A significant mutual inclination between the planets within the π Mensae system
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2020July 20, 2020
A significant mutual inclination between the planets within the π Mensae system
Robert J. De Rosa , Rebekah Dawson , and Eric L. Nielsen , European Southern Observatory, Alonso de Córdova 3107, Vitacura, Santiago, Chilee-mail: [email protected] Department of Astronomy and Astrophysics, Center for Exoplanets and Habitable Worlds, The Pennsylvania StateUniversity, University Park, PA 16802, USA Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94305, USA Department of Astronomy, New Mexico State University, P.O. Box 30001, MSC 4500, Las Cruces, NM 88003, USA26 May 2020
ABSTRACT
Context.
Measuring the geometry of multi-planet extrasolar systems can provide insight into their dynamical historyand the processes of planetary formation. Such measurements are challenging for systems detected through indirecttechniques such as radial velocity and transit, having only been measured for a handful of systems to-date.
Aims.
We aimed to place constraints on the orbital geometry of the outer planet in the π Mensae system, a G0V star at18.3 pc host to a wide-orbit super-jovian ( M sin i = 10 . ± . M Jup ) with a 5.7-year period and an inner transitingsuper-earth ( M = 4 . ± . M ⊕ ) with a 6.3-d period. Methods.
The reflex motion induced by the outer planet on the π Mensae star causes a significant motion of thephotocenter of the system on the sky plane over the course of the 5.7-year orbital period of the planet. We combinedastrometric measurements from the
Hipparcos and
Gaia satellites with a precisely determined spectroscopic orbit in anattempt to measure this reflex motion, and in turn constrain the inclination of the orbital plane of the outer planet.
Results.
We measured an inclination of i b = 49 . +5 . − . deg for the orbital plane of π Mensae b, leading to a directmeasurement of its mass of . +1 . − . M Jup . We found a significant mutual inclination between the orbital planes ofthe two planets; a 95% credible interval for i mut of between ◦ . and ◦ . after accounting for the unknown positionangle of the orbit of π Mensae c, strongly excluding a co-planar scenario for the two planets within this system. Allorbits are stable in the present-day configuration, and secular oscillations of planet c’s eccentricity are quenched bygeneral relativistic precession. Planet c may have undergone high eccentricity tidal migration triggered by Kozai-Lidovcycles, but dynamical histories involving disk migration or in situ formation are not ruled out. Nonetheless, this systemprovides the first direct evidence that giant planets with large mutual inclinations have a role to play in the origins andevolution of some super-Earth systems.
Key words. astrometry – planets and satellites: dynamical evolution and stability – stars: individual: π Mensae
1. Introduction π Mensae ( π Men) is a G0V star (Gray et al. 2006) at a dis-tance of 18.2 pc. The star is known to host two planetary-mass companions; a massive super-jovian with a 5.7-yearperiod discovered via Doppler spectroscopy (Jones et al.2002), and a transiting super-earth with a 6.3-day period(Gandolfi et al. 2018; Huang et al. 2018), the first discov-ered by the
Transiting Exoplanet Survey Satellite ( TESS ;Ricker et al. 2015). Combined radial velocity and transitsurveys (Zhu & Wu 2018; Bryan et al. 2019) and the iden-tification of long-period transiting planets (Herman et al.2019) have shown that such system configurations are com-mon; the majority of systems with a long-period gas giant( a > au) also harbor short-period super-earths (1–4 R ⊕ ),implying potential links between their formation and dy-namics. Outer gas giants – particularly those with largeeccentricities and/or mutual inclinations – can limit themasses and orbits super-Earths form on (e.g., Walsh et al.2011; Childs et al. 2019), stir up their eccentricities and mu-tual inclinations (e.g., Hansen 2017; Huang et al. 2017; Lai & Pu 2017), drive secular chaos (e.g., Lithwick & Wu 2011),and in extreme cases excite their eccentricities to valueshigh enough for tidal migration (e.g., Dawson & Johnson2018, Section 4.4). Masuda et al. (2020) presented statis-tical evidence for a population of high mutual inclinationgas giants in systems with single transiting super-Earthsbut due to the limits of the radial velocity and transit tech-niques, we have yet to directly measure the mutual inclina-tion for any individual systems.Full orbital characterization of systems containing bothouter giant planets and inner super-Earths can also aid ourunderstanding of the origin of short period ( < day)super-Earths. Like hot Jupiters (see Dawson & Johnson2018 for a review), they may have formed in situ or ar-rived close to their star via disk or tidal migration. The“Hoptune" population (planets with orbital periods lessthan 10 days and sizes between 2 and 6 Earth radii) inparticular exhibits similarities to hot Jupiters in host starmetallicity dependence, lack of other transiting planets inthe same system, and occurrence rates (Dong et al. 2018). Article number, page 1 of 24 a r X i v : . [ a s t r o - ph . E P ] J u l &A proofs: manuscript no. main Dawson & Johnson (2018) argued that high eccentricitytidal migration could account for the fact that these planetsdid not lose their atmospheres during the star’s young, ac-tive stage. Characterizing the full three-dimensional orbitalarchitecture of individual systems containing a super-Earthaccompanied by an outer gas giant can aid us in testing thehigh eccentricity tidal migration scenario.The orbital inclination of the outer planet had previ-ously been investigated using
Hipparcos astrometric mea-surements of the host star to detect the astrometric reflexmotion induced by the planet (Reffert & Quirrenbach 2011).This analysis yielded a plausible range for the orbital incli-nation of ◦ . – ◦ . , corresponding to a maximum massfor the companion of 29.9 M Jup . This analysis was not ableto conclusively differentiate between a co-planar and a mu-tually inclined configuration for the system. Efforts havealso been made to resolve the outer planet via direct imag-ing, but the contrast achieved was not sufficient to detectthe companion (Zurlo et al. 2018). A single epoch of rela-tive astrometry between the planet and the host star wouldlikely lead to an immediate determination of the inclinationof the orbital plane.In this paper we present an analysis of spectroscopicand astrometric measurements of the motion of the starinduced by the orbit of π Men b. We describe the acquisi-tion of the data in Section 2. We repeat the analysis pre-sented in (Reffert & Quirrenbach 2011) to demonstrate theimprovement of the inclination constraints achieved with arevised spectroscopic orbit in Section 3. We incorporate thelatest astrometric measurements from
Gaia into our jointastrometric-spectroscopic model in Section 4. We discussthe measured mutual inclination in the context of the dy-namical history of the system in Section 5, and conclude inSection 6.While this manuscript was being reviewed an indepen-dent analysis of the same datasets by Xuan & Wyatt (2020)was published. We derive consistent results on the mutualinclination of the two planets despite using a different ap-proach for incorporating the various astrometric measure-ments of the system.
2. Data Acquisition
We collected 359 literature and archival radial velocities ofthe π Men primary star were obtained between 1998 Jan-uary 16 and 2020 January 8 using the University CollegeLondon Echelle Spectrograph (UCLES; Diego et al. 1990)and the High Accuracy Radial velocity Planet Searcher(HARPS; Mayor et al. 2003). The 77 velocities from UCLESspan from 1998 January 16 until 2015 November 22 (R.Wittenmyer, priv. comm. ), the same record used for thediscovery and characterization of π Men c (Gandolfi et al.2018; Huang et al. 2018). The 282 velocities from HARPSspan from 2003 December 28 until 2020 January 08 andwere obtained from the reduced data products publiclyavailable on the ESO Archive Facility , an additional sevenvelocities from HARPS were available but were rejecteddue to a deterioration of the thorium-argon lamp on 2018November 26 and 27. The HARPS instrument underwentan intervention on 2015 June 03 that led to a significant https://archive.eso.org/ Table 1.
Astrometric measurements of the photocenter of the π Mensae system
Property Value Error Unit
Hipparcos (1991.25) - HIP 26394 α . . deg, mas a,b δ − . . deg, mas a (cid:36) .
