The Anglo-Australian Planet Search XXV: A Candidate Massive Saturn Analog Orbiting HD 30177
Robert A. Wittenmyer, Jonathan Horner, M.W. Mengel, R.P. Butler, D.J. Wright, C.G. Tinney, B.D. Carter, H.R.A. Jones, G. Anglada-Escude, J. Bailey, Simon J. O'Toole
TThe Anglo-Australian Planet Search XXV: A Candidate MassiveSaturn Analog Orbiting HD 30177
Robert A. Wittenmyer , , Jonathan Horner , , M.W. Mengel , R.P. Butler , D.J. Wright ,C.G. Tinney , B.D. Carter , H.R.A. Jones , G. Anglada-Escud´e , J. Bailey , SimonJ. O’Toole [email protected] ABSTRACT
We report the discovery of a second long-period giant planet orbiting HD 30177,a star previously known to host a massive Jupiter analog (HD 30177b: a=3.8 ± i = 9 . ± Jup ). HD 30177c can be regarded as a massive Saturnanalog in this system, with a=9.9 ± i = 7 . ± Jup . Theformal best fit solution slightly favours a closer-in planet at a ∼ n -body dynamical simulations show that configuration to be unstable. A shallowlocal minimum of longer-period, lower-eccentricity solutions was found to bedynamically stable, and hence we adopt the longer period in this work. Theproposed ∼
32 year orbit remains incomplete; further monitoring of this and otherstars is necessary to reveal the population of distant gas giant planets with orbitalseparations a ∼
10 au, analogous to that of Saturn.
Subject headings: planetary systems — techniques: radial velocities – stars: in-dividual (HD 30177) Computational Engineering and Science Research Centre, University of Southern Queensland,Toowoomba, Queensland 4350, Australia School of Physics and Australian Centre for Astrobiology, University of New South Wales, Sydney 2052,Australia Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road,NW, Washington, DC 20015-1305, USA Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, Herts AL10 9AB,UK School of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E14NS, UK Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia a r X i v : . [ a s t r o - ph . E P ] D ec
1. Introduction
Prior to the dawn of the exoplanet era, it was thought that planetary systems aroundother stars would likely resemble our own - with small, rocky planets close to their hoststars, and the more massive, giant planets at greater distances. With the discovery of thefirst exoplanets, however, that paradigm was shattered - and it rapidly became clear thatmany planetary systems are dramatically different to our own. But to truly understandhow unusual (or typical) the Solar system is, we must find true Jupiter and Saturn analogs:massive planets on decade-long orbits around their hosts. The only way to find such planetsis to monitor stars on decade-long timescales, searching for the telltale motion that mightreveal such distant neighbours.Nearly three decades of planet search have resulted in a great unveiling, at every stageof which we are finding our expectations consistently upturned as the true diversity of worldsbecomes ever more apparent. Much progress has been made in understanding the occurrencerates and properties of planets orbiting within ∼ Kepler revolution (e.g. Borucki et al. 2011; Rowe et al. 2014; Coughlin et al. 2016) and the adventof Doppler velocimetry at precisions of 1 m s − (Fischer et al. 2016). While Kepler has beenhugely successful in exploring the frequency of planets close to their stars, such transit surveysare not suited to search for planetary systems like our own - with giant planets moving onorbits that take decades to complete. To understand the occurrence of such systems requiresa different approach - radial velocity monitoring of individual stars on decadal timescales.Sometimes overshadowed by the
