arXiv:0705.1858v2 [astro-ph] 5 Jan 2008 o.Nt .Ato.Soc. Astron. R. Not. Mon. 9Otbr2018 October 29 ryzo Go Krzysztof system planetary Her 14 the of analysis dynamical A

trwsas oioe yohrpae-utn em.TeJ The teams. planet-hunting in other companion by (1998) monitored talk also conference was a star in Ge announced the was by team discovered search Herculis planet 14 around planet extrasolar An INTRODUCTION 1 20) nternwppr h icvr em(afe l 20 al. et (Naef of trend team RV linear discovery a revealing the observations 119 paper, published new their In (2003). oun oad bewtru srnmcn,Uniwersyte Pozna´n, Poland Astronomiczne, iewicza, Obserwatorium Academy Poland; Polish Toru´n, Center, Astronomical Copernicus Nicolaus ‡ Poland Astronomy, for Toru´n Centre † ⋆ d RV the with consistent are configurations which stable systems for two-planet searched tative we algorithm, con fitting stability the incorporating optimization the i.e., 2003), enlzdtefl aasto 5 esrmns oeiga covering measurements, 154 W window. of days, observational set 3400 data combined full the the of reanalyzed part middle the ning uncertainty 20) emre hs esrmnswt cuae 5obse 35 accurate, of mean with (the measurements vations these Go´zdziews merged in we bodies, distant (2006), more other, indicate may data rf a cone o,terso h igepae+rf m planet+drift single the of rms the if was for, Even meas m/s. accounted 14 these about was to of drift rms solution large abnormally Keplerian an planet yielded ments single The year. per c ounCnr o srnm,Poland Astronomy, for Toru´n Centre 00RAS 0000 ∼ olwn h yohssta h iertedi h RV the in trend linear the that hypothesis the Following 11 . /,sgicnl agrta h enobservational mean the than larger significantly m/s, 3 h ∼ σ ∼ m 2 ∼ i P 70dy ri a etcnre yBte tal. et Butler by confirmed next was orbit days 1700 b sn u APmto G´diwk tal. (Go´zdziewski et method GAMP our Using . 7 . zdziewski / utdi htpaper. that in quoted m/s 2 σ ´ m 000 ∼ 0–0 00)Pitd2 coe 08(NL (MN 2018 October 29 Printed (0000) 000–000 , 3 . /)b ulre l 20) span- (2003), al. et Butler by m/s) 1 ue oincmaini nobtwithin orbit o an ex the in be to companion can limits Jovian dynamical trend derive outer the We that planet. obje hypothesis giant distant additional a second test the ( indicating al we et trend considerations, Wittenmyer a and and (2006) orbit al day et. 1760 Butler (2004), Sun-lik al the et. of Naef measurements (RV) velocity radial Precision ABSTRACT h ue lnti Uobtwt oeaeeccentricity moderate a with orbit config AU edge-on 9 an in to planet s fit outer long-term Newtonian the the stable for important and are self-consistent planets Jovian the between xso h ue lnti h ag f(,3 Uadteeccen the it when and but AU determined (6,13) be approach of yet range cannot the inclination in orbital The planet outer the of axis pne ytelwodrma oinrsnne : n 6:1. and 5:1 of resonances minimum motion mean low-order the by spanned ofiuain ihn1 within configurations e words: Key 1 ⋆ eayMigaszewski Cezary , ∼ 10m xrslrpaesrda eoiysasidvda 14 velocity—stars:individual planets—radial extrasolar ( χ ν 2 tansinto straints J ) .Mick- A. t fSciences, of 1 n for and ∼ / 2 ie al. et ki 3 fpu- of . n esssoe ierneo h ytmiciain Oth inclination. system the of range wide a over persists and ovian m/s 6 The . neva bout odel ure- σ ata. 04) the r- i e ofiec nevlo h etfi r osbefrtesemi-m the for possible are fit best the of interval confidence ∼ 30 o a jitter lar sntaeut opoel nepe h Vobservations. RV the interpret properly to ana adequate dynamical kinemati not the earlier is How- that suspect our (MMR). can of we resonance 2006), results al. motion (Go´zdziewski the et mean mind 4:1 in having the ever, sol of best-fit vicinity the found the have in (2007) al. et Wittenmyer RV, the of xesoe h mean the over excess hywr oaie ntepoiiyo o-re enmotio mean low-order Her of (14 MMR proximity 3:1 the (MMRs), resonances in localized were They ottepeec fascn lntr bet h single-p The object. rms planetary an second has a solution of presence do still the and days port 4463.2 n spanning is observations set 203 data of available mean prised publicly full a The have m/s. They 7.5 Observatory. (200 of accuracy McDonald al. the et precisio at Wittenmyer independent made Recently, and surements additional m/s. 35 1 published have as also points have small data as single window, errors observational mal the of end the tutr ftepaesae efudacerrlto fcha configurati 10 of of relation self-disrupting time-scale clear short unstable, a strongly comp found to We a revealed space. motions fits phase selected the the co of of structure maps neighborhood dynamical the The dynamically. in active puted the be but still separated, would well tem be would companions the Apparently, pcr n ulse eie e f5 bevtoso 4H 14 t of observations re-analyzed 50 have of accuracy (2006) set mean revised Their al. a et Butler published and pipeline, spectra data the to ic n qal odbs t with fits t best particular good in equally solutions, and stable tinct many revealed search This h irrh ftemse sreversed. is masses the of hierarchy the 1 n aijKonacki Maciej and † fe h prd fHRSsetorp n improvements and spectrograph HIRES of upgrade the After A T E ∼ tl l v2.2) file style X 3 . ∼ / Bte ta.20) sn h elra model Keplerian the Using 2006). al. et (Butler m/s 5 3A.I hscs,temta interactions mutual the case, this In AU. 13 ∼ 4 ∼ σ 3ms n hw e ee e second per meter few a shows and m/s, 13 eid fteotrotplanet. outermost the of periods m 2 . ∼ / sarayvr od e,towards Yet, good. very already is m/s 4 /,addi udauet h stel- the to quadrature in added m/s, 6 wr 4Hrui ulse by published Herculis 14 dwarf e erae,bt lntr masses planetary both decreases, btlprmtr fteputative the of parameters rbital t ntegonso dynamical of grounds the On ct. 07 eelaJva lnti a in planet Jovian a reveal 2007) aiiyo h ytm h best The system. the of tability lie ytepeec fan of presence the by plained ∼ rto fJva lnt has planets Jovian of uration hsslto isi shallow a in lies solution This rct nterneo (0,0.3). of range the in tricity 0 . a n ofie oazone a to confined and 2 c a e—-oyproblem Her—N-body 2 ∼ n : M 1 Her (14 MMR 6:1 and ) ‡ 5 . Uand AU 8 a c rstable er n na on ons ∼ wcom- ow model c odis- wo mea- n ssup- es AU. 9 ution lanet lysis ajor sys- heir otic for- lex m- b er. 7) ). n 2 K. Gozdziewski,´ C. Migaszewski and M. Konacki

