arXiv:0811.0001v1 [astro-ph] 31 Oct 2008 pc cecsCne,Gievle L311 USA 32111, FL Gainesville, Center, Sciences Space ysclrasdllc o ra ag finclinations. of range broad a stabilized for be lock could that apsidal very MMR secular system 6:1 the by the placed in) that likely fit (and orbital veloci- near radial better (2003) a al. find to et inde- ties Fischer the an of analysis configuration. performed pendent antialigned (2003) an for Maciejewski about Go´zdziewski stable & lock still apsidal but more secular system thus chaotic, the and be incli- found con- Go´zdziewski chaotic (2003) could of aligned stable. be range be that to to broad claimed likely likely a (2003) less for were Sun that figurations & as for and Zhou well libration MMR as nations. 11:2 amplitude case, the large edge-on to underwent the close periapses was found the (MMR). system (2003) Peale the resonance & that mean-motion Lee close and to 9:2 Go´zdziewski (2003) due Both the likely is to that proximity behavior best-fit chaotic, then-current the was that solution found coplanar and considered systems (2002) edge-on al. configura- orbital et con- current Kiseleva-Eggleton of the tion. in importance sur- uncertainty the brief the demonstrates sidering A results sys- their studies. This of dynamical 2003). about vey several al. orbit inspired et circular has (Fischer nearly tem away a further follows times c three 12661 HD planet of axis jor uino h w in lnt riigH 26,a 12661, HD orbiting evo- planets secular giant the two investigate 2006b; ≃ the We Greenberg of & con- lution Barnes 2007). for 2006; al. 2005; importance S´andor et al. Laughlin et considerable & (Ford Adams theories of formation planetary topic straining a become lcrncades [email protected]fl.edu address: Electronic 1 h eua vlto fmlipae ytm has systems multi-planet of evolution secular The rpittpstuigL using typeset Preprint ApJL press, in srnm eatet nvriyo lrd,21Bryant 211 Florida, of University Department, Astronomy 1 . 136 ial,w oeta h eua vlto mle ytecretorbit current the by implied evolution was spends secular that c the planet planet that additional c an note planet ne of we if scattering for Finally, arise strong would oscillations to evolution secular due large im of perhaps type undergoes significantly This amplitud planet not large solutions. outer do orbital or the and circulation of weak indicate eccentricity observations quite current n are that direct resonances for and conditions motion e initial I HD mean of secular of ensembles any evolution. observations of generate velocity to secular range radial sampling the full Carlo combine of we the Thus, nature th describes observations. the in inadequately in uncertainties elements ambiguities inevitable orbital significant the to Unfortunately, lead models. formation h bevdrda eoiisado h ytmssclreccent o secular state, system’s rare the relatively headings: and/or Subject a velocities during radial observed observed being the is system this Either M h cetiiyeouino utpepae ytm a rvd valu provide can systems planet multiple of evolution eccentricity The ≃ ⊙ EUA VLTO FH 26:ASSE AGTA NUNLIKELY AN AT CAUGHT SYSTEM A 12661: HD OF EVOLUTION SECULAR tr lntH 26 a semima- a has b 12661 HD Planet . 0 . 3Uadamdrt cetiiy and eccentricity, moderate a and 83AU 1. INTRODUCTION A T E tl mltajv 03/07/07 v. emulateapj style X ∼ eeta ehnc tr:idvda H 26)—paeaysyst planetary — 12661) (HD individual : — mechanics celestial ehd:nbd iuain,statistical simulations, n-body methods: — 6 ftetm olwn nobtmr ceti hnta presently that than eccentric more orbit an following time the of 96% ∼ y hnst a to thanks Gyr iir Veras Dimitri ABSTRACT npes ApJL press, in 1 rcB Ford B. Eric , tblt.Orsuyipoe pnpeiu tde by studies previous upon improves long-term study of Our requirement the stability. and observations with tent in- dynamics, MMRs. full possible the all for direct cluding account use to we 6:1 integrations evolution, terms and secular n-body near-resonant the 11:2, expect affect 5:1, not significantly the to do we by dynam- Although affected secular high-order significantly the MMRs. that not or and is well ics octupole 12661 secular HD the the of Henrard evolution (2003), described & either Peale approximation Libert & that Laplace-Lagrange Lee and found (2007) behavior. (2005), system’s theories Henrard Gallardo & secular Rodr´ıguez the & Libert low-order model using and