Kepler dicoveries, but equally important for a completepicture of planetary system properties, are the results from ongoing “legacy” Doppler surveys,which are now sensitive to giant planets in orbits approaching 20 years. Those surveysinclude, for example, the McDonald Observatory Planet Search (Robertson et al. 2012;Endl et al. 2016), the California Planet Search (Howard et al. 2010; Feng et al. 2015), theAnglo-Australian Planet Search (Tinney et al. 2001; Wittenmyer et al. 2011, 2014b), andthe Geneva Planet Search (Marmier et al. 2013; Moutou et al. 2015).The emerging picture is that the Solar System is not typical of planetary systems inthe Solar neighbourhood. For example, super-Earths, planets with masses ∼ ⊕ , areextremely common yet are completely absent from our Solar System. Jupiter-like planets inJupiter-like orbits appear to be relatively uncommon, orbiting only about 6% of solar-typestars (Wittenmyer et al. 2016b).Such a low incidence of true Solar system analogs is of particular interest in the contextof astrobiology, and the search for truly Earth-like planets beyond the Solar system. In theSolar System, Jupiter has played a key role in the formation and evolution of the planetary 3 –system - variously corralling, sculpting and destabilising the system’s smaller bodies (andthereby likely contributing significantly to the introduction of volatiles, including water, tothe early Earth). Over the system’s more recent history, Jupiter has managed the flux ofsmaller objects towards the Earth, influencing (but not necessarily reducing) the frequencyof impacts on the terrestrial planets. It has long been argued that the presence of a trueJupiter analog would be an important selection factor for an Earth-like planet to be trulyhabitable - although many recent studies have suggested that this might not be the case(e.g. Horner & Jones 2008; Horner et al. 2010; Horner & Jones 2010b; Lewis et al. 2013).The Anglo-Australian Planet Search (AAPS) monitored ∼
250 solar-type stars for 14years. Of these, a subset of ∼
120 stars continued to be observed for a further three years,with the primary aim of detecting Jupiter-mass planets in orbits
P > − velocity precision since its inception,enabling the discovery of several Jupiter analogs (e.g. Jones et al. 2010; Wittenmyer et al.2012a, 2014a).This paper is organised as follows: Section 2 outlines the AAT and HARPS observationsof HD 30177 and gives the stellar parameters. Section 3 describes the orbit-fitting proceduresand gives the resulting planetary parameters for the HD 30177 system. In Section 4 weperform a detailed dynamical stability analysis of this system of massive planets. Then wegive our conclusions in Section 5.
2. Observational Data and Stellar Properties
HD 30177 is an old, Sun-like star, with a mass within 5% of Solar. It lies approximately54.7 parsecs from the Sun, and has approximately twice Solar metallicity. The stellar param-eters for HD 30177 can be found in Table 1. We have observed HD 30177 since the inceptionof the AAPS, gathering a total of 43 epochs spanning 17 years (Table 6). Precise radialvelocities are derived using the standard iodine-cell technique to calibrate the instrumentalpoint-spread function (Valenti et al. 1995; Butler et al. 1996). The velocities have a meaninternal uncertainty of 3.9 ± − .HD 30177 has also been observed with the HARPS spectrograph on the ESO 3.6mtelescope in La Silla. At this writing, 20 epochs spanning 11 years are publicly available atthe ESO Archive. Velocities were derived using the HARPS-TERRA technique (Anglada-Escud´e & Butler 2012), and are given in Table 6. 4 –
3. Orbit Fitting and Results
The inner planet, HD 30177b was first announced in Tinney et al. (2003), with a rel-atively unconstrained period of 1620 ±
800 days and m sin i = 7 . ± Jup . Its orbit wasupdated in Butler et al. (2006) on the basis of observations that clearly spanned one fullorbital period, to P = 2770 ±
100 days and m sin i = 10 . ± Jup . We now present afurther 10 years of AAT data, along with 11 years of concurrent HARPS data, to refine theorbit of this planet. As a result of this additional data, we now find that the single-planet fitexhibits significant residuals, suggesting the presence of a second, very long-period object inthis system. We have added 6 m s − of jitter in quadrature to both data sets; this brings thereduced χ close to 1 for two-planet models. A single-planet model now has a reduced χ of7.1 and an rms of 17.3 m s − . As in our previous work (e.g. Tinney et al. 2011; Wittenmyeret al. 2013; Horner et al. 2014; Wittenmyer et al. 2016a), we have used a genetic algorithmto explore the parameter space for the outer planet, fitting a simultaneous two-Keplerianmodel that allows the outer planet to take on periods 4000-8000 days and eccentricities e < .
