A Detailed Analysis of the HD 73526 2:1 Resonant Planetary System
Robert A. Wittenmyer, Xianyu Tan, Man Hoi Lee, Jonathan Horner, C.G. Tinney, R.P. Butler, G.S. Salter, B.D. Carter, H.R.A. Jones, S.J. O'Toole, J. Bailey, D. Wright, J.D. Crane, S.A. Schectman, P. Arriagada, I. Thompson, D. Minniti, M. Diaz
aa r X i v : . [ a s t r o - ph . E P ] N ov A Detailed Analysis of the HD 73526 2:1 Resonant PlanetarySystem
Robert A. Wittenmyer , , Xianyu Tan , , Man Hoi Lee , , Jonathan Horner , ,C.G. Tinney , , R.P. Butler , G.S. Salter , , B.D. Carter , H.R.A. Jones , S.J. O’Toole ,J. Bailey , , D. Wright , , J.D. Crane , S.A. Schectman , P. Arriagada , I. Thompson ,D. Minniti , , & M. Diaz [email protected] ABSTRACT
We present six years of new radial-velocity data from the Anglo-Australian andMagellan Telescopes on the HD 73526 2:1 resonant planetary system. We investi-gate both Keplerian and dynamical (interacting) fits to these data, yielding fourpossible configurations for the system. The new data now show that both reso-nance angles are librating, with amplitudes of 40 o and 60 o , respectively. We thenperform long-term dynamical stability tests to differentiate these solutions, whichonly differ significantly in the masses of the planets. We show that while there is School of Physics, University of New South Wales, Sydney 2052, Australia Australian Centre for Astrobiology, University of New South Wales, Sydney 2052, Australia Department of Earth Sciences, The University of Hong Kong, Pokfulam Road, Hong Kong Department of Planetary Sciences and Lunar and Planetary Laboratory, The University of Arizona, 1629University Boulevard, Tucson, AZ 85721, USA Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road,NW, Washington, DC 20015-1305, USA Faculty of Sciences, University of Southern Queensland, Toowoomba, Queensland 4350, Australia University of Hertfordshire, Centre for Astrophysics Research, Science and Technology Research Insti-tute, College Lane, AL10 9AB, Hatfield, UK Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, CA91101, USA Institute of Astrophysics, Pontificia Universidad Catolica de Chile, Casilla 306, Santiago 22, Chile Vatican Observatory, V00120 Vatican City State, Italy i = 90 o providesthe best combination of goodness-of-fit and long-term dynamical stability. Subject headings: planetary systems: individual (HD 73526) – techniques: radialvelocities – methods: N-body simulations
1. Introduction
The ever-growing population of known multiple-planet systems has proven to be an ex-ceedingly useful laboratory for testing models of planetary system formation and dynamicalevolution. Of particular interest are the systems which are in, or near, resonant configu-rations. A number of such systems have been identified from radial-velocity surveys, withsome notable examples including GJ 876 (Marcy et al. 2001), HD 128311 (Vogt et al. 2005),HD 82943 (Mayor et al. 2004), and HD 200964 (Johnson et al. 2011). Wright et al. (2011)noted that about 1/3 of well-characterized multi-planet systems were in such low-order pe-riod commensurabilities. The
Kepler mission has revealed hundreds of candidate multiply-transiting planetary systems (Borucki et al. 2010; Batalha et al. 2013), some of which are inor near low-order resonances (Lissauer et al. 2011; Steffen et al. 2013). One emerging trendfrom the
Kepler results is that a significant number of such near-resonant planet pairs areoutside of the resonance (Fabrycky et al. 2012; Veras & Ford 2012; Lee et al. 2013), with anexcess population slightly wide of the resonance, and a deficit of planet pairs just inside theresonance (Lithwick & Wu 2012).Marti et al. (2013) recently showed that the 4:2:1 Laplace resonance in the GJ 876 sys-tem (Rivera et al. 2010; Baluev 2011) acts to stabilise the three outer planets, constrainingtheir mutual inclinations to less than 20 degrees and e < ∼ ± o tothe sky plane, which was dynamically stable despite the high planetary masses implied bythat inclination. Interestingly, Herschel debris disk observations reported by Kennedy et al.