to against (2007) MMR. caution 11:2 secular Armitage (2007) the & the and Veras to eccentricities Both approximation large to crude due a evolution only gave theory determinations. orbital ac- in properly uncertainties of for orbital counting importance of the range regarding illustrates historical consensus configurations the updated of Further, an lack motivate study. the both dynamical and evolution c secular system’s & solutions the b orbital 12661 revised HD The very circulating for evolution. system and secular librating the of (2006a) between modes places boundary Greenberg solution the & the orbital pe- Barnes near orbital this of that ratio 13:2. found a approaching (2006) with al. solution riods et orbital Butler an Subsequently, published likely. equally were (2003) configurations nearly al. antialigned and et aligned Ji that However, reported for inclinations. particularly an large an possible, with about systems about also librations were librate configuration that to aligned likely but found more configuration, were (2003) antialigned periapses Maciejewski Peale the Go´zdziewski & & that Lee and both solutions, (2003) orbital different the Despite edtrietemd()o eua vlto consis- evolution secular of mode(s) the determine We secular classical the that found (2003) Go´zdziewski iiyevolution. ricity 1 usqetyacee notestar. the onto accreted subsequently ryalo h loe two-planet allowed the of all arly ouin ossetwt current with consistent volutions 26 ihMro hi Monte Chain Markov with 12661 irto fteprass The periapses. the of libration e diinlpaesaeaffecting are planets additional r bd nertos efidthat find We integrations. -body a eniplieyperturbed, impulsively been had urn ria lmnscan elements orbital current e lcngrto mle that implies configuration al tgaigaysnl e of set single any ntegrating attesclrevolution, secular the pact becntanso planet on constraints able m:formation ems: TIME observed. 2 Veras and Ford combining a Bayesian analysis of an updated set of ra- of stable systems in our 7 bins are: 1) 100%, 2) 99.7%, 3) dial velocity observations with direct n-body integrations 93.3%, 4) 16.4%, 5) 0.2%, 6) 16.7%, and 7) 98.6%. We to examine HD 12661’s stability and secular evolution. independently affirmed the trend exhibited by these sta- We characterize the full range of orbital histories that bility percentages by performing a smaller, additional set are consistent with observations and reject solutions that of simulations by using an ensemble of initial conditions do not exhibit long-term orbital stability. Thus, we can generated from a different MCMC code that accounted make quantitative statements about the relative proba- for planet-planet interactions with a Hermite integrator. bility of different modes of secular evolution. 3. RESULTS 2. METHODS Several previous studies of HD 12661 (e.g. Ji et al. We reanalyze an updated set of ob- 2003; Lee & Peale 2003; Zhou & Sun 2003) and other servations (Wright et al. 2008) assuming that the stel- planetary systems (Chiang et al. 2001; Malhotra 2002; lar velocity is the superposition of two Keplerian or- Ford et al. 2005; Barnes & Greenberg 2006b) have high- bit plus noise. First, we perform a brute force search lighted the importance of the apsidal angle (∆̟ = over parameter space to verify that there are no qualita- Ωb + ωb − Ωc − ωc), which is useful for describing the tively different and comparably likely solutions. Work- secular dynamics of systems with small relative inclina- ing in a Bayesian framework, we generate an ensemble of tions. Since we consider systems with a wide range of ≃ 5 × 105 orbital solutions using Markov Chain Monte relative inclinations, we instead focus on ∆̟′, the angle Carlo (MCMC; Ford 2005, 2006; Gregory 2007a,b). Each between the periastron directions projected onto the in- state of the Markov chain includes the variable plane. Typically, the apsidal angle is classified as (P ), velocity amplitude (K), eccentricity (e), argument circulating, librating about 0◦, or librating about 180◦. of pericenter measured from the plane of the sky (ω), and By inspecting plots of apsidal angle evolution for indi- mean anomaly at a given (u) for each planet. We vidual systems, we find that some systems spend most use the priors, candidate transition functions, automated of the time librating about one center, but occasionally step size control, and convergence