3. The best fit from the genetic algorithm results was then used as a starting point fora two-Keplerian fit (downhill simplex minimisation) within the
Systemic Console (Meschiariet al. 2009).The results of these fits are given in Table 4. The precision with which the parametersfor the inner planet are known are now improved by a factor of ten, or more, over thepreviously published values (Butler et al. 2006). The model fit for the inner planet is shownin Figure 1. The nominal best fit solution features a second planet, HD 30177c, with period of6921 ±
621 days and m sin i = 3 . ± Jup . We present both a “best fit” and a “long period”solution in recognition of the fact that for an incomplete orbit, the period can be wildlyunconstrained and allow for solutions with ever-longer periods by adjusting the eccentricity.Figure 2 shows the χ contours as a function of the outer planet’s period and eccentricity,based on the results from the best-fit solution given in the left columns of Table 4. The bestfit solution appears to be a shallow minimum in the χ space, with a secondary minimumat lower eccentricity and longer period ( P ∼ Systemic simplexalgorithm into the apparent secondary χ minimum seen in Figure 2. The results are givenin the right columns of Table 4, labelled “long period.” This fit results in an outer planetwith period 11640 ± i = 6 . ± Jup ; the uncertainties are of coursemuch larger since the available data only cover ∼
60% of the orbital period. The best-fit andlong-period solutions are plotted in Figure 3.One might argue that the outer planet hypothesis relies heavily on the presumption of avelocity turnover in the first few epochs, in particular the point at BJD 2451119, which lies 5 –about 30 m s − below the previous night’s velocity. To check the effect of this potentially badobservation, we repeated the orbit fitting described above after removing that point. Theresults are given in Table 5, again expressed as a “best fit” and a “long period” solution. Wenow find a best fit at a period of 7601 ± i = 3 . ± Jup . Removing thesuspected outlier resulted in a slightly longer period that remains in formal agreement withthe original solution in Table 4. For the long-period solution, we again started the
Systemic fitting routine at a period of 10,000 days for the outer planet. This results again in a longperiod consistent with the long period solution obtained from the full set of velocities: weobtain a period of 11613 ± i = 7 . ± Jup . We thus have two possiblesolutions for the HD 30177 two-planet system, which are virtually indistinguishable in termsof the RMS about the model fit or the χ , due to the shallow minima and complex χ space(Figure 4).For the old, solar-type stars generally targeted by radial velocity surveys, stellar mag-netic cycles like the Sun’s 11-year cycle are a concern when claiming detection of planets withorbital periods ∼
10 years and longer. While our AAT/UCLES spectra do not include the CaII H and K lines most commonly used as activity proxies, the HARPS spectra used in thiswork do (e.g. Dumusque et al. 2011; Lovis et al. 2011; H´ebrard et al. 2014). Figure 5 showsthe Ca II activity log R (cid:48) HK versus the HARPS radial velocities. No correlation is evident.Clearly, a long-period body is present, but a longer time baseline is necessary to observea complete orbit and better constrain its true nature. In the next section, we explore thedynamical stability of the two candidate orbital solutions.
4. Dynamical Stability Simulations
In order to understand the dynamical context of the two distinct orbital solutions pre-sented above, and to see whether they yield planetary systems that are dynamically feasible,we followed a now well-established route (e.g. Marshall et al. 2010; Robertson et al. 2012;Horner et al. 2013). For each solution, we performed 126,075 unique integrations of the sys-tem using the Hybrid integrator within the n -body dynamics package, Mercury (Chambers1999). In each of those simulations, we held the initial orbit of the innermost planet fixedat its nominal best-fit values (as detailed in Table 4). We then proceeded to systematicallymove the orbit of the outermost planet through the full ± σ uncertainty ranges for semi-major axis, a , eccentricity, e , argument of periastron, ω , and mean anomaly, M . In thismanner, we created a regular grid of solutions, testing 41 unique values of a and e , 15 uniquevalues of ω , and 5 unique values of M .These simulations make two assumptions: first, that the two planets move on coplanar 6 –orbits (as is essentially the case in the Solar system), and second, we assign the planets theirminimum masses (m sin i ) as derived from the radial velocity data. In a number of previousstudies (e.g. Horner et al. 2011, 2014; Hinse et al. 2014), we have examined the impact ofmutual inclination on system stability. However, for widely separated planets, the inclinationbetween the two orbits seems to play little role in their stability. It seems most likely thatthere would not be large mutual inclination between the orbits of the planets; from a dynam-ical point of view, given the assumption that the two planets formed in a dynamically cooldisk, it is challenging to imagine how they could achieve significant mutual inclination with-out invoking the presence of a highly inclined distant perturber (i.e. an undetected binarycompanion, driving excitation through the Kozai mechanism). It is certainly reasonable toassume that the orbits are most likely relatively coplanar, as is seen in the Solar system giantplanets, and also in those multiple exoplanet systems with orbital inclinations constrainedby transits (Fang & Margot 2012) or by resolved debris disk observations (Kennedy et al.2013).Regarding the use of minimum masses, one would expect increased planetary masses todestabilise the systems. The reason for this can be seen when one considers the “gravitationalreach” of a planet, which can be defined in terms of its Hill radius. The Hill radius isproportional to the semi-major axis of the planet’s orbit, but only increases as the cuberoot of the planet’s mass. As a result, doubling the mass of a planet only increases itsgravitational reach, and therefore its Hill radius, by a factor of 2 ( /
3) = 1 .