60 0 . mas µ (cid:63)α .
01 0 . mas yr − µ δ .
38 0 . mas yr − Gaia (2015.50) - Gaia DR2 4623036865373793408 α . . deg, mas a,b δ − . . deg, mas a (cid:36) . . mas µ (cid:63)α .
246 0 . mas yr − µ δ .
908 0 . mas yr − Notes. ( a ) Value in degrees, uncertainty in milliarcseconds ( b ) Uncertainty in α (cid:63) = α cos δ chi2AL gofAL . . . G magnitude10 − epsiAL ( m a s ) Fig. 1.
Quality of fit metrics for π Men (yellow cross) and asample of stars within a factor of two brightness (points) fromthe
Gaia catalogue. The parameters shown are the the χ of thefit of the five-parameter astrometric model to the measurements(top panel), an analogue of a reduced χ (middle panel), and thesum of the residuals (bottom panel). The quality of the fit for π Men is consistent with that of other stars of a similar brightness. change in the radial velocity zero point of the instrument(Lo Curto et al. 2015). To correct for this in our analysiswe treat the 129 velocities from before this date as comingfrom a different instrument (“HARPS1”) to the remaining153 (“HARPS2”). A complete listing of the radial velocitiesused in this study is given in Table 5.
Article number, page 2 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Astrometric measurements of the π Men system were ob-tained from
Hipparcos catalogue re-reduction (van Leeuwen2007b) and from the second
Gaia data release (DR2; GaiaCollaboration et al. 2018). The coordinates are expressedin the ICRS reference frame at the 1991.25 epoch for
Hip-parcos and the 2015.5 epoch for
Gaia . A minor correctionto the
Gaia astrometry was applied to account for the ori-entation and rotation of the bright star reference (for π Men: ∆ α (cid:63) = − . ± . mas, ∆ δ = 0 . ± . mas, ∆ µ α (cid:63) = ∆ µ δ = 0 . ± . mas yr − ; Lindegren et al. 2018;Lindegren 2019) , and the ∼ error inflation terms de-scribed in Arenou et al. (2018). We use the revised orien-tation and spin parameters presented in Lindegren (2020); (cid:15) X = − . ± . mas, (cid:15) Y = 1 . ± . mas, (cid:15) Z =0 . ± . mas, ω X = − . ± . mas yr − , ω Y = − . ± . mas yr − , ω Z = − . ± . mas yr − ),applying them to the catalogue values using a Monte Carlo-based approach. The goodness of fit metrics given in the Gaia catalogue are better than for a typical star of a sim-ilar magnitude (Figure 1). The astrometric measurementsfrom both catalogues, after applying the correction for the
Gaia values, are given in Table 1.In addition to the
Hipparcos catalogue values, we alsoobtained the individual measurements made by the satel-lite of π Men that were used to derive the five astrometricparameters given in Table 1. These measurements are ef-fectively one-dimensional, constraining the position of thestar to within 1–2 mas along a great circle on the celestialsphere with a scan orientation angle ψ (a north-south scanhas ψ = π/ ). Instead of publishing the abscissa data Λ —the raw scan measurements—the re-reduction of the Hip-parcos data presented the abscissa residuals δ Λ after thebest-fit astrometric model had been subtracted. These arereferred to as the Hipparcos intermediate astrometric data(IAD; van Leeuwen 2007a).The IAD for π Men contain the epoch t , parallax factor Π , scan orientation ψ , abscissa residual δ Λ and correspond-ing uncertainty σ Λ for each of the 137 measurements of thestar (Table 6). The measurements span from 1989 Novem-ber 05 until 1993 January 29, have an average uncertainty of1.6 mas, and none were rejected when fitting the astrometricmodel presented in the catalogue. The abscissa Λ HIP can bereconstructed using the formalism described in Sahlmannet al. (2011) as Λ HIP = ( α (cid:63) + µ α (cid:63) t ) cos ψ + ( δ + µ δ t ) sin ψ + (cid:36) Π + δ Λ (1)We also decomposed four of the astrometric parameters intoa constant term and an offset (e.g., α (cid:63) = α (cid:63) H +∆ α (cid:63) ) to avoidprecision loss. When reconstructing the abscissa both theconstant and offset terms were set to zero: Λ HIP = (cid:36) Π + δ Λ (2)which we can compare to a noiseless model abscissa gener-ated with a small perturbation of the catalogue parameters Λ = (∆ α (cid:63) + ∆ µ α (cid:63) t ) cos ψ + (∆ δ + ∆ µ δ t ) sin ψ + (cid:36) Π (3)in order to compute a goodness of fit, for example.The individual scan measurements from the Gaia satel-lite are only due to be published at the conclusion of the We use the notation α (cid:63) = α cos δ mission, limiting us to the astrometric parameters pre-sented in the catalogue. However, the predicted date, par-allax factor, and scan orientation θ of the individual Gaia measurements are available (Table 7), giving us the neces-sary information to perform a simplistic simulation of the Gaia measurement of the position and motion of the pho-tocenter. Unlike for
Hipparcos , here the scan orientationdescribes the position angle of the scan direction relative tonorth (i.e. a north-south scan has θ = 0 ). These simulatedmeasurements can be compared with the catalogue valuesto constrain the photocenter motion during the time spanof the Gaia measurements. We generated a forecast usingthis tool on 2020 January 7, resulting in a prediction of 26measurements between 2014 August 26 and 2016 May 20,within the range of dates for which measurements were usedto construct the DR2 catalogue. This is more than the 21transits used to compute the astrometric parameters in thecatalogue, consistent with the warning on the gost utilitywebsite that only 80% of the measurements are usable.