In this paper, we re-analyse the updated high-precision RV that helps us to detect local minima of χ2, even if they are distant in data to study and extend the kinematic model by Wittenmyer et al. the parameter space. We can also obtain reliable approximation to (2007). We also revise and correct the conclusions from our previ- the errors of the best-fit parameters (Bevington & Robinson 2003) ous paper (Go´zdziewski et al. 2006) that were derived on the basis within the 1σ, 2σ and 3σ confidence levels of χ2 at selected 2-dim of a smaller and less precise RV data set. The goal of our work is parameter planes, as corresponding to appropriate increments of to derive dynamical characteristics of the putative planetary system χ2. The hybrid approach will be called the algorithm I from here- and to place limits on its orbital parameters. Assuming a putative after. configuration close to the 4:1 MMR, the outermost orbital period However, in this paper we deal with a problem of only a partial would be ∼ 18–20 yr. Hence, the currently available data would coverage of the longest orbital period by the data. In such a case χ2 cover only a part of this period and a full characterization of the does not have a well defined minimum or the space of acceptable orbits would still require many years of observations. One cannot solutions is very “flat” so the confidence levels may cover large expect that in such a case the orbits can be well determined. Nev- ranges of the parameters. In such a case, to illustrate the shape of ertheless, we show in this work, that the modeling of the RV data χ2 in the selected 2-dimensional parameter planes, we use a com- when merged with a dynamical analysis and mapping of the phase plementary approach that relies on systematic scanning of the pa- space of initial conditions with fast-indicators enable us to derive rameter space with the fast Levenberg-Marquardt (L-M) algorithm meaningful limits to the orbital parameters. Such an extensive dy- (Press et al. 1992). Usually, as the parameter plane for represent- namical study of this systems has not yet been done. ing such scans, we choose the semimajor-axis—eccentricity (a,e) In order to take into account the RV variability induced by plane of the outermost planet. We fix (a,e) and then search for the the stellar activity, the internal errors of the RV measurements are best fit, with the initial conditions selected randomly (but within σ2 σ2 σ2 rescaled by the jitter added in quadrature: = m + j , where reasonably wide parameter bounds). Then the L-M scheme ensures σm and σj are the mean of the internal errors and the adopted a rapid convergence. This approach is called the algorithm II from estimate of the stellar jitter. The 14 Herculis is a quiet star with hereafter. In fact, we already used it in Go´zdziewski et al. (2003). ′ logRHK = −5.07 (Naef et al. 2004) thus it is reasonable to keep the The method is CPU-time consuming and may be effectively ap- safe estimate of σj = 3.5 m/s (Butler et al. 2006). The mass of the plied in low-dimensional problems but in reward it can provide a parent star varies by ∼ 10% in different publications. In this work, clear and global picture of the parameter space. we adopt the canonical mass 1M⊙, although we also did some cal- Moreover, the RV measurements can be polluted by many culations for the mass of 0.9M⊙ and 1.1M⊙. sources of error, like complex systematic instrumental effects, short time-series of the observations, irregular sampling due to observing conditions, and stellar noise. The problem is even more complex when we deal with models of resonant or strongly interacting plan- 2 MODELING THE RV DATA – AN OVERVIEW etary systems. It is well known that the kinematic model of the The standard formulae by Smart (1949) are commonly used to RV is not adequate to describe the observations in such instances. model the RV signal. Each planet in the system contributes to the Instead, the self-consistent N-body Newtonian model should be ap- reflex motion of the star at time t with: plied (Rivera & Lissauer 2001; Laughlin & Chambers 2001). (Note that the hybrid optimization can be used to explore the param- ω ν ω Vr(t) = K[cos( + (t)) + ecos ] +V0, (1) eter space for both models). Still, the best-fit solutions may be where K is the semi-amplitude, ω is the argument of pericenter, related (and often do) to unrealistic quickly disrupting configura- ν(t) is the true anomaly (involving implicit dependence on the or- tions (Ferraz-Mello et al. 2005; Lee et al. 2006; Go´zdziewski et al. bital period P and time of periastron passage Tp), e is the eccen- 2006). In such a case one should explore the dynamical stability tricity, V0 is the velocity offset. We interpret the primary model of the planetary system in a neighborhood of the best-fit configura- parameters (K,P,e,ω,Tp) in terms of the Keplerian elements and tion. One can do that with the help of direct numerical integrations minimal masses related to Jacobi coordinates (Lee & Peale 2003; or by resolving the dynamical character of orbits in terms of the Go´zdziewski et al. 2003). maximal Lyapunov exponent or the diffusion of fundamental fre- In our previous works, we tested and optimized different tools quencies. Hence, the most general way of modeling the RV data used to explore the multi-parameter space of χ2. In the case when should involve an elimination of unstable (for instance, strongly χ2 has many local extrema, we found that the hybrid optimization chaotic) solutions during the fitting procedure. We described such provides particularly good results (Go´zdziewski & Konacki 2004; an approach (called GAMP) in our previous works. In particular, Go´zdziewski & Migaszewski 2006). A run of our code starts the we made attempts to analyze the old data set of 14 Her as well genetic algorithm (GA, Charbonneau 1995) that basically has a as other stars hosting multiple planet systems (Go´zdziewski et al. global nature and requires only the χ2 function and rough pa- 2006). rameter ranges. GA easily permits for a constrained optimization within prescribed parameter bounds or for a penalty term to χ2 (Go´zdziewski et al. 2006). Because the best fits found with GAs 3 KEPLERIAN FITS are not very accurate in terms of χ2, we refine a number of “fittest individuals” in the final “population” with a relatively fast local In order to compare our fits with the results of Wittenmyer et al. method like the simplex of Melder and Nead (Press et al. 1992). (2007), we carried out the hybrid search for the two-planet edge- The use of simplex algorithm is a matter of choice and in princi- on configuration. In the experiment, we limited the orbital peri- ple we could use other fast local methods. However, a code using ods to the range of [1000,14000] days and the eccentricities to the non-gradient methods works with the minimal requirements for the range of [0,0.9]. The choice of the upper limit of orbital periods user-supplied information, i.e., the model function, usually equal to followed the conclusions of Wittenmyer et al. (2007). According to χ2 [its inverse 1/χ2 is the fitness function required by GAs]. The re- the results of adaptive-optics imaging by Luhman & Jayawardhana peated runs of the hybrid code provide an ensemble of the best-fits (2002) and Patience et al. (2002), the parent star has no stellar com-