26 - a relativelyminor change.The simulations were set to run for a maximum of 100 million years, but were broughtto a premature end if one or other of the planets were ejected from the system or collidedwith the central body. Simulations were also curtailed if the two planets collided with oneanother. For each of these conditions, the time at which the simulation finished was recorded,allowing us to create dynamical maps of the system to examine the dynamical context ofthe orbits presented above, and to see whether the system was dynamically feasible. Suchmaps have, in the past, revealed certain systems to be dynamically unfeasible (e.g. Horneret al. 2011; Wittenmyer et al. 2012b; Horner et al. 2013, 2014). In other cases, dynamicalmappings have resulted in stronger constraints for a given system’s orbits than was possiblesolely on the grounds of the observational data (e.g. Wittenmyer et al. 2012c; Robertsonet al. 2012; Wittenmyer et al. 2014c). Dynamical simulations therefore offer the potentialto help distinguish between different solutions with similar goodness of fit, such as thoseproposed in this work, as well as yielding an important dynamical ’sanity check’.To complement these dynamical simulations, we also chose to trial a new technique forthe dynamical analysis of newly discovered systems. Rather than populate regular grids in 7 –element space, whilst holding the better constrained planet’s initial orbit fixed, we insteadperformed repeated fits to the observational data. In our fitting, we required solely thatthe solutions produced lie within 3 σ of the best-fit solution, allowing all parameters to varyfreely. This created clouds of ’potential solutions’ distributed around the best fit out to arange of χ best + 9. We then randomly selected solutions to evenly sample the phase spacebetween the best-fit solution (at χ best ) and χ best + 9. As before, we generated 126,075 uniquesolutions for each of the two scenarios presented above.Where our traditional dynamical maps explore the dynamical context of the solutionsin a readily apparent fashion, these new simulations are designed to instead examine thestability of the system as a function of the goodness of fit of the orbital solution. In addition,they allow us to explore the stability as a function of the masses assigned to the two planetsin question. As such, they provide a natural complement to the traditional maps, as can beseen below.
5. Dynamical Stability Results
Figure 6 shows the dynamical context of the short period solution for the two planetHD 30177 system, as described in Table 4. The best fit solution lies in an area of strongdynamical instability, with the majority of locations within the 1 σ uncertainty range beingsimilarly unstable. There is, however, a small subset of solutions in that range that arestable, marking the inner edge of a broad stable region seen towards larger semi-major axesand smaller eccentricities. The small island of stability at a ∼ .
687 au is the result of theplanets becoming trapped in mutual 2:1 mean-motion resonance, whilst the narrow curvedregion of moderate stability at high eccentricities is caused by orbits for HD 30177c withperiastron located at the semi-major axis of HD 30177b. Dynamical stability for the systemon near-circular, non-resonant orbits is only seen in these simulations exterior to the planet’smutual 3:1 mean-motion resonance, located at a ∼ .
453 au (and the cause of the small islandof stability at non-zero eccentricities at that semi-major axis). As a result, these simulationssuggest that the short-period solution is not dynamically favoured, unless the orbital periodfor HD 30177c is significantly longer than the best fit, the orbit markedly less eccentric, orif the two planets are trapped in mutual 3:1 mean motion resonance.These results are strongly supported by our subsidiary integrations - the results of whichare shown in Figures 7-8. Figure 7 shows the stability of the candidate HD 30177 planetarysystems as a function of the eccentricities of the two planets, their period ratio, and thegoodness of the fit of the solution tested. In Figure 7, the upper plots show all solutionswithin 3 σ of the best fit, whilst the lower show only those solutions within 1 σ of the best fit. 8 –It is immediately apparent that truly stable solutions are limited to only a very narrow rangeof the plots - namely two narrow regions with low eccentricities, and widely separated orbits.In fact, these solutions all lie at, or somewhat outside, the location of the 3:1 mean motionresonance between the two planets (P1/P2 ∼ . ∼ .