tests described in Ford the apsidal angle circulates for a short period of time. (2006). In particular, we assume a prior for the velocity Therefore, we consider two summary statistics describ- −1 −1 “jitter” (σj ) of p(σj ) ∝ [1 + σj /(1ms )] . ing the secular evolution: the root mean square (RMS) We randomly select subsamples of orbital solutions to of ∆̟′ and the mean absolute deviation (MAD) of ∆̟′ be investigated with long-term n-body integrations and about the libration center, where the “libration center” consider a range of possible inclinations (i) and ascend- is defined as the angle about which the RMS ∆̟′ is min- ing nodes (Ω). We generate each orbit from an isotropic imized. For a system undergoing small amplitude libra- distribution (i.e., uniform in cos i and Ω) and divide the tion about either center, the MAD and RMS deviation resulting systems into bins based on the relative incli- are of order the libration amplitude. For a system under- ◦ nation between the two orbital planes (irel). The bins going uniform circulation, the MAD approaches 90 and ◦ ◦ ◦ ◦ ◦ are 1) irel = 0 (coplanar), 2) 0 ≤ irel ≤ 30 , 3) 30 ≤ the RMS deviation approaches 103.92 . These statistics ◦ ◦ ◦ ◦ ◦ irel ≤ 60 , 4) 60 ≤ irel ≤ 90 , 5) 90 ≤ irel ≤ 120 , 6) can be used to identify the mode of secular evolution in ◦ ◦ ◦ ◦ 120 ≤ irel ≤ 150 , and 7) 150 ≤ irel ≤ 180 . We inte- clear cut cases and provide a quantitative measure that grate 2,000 systems in each of the first four bins and is well-defined even for systems with complex evolution. 500 in each of the last three bins. We can reweight We calculate both measures using orbital elements our simulations so as to determine the probability of measured every 1000 yr. Although the libration am- the different modes of secular evolution for various as- plitude (maximum absolute deviation of ∆̟′) can be sumed distributions of relative inclinations. From each sensitive to the frequency of sampling and leads to ambi- set (P, K, e, ω, i, Ω,u), we generate the planet (m) guities, both RMS ∆̟′ and MAD ∆̟′ are more robust and semi-major axis (a) using a Jacobi coordinate system statistics that allows us to describe the secular evolu- (Lee & Peale 2003). tion with a simple quantitative measure, regardless of We used the hybrid symplectic integrator of Mercury whether the system is librating, circulating, or switching (Chambers 1999) to integrate each set of initial condi- between regimes due to short-term perturbations. This tions for at least 1 Myr. Based on a smaller series of robustness is particularly important for studying systems 10 Myr integrations, we found that the vast majority of where there is little distinction between the librating and instabilities are manifest within 1 Myr. We classified sys- circulating regimes, due to one planet’s orbit periodically tems as “unstable” if, for either planet, amax − amin > becoming nearly circular. τai, where amax and amin represent the maximum and For each n-body integration, we determine the libra- minimum values of the semimajor axis, ai is the ini- tion center and calculate both the RMS and MAD of tial value of the semimajor axis, and τ = 0.3. We dis- ∆̟′ about that center. We discover that the RMS carded each set of initial conditions that was found to be ∆̟′ ranges from 68◦ − 85◦ and MAD ∆̟′ ranges from unstable, and analyzed the properties of the remaining 63◦−82◦, depending on the relative inclination (see Table “stable” systems. Although a small fraction of our “sta- 1). The libration centers are preferentially anti-aligned ble” systems might exhibit instability if integrated for for prograde, nearly-coplanar systems, and transition to much longer timescales, our criteria avoids miscategoriz- almost entirely aligned for near-coplanar retrograde sys- ing a system exhibiting bounded chaos (e.g., Go´zdziewski tems. The stability is highly dependent on initial relative 2003) as unstable. We manually verified that the above inclination; very few highly inclined retrograde systems criteria gives reasonable results and that various choices were stable, indicating that non-secular perturbations in- of τ ∈ [0.10, 0.30] make just a few percent difference in fluence the long-term dynamics for some relative inclina- the number of simulations labeled as stable. The percent tions. Although large values of RMS ∆̟′ and MAD ∆̟′ HD 12661 Unexpected Evolution 3