c 0000 RAS, MNRAS 000, 000–000 A dynamical analysis of the 14 Her planetary system 3

0.37 90.0 89.8 89.6 b

0.36 [m/s] e b 89.4 K 89.2 0.35 89.0 1762 1764 1766 1768 1770 1772 1762 1764 1766 1768 1770 1772 Pb [days] Pb [days]

250 0.60 200

0.40 150 c [m/s] e c 100 0.20 K 50 0.00 0 4000 6000 8000 10000 12000 14000 4000 6000 8000 10000 12000 14000 Pc [days] Pc [days]

Figure 1. The best fit solutions to the two-planet model of the RV data of 14 Her in the (P,e)- and (P,K)-plane. In the search, a distant companion to the known Jovian planet in ∼ 1760 days orbit is assumed. Upper plots are for the inner planet, middle plots are for the outer one. The best-fit found with the hybrid code in 2 1/2 the range of moderate orbital periods of the outer planet, with (χν) ∼ 1.085, is marked with intersecting lines. Its elements, in terms of the parameter tuples of Eq.1, (K[m/s],P[days],e,ω[deg],Tp − T0[days]), are (89.460, 1765.743, 0.360, 22.108, 1375.03) for the inner and (226.2263, 11675.333, 0.395, 182.006, 6115.352 ) for the outer planet, respectively, with the velocity offsets (VN2004,VB2006,VW2007) = (-129.105, -132.345, -108.384) m/s for the data in (Naef et al. 2 1/2 2004), (Butler et al. 2006) and (Wittenmyer et al. 2007), respectively; the epoch T0 = JD2,450,000. Dots are for 1σ solutions with (χν) ∈ [1.0848,1.090) 2 1/2 and an rms ∼ 8.17 m/s, red crosses are for the the best fit solutions with (χν) ∼ 1.0848. The red curve in (Pc,ec)-plane marks an approximate collision 2 1/2 line of the orbits determined from ab(1 + eb)= ac(1 − ec), with ab,eb fixed at their best-fit values. Bottom panels are for the scans of (χν) at the selected 2 1/2 parameter planes. Contours (also shown in the upper panels, for a reference) mark the limits of (χν) corresponding to 1σ,2σ and 3σ of the best fit solution. Circle mark the position of the best N-body solution (see Sect. 4). See the text for remaining details. panion beyond 9 — 12 AU, so we try to verify the hypothesis about best fit may be as low as ∼ 3. The orbital period in the best fit is a short-period, low-mass companion 14 Her c up to that distance. ∼ 12000 days and Pc/Pb ∼ 7. This is roughly consistent with the The RV offsets are different for each of the three instruments and work of Wittenmyer et al. (2007) who quote the best-fit configura- are included as free parameters in the model together with the tu- tion close to the 4:1 MMR and with a similar rms 8.17 m/s. Yet ples of (K,P,e,ω,Tp) for each companion. Note that we calculate the fits are not constrained with respect to Pc. Many values between the RV offset VN2004 with respect to the simple mean of all RV mea- [5000,14000] days are equally good in terms of 1σ confidence in- surements from the set published by Naef et al. (2004). terval of the best-fit solution. Moreover, it means that the system The results of the hybrid search (algorithm I) are illustrated in may be confined to a zone spanned by many low-order mean mo- upper panels Fig. 1 (note that before plotting, the fits are sorted with tion resonances of varying width in the (Pc,ec)-plane. In such cases, respect to the smaller semi-major axis). The best-fit found with this the mutual interactions between the planets are important for the 2 1/2 method has (χν) ≃ 1.085 and an rms ≃ 8.17 m/s. Its param- long-term stability. 2 1/2 eters are given in the caption to Fig. 1. Clearly, there is no iso- In order to illustrate the shape of (χν) in more detail, we lated best-fit solution but rather a huge number of acceptable fits. also applied the algorithm II for the systematic scanning of that The ratio of the orbital periods within 1σ confidence interval of the function in selected parameter planes (see lower panels in Fig. 1).