25 orgreater to either planet prove strongly unstable.Figure 8 shows the influence of the mass of the two planets on the stability of thesolution. The resonant and extra-resonant stable regions are again clearly visible, and it isapparent that the masses of the two planets seem to have little influence on the stabilityof the solution. A slight influence from the planetary mass can be seen in the middlerow of Figure 8, which shows that stable solutions with the lowest cumulative planet mass(i.e. M b + M c ) have slightly higher mean eccentricities than those for larger cumulativemasses. This effect is only weak, and is the result of the least massive solutions veeringaway from lower eccentricities. Given that the eccentricities of planetary orbits tend to besomewhat over-estimated when fitting radial velocity data (O’Toole et al. 2009), this maybe an indication that the lower-mass solutions are slightly less favourable than their highermass counterparts.Finally, the bottom row of Figure 8 shows the stability of the solution clouds as afunction of the maximum eccentricity between the two planets (i.e. the value plotted onthe y-axis is whichever is greater of e b and e c ). This reinforces the result from Figure 7that solutions with either of the two planets moving on orbits with e ≥ .
25 are unstableregardless of their separation, or the mass of the planets involved.Taken in concert, our results show that, whilst short-period solutions for the HD 30177system can prove dynamically stable, they require the two planets to either be trapped inmutual 3:1 mean motion resonance, or to be more widely spaced, and further require thatneither planet move on an orbit with eccentricity greater than 0.25.But what of our alternative, longer-period solution for the planetary system? Figure 9shows the dynamical context of that solution. Unlike the short period solution, the twoplanets are now sufficiently widely separated that the great majority of orbits around thebest-fit solution are now dynamically stable for the full 100 Myr of our simulations. Atthe very inner edge of the plot, the cliff of instability exterior to the planet’s mutual 3:1mean-motion resonance can again be seen, as can hints of the moderate stability affordedby the periastron of HD 30177 c falling at the semi-major axis of HD 30177 b (top left ofthe plot). Purely on the basis of this plot, the longer-period solution seems markedly more 9 –dynamically feasible, a result once again borne out by the plots of our subsidiary simulationsof the long-period solution (Figures 10-13).Figure 10 reveals many of the same features as Figure 9 - a significant proportionof the solutions are dynamically stable - particularly those within 1 σ of the best fit (lowerpanels). The greater the orbital separation of the two planets, the greater can be their orbitaleccentricities before destabilising the system. In addition, however, the destabilising influenceof distant mean-motion resonances is revealed in these plots, as the notches of instabilitycarved into the distribution at specific period ratios. Aside from these few unstable regions,however, the great majority of solutions within 1 σ of the best fit are stable.Figure 11 shows that the mass ratio of the planets (left hand plots) has little or noinfluence on their stability. Interestingly, though, the lower-right hand panel reveals anapparent relationship in the fitting between the cumulative mass of the planets and theirmutual separation. The more widely separated the two planets (and hence the more distantis HD 30177c), the greater their cumulative mass. This is not at all surprising: the moredistant HD 30177c is, the greater its mass would have to be to achieve a radial velocity signalof a given amplitude. This feature is therefore entirely expected, but nevertheless serves tonicely illustrate the relationship between different parameters in the radial velocity fittingprocess.Figure 12 again reveals that the more eccentric the orbits of the planets, the morelikely they are to prove unstable - although once again, the great majority of the sampledphase-space proves dynamically stable. More interesting, however, are the results shown inFigure 13. The left-hand panels of that plot, which show the stability of the solutions as afunction of the maximum eccentricity between the two panels (y-axis) and the mass ratioof the two planets (x-axis) suggest that, the closer the two planetary masses are to parity,the more likely eccentric orbits are to be stable. By contrast, the lower-right hand