator log RHK of −4.94. No significant radial-velocity Fig. 1. HARPS radial velocities for HD45364, superimposed on jitter is thus expected for the star. From the activity indicator, a 3-body Newtonian orbital solution (Table2). we also derive a period Prot = 32day (following Noyes et al. 1984). HD45364 has a subsolar metallicity with [Fe/H] = −0.17 un- the first stages of the observations, peculiar variations in the like most of the gaseous giant-planet host stars (Santos et al. radial velocities (Fig.1) have shown the presence of one or 2004). According to simulations of planet formation based more companions in the system. After 58 measurements, we on the core-accretion paradigm, moderate metal-deficiency are now able to determine the nature of these bodies. Using does not, however, prevent planet formation (Ida & Lin 2004; a genetic algorithm combined with the iterative Levenberg- Mordasini et al. 2007). Taking into account its subsolar metal- Marquardt method (Press et al. 1992), we first attempted to fit licity, Sousa et al. (2008) derived a mass of 0.82M⊙ for the star. the complete set of radial velocities using a model with two From the , the derived effective temperature, and the Keplerian orbits. This fit yields an adjustment of pχ2 = 2.71 corresponding bolometric correction, we estimated the star lu- and rms = 1.38ms−1 with two planets, one at P = 225.8day, minosity to be 0.57L⊙. e = 0.174, and a minimum mass of 0.186 MJup, and another at P = 343.9day, e = 0.097, and a minimum mass of 0.659 MJup. 3. Orbital solution for the HD 45364 system Due to the proximity of the two planets and their high min- imum masses, the gravitational interactions between these two The HARPS observationsof HD 45364 started in December 2003 bodies are strong. This prompts us to fit the observational data and have now been going for about four and a half. From using a 3-body Newtonian model, assuming co-planar motion A.C.M. Correia et al.: HD 45364, a pair of planets in a 3:2 mean motion resonance 3