c 0000 RAS, MNRAS 000, 000–000 4 K. Gozdziewski,´ C. Migaszewski and M. Konacki

2 1/2 This has lead to even better statistics of the best fits and a very clear Overall, the topology of (χν) does change, nevertheless 2 1/2 2 1/2 illustration of the (χν) -surface. The panels reveal that (χν) qualitatively certain features are common. Contrary to the Keple- 2 1/2 of 2-Kepler model has no minimum in the examined ranges of the rian case, we can detect a shallow minimum of (χν) close to outermost periods, Pc < 14,000 days. (9 AU,0.2) and a hint of another minimum localized far over the Along a wide valley, the best fits with an increasing Pc have collision line. The white thick lines on the panels mark mass lim- also rapidly increasing ec. For Pc ∼ 14,000 days, they reach the its of mc. Clearly, for ac > 9 AU and for moderate eccentricity, mc collision zone of the orbits. In this strongly chaotic region a num- approaches the brown-dwarf mass limit and substellar mass 80 mJ ber of MMRs of the type p : 1 with p > 7 overlap. It justifies a- for ∼ 12 AU. Thus a brown-dwarf could exist in the system beyond posteriori our choice of the search limit Pc ∼ 14,000 days. Note, 10-11 AU depending on the inclination and eccentricity — and the that the two searches are in an excellent accord that is illustrated fits preclude a low-mass planetary object over the 14 mJ level. 2 1/2 with the (χν) smooth contour levels overlayed on the results of Note, that the best-fit for the nominal edge-on configuration the hybrid search. In that case, the scanning of parameter space with i = 90◦ turns out to be unstable but its parameters are very 2 1/2 helps to understand the shape of (χν) in detail. close to a stable solution given in Table 1; its synthetic curve is The 1σ scatter of solutions around the best-fit (Fig. 1) pro- illustrated in Fig. 5. vides realistic error estimates. Although the parameters of the inner To characterize the stability of the best-fit solutions, we com- planet are very well fixed, the elements of the outer Jovian planet puted their MEGNO fast indicator (Cincotta & Sim´o2000), hY i, have much larger formal errors. The kinematic fits do not provide (i.e., an approximation of the maximal Lyapunov exponent) up 5 an upper bound to the semi-major axis of the outer planet. Both to ∼ 2 × 10 Pc using a fast and accurate symplectic algorithm methods reveal exactly the same parameters of the inner companion (Go´zdziewski et al. 2005). The total integration time 3-4 Myr is (see Fig. 1). Obviously, certain discrepancy between the outcomes long enough to detect even weak unstable MMRs. 2 1/2 of algorithm I and algorithm II is related to a “flat” shape of (χν) The best-fits that passed the MEGNO test (i.e. that are reg- in the (ac,ec)-plane and to a worse efficiency of simplex in detect- ular over the integration time) are marked with black dots in the 2 1/2 ing shallow minima compared to the gradient-like L-M algorithm. right panels of Fig. 3. For a reference, the σ-levels of (χν) from 2 1/2 When (χν) does not have a well localized minimum, it is very the respective left-panels are marked in these plots as well. The difficult to resolve its behaviour with any non-gradient method. red curves are for the collision line of the orbits computed for well constrained and fixed parameters of the inner planet. Over- all, the stable solutions reveal the structure of the MMRs that we have already seen in the dynamical maps computed for fixed or- 4 SELF-CONSISTENT NEWTONIAN FITS bital parameters (see Fig. 2). The semi-major axis of the outer In order to compare the outcome of kinematic modeling with the planet cannot be yet constrained well and may vary between 6 AU self-consistent N-body one, at first, we transformed the ensemble and (at least) 13 AU. Moreover there is a rather evident although of Keplerian fits to astro-centric osculating elements (Lee & Peale irregular border ec ∼ 0.3 of all solutions, up to 3σ. Besides, we 2003; Go´zdziewski et al. 2003). These fits became the initial con- found that in the relevant range of ac, the mean longitude λc = ◦ ◦ ditions for the L-M algorithm employing the N-body model of the ϖc + Mc ∼ (200 ,300 ) within the 1σ interval of the best fit, in- RV. In this experiment, the mass of the star was M⋆ = 1 M⊙. Curi- creasing along quasi-parabolic curve as the function of ac. Simul- ◦ ◦ ously, we found that the procedure often converged to the two best- taneously, λb ∼ 352.6 ± 1.5 . From the dynamical point of view, 2 1/2 fits, both having similar (χν) ∼ 1.083, an rms ∼ 8.15 m/s, and this provides significant limits on the relative mean phases of the ac ∼ 7.8 AU and ac ∼ 8.4 AU, respectively. To shed some light on planets. the orbital stability of these solutions, we computed the dynamical The results for two-planet 14 Her system in Go´zdziewski et al. maps in the neighborhood of the best-fits in terms of the Spectral (2006) are basically in accord with the present work although not 2 1/2 Number (logSN, Michtchenko & Ferraz-Mello 2001). We chose ac all conclusions are confirmed. The two isolated minima of (χν) and ec as the map coordinates, keeping other orbital parameters at found in that paper can be identified in this work too. In particu- their nominal values. The results are shown in Fig. 2. The left panel lar it concerns the narrow 3:1 MMR island. The second solution is for the unstable fit located in the separatrix of the 5:1 MMR. The close to the 6:1 MMR has also been found close to the present right panel is for the stable solution in the center of 9:2 MMR. best fit solution. Nevertheless, the more extensive current search These dynamical maps prove that the mutual interactions between reveals many other stable solutions between 3:1 and 9:1 MMRs planets are significant — even small changes of initial parameters equally good in terms of 1σ. Thus the results of our previous work modify the shapes of low-order MMRs which overlap already for are not wrong in general but rather incomplete — it appears that ec ∼ 0.2 − 0.3. Besides, the chaotic motions are related to strongly the search performed was not exhaustive enough to find all accept- unstable configurations (see also Fig. 3, 4). able solutions. Still, some of the unstable solutions can be modi- As in the case of the Keplerian model, to resolve the topology fied in order to obtain stable fits, possibly with slightly increased 2 1/2 2 1/2 of (χν) in detail, we performed similar searches for the best-fit (χν) . For instance, the stability maps in Fig. 3 seem to rule out configurations using the N-body model of the RV. Again, the re- the 4:1 MMR in the system what may be related to specific orbital sults of the hybrid algorithm are consistent with the algorithm II. phases. After an appropriate change of these phases [still keeping 2 1/2 2 1/2 However, to conserve space, we only show the (χν) -maps ob- (χν) at acceptable level] we could find also stable solutions close tained by scanning the (ac,ec)-plane. The results are illustrated in to the 4:1 MMR. In order to do this in a self-consistent manner, the left column panels of Fig. 3. a GAMP-like method would be necessary. However, as we have 2 1/2 These scans are computed for three different and fixed values shown, (χν) is very flat in the interesting range of orbital pa- of the inclination of the putative orbital plane of the system, i = rameters. Then, the extensive enough GAMP search would require 90◦, 60◦ and 30◦, respectively. In the maps, we mark a few contour a very significant amount of CPU time, so we decided not to do it. 2 1/2 levels of (χν) that correspond to 1σ,2σ and 3σ-levels of the best Finally, using the algorithm II, we try to resolve the topology 2 1/2 fit solution denoted with a circle in every panel. of (χν) with respect to a varying mass of the star and inclina-