(2013) show that the disk has a similar line-of-sight inclination of 27 ± o . These examplesshow how planetary systems can be characterized with multiple complementary approaches.HD 73526 is one of 20 stars added to the Anglo-Australian Planet Search (AAPS) inlate 1999, based on high metallicity and the then-emerging planet-metallicity correlation(Laughlin 2000; Valenti & Fischer 2005). The first planet, HD 73526b (Tinney et al. 2003),was reported to have period P = 190 . ± e = 0 . ± m sin i = 3 . ± Jup . Tinney et al. (2006) reported a second planet with P = 3 –376 . ± θ librating around 0 o and θ circulating(Tinney et al. 2006), where θ and θ are the lowest order, eccentricity-type 2:1 MMR angles: θ = λ − λ + ̟ , (1) θ = λ − λ + ̟ . (2)Here, λ is the mean longitude, ̟ is the longitude of periapse, and subscripts 1 and 2 representthe inner and outer planets, respectively. This type of 2:1 MMR configuration is dynamicallyinteresting as it cannot be produced by smooth migration capture alone (Beauge et al. 2003;Ferraz-Mello et al. 2003; Lee 2004; Beauge et al. 2006; Michtchenko et al. 2008), andalternative mechanisms have been suggested to produce such configuration.The resonant property of the HD 73526 planets makes this an interesting system interms of its dynamical evolution. Subsequent work has focused on how planets get into the2:1 resonance in this and other exoplanetary systems. S´andor et al. (2007) proposed thatthe HD 73526 system experienced both migration and a sudden perturbation (planet-planetscattering or rapid dissipation of the protoplanetary disk) which combined to drive the systeminto the observed 2:1 resonance. Similarly, Zhang et al. (2010) suggested that the HD 73526and HD 128311 systems, both of which are in 2:1 librating-circulating resonances, arrivedin that configuration via a hybrid mechanism of scattering and collisions with terrestrialplanetesimals. Scattering into low-order resonances was also implicated by Raymond et al.(2008) as a likely formation mechanism, where scattering events drive the two larger planetsinto a resonance while ejecting the smaller planet. In summary, there is general agreementthat the HD 73526 system did not arrive in the 2:1 resonance by smooth migration alone.The aim of this work is to provide an updated set of parameters for the HD 73526system, based on a further 6 years of AAPS observations, as well as new data from Magellan(Section 2). In addition, we perform Keplerian and full dynamical fits to the complete dataset (Section 3). In Section 4, we present detailed dynamical stability maps of the system,using both the parameters from the Keplerian and dynamical fits. Finally, in Section 5 weoffer conclusions on the architecture of the system based on the combination of our orbitfitting and dynamical stability analysis. 4 –
2. Observations2.1. Anglo-Australian Telescope
AAPS Doppler measurements are made with the UCLES echelle spectrograph (Diego et al.1991). An iodine absorption cell provides wavelength calibration from 5000 to 6200 ˚A. Thespectrograph point-spread function and wavelength calibration are derived from the iodineabsorption lines embedded on every pixel of the spectrum by the cell (Valenti et al. 1995;Butler et al. 1996). The result is a precision Doppler velocity estimate for each epoch, alongwith an internal uncertainty estimate, which includes the effects of photon-counting uncer-tainties, residual errors in the spectrograph PSF model, and variation in the underlyingspectrum between the iodine-free template, and epoch spectra observed through the iodinecell. All velocities are measured relative to the zero-point defined by the template observa-tion. A total of 36 AAT observations have been obtained since 1999 Feb 2 (Table 1) and usedin the following analysis, representing a data span of 4836 days. The exposure times rangefrom 300 to 900 sec, and the mean internal velocity uncertainty for these data is 4.1 m s − . Since HD 73526 is among the faintest AAPS targets ( V = 9 . R ∼ , − .