plotsuggests that the greater the sum of the planetary masses, the more likely solutions withhigh eccentricities are to be stable. Taken in concert, these results are once again a reflectionof the relationship between cumulative mass and orbital separation. That is, the greater theorbital separation of the two planets, the greater their cumulative mass, and the closer toparity their masses become (since our fits suggest that HD 30177c is the less massive of thetwo). At the same time, we saw from Figure 10 that, the greater the separation of the twoplanets, the more stable are those orbital solutions at higher eccentricity. So once again, weare looking at the same thing - these two apparent trends are the result of the requirementthat a more distant HD 30177c must be more massive in order to generate the observedradial velocity amplitude. 10 –
6. Conclusions
We present the results of new radial velocity observations of HD 30177, which revealfor the first time the presence of a second, long-period planet in that system. Two possibleorbital solutions for the planetary system are presented - one with a shorter-period orbitfor HD 30177 c, and one with the two planets more widely spaced, and HD 30177 c on alonger period orbit. The two solutions are virtually indistinguishable from one another interms of the quality of fit that they provide to the data. However, the short-period solutionplaced the two planets on orbits sufficiently compact that they lie closer than their mutual3:1 mean-motion resonance.Although several highly compact multi-planet systems have been discovered in recentyears, it has become apparent that such compact systems rely on dynamical stability con-ferred by mutual resonant planetary orbits. As such, it seemed prudent to build on ourearlier work, and carry out detailed n -body simulations of the two potential solutions for theHD 30177 system, to see whether it was possible to rule either out on dynamical stabilitygrounds.Our results reveal that, although some stable solutions can be found for the short-periodvariant of the HD 30177 system, those solutions require orbital eccentricities for the planetsthat are typically smaller than given by the best fit solution, and require HD 30177 c tobe somewhat more distant than the best fit. In other words, the require relatively loweccentricity orbits for that planet exterior to its mutual 3:1 mean-motion resonance withHD 30177 b. By contrast, the great majority of the longer-period solutions tested proveddynamically stable - and across a much greater range of potential semi-major axes and orbitaleccentricities.As a result, we consider that the most likely solution for the orbit of HD 30177c isthe longer period option: an m sin i = 7 . ± Jup planet with a = 9 . ± .
04 au, e = 0 . ± .
14, and an orbital period of P = 11613 ± i < ∼ o , the two orbiting bodies in the HD 30177 system fall into the browndwarf regime. With an orbital separation of a ∼
10 au, one can consider HD 30177c to beone of the first members of an emerging class of “Saturn analogs,” referring to planets withorbital separations similar to Saturn. Just as long-term radial velocity surveys have begunto characterize “Jupiter analogs” (Wittenmyer et al. 2011; Rowan et al. 2016; Wittenmyeret al. 2016b), the continuation of legacy surveys such as the Anglo-Australian Planet Searchwill enable us to probe the population of planets in Saturn-like orbits in the coming decade.JH is supported by USQ’s Strategic Research Fund: the STARWINDS project. CGT 11 –is supported by Australian Research Council grants DP0774000 and DP130102695. Thisresearch has made use of NASA’s Astrophysics Data System (ADS), and the SIMBADdatabase, operated at CDS, Strasbourg, France. This research has also made use of theExoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org (Wright etal. 2011; Han et al. 2014).
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14 –Fig. 1.— Data and Keplerian model fit for the inner planet HD 30177b. AAT – blue; HARPS– green. The orbit of the outer planet has been removed. We have added 6 m s − of jitter inquadrature to the uncertainties, and this fit has an rms of 7.07 m s − . 15 – Period of planet 2 E cce n t r i c i t y o f p l a n e t . . . . . Period of planet 2 vs Eccentricity of planet 2