0.40

1.00

0.75 HD 12661 c

0.50 Time Fraction 0.25

0.20 0.00

Unitless 1.00

0.75

0.50 HD 12661 b Time Fraction 0.25

0.00 0.00 36 45 54 63 72 81 90 0.00 0.20 0.40 0.59 0.79 0.99 Eccentricity Inclination of Invariable Plane (deg)

Fig. 1.— Values of c1 (dots), c2 (triangles), c3 (squares), and Fig. 2.— Fraction of time for which each planet’s eccentricity c4 (crosses) and their standard deviations, for coplanar systems, exceeded a given threshold (x-axis) averaged over allowed orbital binned according to the sine of the inclination of the invariable solutions. The curves with dots show results for the coplanar case, plane. but assuming an isotropic distribution of inclinations relative to the plane of the sky. The dashed vertical lines indicate the me- dian eccentricity values (0.031 and 0.361) for the planets’ initial eccentricities, and their 5%-95% uncertainty ranges are indicated may be consistent with non-uniform circulation, visual by the horizontal solid segments. HD 12661 c’s spot checks of individual systems all suggest libration. exceeds its current value for approximately 96% of its evolution. If a system of two planets on nearly circular orbits is excited impulsively, then the secular evolution will fol- culation. For the vast majority of systems, c3 is over low the boundary between the circulating and librating one order of magnitude greater than 0.003 (as found by regimes and cause one orbit to repeatedly return to a Barnes & Greenberg 2006b). nearly circular orbit (Malhotra 2002). In some systems, The finding that planet c undergoes large eccentric- this secular evolution can be used to constrain the sys- ity oscillations is remarkable since it currently has an tem’s formation (Ford et al. 2005). Barnes & Greenberg eccentricity of only ≃ 0.02. We investigate this observa- (2006b) found that the Butler et al. (2006) orbital solu- tion further by plotting the fraction of simulation time tion implies the eccentricity of HD 12661 c comes par- that each planet’s eccentricity exceeded a given value for ticularly close to zero, as parameterized by their ǫ pa- the coplanar systems (Fig. 2). The dotted vertical lines rameter (Eq. 1 of Barnes & Greenberg 2006b). Because indicate representative values for the planets’ initial ec- they considered only the published orbital solution in an centricities. HD 12661 c’s orbital eccentricity exceeds edge-on, coplanar orientation, they were unable to asses its current value for approximately 96% of its evolution. the finding’s robustness to uncertainties in the orbital pa- Thus, the near-circular orbit we now observe is a rare rameters or non-edge-on systems. In order to determine occurrence. what fraction of stable systems consistent with observa- tions result in one planet returning to a nearly circular 4. CONCLUSION orbit, we calculate four statistics from each of our n-body Several previous studies had suggested HD 12661 b & c integrations. The first two are c1 ≡ emin,c/emax,c and are in a secular apsidal lock and are undergoing large am- c2 ≡ emin,b/emax,b, which represent the ratio of the mini- plitude librations. However, previously, even the orbital mum to maximum eccentricity for each planet. The third period of HD 12661 c was poorly determined. Now that (c3) is equal to [2min(ebec)] /(xmax−xmin +ymax−ymin), radial velocity observations span over two orbital peri- as defined by Barnes & Greenberg (2006b), where x ≡ ods of HD 12661 c, the current orbital period of planet ebec sin (∆̟) and y ≡ ebec cos(∆̟). The fourth is c is well-constrained. An exhaustive search for libration c4 ≡ min(ebec)/max(ebec), which is a similar measure of all possible 11:2, 6:1, and 13:2 resonant angles found that includes the eccentricities of both planets in a ro- that in only a few percent of all simulations performed, tationally symmetric manner that is independent of the large amplitude libration lasts over several 105 yr. The systems’ orientation (unlike c3). current orbital elements of the two giant planets imply Figure 1 plots the mean and standard deviation of that the system lies near the boundary of circulating and each measure (c1 - dots, c2 - triangles, c3 - squares, c4 - librating regimes, regardless of the unknown orbital incli- crosses) for coplanar systems which have been binned nations. This behavior causes the eccentricity of planet according to the sine of the inclination of the invari- c to spend most of its time with a significant eccentricity able plane. The values c1, c3 and c4 all maintain values (ec ≥ 0.2). By computing the projected apsidal angle on < 0.1 for all bins of relative inclination for prograde sys- the invariable plane, we find that the RMS ∆̟′ ranges tems, and < 0.15 for retrograde systems. The large val- from 68◦ − 85◦ and MAD ∆̟′ ranges from 63◦ − 82◦, ues of c2 (triangles) imply that HD 12661 b maintains a for any stable initial relative inclination, and planet c’s moderately eccentric orbit throughout the planet’s evo- eccentricity would be greater than its current value 96% lution. Regardless, the measure c4 demonstrates that of the time. the system lies near the boundary of libration and cir- The most straightforward interpretation is that the 4 Veras and Ford eccentricity of HD 12661 b was excited by an impul- dial velocity observations, there are few alternatives. We sive perturbation and planet c’s eccentricity is peri- consider it unlikely that our global search missed a qual- odically excited via secular perturbations, similar to itatively different pair of orbits due to our assumption of the history of the υ And c & d system (Malhotra non-interacting Keplerian orbits. More plausibly, one or 2002; Ford et al. 2005). Such an impulse could be more undetected planets could be affecting the orbital so- the result of planet-planet scattering involving at lution and/or the secular evolution of the system. After least one additional planet (e.g. Rasio & Ford 1996; accounting for planets b & c, we find no statistically sig- Weidenschilling & Marzari 1996). The current eccentric- nificant periodicities in the current radial velocity obser- ity of planet b suggests that the putative scattered planet vations. Current observations are consistent with a small would have had a mass roughly half that of planet b long-term acceleration, but zero acceleration falls within (Ford & Rasio 2008). If the additional planet had been the 95% credible interval. Further, models with a long- scattered outwards, then there is a significant chance that term acceleration do not result in significantly different it would have altered the secular evolution of planet c orbital parameters. Our Bayesian analysis assuming two (Barnes & Greenberg 2006b). Given the semi-major axis planets and uncorrelated Gaussian noise constrains the of planet b, it is possible that the scattered planet was jitter to be ≃ 3.4 ± 0.7m/s. Further radial velocity ob- accreted onto the star. If the instability occurred after servations can test whether a portion of this jitter is due the star had reached the , then this in- to additional planets. stability could contribute to the star’s large surface high metalicity. One potential concern for this mechanism is whether the timescale for the apoastron of the scattered We thank the referee for insightful comments, and De- planet to be lowered would be significantly shorter than bra Fischer and Geoff Marcy for providing updated radial the ≃ 104 −105yr timescale for secular evolution of plan- velocity observations prior to publication. This research ets b & c. was supported by NASA/JPL RSA1326409. Some of There is an alternative interpretation of our results. the data analyzed were obtained at the W. M. Keck Ob- One might think that we are unlikely to observe a sys- servatory, which is operated as a scientific partnership tem at such a rare time, casting doubt upon the orbital among the California Institute of Technology, the Uni- fit and thus the inferred secular evolution. One would versity of California, and the National Aeronautics and expect that one of the 30 currently observed multiple Space Administration. The Observatory was made pos- planet systems is transiently passing through such an un- sible by the generous financial support of the W. M. Keck usual state. However, there are several ways in which a Foundation. We acknowledge the University of Florida system could be “unusual” so that the standard caveats High-Performance Computing Center for providing com- of a posteriori statistics apply. Given the current ra- putational resources and support.