3. Orbit Fitting3.1. Noninteracting Keplerian Fit
New radial-velocity observations of exoplanetary systems can sometimes result in sub-stantial modification of the best-fit planetary orbits. For example, the two planets in theHD 155358 system were initially reported to be in orbital periods of 195 and 530 days(Cochran et al. 2007). A further five years of observations revealed that the outer planetactually has an orbital period of 391.9 days, and is trapped in the 2:1 mean-motion resonance 5 –(Robertson et al. 2012a). In light of the possibility that the best-fit orbits of the two planetsmay be significantly different than initially presented in Tinney et al. (2006), we begin ourorbit fitting process with a wide-ranging search using a genetic algorithm. This approachis often used when the orbital parameters of a planet candidate are highly uncertain (e.g.Tinney et al. 2011; Wittenmyer et al. 2012a; Horner et al. 2012a), or when data are sparse(Wittenmyer et al. 2011). We allowed the genetic algorithm to search a wide parameterspace, and it ran for 50,000 iterations, testing a total of about 10 possible configurations.We then fit the two data sets simultaneously using GaussFit (Jefferys et al. 1987), a gener-alized least-squares program used here to solve a Keplerian radial-velocity orbit model. The
GaussFit model has the ability to allow the offsets between multiple data sets to be a freeparameter. The parameters of the best 2-planet solution obtained by the genetic algorithmwere used as initial inputs to
GaussFit , and a jitter of 3.3 m s − was added in quadratureto the uncertainty of each observation (following Tinney et al. 2006). The best-fit Kepleriansolutions are given in Table 3; planetary minimum masses m sin i are derived using a stellarmass of 1.014 ± ⊙ (Takeda et al. 2007). This fit has a reduced χ of 1.63 and a totalRMS of 6.32 m s − (AAT – 7.67 m s − ; PFS – 2.75 m s − ). Because the two planets are massive enough and orbit close enough to each other to beinteracting, we also apply a full dynamical model to these data. This model includes theeffects of planet-planet interactions, and can be used to place constraints on the system’sinclination to the sky plane, i , a quantity which cannot be determined from Keplerian fittingalone. The system inclination then sets the true masses of the planets. The technique isdescribed fully in Tan et al. (2013) for the HD 82943 two-planet system. The Levenberg-Marquardt (Press et al. 1992) method is adopted as our fitting method. Using the Keplerianbest fit as an initial guess, assuming coplanar edge-on orbits, the Levenberg-Marquardtalgorithm converges to a local minimum with χ ν of about 1.70 and RMS of about 6.54m s − . Based on this local minimum, we conduct a parameter grid search (Lee et al. 2006;Tan et al. 2013) to ensure a global search for the best fit. This minimum is indeed a globaldynamical best fit assuming coplanar edge-on orbits; two other local minima with slightlylarger χ ν have been found. The coplanar edge-on best-fit parameters are listed in Table 4,with their error bars determined by the covariance matrix. This fit and its residuals areshown in the left panel of Figure 1. The right panel of Figure 1 shows that both resonanceangles are librating, with amplitudes of ± o ( θ ) and ± o ( θ ).Assuming the planets are in coplanar orbits, we then allow the inclination to the sky 6 –plane to vary along with other fitting parameters. Figure 2 shows χ and RMS as a functionof sin i , and Figure 3 shows best-fit parameters as a function of sin i . The χ curve is shallowin the range of sin i & .
6, but then shows a clear local minimum at sin i ∼ .
36 ( i = 20 . o ).Two further local minima were found, at inclinations of i =90 o and 40.2 o . The parameters ofthese three solutions are given in Table 4; the planetary masses scale accordingly as 1/sin i ,resulting in more massive planets for the low-inclination solutions.The χ curve and fitting parameters (Fig. 3) show discontinuities along the sin i axis,especially those near sin i ∼ .
43. To understand these discontinuities, we explore grids indifferent fixed sin i , to see the evolution of the parameter space along different inclinations.Figure 4 shows K - K grids for different sin i . Initially when the orbits are at sin i ∼ . χ minimum ( K ∼ K ∼
82) appearsaround sin i = 0 . K ∼ i drops down to 0.425, the original minimum vanishes and thenew one becomes a single minimum in parameter space. The appearance of the additional χ ν minimum results in the big “jump” of fitting parameters at about sin i = 0 .
43 (see Fig. 3).In summary, we have four possible configurations for this system (one Keplerian fit andthree dynamical fits). The four solutions are not substantially different from one another,apart from the sin i factor for the three solutions in Table 4, which serves to increase theplanetary masses relative to the Keplerian scenario in which we have assumed the planetsto be at their minimum masses (m sin i ). As a first-order check of dynamical stability, thebest-fit system configuration at each inclination was integrated for 10 yr. For all fits sotested, at inclinations of 26.7, 30.0, 33.4, 36.9, 40.0, and 90.0 degrees (sin i =0.45, 0.5, 0.55,0.6, 0.64, 1.0), the systems remained stable for 10 yr. However, since dynamical stability ishighly dependent on the initial conditions, we expand on these tests in the next section toobtain a more robust and complete picture of the stability of the various configurations.