Fig. 2.— Contours of χ as a function of the outer planet’s eccentricity and orbital period.Contours are labeled with confidence intervals around the best fit (red dot). Hints of a secondlocal χ minimum can be seen in the lower right, at long periods and low eccentricities. 16 –Table 1. Stellar Parameters for HD 30177 Parameter Value ReferenceSpec. Type G8 V Houk & Cowley (1975)Distance (pc) 54.7 ± (cid:12) ) 0.951 +0 . − . Takeda et al. (2007)1.05 ± ± i (km s − ) 2.96 ± F e/H ] +0.33 ± ± ± ± T eff (K) 5580 ±
12 Franchini et al. (2014)5601 ±
73 Adibekyan et al. (2012)5595 ±
50 Ghezzi et al. (2010)5607 ±
44 Butler et al. (2006)log g ± ± ± ± +1 . − . Takeda et al. (2007)
Fig. 3.— Data and Keplerian model fit for the outer planet HD 30177c. AAT – blue; HARPS– green. The orbit of the inner planet has been removed. We have added 6 m s − of jitterin quadrature to the uncertainties. Left panel: Nominal best fit, with P = 6921 d. Rightpanel: Long-period solution, with P = 11640 d. 17 –Table 2. AAT Radial Velocities for HD 30177 BJD-2400000 Velocity (m s − ) Uncertainty (m s − )51118.09737 227.2 4.551119.19240 188.6 6.951121.15141 210.7 6.151157.10219 223.5 4.551211.98344 234.6 5.051212.96597 235.8 4.251213.99955 245.4 4.051214.95065 237.3 3.651525.99718 177.1 3.451630.91556 144.9 4.651768.32960 73.4 4.251921.10749 14.8 4.652127.32049 -9.2 8.552188.25324 -41.3 3.652358.91806 -45.6 3.852598.18750 -49.8 2.052655.02431 -57.6 4.452747.84861 -49.0 2.252945.18132 -12.7 2.653044.03464 8.2 3.653282.26188 58.2 2.853400.99440 91.4 2.554010.25007 137.4 1.854038.21420 126.4 3.454549.93698 -47.3 2.254751.25707 -83.8 3.854776.17955 -79.6 2.254900.95132 -78.0 3.455109.18072 -77.0 3.255457.26529 -32.2 3.955461.28586 -25.3 4.355519.17942 -2.1 3.355845.21616 82.2 4.755899.10987 79.0 3.256555.28257 149.0 4.156556.25219 152.0 3.656746.90702 97.7 5.156766.86295 66.2 4.056935.25257 10.2 4.056970.23271 5.6 3.057052.02821 -2.2 3.057094.90039 -28.0 4.657349.14648 -34.5 3.1
18 –Table 3. HARPS-TERRA Radial Velocities for HD 30177
BJD-2400000 Velocity (m s − ) Uncertainty (m s − )52947.76453 -70.7 1.653273.88347 0.0 1.953274.88548 4.6 1.953288.84830 4.6 1.553367.68146 21.6 1.853410.60057 32.3 1.553669.80849 95.9 2.054137.58873 31.0 1.554143.51190 28.9 1.454194.47989 8.6 1.554315.91894 -38.8 2.354384.87123 -60.3 3.254431.68520 -75.4 1.955563.54385 -63.1 1.055564.57743 -66.2 0.855903.70118 30.9 2.256953.81794 -43.4 0.756955.78182 -45.2 0.756957.88054 -46.5 1.156959.68147 -47.8 0.8 Table 4. HD 30177 Planetary System Parameters (all data)
Parameter Nominal Best Fit Long-Period SolutionHD 30177b HD 30177c HD 30177b HD 30177cPeriod (days) 2532.5 ± ±
621 2520.6 ± ± ± ± ± ± ω (degrees) 32 ± ±
13 30 ± ± K (m s − ) 126.1 ± ± ± ± T (BJD-2400000) 51428 ±
26 51661 ±
573 51434 ±
24 48426 ± i (M Jup ) 8.07 ± ± ± ± a (au) 3.58 ± ± ± ± − ) 7.04 7.17 χ ν (51 d.o.f.) 0.98 1.01
19 –
Period of planet 2 E cce n t r i c i t y o f p l a n e t . . . . . Period of planet 2 vs Eccentricity of planet 2
Fig. 4.— Same as Figure 2, but for the long-period solution where one outlier data point hasbeen removed. Two local χ minima are evident, with the longer-period solution at lowereccentricity (red dot). 20 –Fig. 5.— Ca II activity index log R (cid:48) HK as a function of radial velocity for the HARPS spectraof HD 30177. No correlation is evident from the 11 years of data, and hence we concludethat a stellar magnetic cycle is not responsible for the observed radial velocity variations. 21 –Table 5. HD 30177 Planetary System Parameters (outlier removed) Parameter Nominal Best Fit Long-Period SolutionHD 30177b HD 30177c HD 30177b HD 30177cPeriod (days) 2531.3 ± ± ± ± ± ± ± ± ω (degrees) 32 ± ±
16 31 ± ± K (m s − ) 125.8 ± ± ± ± T (BJD-2400000) 51430 ±
27 52154 ± ±
29 48973 ± i (M Jup ) 8.06 ± ± ± ± a (au) 3.58 ± ± ± ± − ) 5.98 6.01 χ ν (50 d.o.f.) 0.76 0.77 Fig. 6.— The dynamical stability of the short-period solution for the orbit of HD 30177c,as a function of semi-major axis and eccentricity. The red box, to the centre of the plot,denotes the location of the best-fit solution, whilst the lines radiating from that point showthe 1 − σ uncertainties. It is immediately apparent that the best-fit solution lies in a regionof significant dynamical instability. 22 – P / P M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =1.0 12345678 χ P / P M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =1.0 12345678 χ Fig. 7.— The stability of the short-period solution for HD30177, as a function of the mean(left) and maximum (right) eccentricity of the two planets in the system. The colour axisshows the goodness of fit for each of the solutions tested, with the vertical axis showingthe lifetime, and the y-axis the ratio of the two planetary orbital periods. The upper plotsshow the results for solutions within 3 σ of the best fit, whilst the lower show only thosesimulations within 1 σ of that solution. We note that animated versions of the figures areavailable in the online edition of this work, which may help the reader to fully visualise therelationship between the stability and the various variables considered. 