c 0000 RAS, MNRAS 000, 000–000 A dynamical analysis of the 14 Her planetary system 5

Figure 2. The dynamical maps in terms of the Spectral Number (logSN) in the (ac,ec)-plane for the two-planet Newtonian models of the 14 Her system. Colors used in the log SN map classify the orbits — black indicates quasi-periodic regular configurations while yellow strongly chaotic ones. The maps are computed for the following osculating elements of the Jovian planets at the epoch of the first observation t0, in terms of tuples (m [mJ],a [AU],e,ω [deg],M (t0) [deg]): the left panel is for (4.975, 2.864, 0.359, 22.281, 330.365) of the inner planet, and (5.618, 8.343, 0.174, 188.543, 72.091) of the outer planet, respectively; the right panel is for (4.974, 2.863, 0.359, 22.310, 330.314) of the inner planet, and (4.558, 7.863, 0.173, 187.674, 64.417) of the outer planet, respectively. The large circles mark the parameters of the best fits. The thin line marks the collision curve for planets b and c, as determined by ab(1 + eb)= ac(1 − ec). The 4 low-order MMRs of the planets b and c are labeled. The integrations are conducted over ∼ 10 Pc. The resolution is 500 × 120 data points.

200 Table 1. Astrocentric, osculating Keplerian parameters of the edge-on, Vr [m/s] 14 Herculis best-fit, self-consistent configuration of 14 Her system at the epoch of the 150 first observation, t0 = JD 2459464.5956. The mass of the parent star is 1.0 M⊙. See the text and Fig. 3 for the error estimates. The RV offsets 100 VN2004, VB2006, and VW2007 label the data sets published in (Naef et al. 2004; 50 Butler et al. 2006; Wittenmyer et al. 2007), respectively; the offset of VN2004 M is given with respect to the mean of RV in (Naef et al. 2004). denotes 0 the mean anomaly at the epoch t0. -50 2-planet Newtonian fit (rms=8.15 m/s) parameter planet b planet c -100 10000 12000 14000 m [mJ] 4.975 7.679 JD-2,450,000 [days] a [AU] 2.864 9.037 P [days] 1766 9886 Figure 5. Synthetic RV curve of the self-consistent best fit to the two- e 0.359 0.184 planet model. The osculating elements at the epoch of the first observation ω 2 1/2 [deg] 22.230 189.076 are given in Table 1. This fit yields (χν) ∼ 1.082 and an rms ∼ 8.15 m/s. M (T0) [deg] 330.421 81.976 Observations from Naef et al. (2004), Butler et al. (2006) and Wittenmyer −1 VN2004 [m s ] -52.049 et al. (2007) are marked with red, blue and green circles, accordingly. −1 VB2006 [m s ] -55.290 −1 VW2007 [m s ] -31.368 χ2 1/2 ( ν) 1.0824 tricity grows quickly to 1, implying ejections or collisions between −1 rms [m s ] 8.15 the planets or with the star. The nominal system lies in a stable zone between 5:1 and 6:1 MMRs (see Fig. 4). However, for the inclination of ∼ 60◦, the best fit configuration appears locked in the 5:1 MMR. Overall, the tion of the system (both can be free parameters in the model) but phase space does not change significantly compared to the edge-on assuming that the system remains coplanar. Adopting the uncer- system (Fig. 4, also Fig. 2). For the inclination of 30◦, the stable tainty of the 14 Her mass ∼ 10% (Butler et al. 2006), we choose zone shrinks due to strong interactions caused by large masses of ◦ 0.9M⊙,1.0M⊙,1.1M⊙, and inclinations of orbital plane as 90 the planets ∼ 10mJ (see caption to Fig. 6). Moreover, the masses (the edge-on system), 60◦,45◦,30◦. Apparently, the distributions of are not simply scaled by factor 1/sini, as one would expect, assum- ◦ the best-fits are similar for all (M⋆,i)-pairs. Their quality in terms ing the kinematic model of the RV. Instead, already for i ∼ 30 , the 2 1/2 of (χν) is comparable. mass hierarchy is reversed – the inner planet becomes more massive The significant differences between the best-fit configurations than the outer one. It is yet another argument against the kinematic are related to their stability. To address this issue, we calculated model of the RV data. The best-fits derived for small inclinations the dynamical maps for the best fits derived for the nominal mass are confined to a strongly chaotic zone. Comparing the dynamical ◦ o of the parent star (1 M⊙) and inclinations 90 (the edge-on sys- map for i ∼ 30 with Fig. 4, we conclude that is unlikely to find ◦ ◦ tem, Table 1), 60 , and 30 (see Fig. 4,6). The logSN maps are stable systems close to the formal best fit with ac ∼ 9 AU unless accompanied by the maximal eccentricity maps, maxec, i.e., the they are trapped in 5:1 MMR. The resonance island can be seen at eccentricity attained by the orbit during the total integration time. the edge of the bottom panels in Fig. 6. The planetary masses close 2 1/2 The maxe indicator makes it possible to resolve a link between to the minimum of (χν) remain smaller from the brown-dwarf formally chaotic character of orbits with their physical, short-term limit even for relatively small inclinations of the orbital plane. instability. In the regions where chaos is strong, the maximal eccen- Still, the parameters of the outer planet are not well con-