4. Dynamical Stability Testing
We have found four possible solutions for the HD 73526 system, which significantly differin inclination (and hence the planetary masses). It is therefore critical to perform dynamicalstability tests on these configurations, as the solution with the absolute χ minimum mayprove dynamically unfeasible. 7 – When analyzing any multiple-planet system, it is prudent to investigate the long-termdynamical stability of the system. As more multi-planet systems are discovered, the an-nouncement of planetary systems which turn out to be dynamically unfeasible has becomeincreasingly common. Detailed N-body simulations can be used to test the veracity of planetclaims. Sometimes the results of such tests have shown that some systems simply cannotexist in their proposed configuration on astronomically relevant timescales (e.g. Horner et al.2011; Wittenmyer et al. 2012a; Horner et al. 2012b; Wittenmyer et al. 2013). In other cases,dynamical testing can place additional constraints on planetary systems, particlarly when theplanets are in or near resonances (e.g. Robertson et al. 2012a,b; Wittenmyer et al. 2012c).In this section, we examine the various solutions for the HD 73526 system, performing de-tailed dynamical tests of the planetary system configurations as given in Tables 3 and 4.Given that the four solutions are not substantially different from each other in terms ofgoodness-of-fit, these dynamical stability tests can serve to discern which scenario is mostplausible: a solution which is favored by the fitting process may prove to be unstable andhence unfeasible.
As in our previous dynamical work (e.g. Marshall et al. 2010; Wittenmyer et al. 2012b;Horner et al. 2012a), we used the Hybrid integrator within the N -body dynamics package Mercury (Chambers 1999) to perform our integrations. We held the initial orbit of theinner planet fixed at its best-fit parameters, as given in Table 3, and then created 126,075 testsystems. In those test systems, the initial orbit of the outer planet was varied systematicallyin semi-major axis a , eccentricity e , periastron argument ω , and mean anomaly M , resultingin a 41x41x15x5 grid of “clones” spaced evenly across the 3 σ range in those parameters. Weassumed the planets were coplanar with each other and, for the Keplerian case, we assignedmasses equivalent to their minimum mass, m sin i (Table 3). We then followed the dynamicalevolution of each test system for a period of 100 million years, and recorded the times atwhich either of the planets was removed from the system. Planets were removed if theycollided with one another, hit the central body, or reached a barycentric distance of 10 AU.We performed these dynamical simulations for the Keplerian fit ( i = 90 o ), the dynamicalfit at i = 90 o , and the lowest-inclination dynamical fit: the configuration given in Table 4at i = 20 . o . For the latter scenario, the planet masses were scaled according to the derivedsystem inclination i . Clearly, the masses of the planets are a proxy for the expected dynamicalstability – systems containing more-massive planets are likely to be less stable. Hence, the 8 –three scenarios we have tested, at i = 90 o and i = 20 . o , represent the extremes of dynamicalstability (or instability) for the HD 73526 system.To explore the effects of mutual inclinations between the planets, we performed fiveadditional N-body simulations, for scenarios in which the two planets were inclined withrespect to each other. These simulations were set up exactly as described above, usingthe parameters of the Keplerian solution (Table 3), except at a lower resolution due tocomputing limitations: a 21x21x5x5 grid in a , e , ω , and M . Five runs were performed,at mutual inclinations of 5, 15, 45, 135, and 180 degrees. The latter two cases representretrograde scenarios, which can sometimes allow for a larger range of dynamically stableorbits (Eberle & Cuntz 2010; Horner et al. 2011). The results of our dynamical stability simulations for the Keplerian solution are shownin Figure 5. We show six panels, for the coplanar and five mutually-inclined scenarios asdescribed above. For the coplanar and 5-degree cases (panels a and b), the best-fit setof parameters (shown by the open square with 1 σ crosshairs) lies in a region of moderatestability, with mean system survival times of ∼ years. The stability rapidly degrades asthe inclination between the planets becomes significant, and even for retrograde cases (panelse and f), the nominal best-fit system destabilizes within 10 yr. From these simulations, wecan conclude that the HD 73526 planets are most likely coplanar with each other. Panels(a) and (b) also show that the stability of the system increases as the outer planet takes onlower eccentricities. For e < ∼ yr. This is not a surprisingresult, as high eccentricities generally increase the possibility of strong interactions or evenorbit crossings (though systems in protected resonances may remain stable for some valuesof M and ω ). Indeed, the statistics of multi-planet systems show that planets in multiplesystems tend to have lower eccentricities (Wright et al. 2009; Wittenmyer et al. 2009).As shown in Figure 5, the Keplerian best-fit solution is stable on million-year timescales.However, the colored squares in Figure 5 represent the mean survival times across the rangeof mean anomalies and ω tested. As the best-fit solution for the HD 73526 system places theplanets on resonant orbits, their stability will naturally be highly sensitive to the values ofthese angles. Hence, Figure 6 shows the outcomes of the 75 individual simulations performedat the best-fit a and e spanning a 5x15 grid in M and ω . We see that the nominal solution(where a and e are fixed at the best-fit values) lies at the point of maximum stability.While this is reassuringly consistent with our expectations of enhanced stability within theresonance, we caution that each colored square in Figure 6 represents only a single run, and 9 –dynamical evolution is known to be a chaotic process (e.g. Horner et al. 2004a,b).The long-term stability results for the dynamical fit with i = 90 o are shown in Figure 7.It is immediately apparent that this solution results in a higher degree of stability, withthe entire one-sigma region exhibiting mean lifetimes exceeding 10 yr. The right panel ofFigure 7 shows the results from the 75 individual runs in the central best-fit square, as inFigure 6. For this case, when we examine the dependence on M and ω , we see that the entirediagonal region (including the best fit) is stable for 10 years. In contrast, Figure 8 showsthe dynamical stability of the i = 20 . o solution from Table 4. Though this fit is formallyalmost as good as the Keplerian fit, the increased masses implied by the inclination renderthe system unstable on short timescales (1000 yr).