15

10

5

0 radial velocity (m/s) -5

-10

-15 2004 2006 2008 2010 2012 2014 2016 2018 2020 Fig. 2. Radial velocity differences between the two independent Keplerian model and the 3-body Newtonian model (Table2). Data coincides at JD=2453776 (Feb. 9th 2006). We also plot the velocity residuals of the 3-body fit (Fig.1). With the cur- rent HARPS’s precision for this star, we expect to observe these differences within ten years. perpendicular to the plane of the sky, similarly to what has been done for the system HD202206 (Correia et al. 2005). The or- bital parameters corresponding to the the best fitted solution are listed in Table 2. We get identical results for pχ2 and veloc- ity residuals as obtained with the two-Keplerian fit. Although Fig. 3. Variation in the resonant argument, θb = 2λb − 3λc + ωb (a) and in the secular argument, ∆ω = ωb − ωc (b), with there is no significant improvement in the fit, an important dif- ◦ time. θb is in libration around 0 , with a libration period Plθ ≃ ference exists: the new orbital parameters for both planets show ◦ some deviations from the two-Keplerian case. In particular, the 18.16yr, and a principal amplitude of about 68.4 (Table 4). ∆ω ◦ arguments of the periastrons (ω) show a difference of some de- is in libration around 180 , with a libration frequency g∆ω = g1 − g2 (corresponding to a period P∆ω ≃ 413.9yr), and a maximum grees. We then conclude that, despite being unable to detect the ◦ planet-planet interaction in the present data, the orbits undergo amplitude of about 36.4 . important perturbations, and we expect to detect this gravita- tional interaction in the future. In Fig.2, we plot the two best fit models evolving in time and clearly observe detectable devi- as 5 or 18 days, frequently with very high eccentricity values. ations between the two curves that appear within ten years. The Therefore, no other companion can be conclusively detected in 3-body Newtonian fit provides a more accurate approximation of the residuals from the orbital solution listed in Table 2. The best fit solution was obtained by adding a linear drift to the data, with the HD45364 planetary system, and the orbital parameters thus −1 determined will be adopted as reference henceforward (Table2). slope = −0.86 ± 0.09 ms /yr, allowing us to reduce the value We also fitted the data with a 3-body Newtonian model for of pχ2 to 2.42 and the rms = 1.21m/s, while the orbital param- which the inclination of the orbital planes, as well as the node of eters of the two planets remain nearly the same. The solution the outer planet orbit, were free to vary. We were able to find a in Table2 must then be considered to be the best determination wide variety of configurations, some with low inclination values achievable so far, and a longer tracking of the system will pro- for one or both planets, that slightly improved our fit to a mini- vide more accurate orbital parameters for the HD45364 system. mum pχ2 = 2.32 and rms = 1.14m/s. However, all of these de- terminations are uncertain, and since we also increase the num- 4. Dynamical analysis ber of free parameters by three, we cannot say that there has been an improvement with respect to the solution presented in We now briefly analyze the dynamics and stability of the plane- Table2. The inclination of the orbits therefore remain unknown, tary system given in Table 2. Due to the two planets’ proximity as do their true masses. and high values of the masses, we expect that both planets are The residuals around the best fit solution are small, but re- affected by strong planetary perturbations from each other. The main slightly larger than the internal errors (Fig.1). We may present orbits of the two planets almost cross (Fig.6), and unless then ask if there are other companions with different orbital pe- a resonant mechanism is present to avoid close encounters, the riods. For this purpose, we used a genetic algorithm, since we system cannot be stable. were unable to clearly isolate any significant peak in the fre- quency analysis of the residuals. The inclusion of additional 4.1. The 3:2 mean motion resonance companions in the system allows us to reduce the value of pχ2 slightly, although this can be justified as a natural consequence The ratio between the orbital periods of the two planets deter- of increasing the number of free parameters. Identical adjust- mined by the fitting process (Table 2) is Pc/Pb = 1.511, sug- ments can be obtained with many orbital periods, as different gesting that the system may be trapped in a 3:2 mean motion 4 A.C.M. Correia et al.: HD 45364, a pair of planets in a 3:2 mean motion resonance

Fig. 4. Stability analysis of the nominal fit (Table2) of the HD45364 planetary system. For a fixed initial condition of the outer (a) and inner planet (b), the phase space of the system is explored by varying the semi-major axis ak and eccentricity ek of the other planet, respectively. The step size is 0.001 AU in semi-major axis and 0.005 in eccentricity. For each initial condition, the full system is integrated numerically over 10 kyr and a stability criterion is derived with the frequency analysis of the mean longitude (Laskar 1990, 1993). As in Correia et al. (2005), the chaotic diffusion is measured by the variation in the frequencies. The “red” zone corresponds to highly unstable orbits, while the “dark blue” region can be assumed to be stable on a billion-years timescale. The contour curves indicate the value of χ2 obtained for each choice of parameters. It is remarkable that in the present fit, there is perfect correspondence between the zone of minimal χ2 and the 3:2 stable resonant zone, in “dark blue”.