3. Photocenter motion measured by Hipparcos
The presence of a massive companion to π Men makes inter-pretation of astrometric measurements of this object morechallenging. In a single star system the barycenter and pho-tocenter of the system are coincident with the location ofthe star. In an unresolved multiple system with two or moreobjects of unequal brightness—as is the case for the π Mensystem—the photocenter, barycenter, and the location ofthe primary star are no longer coincident. Astrometric mea-surements of such a system are therefore of the position andmotion of the photocenter of the system, a combinationof the constant velocity of the system barycenter throughspace and the orbital motion of the photocenter around thebarycenter.We first investigated whether a combination of the
Hip-parcos
IAD and the best fit spectroscopic orbit can provideany significant constraint on the geometry of the orbit of π Men b. We implemented the grid-based approach presentedin Sahlmann et al. (2011) where for each combination of theorbital inclination i and longitude of the ascending node Ω ,the orbital elements measured from the spectroscopic orbitwere fixed, and the astrometric parameters were calculatedusing a least-squares based approach. A Keplerian fit to the radial velocities provides a measure-ment of the period P , velocity semi-amplitude K , eccen-tricity e , argument of periastron ω (cid:63) and the time of perias-tron T of the orbit of the star around the system barycen-ter. Here we use the notation ω (cid:63) to refer to the argumentof periastron for the primary star, and ω = ω (cid:63) + π as theargument of periastron for the companion. These spectro-scopic elements can be combined with an assumed mass ofthe primary star M and the inclination of the orbit i to de-termine the mass ratio q = M /M by solving the followingexpression for the mass function P K πG (cid:0) − e (cid:1) / = q (1 + q ) M sin i (4) https://gaia.esac.esa.int/gost/ Article number, page 3 of 24 &A proofs: manuscript no. main where G is the gravitational constant. The total semi-majoraxis a in astronomical units can then be calculated usingKepler’s third law as a = P ( M + M ) (5)when the period and masses are expressed in years and solarmasses.The semi-major axis of the photocenter orbit a p aroundthe barycenter of a binary system can be calculated fromthe total semi-major axis of the orbit a , the masses of thetwo components M and M , and their magnitude differ-ence ∆ m = − . F /F ) . We define the fractional mass B as B = M M + M (6)and the analogous fractional flux β as β = F F + F = (cid:0) . m (cid:1) − , (7)from which a p can be calculated as a p = a ( B − β ) (e.g.,Heintz 1978; Coughlin & López-Morales 2012). In a binarysystem where the companion is much fainter than the pri-mary star, the value of β asymptotes to zero and the photo-center orbit can be considered as coincident with the orbitof the primary around the barycenter (i.e. a p ≈ a ). Weinclude this term for completeness; it is only relevant forextremely low inclinations where the mass of the secondaryexceeds the hydrogen burning limit. The value of β is al-most always wavelength dependent unless the spectral en-ergy distributions of the two objects are identical. We referto fractional fluxes computed using the Hipparcos H p filteras β H , and those using the Gaia G filter as β G . We usedan empirical mass-magnitude relationship for stars to de-termine the flux ratio for solutions where M > . M (cid:12) (Pecaut & Mamajek 2013), and set β H = β G = 0 otherwise.The position of the photocenter relative to the barycen-ter in the sky plane can be calculated from the elementsdiscussed previously, the position angle of the ascendingnode Ω and the epoch of the observation. The radius of aphotocenter orbit r p at a given epoch is defined as r p = a p (cid:0) − e (cid:1) e cos ν (8)where the true anomaly ν is computed from the mean andeccentric anomalies M and EM = 2 πP ( t − T ) = E − e sin E,ν = cos − (cid:18) cos E − e − e cos E (cid:19) . (9)The offset between the photocenter and barycenter in theright ascension x and declination y directions can be calcu-lated as x = r p [cos ( ω (cid:63) + ν ) sin Ω + sin ( ω (cid:63) + ν ) cos Ω cos i ] y = r p [cos ( ω (cid:63) + ν ) cos Ω − sin ( ω (cid:63) + ν ) sin Ω cos i ] (10)We modeled a Hipparcos abscissa by combining the fivestandard astrometric parameters described previously withthe motion of the photocenter in the right ascension and declination directions. As in Sahlmann et al. (2011) we de-fined a new variable
Υ = x cos ψ + y sin ψ (11)that converts the two-dimensional measurement into a one-dimensional measurement along the orientation of the scan.This variable is expressed in astronomical units, allowing usto account for a potential change in the best fit parallax ofthe star. We combined this new variable with our modelabscissa given in Equation 3 as Λ = (∆ α (cid:63) + ∆ µ α (cid:63) t ) cos ψ + (∆ δ + ∆ µ δ t ) sin ψ + (cid:36) (Π + Υ) (12)The least-squares solution to this linear equation can befound numerically via a matrix-based approach: ∆ α (cid:63) ∆ δ ∆ µ α (cid:63) ∆ µ δ (cid:36) = (cid:0) A T · A (cid:1) − · (cid:0) A T · Λ HIP (cid:1) , (13)where A is a (5,137) matrix constructed using the valuesfrom the Hipparcos
IAD and the variable Υ defined previ-ously; A = cos ψ sin ψ t cos ψ t sin ψ Π + Υ cos ψ sin ψ t cos ψ t sin ψ Π + Υ ... ... ... ... ... cos ψ n sin ψ n t n cos ψ n t n sin ψ n Π n + Υ n . (14)The least-squares solution can be used to construct a modelabscissa using Equation 12 which can be compared with thereconstructed Hipparcos abscissa to calculate the goodnessof fit χ = (cid:88) (cid:18) Λ − Λ HIP σ Λ (cid:19) . (15)This calculation was performed on a finely sampled i - Ω grid,with 500 elements between i = 0 deg and deg and 1000elements between Ω = 0 deg and deg.
We verified our model implementation by comparing to theresults of a previous study combining radial velocity and
Hipparcos measurements by Reffert & Quirrenbach (2011).We first applied it to HD 168443, a brown dwarf with aminimum mass of M sin i = 18 M Jup on a 4.8 yr orbit. Toensure consistency with the previous study we use the samespectroscopic orbital elements from Butler et al. (2006). Wefound χ = 65 . at i = 36 ◦ . ◦ . and a 3- σ rangefor i of ◦ . – ◦ . , consistent with their results for thissystem. Our χ surface for this example is a good matchto their Figure 3. We found similarly consistent results forother systems in their study. Article number, page 4 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system i ( d e g ) Butler et al. 2006 0 90 180 270Ω (deg)This work195 200 205 210 215 220 225 230 χ Fig. 2.
Goodness of fit ( χ ) as a function of the inclination i andthe position angle of the ascending node Ω for the photocenterorbit of the π Men system using the spectroscopic orbit fromButler et al. (2006) (left), and the revised orbit presented inTable 2 (right). Dashed lines (black and white) denote areasenclosing 68, 95, and 99.7 % of the probability. The value of χ calculated using an unperturbed five-parameter fit is alsoplotted (red dashed line). The lowest χ is indicated by thecross, and the inclination of π Men c from Huang et al. (2018)is plotted for reference (horizontal dashed line). . π Mensae
We first applied this model to π Men using the spectroscopicorbital elements from Butler et al. (2006) both to comparewith previous results from Reffert & Quirrenbach (2011)and to demonstrate the effect of a revised spectroscopicorbit on the determination of the astrometric orbit of thephotocenter. With data spanning slightly more than oneorbital period, Butler et al. (2006) measured P = 2151 d, K = 196 . m s − , e = 0 . , ω (cid:63) = 330 ◦ . and T =47819 . MJD. We used these spectroscopic elements andtheir estimate for the mass of the primary of . M (cid:12) . Weused the same size and range for the i - Ω grid describedpreviously. The small subset of inclinations that correspondto orbital configurations where M > M are ignored bysetting χ = ∞ .The resulting χ surface is shown in Figure 2 (leftpanel). We found a minimum χ of . at i = 55 ◦ . and Ω = 219 ◦ . ( χ ν = 1 . given 137 measurements and fivefree parameters), with marginalized σ confidence intervalsof ◦ . – ◦ . for i and ◦ . – ◦ . for Ω . The minimum χ is a slight improvement over the single star five-parameterastrometric model ( χ = 203 . , χ ν, null = 1 . ), howeverthe reduced χ in both cases suggesting a slight underes-timate in the Hipparcos abscissa uncertainties. Our resultsare consistent with the σ joint confidence interval on theinclination in Reffert & Quirrenbach (2011); we found a range of ◦ . – ◦ . , compared to ◦ . – ◦ . within theirstudy.This technique is dependent on having a good estimateof the motion of the photocenter of the system duringtime span of the Hipparcos observations. The radial velocityrecord used by Butler et al. (2006) covered one and a thirdorbital periods, but only sampled the part of the orbit withthe maximum velocity change near periastron once. This re-sulted in quite a large uncertainty in the period ( σ P = 85 d)which translated into an uncertainty of the epoch of peri-astron in late 1989/early 1990 of σ T = 170 d. This causessignificant uncertainty in the model of the astrometric mo-tion of the photocenter for a given M , i , and Ω .We used the radial velocity record that we compiled(Section 2.1) to revise the measurements of the spectro-scopic orbital elements. We used maximum likelihood esti-mation to determine the Keplerian orbit that best fit thedata. The radial velocity of the star v is calculated from thespectroscopic orbital elements as v = K [ e cos ω (cid:63) + cos ( ν + ω (cid:63) )] + γ. (16)Here γ is the time-dependent apparent radial velocity of thesystem barycenter; secular acceleration due to the chang-ing perspective between us and the π Mensae system causea significant change in the apparent radial velocity of thebarycenter, which is assumed to be moving linearly throughspace, over the twenty-year radial velocity record. We usedthe coordinate transformations described in Butkevich &Lindegren (2014) and the
Hipparcos astrometry in Table 1to account for this effect (see also Section 4.1). We evalu-ated the likelihood L using the predicted velocities derivedfrom of a given set of orbital parameters as ln L i = − n i (cid:88) j (cid:20) (cid:0) v model j − (cid:2) v obs j + ∆ v i (cid:3)(cid:1) σ j + (cid:15) i + ln (cid:0) π (cid:2) σ j + (cid:15) i (cid:3)(cid:1) (cid:21) (17)where v model j and v obs j are the predicted and measured ve-locities for the j th epoch, and σ j is the uncertainty of themeasurement. The likelihood is evaluated for each instru-ment i independently. The summation is performed overthe n i measurements for the i th instrument, with two addi-tional terms describing the radial velocity offset for that in-strument ∆ v i , and an error inflation term (cid:15) i to account forboth underestimated uncertainties and stellar jitter (e.g.,Price-Whelan et al. 2017; Fulton et al. 2018). The finallikelihood is then simply ln L = (cid:80) i ln L i . We used theHARPS measurements taken prior to the replacement ofthe fibres (“HARPS1”) to define our absolute radial veloc-ity zero-point by fixing ∆ v i to zero for this instrument.Table 2 contains the maximum likelihood estimate ofthe set of spectroscopic orbital elements for the orbit of π Men b. We used a typical parameterization for the spec-troscopic elements where the eccentricity e and argumentof periastron ω (cid:63) are combined into √ e sin ω (cid:63) and √ e cos ω (cid:63) to avoid angle wrapping. The effect of the inner planet onthese orbital parameters is negligible due to the significantlysmaller velocity semi-amplitude and shorter orbital period.We repeat the analysis of the Hipparcos
IAD using thisupdated set of orbital elements. Here we used the mass es-timate of M = 1 . ± . M (cid:12) from Huang et al. (2018). Article number, page 5 of 24 &A proofs: manuscript no. main . . . . . v ( k m s − ) . . . . . v ( k m s − ) Fig. 3.