23 – P / P M / M l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M + M l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M / M M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M + M M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M / M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M + M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ Fig. 8.—
Upper row:
The stability of the short-period solution for HD 30177c, as a functionof the mass ratio (left) and total mass (right) of the two planets in the system. The color scaleshows the goodness of fit for each of the solutions tested, with the vertical axis showing thelifetime, and the y-axis the ratio of the two planetary orbital periods. Results for solutionswithin 3 σ of the best fit are shown. Middle row:
Same, but the x-axis now denotes themean eccentricity of the planetary orbits.
Bottom row:
Same, but the x-axis now denotes themaximum eccentricity of the planetary orbits. Animated versions of the figures are providedin the online version of the paper. 24 –Fig. 9.— The stability of the long-period solution for the orbit of HD 30177c, as a functionof semi-major axis and eccentricity. As with Figure 6, the red box marks the location ofthe best-fit solution, with the red lines radiating showing the 1 − σ uncertainties. Unlikethe short-period solution, the best-fit orbit now lies in a broad region of dynamical stability,with most solutions within 1 − σ proving stable for the full 100 Myr of our integrations. P / P M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ Fig. 10.— The stability of the long-period solution for HD 30177, as a function of the mean(left) and maximum (right) eccentricity of the two planets in the system. The color scale andaxes have the same meaning as in Figure 7. The upper plots show the results for solutionswithin 3 σ of the best fit, whilst the lower show only those simulations within 1 σ of thatsolution. Animated versions of the figures are provided in the online version of the paper. 25 – P / P M / M l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M + M l o g ( L i f e t i m e ) χ max =9.0 12345678 χ P / P M / M l o g ( L i f e t i m e ) χ max =1.0 12345678 χ P / P M + M l o g ( L i f e t i m e ) χ max =1.0 12345678 χ Fig. 11.— The stability of the long-period solution for HD 30177, as a function of the massratio (left) and total mass (right) of the two planets in the system. As before, the colouraxis shows the goodness of fit for each of the solutions tested, with the vertical axis showingthe lifetime, and the y-axis the ratio of the two planetary orbital periods. Solutions within3 σ of the best fit are shown in the upper panels, and only those within 1 σ are shown inthe lower panels. Animated versions of the figures are provided in the online version of thepaper. 26 – M / M M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M + M M e a n E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ Fig. 12.— The stability of the long-period solution for HD 30177, again as a function ofthe mass ratio (left) and total mass (right) of the two planets in the system. Again, thecolour axis shows the goodness of fit for each of the solutions tested, with the vertical axisshowing the lifetime, and the x-axis the mean eccentricity of the planetary orbits. Results forsolutions within 3 σ of the best fit are shown. Animated versions of the figures are providedin the online version of the paper. 27 – M / M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M + M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =9.0 12345678 χ M / M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =1.0 12345678 χ M + M M a x E cc e n t r i c i t y l o g ( L i f e t i m e ) χ max =1.0 12345678 χ Fig. 13.— The stability of the long-period solution for HD 30177, again as a function of themass ratio (left) and total mass (right) of the two planets in the system. The colour axisshows the goodness of fit for each of the solutions tested, with the vertical axis showing thelifetime, and the x-axis the maximum eccentricity of the planetary orbits. The upper plotsshow the results for solutions within 3 σ of the best fit, whilst the lower show only thosesimulations within 1 σσ