5. Discussion and Conclusions
We have fit the HD 73526 system using both kinematic and dynamical techniques, yield-ing four possible solutions. There are no compelling differences between the four models interms of their goodness-of-fit statistics or derived planetary parameters. The only significantdistinguishing characteristics are the planet masses derived from the system inclinations inthe dynamical fits (Table 4). We thus turned to a detailed dynamical stability mappingprocedure in which we tested a broad range of parameters about the best-fit solutions.Our dynamical stability testing showed that the Keplerian model yielded a system whichwas stable on million-year time scales, with stability increasing for lower eccentricities (Fig-ure 5). The interacting dynamical fitting procedure gave three “best” solutions, one of whichwas at a system inclination of 90 o (giving planet masses equal to the m sin i minimum massesused in the Keplerian model). Our stability testing for the inclined solution at i = 20 . o resulted in severe instability throughout the allowed 3 σ parameter space. The increasedplanetary masses for the low-inclination solutions appear to destabilize the system on astro-nomically short timescales ( < i = 20 . o scenario.While the individual best-fit solutions proved stable for i > . o (as noted in Section 3.2), itis clear that the region of long-term stability expands as the system inclination increases. Wethus adopt the i = 90 o dynamical fit for two primary reasons: first, the planets are massiveenough that they are certainly interacting with each other, as evidenced by the 2:1 resonantconfiguration; and second, this fit proved to be significantly more stable than the Keplerianfit (Figure 5). We note in passing that if the system’s inclination is indeed near 90 o , thereis the possibility that one or both planets transit.This work has shown how dynamical stability considerations can serve to constrain the 10 –configuration of a planetary system when the χ surface is such that a clear minimum is notevident (e.g. Campanella 2011). We have combined two fitting methods with the detaileddynamical simulations to present an updated view of the interesting 2:1 resonant planetarysystem orbiting HD 73526.This research has made use of NASA’s Astrophysics Data System (ADS), and the SIM-BAD database, operated at CDS, Strasbourg, France. This research has also made use of theExoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org (Wright et al.2011). M.H.L. was supported in part by the Hong Kong RGC grant HKU 7024/13P. DMacknowledges funding from the BASAL CATA Center for Astrophysics and Associated Tech-nologies PFB-06, and the The Milky Way Millennium Nucleus from the Ministry for theEconomy, Development, and Tourism’s Programa Iniciativa Cient´ıfica Milenio P07-021-F. REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
14 –Table 1. AAT/UCLES Radial Velocities for HD 73526
JD-2400000 Velocity (m s − ) Uncertainty (m s − )51212.13020 7.91 5.4051213.13145 0.32 5.3751214.23895 5.52 6.5451236.14647 15.70 6.6251630.02802 3.91 5.1251717.89996 -190.60 7.0951920.14186 -77.28 6.4551984.03780 10.04 4.8752009.09759 12.38 4.2152060.88441 -105.26 3.7352091.84653 -223.76 6.7652386.90032 -2.62 3.4652387.89210 1.72 2.8652420.92482 -66.78 3.2852421.91992 -64.87 3.2352422.86019 -66.65 3.3452424.92369 -77.31 7.0252454.85242 -151.57 3.4452655.15194 -81.59 3.5353008.13378 0.13 2.4253045.13567 -95.56 3.2053399.16253 -52.76 2.9753482.87954 20.95 2.0253483.88740 26.55 2.5953485.96240 22.83 3.6553488.93814 14.81 2.3353506.88650 5.03 2.2553508.91266 11.94 2.0353515.89441 -4.01 2.8953520.91025 -4.97 3.2754041.18613 -14.76 7.2854549.03413 -97.63 2.8354899.03133 -7.35 4.2455315.92532 -91.43 3.0655997.03979 62.28 4.1056048.94441 -57.19 4.16