c 0000 RAS, MNRAS 000, 000–000 6 K. Gozdziewski,´ C. Migaszewski and M. Konacki

1.077 1.078 1.079 1.08 1.081 1.082 ec ec mc>80 MJ σ σ 3 0.6 0.6 3 i=90o 2σ 2σ 1σ 0.5 mc>14mJ mc>14 MJ σ 0.4 0.4 1 5:1 7:1 o 8:1 i=90 0.3 6:1 0.2 0.2 ••• 3:1 ••• 2 √χν<1.0768 0.1 0 0 5 7 9 11 ac [AU] 5 6 7 8 9 10 11 12 ac [AU]

1.077 1.078 1.079 1.08 1.081 1.082 ec ec 1σ m >80 M σ c J 3σ o 3σ 2 0.6 0.6 i=60 2σ 0.5 mc>14mJ σ mc>14 MJ 0.4 0.4 1 7:1 5:1 o 0.3 8:1 i=60 6:1 0.2 0.2 ••• 3:1 ••• 2 √χν<1.0764 0.1 0 0 5 7 9 11 ac [AU] 5 6 7 8 9 10 11 12 ac [AU]

1.074 1.076 1.078 1.08 1.082 e ec c σ σ 3σ o 3 2 0.6 0.6 2σ i=30 1σ m >80m mc>80 MJ 0.5 c J

0.4 mc>14mJ 0.4 7:1 5:1 m >14 M 0.3 c J 1σ 6:1 8:1 i=30o 0.2 0.2 •• 3:1 ••• • 0.1 2 √χν<1.0741 0 0 a [AU] 5 7 9 11 ac [AU] 5 6 7 8 9 10 11 12 c

Figure 3. The panels in the left column are for the Newtonian best-fit model with varying fixed inclination of the coplanar system i and illustrated as projections onto the (ac,ec)-plane of initial osculating elements at the epoch of the first observation. In each panel, the best fit is marked with a circle. The quality of the 2 1/2 fits within formal 1σ,2σ and 3σ confidence interval of the best fit, in terms of their (χν) is marked with contours and labeled in subsequent panels. White thick lines corresponds to mass levels of a brown-dwarf and sub-stellar companion, respectively. These solutions are derived with algorithm II. The panels in the right column are for the stability analysis of the best-fit configurations. See the text for more explanation. strained and it is clear that many years of new observations are Curiously, after the additional ∼ 3 yr observations, the parameters required to fix the elements of the putative outer companion. To of the outer planet cannot be still determined without doubt. After 2 1/2 simulate the future RV analysis of the system, we prepared two syn- ∼ 6 yr, the space of (χν) shrinks significantly and we may expect thetic sets of observations, choosing the elements of Table 1 as the that after such a time, the semi-major axis of the outer companion parameters of the “real” system. We computed the synthetic RV for can be fixed with 1σ range of ∼ 1 AU. that system and added Gaussian noise to these data with parame- ters as from the real observations. Next, we selected the data points sampling them with a similar frequency to that in the real measure- ments. We analyzed two sets: one extended over +1000 days and 5 SECULAR DYNAMICS OF THE BEST-FIT SYSTEM the second one extended over +2000 days after the last moment in Although the RV data permit for different dynamical states of the the real observations. Next, for both synthetic data sets (Fig. 7), we 2 1/2 system, we can analyse the secular evolution of the best fit coplanar scanned the (χν) space with the algorithm II in order to resolve and edge-on configuration (Table 1) and the solutions in its neigh- the best-fit system, moreover, in this experiment, no stability con- borhood. Due to the significant uncertainty of the orbital elements straints were imposed. The results are shown in two panels in Fig. 8. of outer planet, such analysis cannot provide exhaustive conclu-

c 0000 RAS, MNRAS 000, 000–000 A dynamical analysis of the 14 Her planetary system 7