Table 3. Fundamental frequencies for the orbital solution in of about 68.44 degrees (Fig.3a, Table4). For the complete so- Table 2. lution, the libration amplitude can reach more than 80 degrees because additional periodic terms are present. In Table4, we pro- Frequency Period vide a quasi-periodic decomposition of the resonant angle θb in ◦ /yr yr terms of decreasing amplitude. All the quasi-periodic terms are nb 576.429624 0.624534 easily identified as integer combinations of the fundamental fre- nc 384.033313 0.937419 quencies (Table3). Since the resonant angle is modulated by a − g1 0.759309 474.115 relatively short period of about 18 years, the observation of the g2 0.110472 3258.74 l 19.820696 18.1628 system over a few additional decades may provide an estimate θ of the libration amplitude and thus a strong constraint on the pa- nb and nc are the mean motions, g1 and g2 are the secular frequencies rameters of the system. of the periastrons, and lθ is the libration frequency of the resonant angle Although the mean motions nb and nc can be associated with θb = 2λb − 3λc + ωb. Indeed, we have 2nb − 3nc + g1 = 0. the two planets b and c, respectively, it is not the case for the secular frequencies g1 and g2, and incidentally, both periastrons resonance. To test the accurancy of this scenario, we performed precess with mean frequency g1 that is retrograde, with a period a frequency analysis of the orbital solution listed in Table 2 com- of 474.115 years. The two periastrons are thus locked in an an- puted over 100 kyr. The orbits of the planets are integrated with tipodal state, and the difference ∆ω = ωb − ωc is in libration around 180◦ with an amplitude of about 36.4◦ (Fig.3b). As a re- the symplectic integrator SABAC4 of Laskar & Robutel (2001), ◦ using a step size of 0.02 years. We conclude that in the nomi- sult, the argument θc = 2λb − 3λc + ωc librates around 180 with nal solution of Table 2, the two planets in the HD 45364 system the same libration frequency lθ. indeed show a 3:2 mean motion resonance, with resonant argu- ment: 4.2. Stability analysis θb = 2λb − 3λc + ωb . (1) To analyze the stability of the nominal solution and confirm the The fundamental frequencies of the systems are the two mean presence of the 3:2 resonance, we performed a global frequency motions nb and nc, the two secular frequencies of the periastrons analysis (Laskar 1993) in the vicinity of the nominal solution g1 and g2, and the libration frequency of the resonant argument (Fig.4), in the same way as achieved for the HD202206 sys- lθ (Table3). These frequencies are not independent because, due tem by Correia et al. (2005). For each planet, the system is inte- to the 3:2 resonance, we have up to the precision of the determi- grated on a regular 2D mesh of initial conditions, with varying nation of the frequencies (≈ 10−10), semi-major axis and eccentricity, while the other parameters are 2n − 3n + g = 0 . (2) retained at their nominal values. The solution is integrated over b c 1 10 kyr for each initial condition and a stability indicator is com- ◦ The resonant argument θb is in libration around 0 , with a puted to be the variation in the measured mean motion over the libration period 2π/lθ = 18.16 yr, and an associated amplitude two consecutive 5 kyr intervals of time. For regular motion, there A.C.M. Correia et al.: HD 45364, a pair of planets in a 3:2 mean motion resonance 5

Fig. 5. Evolution of the HD45364 planetary system over 100 kyr in the co-rotating frame of the inner planet (left) and outer planet (right). x and y are spatial coordinatesin a frame centered on the star and rotating with the frequency nb (left) or nc (right). Due to the 3:2 mean motion resonance the planets never get too close, the minimal distance of approach being 0.371 AU, while the maximum distance can reach 1.811 AU. We also observe that the trajectories are repeated every 3 orbits of the inner planet and every 2 orbits of the outer planet.

Table 4. Quasi-periodic decomposition of the resonant angle 1 θb = 2λb − 3λc + ωb for an integration over 100 kyr of the orbital solution in Table 2.