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TABLE 1 Results of N-body Integrations

′ ′ irel (RMS) aligned fraction (MAD) aligned fraction RMS ∆̟ MAD ∆̟ Stable Isotropic 0◦ 35.9% 31.7% 84.9◦ ± 8.5◦ 81.1◦ ± 7.8◦ 100% ··· 0◦ − 30◦ 39.3% 35.2% 83.8◦ ± 9.4◦ 79.5◦ ± 8.6◦ 99.7% 6.7% 30◦ − 60◦ 56.8% 51.8% 84.2◦ ± 10.4◦ 79.7◦ ± 9.8◦ 93.3% 18.3% 60◦ − 90◦ 56.0% 52.0% 83.8◦ ± 10.5◦ 79.6◦ ± 9.9◦ 16.4% 25.0% 90◦ − 120◦ — — ——0.2% 25.0% 120◦ − 150◦ 95.2% 95.2% 74.7◦ ± 12.1◦ 69.3◦ ± 11.1◦ 16.7% 18.3% 150◦ − 180◦ 99.6% 99.2% 68.4◦ ± 12.8◦ 63.0◦ ± 12.1◦ 98.6% 6.7% Prograde, isotropic 69.8% 80.0% Retrograde, isotropic 38.5% 50.0%

◦ Note. — Columns 2 and 3 each list the RMS and MAD percent of stable systems with a libration center of 0 instead of 180◦ for the range of relative inclinations in column 1. Columns 4 and 5 report the RMS and MAD variations for systems which librate around the most common center for that range of initial relative inclinations. Column 6 lists the percent of stable systems, and Column 7 lists the percent of systems with the given range of inclination for an isotropic distribution. The bottom two rows present aggregated stability statistics for the isotropic prograde and retrograde distributions of orbits. 2000 simulations were performed for each prograde relative inclination bin, and 500 simulations for each retrograde bin.