Spectroscopic orbit of the star π Men around the system barycenter using the radial velocity record presented in Butleret al. (2006) (top panel) and this work (bottom panel). Measurements are plotted from UCLES (black circles), HARPS1 (pre-upgrade, pink squares), and HARPS2 (post-upgrade; yellow diamonds). The maximum likelihood orbit is plotted as a dashed line,with draws from an MCMC fit shown as light solid lines. The orbit is now so well constrained that the uncertainty of the predictedradial velocity at any given epoch is smaller than the thickness of the dashed line. In both panels the best fit orbit computed byButler et al. (2006) and used by Reffert & Quirrenbach (2011) is shown as a solid gray curve. Dates of the individual
Hipparcos and
Gaia measurements are denoted by orange and green vertical lines, respectively.
Table 2.
Maximum likelihood estimate of the spectroscopic or-bit of π Mensae b
Property Value Unit P . d K . m s − √ e cos ω (cid:63) . - √ e sin ω (cid:63) − . - T . MJD γ . . m s − ∆ v H1 ≡ m s − ∆ v H2 − . m s − ∆ v U . m s − (cid:15) H1 . m s − (cid:15) H2 . m s − (cid:15) U . m s − Derived parameters e . - ω (cid:63) . degThe resulting χ surface is shown in Figure 2 (right panel),showing a significant improvement of the constraint of theinclination of the orbit. We found a minimum χ of 193.4 at i = 45 ◦ . and Ω = 253 ◦ . ( χ ν = 1 . ), an improved goodnessof fit with the more accurate model of the photocenter mo-tion during the time span of the Hipparcos measurements.The marginalized σ confidence intervals for i and Ω are ◦ . – ◦ . and ◦ . – ◦ . . An edge-on orbit of i ∼ degfor π Men b is weakly excluded when only using the
Hip-parcos
IAD at the ∼ . σ level.
4. Incorporating Gaia astrometry
Astrometric measurements of bright stars such as π Menwith
Gaia have a typical per-scan measurement error of ∼ Hipparcos , andworse than predicted from models of the instrument (Lin-degren et al. 2018). Despite this, the combination of two ex-tremely precise measurements of the photocenter locationseparated by approximately 24 years, the
Hipparcos
IAD,and the instantaneous proper motion measured by
Gaia canbe used to measure deviations from linear motion caused bythe presence of an orbiting companion (e.g., Kervella et al.2019).Without access to the individual astrometric measure-ments used to fit the astrometric parameters given withinthe catalogue it is not possible to perform the same analy-sis as applied to the
Hipparcos data described in Section 3.Instead, we used the predicted scan timings, orientations,and parallax factors to forward model the
Gaia measure-ment of the position and motion of the photocenter at the
Gaia reference epoch of 2015.5. This method will be mostreliable with a relatively constant photocenter motion overthe time span of the
Gaia scans of the star, with only asmall deviation from linear motion. The
Gaia scans of π Men are sampling a relatively slow part of the orbit (Fig-ure 3, bottom panel), with a constant rate of astrometricacceleration of ∼ − predicted from the best fit or-bit from Section 3, depending on the orbital inclination.This is comparable in magnitude to the very smallest ac-celerations detected by Hipparcos with a longer baseline ofobservations; 99% of the stars where the accelerating model
Article number, page 6 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system was a better fit than the linear motion model had acceler-ations of > . mas yr − . The relatively constant motion ofthe photocenter also mitigates the effect of the discrepancybetween the number of predicted scans (26) and the actualnumber of scans used in the Gaia catalogue (21) as de-scribed in Section 2.2, especially if these five missing scansare randomly distributed throughout the time span of the
Gaia measurements. The upcoming
Gaia data releases willcontain measurements of astrometric accelerations whereappropriate with which it will be possible for a more pre-cise simulation of the
Gaia measurement without access tothe individual measurements themselves.