15 –Table 2. Magellan/PFS Radial Velocities for HD 73526
JD-2400000 Velocity (m s − ) Uncertainty (m s − )55582.79672 14.7 1.255584.75698 20.6 1.255585.74045 22.1 1.255587.77487 28.3 1.055588.71850 28.4 0.955663.53102 -60.3 1.155668.54537 -73.8 0.855672.50855 -94.5 0.855953.76750 0.4 1.355955.71181 0.0 1.156282.77476 -135.9 1.556292.76731 -101.0 1.356345.67804 23.1 1.256355.63611 33.8 1.356357.65331 33.0 1.256358.70107 39.4 2.456428.46819 -99.9 1.256431.48616 -105.3 1.756434.49819 -110.7 1.156438.46472 -119.0 1.1 Table 3. Keplerian Orbital Solutions
Planet Period T e ω K m sin i a (days) (JD-2400000) (degrees) (m s − ) (M Jup ) (AU)HD 73526 b 188.9 ± ± ± ± ± ± ± ± ±
10 0.28 ± ±
10 65.1 ± ± ±
16 –Table 4. Dynamical Fit Solutions
Parameter Planet b Planet c K [m s − ] 85.4 ± ± ± ± ± ± ω [deg] 198.3 ± ± ± ± a [AU] 0.65 ± ± i [deg] 90.0Mass [M Jup ] 2.35 ± ± χ ν − ] 6.54 K [m s − ] 83.0 ± ± ± ± ± ± ω [deg] 202.3 ± ± ± ± a [AU] 0.65 ± ± i [deg] 40.2Mass [M Jup ] 3.50 ± ± χ ν − ] 6.59 K [m s − ] 81.4 ± ± ± ± ± ± ω [deg] 205.7 ± ± ± ± a [AU] 0.649 ± ± i [deg] 20.8Mass [M Jup ] 6.22 ± ± χ ν − ] 6.76
17 –Fig. 1.— Radial-velocity curves and residuals from the coplanar edge-on dynamical fit inTable 4. Error bars include 3.3m s − of stellar jitter added in quadrature. Red points areAAT data while blue points are PFS data. The right panel shows the dynamical evolution ofthis system. The semimajor axes remain essentially constant, while the eccentricities showsecular variations on timescales of centuries. The fit is in a 2:1 MMR with both θ and θ librating around 0 degrees. 18 –Fig. 2.— χ and RMS as a function of sin i . 19 –Fig. 3.— Best-fit parameters as a function of sin i . 20 –Fig. 4.— Evolution of χ ν contours in K - K space with as a function of sin i . 21 –Fig. 5.— Dynamical stability of the HD 73526 system as a function of the outer planet’sinitial eccentricity and semimajor axis. The best-fit Keplerian parameters (Table 3) aremarked by the open box with 1 σ crosshairs. Each colored square represents the meanlifetime of 75 unique M - ω combinations at that point in ( e , a ) for the outer planet. Panel (a)is the coplanar case, and panels (b)-(f) are the mutually-inclined scenarios, for inclinationsof 5,15,45,135, and 180 degrees, respectively. 22 –Fig. 6.— Dynamical stability of the best-fit Keplerian solution for the HD 73526 systemfor a 15x5 grid of ω and M . The semimajor axis and eccentricity have been fixed to theirbest-fit values. The colors and symbols have the same meaning as in Figure 5; this plotshows results from the 75 individual simulations which were averaged in the center coloredsquare of Figure 5. The best-fit solution lies squarely in the most stable region of this subsetof simulations. 23 –Fig. 7.— Left: Stability of the HD 73526 system as a function of the outer planet’s initialeccentricity and semimajor axis. The colors and symbols have the same meaning as inFigure 5. For this system, we used the i = 90 o solution (Table 4). As compared to theKeplerian solution, this fit results in substantially enhanced stability throughout the 1 σ range. Right: Same as Figure 6, but for the dynamical-fit i = 90 o solution.Fig. 8.— Same as Figure 7, but for the i = 20 . oo