Combination νi Ai φi 0.5 nb nc g1 g2 lθ (deg/yr) (deg) (deg) 0 0 0 0 1 19.8207 68.444 -144.426 0 0 -1 1 0 0.8698 13.400 136.931 0 0 1 -1 1 18.9509 8.606 168.643 0 0 -1 1 1 20.6905 8.094 82.505 0 0 0 -2 2 0 1.7396 2.165 -176.138 y (AU) 0 0 -2 2 1 21.5603 0.622 -50.564 0 0 0 0 3 59.4621 0.540 -73.279 1 -1 0 0 -1 172.5756 0.506 7.504 -0.5 1 -1 0 0 0 192.3963 0.501 -46.923 0 0 -3 3 0 2.6093 0.416 -129.207 0 0 2 -2 1 18.0811 0.420 121.712 1 -1 0 0 1 212.2170 0.416 78.651 -1 0 1 -1 0 0 384.7926 0.451 176.155 -1 -0.5 0 0.5 1 1 -1 0 0 -2 152.7549 0.424 -118.070 x (AU) 0 1 -1 0 -1 364.9719 0.341 50.581 0 1 -1 0 1 404.6133 0.274 121.729 Fig. 6. Long-term evolution of the HD45364 planetary system 0 0 1 -1 3 58.5923 0.212 -120.210 over 5 Gyr starting with the orbital solution from Table 2. The 1 -1 0 0 2 232.0377 0.201 24.225 panel shows a face-on view of the system. x and y are spa- 0 0 -1 1 3 60.3319 0.211 153.652 tial coordinates in a frame centered on the star. Present orbital 1 0 -1 0 -1 557.3682 0.182 -86.342 solutions are traced with solid lines and each dot corresponds to the position of the planet every 100 kyr. The semi-major We have θ = N A cos(ν t + φ ), where the amplitude and phases A , b Pi=1 i i i i axes (in AU) are almost constant (0.676 < a < 0.693 and / b φi are given in degree, and the frequencies νi in degree year. We only 0.893 < a < 0.902), but the eccentricities undergo significant give the first 20 terms, ordered by decreasing amplitude. All terms are c identified as integer combinations of the fundamental frequencies given variations (0.123 < eb < 0.286 and 0.042 < ec < 0.133). The in Table 3. The fact that we are able to express all the main frequencies fundamental periods related to the precession of the periastrons of θb in terms of exact combinations of the fundamental frequencies g1, are 2π/g1 = −474 yr and 2π/g2 = 3259 yr. g2 and lθ is a signature of a very regular motion.

It is quite remarkable that, in contrast to the findings of is no significant variation in the mean motion along the trajec- Correia et al. (2005) for the 5:1 resonancein the HD202206sys- tory, while it can vary significantly for chaotic trajectories. The tem, there is perfect coincidence between the stable 3:2 resonant result is reported in color in Fig.4, where “red” represent the islands, and curves of minimal χ2 obtained in comparing with strongly chaotic trajectories, and “dark blue” the extremely sta- the observations. Since these islands are the only stable zones in ble ones. In both plots (Figs. 4a,b), it appears that the only stable the vicinity, this picture presents a very coherent view of dynam- zone that exists in the vicinity of the nominal solution are the ical analysis and radial velocity measurments, which reinforces stable 3:2 resonant zones. the confidence that the present system is in a 3:2 resonant state. 6 A.C.M. Correia et al.: HD 45364, a pair of planets in a 3:2 mean motion resonance

In Fig.5, we plot the evolution of the HD45364 planetary according to Crida et al. (2008), an inner disc must be present. system over 100 kyr in the rotating frame of the inner and the This singular planetary system may then provide important con- outer planet, respectively. Due to the 3:2 mean motion resonance straints on planetary formation and migration scenarios. trapping, the relative positions of the two planets are repeated The strong gravitational interactions between the planets and never become closer than about 0.37 AU, preventing close may also allow us to model their effect more accurately in the encounters and the consequent destruction of the system. The near future. With the current precision of HARPS, about 1 m/s paths of the planets in the rotating frame illustrate the relation- for HD45364, we expect to detect the signature of planet-planet ship between the resonance and the frequency of conjunctions interactions in data in a few decades. The planet-planet interac- with the internal or external planet. The inner planet is in a 3:2 tions may provide important information about the inclination of resonance with the outer planet, so the orbital configuration of the orbital planes and allow us to determine the precise masses the system is repeated every 3 orbits of the inner planet and ev- of both planets. ery 2 orbits of the outer planet. In this particular frame, we are also able to see the librationof each planet aroundits equilibrium Acknowledgements. We acknowledge support from the Swiss National Research Found (FNRS), the Geneva University, Fundac¸˜ao Calouste Gulbenkian position. (Portugal), and French CNRS.