The model was adapted from the one used in De Rosa et al.(2019) to handle radial velocity data from multiple instru-ments and the lack of of any relevant direct imaging con-straints for this system. The model consists of twenty pa-rameters that allow us to connect our physical model of thesystem to the astrometric and spectroscopic observations.These parameters are described in Table 3. In this modelthe barycenter of the π Men system in 1991.25 is locatedat α = α H + ∆ α (cid:63) , δ = δ H + ∆ δ, (18)with a parallax and proper motion of (cid:36) = (cid:36) H +∆ (cid:36), µ α (cid:63) = µ α (cid:63) , H +∆ µ α (cid:63) , µ δ = µ δ, H +∆ µ δ , (19)where the H subscript denotes values from the Hipparcos catalogue. We followed the same procedure described inSection 3 to generate a model
Hipparcos abscissa Λ usingthe parallax (cid:36) , the offsets to the remaining four astrometricparameters, and the variable Υ that encodes the photocen-ter motion for a given set of orbital parameters. The like-lihood of the set of model parameters given the Hipparcos measurements was evaluated as ln L H = − (cid:88) (cid:34) (Λ − Λ HIP ) σ + (cid:15) + ln (cid:0) π (cid:2) σ + (cid:15) (cid:3)(cid:1)(cid:35) . (20)The error inflation term (cid:15) HIP was added to account for theapparent underestimation of the
Hipparcos
IAD uncertain-ties noted in Section 3.3.The π Men barycenter was then propagated from the
Hipparcos reference epoch (J1991.25) to the
Gaia referenceepoch (J2015.5) using the rigorous coordinate transforma-tion procedure described in Butkevich & Lindegren (2014).This procedure propagates the spherical coordinates, paral-lax, and tangent-plane proper motions, accounting for thenon-rectilinear nature of the ICRS coordinate system. Thistransformation is critical given the declination of the star;applying the simplistic transformation using the tangentplane approximation would result in a 2.9 mas error in theposition and a 0.2 mas yr − error in the proper motion ofthe star at the Gaia reference epoch, mimicking photocen-ter motion due to the orbiting companion. This coordinatetransform also accounts for the secular acceleration of theradial velocity of the system barycenter. As in Section 3.3,we fit for the apparent radial velocity of the system barycen-ter at 1991.25 ( γ . ). With the barycenter propagated to 2015.5, we simu-lated the motion of the photocenter during the time spanof the Gaia measurements using the propagated parallaxand proper motions, and the predicted photocenter motioncaused by the orbiting companion. The photocenter orbitsemi-major axis was calculated using β G due to the differ-ing filter responses of Hipparcos and
Gaia . We then per-formed a simple least-squares fit to the motion of the starto simulate the
Gaia measurement of the five astrometricparameters presented in the catalogue using the scan tim-ings, angles, and parallax factors obtained from the gost utility described previously. We constructed a residual vec-tor r containing the difference between the catalogue andsimulated position and proper motion measurements. Thiswas used to calculate the likelihood of the model parame-ters given the Gaia measurements as ln L G = − r T C − r (21)where C is the covariance matrix of the Gaia cataloguemeasurements constructed using the uncertainties and rel-evant correlation coefficients.The likelihood of the model parameters given the radialvelocity measurements was calculated as in Section 3.3 ln L RV = n (cid:88) i ln L i (22)for each of the n instruments used to construct the radialvelocity record in Table 5. The final likelihood was calcu-lated by summing the three components ln L = ln L H + ln L G + ln L RV (23) π Mensae
We used a combination of maximum likelihood estimationand Markov chain Monte Carlo (MCMC) to determine thebest fit model parameters and their uncertainties and co-variances given the astrometric and spectroscopic measure-ments of the π Men system. This was performed with fourcombinations of the astrometric data;
Hipparcos only tocompare with the results in Section 3, and with the var-ious combinations of the
Hipparcos and the
Gaia positionand proper motion measurements. The maximum likelihoodparameter set was found using the Nelder-Mead simplexalgorithm within the scipy.optimize package (Virtanenet al. 2020). We fixed M = 1 . M (cid:12) for this step as thisparameter is not constrained by any of the measurements.The results for each of the four combinations of astrometricdata are listed in Tables 4.The uncertainties on the model parameters were esti-mated using a Markov chain Monte Carlo algorithm. Weused the parallel-tempering affine-invariant sampler emcee (Foreman-Mackey et al. 2013) to thoroughly explore theposterior distributions of each parameter. We initialized512 MCMC chains at each of 16 “temperatures” (a totalof 8192 chains) near the best fit solution identified via themaximum likelihood analysis described previously. The low-est temperature chains for parallel-tempering MCMC ex-plore the posterior distribution of each parameter, whilstthe highest temperature chains explore the priors. This isespecially advantageous for complex likelihood volumes toensure parameter space is fully explored. Article number, page 7 of 24 &A proofs: manuscript no. main
Table 3.
Joint astrometric-spectroscopic model parameters
Property Unit Description PriorSpectroscopic parameters P d Orbital period U (1 , K m s − Radial velocity semi-amplitude U (0 , − ) √ e cos ω (cid:63) - Paramaterized variable of eccentricity and argument of periastron U ( − , a √ e sin ω (cid:63) - Paramaterized variable of eccentricity and argument of periastron U ( − , a τ - Mean anomaly at reference epoch (51000 MJD) in fractions of P U [0 , γ . m s − Radial velocity of π Mensae system barycenter at 1991.25 (48348.25 MJD) U ( − , v H2 m s − Radial velocity offset for post-upgrade HARPS U ( − , v U m s − Radial velocity offset for UCLES U ( − , (cid:15) H1 m s − Error inflation for HARPS1 U (0 , b (cid:15) H2 m s − Error inflation for HARPS2 U (0 , b (cid:15) U m s − Error inflation for UCLES U (0 , b (cid:15) HIP mas Error inflation for
Hipparcos abscissa measurements U (0 , b Astrometric parameters cos i - Cosine of the orbital inclination U [ − , deg Position angle of the ascending node U [0 , M M (cid:12) Mass of primary star N (1 . , . )∆ α (cid:63) mas R.A. offset between Hipparcos measurement and barycenter at 1991.25 U ( − , δ mas Dec. offset —–"—– U ( − , (cid:36) mas Parallax offset —–"—– U ( − , µ α (cid:63) mas yr − R.A. proper motion offset —–"—– U ( − , µ δ mas yr − Dec. proper motion offset —–"—– U ( − , Notes. ( b ) An additional criterion of ≤ e < was applied. ( a ) A penalty term is added to the likelihood to further constrain the possible values of these error inflation terms.
Each chain was advanced by steps with every tenthstep saved to disk. At each step a probability was calcu-lated by combining the likelihood from Eqn. 23 with theprior probabilities given the distributions listed in Table 3.We discarded the first fifth of the chain as “burn-in” wherethe chain positions were still a function of their initial con-ditions. While we did not apply a statistical check for con-vergence, the chains appeared well-mixed based on a visualinspection of the chain position as a function of step num-ber, with the median and σ ranges of each parameter notsignificantly changing. This process was repeated for eachof the four combinations of astrometric data. The resultingmedian and σ credible intervals are given in Table 4 for thenineteen fitted parameters, along with several derived pa-rameters. A visualization of the fit using all of the availableastrometric data is shown in Figure 4. We found χ ν = 1 . for the maximum likelihood estimate of the fit using all ofthe astrometry available given the 500 measurements and20 model parameters, but this parameters is dominated bythe goodness of fit to the radial velocity measurements andthe Hipparcos
IAD. The quality of the fit to the
Gaia mea-surements is good, although there is some tension betweenthe prediction and measurement of the photocenter propermotion in the declination direction (Fig. 4). A potentialsource of this discrepancy is a residual systematic rotationof the bright star reference frame not corrected using themodel from Lindegren (2019, 2020).
In order to further verify these results, we perform a sec-ond analysis with an independent fitting routine for jointastrometry and RV fits, as previously used in Nielsen et al. (2020) for the directly-imaged planet β Pictoris b. This rou-tine is modified for π Men b, fitting for 20 parameters,including the standard set of seven for visual orbits: semi-major axis ( a ), eccentricity ( e ), inclination angle ( i ), ar-gument of periastron of the planet ( ω ), position angle ofthe ascending ( Ω ), epoch of periastron passage ( T ), andperiod ( P ). Also fitted are seven radial velocity parame-ters, including system radial velocity at 1991.25 ( γ . ), m sin i of the secondary, RV offsets ( ∆ v H2 , ∆ v U ), and jitterterms for each instrument, (cid:15) H1 , (cid:15) H2 , and (cid:15) U , as well as theerror inflation term for the Hipparcos abscissa data (cid:15)
HIP . Fi-nally, we have five astrometric parameters; the offset of thephotocenter from the
Hipparcos catalog position of π Menat 1991.25 ( ∆ α (cid:63) , ∆ δ ), parallax ( (cid:36) ), and proper motionat 1991.25 ( µ α (cid:63) , µ δ ). We assume priors that are uniformin log a , log P , cos i , and uniform in all other parameters,except for an additional prior on the derived mass of theprimary ( M ), as above taken to be a Gaussian with meanof 1.094 M (cid:12) and σ of 0.039 M (cid:12) . As above, we utilize thefull RV dataset for π Men, and the
Gaia
DR2 position andproper motion measurements and errors. However, we usea separate method to convert the
Hipparcos
IAD residu-als and scan directions into individual measurements, as inNielsen et al. (2020). Unlike Nielsen et al. (2020), we userigorous propagation of position and proper motion from1991.25 to 2015.5, by transforming position, proper mo-tion, radial velocity, and parallax from angular coordinateto Cartesian, update all six values assuming linear motion,then transform back to angular coordinates. Fitting is thendone using a Metropolis-Hastings MCMC routine.Despite the different method we find very similar resultsto the
Hipparcos , Gaia , position and proper motion fit givenin Table 4. This second method finds value of [semi-major
Article number, page 8 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 4.
Maximum likelihood estimate and MCMC median and σ credible intervals for the fitted and derived parameters Property Unit
Hipparcos only
Hipparcos , Gaia pos.
Hipparcos , Gaia p.m.