◦ Figure 4. The dynamical maps in the (ac,ec)-plane in terms of the Spectral Number (logSN, the top-left panel), max∆ϖ relative to the libration mode 180 (the top-right panel) and maxeb,c (the bottom panels) for the two-planet edge-on coplanar Newtonian best-fit model of the 14 Her system (Table 1). Colors used in the log SN map classify the orbits — black indicates quasi-periodic regular configurations while yellow strongly chaotic ones. The large crossed circles mark 4 the parameters of the fit. The low-order MMRs of the planets b and c are labeled. The integrations are conducted over ∼ 10 Pc. The resolution is 500 × 120 data points. sions but we can obtain some qualitative view of the secular be- this powerful approach can be found in (Michtchenko & Malhotra haviour of the system. It can be easily extended when more RV 2004; Michtchenko et al. 2006). data will be available. To study the secular dynamics, we follow Michtchenko & Malhotra (2004) and describe the system in As we have demonstrated, the parameters of the inner planet terms of the canonical Poincar´e variables. First, we obtain the as well as the eccentricity and orbital phase of the putative outer time-average of the canonical elements derived from the numerical companion can be relatively well constrained. The most uncon- integration of the full equations of motion with the initial condition in Table 1. The mean canonical semi-major axes of the system are strained parameter in the edge-on, coplanar system is ac. Yet we should remember that the observations constrain very roughly the then habi ∼ 2.868 AU and haci ∼ 8.891 AU. The eccentricities are inclinations; and we did not include the longitudes of nodes as hebi ∼ 0.356 and heci ∼ 0.191. The initial state of the system is free parameters in the model, keeping the planets in coplanar or- also characterized by the difference of arguments of periastron, ∆ϖ ϖ ϖ ◦ σ ◦ bits. Still, we are in relatively a good situation because the care- = b − c ∼ 195 with 1 uncertainty ∼ 27 that is estimated ful analysis of the RV data makes it possible to reduce the di- from the statistics of solutions derived for fixed masses and semi- mension of the phase space. The dynamical map in Fig. 4 cover- major axes (parameters of the secular model), at their respective 2 1/2 values in Table 1. ing the shallow minimum of (χν) and the neighborhood of the formal best fit reveals its proximity to the 11:2 MMR. Several au- Next, we compute the energy levels of the averaged Hamilto- thors argue that in similar cases, (Libert & Henrard 2007, 2005; nian of the system hHseci (Fig. 9, left panel) in the characteristic ◦ ◦ Rodr´ıguez & Gallardo 2005) in the first approximation the near- plane (eb cos∆ϖ,ec) where ∆ϖ is set either to 0 or 180 . Let us resonance effects have a small or negligible influence on the secular note that ∆ϖ always passes through these values during the orbital dynamics. Hence, neglecting the effects of the MMRs, we can re- evolution related to two distinct libration modes of ∆ϖ in the copla- cover its basic properties with the help of the recent quasi-analytical nar system (Michtchenko & Malhotra 2004). The energy level of averaging method by Michtchenko & Malhotra (2004). It general- the nominal system is marked with red line. The black thick lines izes the classic Laplace-Lagrange (L-L) theory (Murray & Dermott mark the exact positions of libration centers (i.e., periodic solu- 2000) or a more recent work by Lee & Peale (2003). It relies tions) of ∆ϖ around 0◦ (mode I) and 180◦ (mode II), respectively. on a semi-numerical averaging of the perturbing Hamiltonian and A green part of the libration curve (on the right half-plane of the en- makes it possible to avoid any series expansions when recovering ergy diagram) is for the nonlinear secular resonance of the system. the secular variations of the orbital elements. Then one obtains a The black dashed lines mark different levels of the total angular very accurate secular model which is valid up to large eccentrici- momentum (expressed by the Angular Momentum Deficit, AMD). ties by far beyond the limits of L-L classical theory. The details of The nominal system (marked with filled dots) is found in the re-

c 0000 RAS, MNRAS 000, 000–000 8 K. Gozdziewski,´ C. Migaszewski and M. Konacki

Figure 6. The dynamical maps in the (ac,ec)-plane in terms of the Spectral Number (logSN, the left panels), and maxec (the right panels) for the two-planet coplanar Newtonian models of the 14 Her system and different inclinations of the orbital plane. Colors used in the logSN map classify the orbits — black indicates quasi-periodic regular configurations while yellow strongly chaotic ones. The maps are computed for the following elements of the Jovian planets in ◦ terms of tuples (m [mJ],a [AU],e,ω [deg],M (t0) [deg]). The top row is for i = 60 and the osculating elements (5.747, 2.864, 0.359, 22.271, 330.378) for the 2 1/2 inner planet, (7.343, 8.675, 0.171, 190.065, 75.540) for the outer planet and the velocity offsets (-43.458, -46.700, -22.780) [m/s]. The fit has (χν) ∼ 1.082 and rms ∼ 8.15 m/s. The bottom row is for i = 30◦ and the elements (9.978, 2.868, 0.357, 22.498, 330.135) of the inner planet, (8.581, 7.974, 0.142, 194.823, 2 1/2 58.154) of the outer planet and the velocity offsets (-29.237, -32.485, -8.585) m/s, (χν) ∼ 1.080 and rms ∼ 8.13 m/s. The large crossed circles mark 4 the parameters of the best fits. The low-order MMRs of the planets b and c are labeled. The SN integrations are conducted over ∼ 10 Pc. The resolution is 500 × 120 data points. Compare with Fig. 4 for the edge-on system.

200 200 Vr [m/s] 14 Herculis Vr [m/s] 14 Herculis 150 150 100 100 50 50 0 0 -50 -50 +1000 days -100 +2000 days -100 -150 10000 12000 14000 10000 12000 14000 16000 JD-2,450,000 [days] JD-2,450,000 [days]

Figure 7. Synthetic RV data and the RV curve of the corresponding best fit to the two-planet model. The left panel is for the time series extended over 1000 days after the last real observation, the right panel is for the synthesized data over 2000 days after the last real data point. Sampling frequency and errors of the artificial observations are similar to those of the real data.

gion of the anti-aligned apsides (close to the thin curve representing the center of mode II with a relatively large amplitude of eb ∼ 0.5. ◦ mode II) with ec in the range ∼ (0.1,0.2). The librations of ∆ϖ are confined to small semi-amplitudes ∼ 30 . The results of the secular theory are in an excellent accord with To illustrate the secular dynamics in more detail, we com- the behavior of the full unaveraged system. Its dynamical maps are puted the phase diagrams of the system in the characteristic plane shown in Fig. 4. The dynamical map of max∆ϖ (i.e., the maximal of (e cos∆ϖ,e sin∆ϖ) (the top-left panel in Fig 9) and in the b b semi-amplitude of ∆ϖ attained during the integration time) reveals (e cos∆ϖ,e sin∆ϖ)-plane (the bottom-left panel in Fig. 9). The c c a similar semi-amplitude of librations ∼ 30◦ around the center of red curves mark the oscillation around libration mode I, blue curves 180◦ (and of course, the presence of mode II). This zone persists are for libration mode II and black curves mark the circulation of over wide ranges of (a ,e ). In fact, almost the entire stable region ∆ϖ. The nominal system (marked with filled circles) lies close to c c

c 0000 RAS, MNRAS 000, 000–000 A dynamical analysis of the 14 Her planetary system 9

2 1/2 Figure 8. The topology of (χν) in the (ac,ec)-plane of the best fits to the synthetic data sets shown in Fig. 7. The left panel is for artificial observations spanning additional 1000 days after the date of the last real data point, the right panel is for the additional 2000 days period. Contours mark 1σ,2σ and 2 1/2 3σ-levels of (χν) . Thick lines (white in the left panel, black in the right panel) are for the mass limits of 14 mJ and 80 mJ, respectively. Circle marks the position of the nominal best-fit in Table 1 (the “real” system).