4.3. Orbital evolution References From the previous stability analysis, it is clear that the HD 45364 Correia, A. C. M., Udry, S., Mayor, M., et al. 2005, A&A, 440, 751 planetary system listed in Table2 is trapped in a 3:2 mean mo- Crida, A., S´andor, Z., & Kley, W. 2008, A&A, 483, 325 tion resonance and stable over a Gyr timescale. Nevertheless, we ESA. 1997, VizieR Online Data Catalog, 1239 tested directly this by performing a numerical integration of the Ida, S. & Lin, D. N. C. 2004, ApJ, 616, 567 orbits over 5 Gyr using the symplectic integrator SABAC4 of Laskar, J. 1990, Icarus, 88, 266 Laskar, J. 1993, Physica D Nonlinear Phenomena, 67, 257 Laskar & Robutel (2001) with a step size of 0.02 years. The re- Laskar, J. & Robutel, P. 2001, Celestial Mechanics and Dynamical Astronomy, sults displayed in Fig. 6 show that the orbits indeed evolve ina 80, 39 regular way, and remain stable throughout the simulation, which Lovis, C., Mayor, M., Pepe, F., et al. 2006, Nature, 441, 305 corresponds to the estimated age of the star. Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20 Mayor, M., Udry, S., Lovis, C., et al. 2008, ArXiv e-prints, 806 Because of the strong gravitational interactions between the Mordasini, C., Alibert, Y., Benz, W., & Naef, D. 2007, ArXiv e-prints, 710 two planets, both orbital eccentricities present significant varia- Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, tions. The eccentricity of the inner planet is within 0.12 < eb < A. H. 1984, ApJ, 279, 763 0.29, while that of the outer planet is within 0.04 < ec < 0.13. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. We also observe rapid secular variations in the orbital parame- 1992, Numerical recipes in FORTRAN. The art of scientific computing (Cambridge: University Press, 2nd ed.) ters, mostly driven by the rapid secular frequency g1, of period Santos, N. C., Israelian, G., & Mayor, M. 2004, A&A, 415, 1153 2π/g1 ≈ 474 yr (Table3). These secular variations in the orbital Santos, N. C., Mayor, M., Naef, D., et al. 2002, A&A, 392, 215 elements occur much more rapidly than in our Solar System, Sousa, S. G., Santos, N. C., Mayor, M., et al. 2008, ArXiv e-prints, 805 which should enable them to be detected directly from observa- Udry, S., Mayor, M., Naef, D., et al. 2000, A&A, 356, 590 tions.

5. Discussion and conclusion We have reported the detection of two planets orbiting the star HD45364, with orbital periods of 228 and 342 days, and mini- mum masses of 0.187 and 0.658 MJup, respectively. A dynamical analysis of the system has further shown a 3:2 mean motion res- onance between the two planets, which ensures its stability over 5 Gyr despite the proximity of the two orbits. This is the first time that such an orbital resonant configuration has been ob- served for extra-solar planets, although an analogue does exist in our Solar System composed by Neptune and Pluto. However, while Neptune evolves in an almost circular orbit and is much more massive than Pluto (which is the largest member of the as- teroid family of the Plutinos), the two planets around HD45364 have masses comparable to those of Saturn and Jupiter, and are evolving in orbits with moderate eccentricity. Dynamically, the system is extremely interesting. In the nominal solution, the resonant angle θb = 2λb − 3λc + ωb is in libration around 0◦, with a libration period of 18.16 years and a dominating amplitude of 68.44 degrees. Such an orbital config- uration may have been reached through the dissipative process of planet migration during the early stages of the system evo- lution. However, after a resonant capture, subsequent migration produces a significant increase in planetary eccentricities, unless a damping mechanism is applied. Since the eccentricities of the two planets around HD45364 are relatively small (Table2), mi- gration may cease shortly after capture in resonance occurs, or,