Hipparcos , Gaia pos. & p.m. L max MCMC L max MCMC L max MCMC L max MCMCSpectroscopic parameters P d .
14 2089 . +0 . − . .
14 2089 . +0 . − . .
14 2089 . +0 . − . .
11 2089 . +0 . − . K m s − .
71 193 . ± .
33 193 .
71 193 . ± .
33 193 .
71 193 . ± .
33 193 .
70 193 . ± . √ e cos ω (cid:63) - . . ± . . . ± . . . ± . . . ± . √ e sin ω (cid:63) - − . − . +0 . − . − . − . +0 . − . − . − . +0 . − . − . − . +0 . − . τ - . . +0 . − . . . +0 . − . . . +0 . − . . . +0 . − . γ . m s − .
25 10700 . ± .
33 10700 .
24 10700 . ± .
33 10700 .
25 10700 . ± .
33 10700 .
23 10700 . ± . v H2 m s − − . − . +0 . − . − . − . +0 . − . − . − . +0 . − . − . − . +0 . − . ∆ v U m s − .
32 10675 . +0 . − . .
32 10675 . ± .
76 10675 .
32 10675 . ± .
76 10675 .
32 10675 . ± . (cid:15) H1 m s − .
12 3 . +0 . − . .
12 3 . +0 . − . .
12 3 . +0 . − . .
12 3 . +0 . − . (cid:15) H2 m s − .
00 2 . +0 . − . .
00 2 . +0 . − . .
00 2 . +0 . − . .
00 2 . +0 . − . (cid:15) U m s − .
54 5 . +0 . − . .
54 5 . +0 . − . .
54 5 . +0 . − . .
55 5 . +0 . − . (cid:15) HIP mas .
83 0 . +0 . − . .
84 0 . +0 . − . .
84 0 . +0 . − . .
87 0 . +0 . − . Astrometric parameters cos i - .
70 0 . +0 . − . .
56 0 . +0 . − . .
72 0 . +0 . − . .
66 0 . +0 . − . Ω deg . . +25 . − . . . +13 . − . . . +10 . − . . . +8 . − . M M (cid:12) .
094 1 . ± .
039 1 .
094 1 . ± .
039 1 .
094 1 . ± .
039 1 .
094 1 . ± . α (cid:63) mas − . − . +0 . − . − . − . ± . − . − . +0 . − . − . − . +0 . − . ∆ δ mas − . − . +0 . − . − . − . +0 . − . − . − . ± . − . − . ± . (cid:36) mas .
23 0 . +0 . − . .
22 0 . ± .
23 0 .
22 0 . ± .
24 0 .
08 0 . ± . µ α (cid:63) mas yr − − . − . +0 . − . − . − . ± . − . − . ± . − . − . ± . µ δ mas yr − − . − . +0 . − . − . − . ± . − . − . ± . − . − . ± . Derived parameters a au .
309 3 . ± .
039 3 .
307 3 . ± .
039 3 .
309 3 . ± .
039 3 .
309 3 . ± . i deg . . +25 . − . . . +15 . − . . . +7 . − . . . +5 . − . e - . . ± . . . ± . . . ± . . . ± . ω (cid:63) deg .
14 331 . ± .
24 331 .
14 331 . ± .
24 331 .
14 331 . ± .
24 331 .
15 331 . +0 . − . T MJD .
11 52123 . +1 . − . .
11 52123 . +1 . − . .
12 52123 . +1 . − . .
19 52123 . +1 . − . M M Jup .
87 11 . +2 . − . .
04 11 . +1 . − . .
29 13 . +1 . − . .
26 13 . +1 . − . Goodness of fit χ - . - . - . - . - χ - . - . - . - . - χ − pos - - - . - - - . - χ − pm - - - - - . - . - (cid:80) χ - . - . - . - . - (cid:80) χ ν - . - . - . - . - axis, period, eccentricity, position angle of nodes, argumentof periastron of the star, secondary mass, inclination angle]of [ . +0 . − . au, . +0 . − . , . ± . , . +8 . ◦− . , . ± . ◦ , . +1 . − . M Jup , . +5 . ◦− . ], compared to thevalues above of [ . ± . au, . +0 . − . d, . ± . , . +8 . ◦− . , . +0 . ◦− . , . +1 . − . M Jup , . +5 . ◦− . ]. For all parameters, the medians and confidence intervalsare almost identical from the two methods. This gives usfurther confidence in the measurement of the parametermost of interest, inclination angle, which is essentially thesame for the two orbit-fitting methods. The joint fit of the spectroscopic and all of the astrometricmeasurements yielded an inclination of i = 49 . +5 . − . deg anda position angle of the ascending node of Ω = 270 . +8 . − . degfor the orbit of π Men b (Fig. 5). We found consistent val-ues for these parameters for each of the four fits shown inTable 4. This inclination corresponds to a mass of π Men bof M = 13 . +1 . − . M Jup , a mass straddling the deuterium-burning limit sometimes used as the planet-brown dwarfboundary. The mutual inclination i mut between the orbitalplane of the two planets in this system cannot be directly measured as the position angle of the orbit of π Men c ( Ω c )is currently unconstrained.We instead estimated a plausible range of the mutualinclination assuming that the position angle of the orbitof the inner planet is randomly orientated relative to thatof the outer planet, p (Ω c ) = U (0 ,
360 deg) . We estimatedthe posterior distribution of i mut by combining the poste-rior distributions for the inclination and position angle for π Men b ( i b , Ω b ) calculated previously, the inclination for π Men c ( i c = 87 . +0 . − . deg; Huang et al. 2018, althoughan inclination of − i c is also consistent with the transitobservations), and our assumption for the posterior distri-bution of Ω c ; cos i mut = cos i b cos i c + sin i b sin i c cos (Ω b − Ω c ) . (24)This yielded a 95th-percentile lower limit on the mutual in-clination of i mut > ◦ . given our assumption for p (Ω c ) .The probability density function of the value of i mut isshown in Figure 6, alongside the prior probability den-sity function to demonstrate the constraints provided bythe measurements. The minimum value of i mut from the4,096,000 samples was ◦ . . We adopted the 95th-percentilecredible interval of i mut from ◦ . to ◦ . as the range ofplausible mutual inclinations for the system given the obser-vational data. We found a similar lower limit of i mut > ◦ . Article number, page 9 of 24 &A proofs: manuscript no. main − α ? − α ? c a t . ( m a s ) − δ − δ c a t . ( m a s ) µ α ? ( m a s y r − ) µ δ ( m a s y r − ) i (deg) Fig. 4.
Position (top two panels) and proper motion (bottomtwo panels) of the photocenter during time span of the
Hippar-cos and
Gaia measurements used in this study from the fit usingall available astrometric data. For the position plots the propermotion of the barycenter was subtracted, and the positions arerelative to the catalogue values. The position of the barycenterrelative to the catalogue value and its proper motion are alsoplotted (horizontal green lines). The significant change in theproper motion of the barycenter in the α (cid:63) direction is due tothe southern declination of the star. Catalogue values are plot-ted as the circle with thick error bars. Simulated measurementsbased on a five-parameter fit to the combined motion of the pho-tocenter and barycenter are shown by the more extended errorbars. when assuming Ω b = Ω c , indicating that the lower limit ofthis value is not strongly affected by the assumption maderegarding the value of Ω c . A co-planar configuration forthe two planets in the π Men system is strongly excludedbased on the spectroscopic and astrometric measurementsused within this analysis.
5. Dynamical constraints on present-day orbit anddynamical history
On its current orbit, the inner planet π Men c is shielded bygeneral relativistic precession from strong dynamical per-turbations by π Men b. Therefore a long-term stability re-quirement does not place any additional constraints on π
60 120 180 240 300 360Ω (deg)0306090120150 i ( d e g ) HipparcosHipparcos & Gaia
Fig. 5.