companion to the known Jovian planet b cannot yet be very well 0 20 40 60 80 100 120 140 160 180 constrained. Nevertheless the available data already reveal a very ec χ2 1/2 0 shallow minimum od ( ν) in the (ac,ec)-plane of the initial os- max ∆ϖ [deg] i=90 0.6 culating elements. The minimum persists for reasonable combina- tions of the parent star mass and the inclination of the system. Its 5:1 7:1 dynamical character strongly depends on that parameter influenc- 0.4 ing the planetary masses. Quite surprisingly, for small inclinations 6:1 the mass hierarchy is reversed and the inner planet becomes more massive than its distant companion. Also the positions of the nearby •• 0.2 3:1 • MMRs are significantly affected. Depending on the inclination, the system may be locked in the low-order 9:2, 5:1 or 6:1 MMR. We can also conclude that the kinematic model of the RV is already not 0 adequate for the characterization of the system and further analysis 5 7 9 11 ac [AU] of the RV would be done best within the self-consistent Newtonian Figure 10. The dynamical map in the (ac,ec)-plane in terms of the max∆ϖ model. indicator for edge-on, coplanar system and assemble of fits shown in Fig. 3, the top-left panel. The large crossed circles mark the parameters of the best The dynamical analysis enables us to derive some limits on fit solution in Table 1. The low-order MMRs of the planets b and c are the orbits elements and the stability of the system. No stable con- 4 labeled. The integrations are conducted over ∼ 10 Pc. figurations are found with a period ratio smaller than ∼ 3. This limit is shifted towards larger values when the inclination grows. For the inclination of 30◦, the best fit solutions are localized in extending at least up to 11 AU is spanned by configurations in- a strongly chaotic and unstable zone. This constitutes a dynami- volved in mode II librations. The error bars in plots of Fig. 9 are cally derived argument against a small inclination of the system. It σ derived from formal estimates of 1 errors of the parameters in Ta- is likely larger than 30◦-40◦. Allowing for some speculation, the ∆ϖ ◦ ble 1 (we estimate the error of ∼ 27 ; according with the analy- presence of the best-fits in a robustly stable zone supports the hy- sis of stability, we adopted the uncertainty of ec as 0.15). The errors pothesis of highly inclined two-planet configuration in the 14 Her are significant but still, in the statistical sense, the libration modes system. Our attempts to fit one more planet to the RV data that would mostly not change within the error ranges. This behaviour is could explain the scatter of the data points close to the minima of ∆ϖ confirmed in Fig. 10 showing for each individual best-fit found the RV curve (Fig. 5), did not lead to a meaningful improvement of (see the top-left panel of Fig. 3). Simultaneously, physically sta- the fit. The third planet would have a weak RV signal at the level ble systems are close to quasi-periodic configurations what follows of a few m/s and a short-period eccentric orbit. Finally, the analysis ∆ϖ from the comparison of the maxe and max indicators. Both of of all available RV data of 14 Her enables us to revise and extend them, as indicators of geometrical or physical behaviour of the sys- some of the conclusions from Go´zdziewski et al. (2006) — an iso- tem in short-time scale, appear tightly corelated to the logSN as the lated and very well defined best-fit solution cannot yet be found to formal measure of the system stability. properly describe the orbital configuration of 14 Her.

Acknowledgements. We thank Eric Ford for a critical review, suggestions and comments that greatly improved the manuscript. 6 CONCLUSIONS This work would not be possible without the access to the precision According to the results from the adaptive-optics imaging RV data which the discovery teams make available to the commu- (Luhman & Jayawardhana 2002; Patience et al. 2002), there is no nity. This work is supported by the Polish Ministry of Science and stellar-mass object in the 14 Her neighborhood beyond ∼ 12 AU. Education through grant 1P03D-021-29. M.K. is also supported by This finding supports the hypothesis that the RV trend may be at- the Polish Ministry of Science and Education through grant N203 tributed to a massive planet c. The orbital period of the putative 005 32/0449.

c 0000 RAS, MNRAS 000, 000–000 10 K. Gozdziewski,´ C. Migaszewski and M. Konacki

e2

0.8 AMD levels

0.6 collision line

0.4

0.2 energy levels

0 -1 -0.5 0 0.5 1 ∆ϖ e1 cos

Figure 9. The secular evolution of the best-fit configuration (see Table 1). The left panel is for the energy levels of the averaged secular Hamiltonian hHisec (blue, continuous curves) and the AMD levels (black, dashed lines) on the representative plane of initial conditions. The energy decreases and AMD ◦ ◦ increases with increasing ec. The sign of eb cos∆ϖ is for initial conditions with ∆ϖ = 0 (+) or ∆ϖ = 180 (-), respectively. The level curves are computed for mb/mc = 0.648 and ab/ac = 0.3226. The red thick curve marks the energy level of the nominal system, and filled circles indicate actual variation of the eccentricities (through the angular momentum integral). The black thick lines mark the centers of librations of ∆ϖ around 0◦ (mode I) or 180◦ (mode II), respectively. A green region (on the right, for positive eb cos∆ϖ) is for the nonlinear secular resonance. Panels in the left column are for the secular phase space of the planets b and c at secular energy hHisec of the nominal system in the (eb cos∆ϖ,eb sin∆ϖ)-plane (the right-top panel) and (ec cos∆ϖ,ec sin∆ϖ)-plane (the right-bottom panel), respectively. Two libration modes of ∆ϖ about the centers ∆ϖ = 0◦ (mode I, red curves) and ∆ϖ = 180◦ (mode II, blue curves) are labeled. Thick black lines are for the transition between the libration and circulation of ∆ϖ. Initial mean parameters of the nominal systems are marked with filled circles. For a reference, we plot formal error bars derived from 1σ errors of the best fit parameters in Table 1.

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