Covariance between the position angle of the ascendingnode Ω and inclination i , and corresponding marginalized dis-tributions, of the photocenter orbit of the π Men system fromthe MCMC calculation using only the
Hipparcos measurements(black) and using all available astrometry (red). Contours de-note areas encompassing 68, 95, and 99.7 % of the probability.Marginalized σ credible intervals are indicated. The values of i and Ω from the maximum likelihood analyses are shown forreference (cross symbols). The orbital inclination of π Men b issignificantly different from that of the inner planet (blue dottedline), indicating that the system is not aligned. P r o b . p (Ω c ) = U (0 , i m (deg) C u m u l a t i v e P r o b . p (Ω c ) = U (0 , . σ : 46 . . σ : 34 . . σ : 26 . . Fig. 6.
Posterior probability (top) and cumulative (bottom)distribution for the value of i mut under the assumption that p (Ω c ) = U (0 ,
360 deg) . The prior probability distribution is plot-ted for reference (dashed histogram).Article number, page 10 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system =0)0.000.050.100.150.200.25 M a x i m u m e cc en t r i c i t y Fig. 7.
Maximum eccentricity of π Men c excited by seculareffects of π Men b over 2.7 Gyr as a function of semi-major axis.We assume an initial eccentricity of 0 for π Men c. π Men c isshielded by general relativistic precession from strong dynamicalperturbations by π Men b. The shaded region indicates the 95%confidence interval on the observed i mut and the dashed linesindicate the most likely values. =0)1.01.21.41.61.82.02.22.4 D e c oup li ng d i s t an c e / a c Fig. 8.
Decoupling distance: semi-major axis of π Men c forwhich general relativistic precession prevents π Men b from ex-citing its eccentricity above 0.1. We assume an initial eccentricityof 0 for π Men c. Dotted horizontal line: π Men c’s present-daysemi-major axis. The shaded region indicates the 95% confidenceinterval on the observed i mut and the dashed lines indicate themost likely values. Men b’s orbital properties. We simulate the system at arange of mutual inclinations by approximating planet c asa test particle and solving Lagrange’s equations of motionusing a secular disturbing potential expanded to octupolarorder (Yokoyama et al. 2003), including general relativisticprecession of both planets’ orbits. Including the octupo-lar order is particularly important for this nearby, eccen-tric outer planet (e.g., Ford et al. 2000; Naoz et al. 2011;Teyssandier et al. 2013; Li et al. 2014). We set planet c’sinitial eccentricity to 0 and use the mass and orbit of planetb from Table 4. For the entire range of possible mutual in-clinations, the system remains stable: planet c’s eccentricityis not excited to a value high enough for tidal disruption(Fig. 7). π Men c’s present-day semi-major axis is close tothe value at which general relativistic precession prevents π Men b from exciting its eccentricity above 0.1 (Fig. 8). π Men c may have undergone high eccentricity tidal mi-gration (e.g., Hut 1981) to reach its present day short or-bital period. In one version of this scenario, planets b andc form on or migrate to well-separated orbits. On a stableorbit, planet c must reside closer to the star than 0.69 AU(Petrovich 2015b). π Men b is disturbed to its observedhigh eccentricity, perhaps by scattering or ejecting anotherplanet (e.g., Jurić & Tremaine 2008). This disturbance cre-ates a sufficient mutual inclination between planets b and c(e.g., Chatterjee et al. 2008) to trigger Kozai-Lidov cycles(e.g., Kozai 1962; Lidov 1962; Naoz 2016), raising planet c’seccentricity and triggering high eccentricity tidal migration.Figure 9 shows an example of this first scenario. We in-clude tidal evolution following Wu & Murray (2003) with atidal quality factor Q p = 10 and initial mutual inclinationof 45 degrees, eccentricity of 0, and semi-major of 0.2 AUfor π Men c, resulting in a mutual final inclination of 38.5 ◦ .If planet c originally had its eccentricity excited by Kozai-Lidov cycles, underwent significant tidal migration, and isfully tidally circularized today, we expect to observe it atmutual inclination near ∼ ◦ or ∼ ◦ (e.g., Fabrycky &Tremaine 2007; Petrovich & Tremaine 2016). Much lowerpresent-day mutual inclinations (i.e., closer to 0 and 180)are possible if planet c began with a large eccentricity. Forexample, in an initially coplanar configuration where planetb has an initial eccentricity of 0.45, longitude of periapseopposite planet b, and semi-major of 0.45 AU, the finalmutual inclination is 10 ◦ .However, we cannot assume that planet c has com-pleted its tidal circularization. Its observed eccentricity isnot well constrained by the radial-velocity or transit lightcurve datasets (Huang et al. 2018; Gandolfi et al. 2018),and other mini-Neptunes at short orbital periods have re-tained their orbital eccentricities (e.g., GJ 436b, Manesset al. 2007). Their tidal quality factors are not well known,with values of Q p spanning to (or an even widerrange) quite plausible. We might observe a higher mutualinclination (i.e., much closer to polar than ∼ or ◦ ) ifplanet c is still in the process of tidally circularizing and hasnot migrated a substantial distance. For example, planet ccould have started at a = 0 . AU and e = 0 and mutualinclination of 65 ◦ , been excited by Kozai-Lidov cycles ontoan orbit with a periapse several times the Roche limit, ex-perienced rather weak tidal dissipation with Q p = 10 , andarrived to its present-day semi-major axis after 1 Gyr with e oscillating between 0.6 and 0.75 and mutual inclinationoscillating between 54 and 63 ◦ as it continues ongoing tidalevolution. At larger initial semi-major axes, near polar mu-tual inclinations lead to either close periapse passages thatresult in tidal disruption or quick tidal circularization tosemi-major axes much smaller than planet c’s current semi-major axis.Another variation on the tidal migration scenario is that π Men c underwent high eccentricity migration initiallytriggered by planet-planet scattering (e.g., Rasio & Ford1996) or secular chaos (e.g., Wu & Lithwick 2011) ratherthan planet-planet Kozai-Lidov cycles. This scenario doesnot require a particular mutual inclination. The evolutioncould involve Kozai-Lidov cycles – or coplanar secular ec-centricity cycles (e.g., Petrovich 2015a) – once tidal evolu-tion separates the planets.Another possibility is that π Men c formed at or nearits current location (e.g., Lee et al. 2014), or arrived viadisk migration (e.g., Cossou et al. 2014). As described
Article number, page 11 of 24 &A proofs: manuscript no. main A U m u t ua l i n c li na t i on ( deg ) Fig. 9.
Example of high eccentricity Kozai-Lidov tidal migrationscenario. Top: Evolution of π Men c’s semi-major axis (black)and periapse (red). Bottom panel: mutual inclination. π Menc circularizes to its present-day semi-major axis with a mutualinclination of 38.5 ◦ . above, general relativistic precession mostly decouples itfrom planet b, with modest oscillations in eccentricity andinclination. Planet c may even reside in a close-in multi-planet system (which would not be easily compatible withthe high eccentricity migration scenario above); radial ve-locity and transit timing constraints do not rule out thepresence of other small planets. Depending on their spacing,coupling among the inner planets may cause them to oscil-late in inclination together as a coplanar set, slightly excitetheir mutual inclinations (e.g., Lai & Pu 2017; Huang et al.2017; Masuda et al. 2020; our Fig. 10), or even allow fora modest eccentricity excitation that results in short scaletidal migration of planet c. The detection of other planetsin the system could allow for stronger stability constraintson mutual inclinations.Therefore a variety of origins scenarios are consistentwith the system’s known properties. The significant mu-tual inclination makes the high eccentricity migration sce-nario appealing and the most likely observed mutual incli-nations coincide with those expected from high eccentricitytidal migration with planet c’s eccentricity originally ex-cited by Kozai-Lidov cycles. Moreover, with a radius of 2.04 R ⊕ (Huang et al. 2018) and period of 6.27 days, π Men cis a member population of short period, 2–6 R ⊕ planetsidentified by Dong et al. (2018), which Dawson & Johnson(2018) argue are likely to have undergone high eccentric-ity tidal migration. However, we cannot rule out a quieterhistory involving in situ formation or disk migration. Cru-cially, this system provides the first direct measurement of amutually inclined outer giant posited for a variety of super-Earth formation and evolution histories. e m u t ua l i n c w r t b ( deg ) Fig. 10.
Example behavior of multi-planet system, with π Menc (black solid) at its present-day semi-major axis, two other plan-ets of the same mass located at 0.103 AU (red dotted) and 0.185AU (blue dashed), and π Men b. Perturbations from planet blead to modest eccentricities (top panel) and mutual inclinationsamong the planets (bottom panel; i.e., each planet has a slightlydifferent mutual inclination with planet b). Dynamical evolutioncomputed in mercury6 ) (Chambers 1999), modified to includegeneral relativistic precession of all planets.
6. Conclusion
We have presented a joint analysis of spectroscopic and as-trometric measurements of the motion of the planet hostingstar π Mensae induced by the massive ( . +1 . − . M Jup )planet π Mensae b orbiting with a period of 5.7 years. Thisanalysis yielded a direct measurement of the inclination ofthe orbital plane of the outer planet of i b = 49 . +5 . − . deg,strongly excluding a co-planar configuration with the innertransiting super-earth. The mutual inclination between theorbital planes of the two planets is constrained to be be-tween ◦ . – ◦ . (95% credible interval), assuming a ran-dom orientation of the position angle of the orbit of π Menc. Outer gas giants on mutually inclined orbits have beeninvoked in shaping the orbits and architectures of innersuper-Earth systems, including stirring up the mutual in-clinations in multi-planet systems (e.g., Huang et al. 2017;Lai & Pu 2017) and driving super-Earths close to theirstars through high eccentricity tidal migration. The π Mensystem is the first directly measured example of a super-Earth accompanied by a mutually inclined giant, and isone of only three multi-planet systems with a measuredand significant mutual inclination: the three super-joviansof the υ Andromedae system ( i mut = 27 ± deg; McArthuret al. 2010) and the two Saturn-mass planets of Kepler-108( i mut = 24 ± deg; Mills & Fabrycky 2017).Histories of in situ formation, disk migration, and higheccentricity migration are all compatible with the system’scurrent configuration. Future tighter upper limits on theinner planet’s eccentricity would allow us to rule out the Article number, page 12 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system scenario in which π Men c is still tidally migrating and to fa-vor tidal eccentricity migration scenarios with a present-daymutual inclination close to 40 or 140 ◦ (while not ruling out in situ formation or disk migration). Detection of nearbyplanets to π Men c – through continuing radial-velocityfollow-up or transit timing variations with future TESSobservations – would favor the in situ formation or diskmigration scenario, rather than long distance tidal migra-tion, and potentially allow for tighter stability constraintson the mutual inclination between the inner planets and π Men b. Constraints on the composition of π Men c’s atmo-sphere (García Muñoz et al. 2020) may provide hints to theformation location (e.g., Rogers & Seager 2010). However,since in situ formation could involve assembly from outerdisk materials (e.g., Hansen & Murray 2012) that couldeven be the in form of large icy cores, atmospheric clues toformation location are not likely to be definitive.A single measurement of the relative astrometry be-tween π Men b and its host star is likely sufficient to mea-sure the orbital inclination without having to rely on abso-lute astrometric measurements such as those used in thisstudy. A previous attempt to directly image this planetfrom the ground did not succeed due to the significant con-trast between the host star and planet in the near-infrared,despite the observations being taken near the time of max-imum angular separation (Zurlo et al. 2018). Developmentsin ground and space-based direct imaging techniques mayenable a detection of this object in the near to medium-term. Indeed, the proximity of π Men to the Earth makesit one of a handful of targets that are amenable to directimaging in reflected light with the coronagraphic instru-ment (CGI; Noecker et al. 2016) on the upcoming
WideField Infrared Survey Telescope . The orbital elements for π Men b presented in this study can be used to constrainthe position and expected brightness of the planet duringthe course of the mission, enabling observations to be timedto maximize the expected signal-to-noise ratio. Future datareleases from the
Gaia consortium will enable a validationof the results presented here. The upcoming DR3 will con-tain measurements of accelerations for sources for whicha model of constant motion is a poor fit; the accelerationof the π Men photocenter over the
Gaia
DR3 time spanshould be detectable at a significant level. The eventual re-lease of the individual astrometric measurements made bythe
Gaia satellite will enable a measurement of the photo-center orbit using
Gaia data alone for this, and a slew ofother long-period planets detected via radial velocity.
Acknowledgements.
We wish to thank both referees for their care-ful review that helped improve the overall quality of this work. Wethank R. Wittenmyer for sharing the UCLES velocities used in thisstudy. RID is supported by NASA Exoplanet Research Program GrantNo. 80NSSC18K0355 and the Center for Exoplanets and HabitableWorlds at the Pennsylvania State University. The Center for Ex-oplanets and Habitable Worlds is supported by the PennsylvaniaState University, the Eberly College of Science, and the Pennsylva-nia Space Grant Consortium. This research made use of computingfacilities from Penn State’s Institute for CyberScience Advanced Cy-berInfrastructure. ELN is supported by NSF grant No. AST-1411868and NASA Exoplanet Research Program Grant No. NNX14AJ80G.This research has made use of the SIMBAD database, operatedat CDS, Strasbourg, France (Wenger et al. 2000), and the VizieRcatalogue access tool, CDS, Strasbourg, France (Ochsenbein et al.2000). This work has made use of data from the European SpaceAgency (ESA) mission
Gaia ( ),processed by the Gaia
Data Processing and Analysis Consortium(DPAC, ).Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the
Gaia
MultilateralAgreement.
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Article number, page 14 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 5.
Radial Velocities of π Mensae
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − − . − . − . − . − . − . − . − . − . − . − . − . − . . . . . . . . . . . . . . . . . . . . . − . . − . . − . − . − . − . − . . . . . . . . . . . . . . . . . . . Article number, page 15 of 24 &A proofs: manuscript no. main
Table 5. continued.
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − . . . . . . . . . . . . . . . . . . . . . . . . . . − . − . − . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . Article number, page 16 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 5. continued.
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − . − . − . . . . − . − . − . − . − . − . − . − . − . . . . . . . . . . . . Article number, page 17 of 24 &A proofs: manuscript no. main
Table 5. continued.
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − . . . . . . . . . . . . . . . . . . . . . . . . . . . . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Article number, page 18 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 5. continued.
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Article number, page 19 of 24 &A proofs: manuscript no. main
Table 5. continued.
BJD
TDB RV ± σ Instrument(BJD-2450000) km s − km s − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Article number, page 20 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 6.
Hipparcos astrometric measurements of the photocenter of the π Mensae system (van Leeuwen 2007a) t − .
25 Π cos ψ sin ψ δ Λ σ Λ yr mas mas − .
405 0 . − . − . .
36 1 . − .
405 0 . − . − . − .
71 1 . − .
405 0 . − . − . − .
91 1 . − .
405 0 . − . − . − .
66 1 . − .
260 0 . − . − . .
23 1 . − .
260 0 . − . − . .
87 0 . − .
260 0 . − . − . .
56 1 . − .
260 0 . − . − . − .
92 0 . − .
260 0 . − . − . − .
40 0 . − .
260 0 . − . − . − .
60 1 . − . − . − . . − .
41 0 . − . − . − . . − .
99 1 . − .
111 0 . − . . − .
86 1 . − .
110 0 . − . . − .
72 1 . − .
110 0 . − . . − .
18 1 . − . − .
682 0 . . .
47 1 . − . − .
682 0 . . − .
33 1 . − . − .
684 0 . . − .
17 1 . − .
949 0 . − . . − .
16 1 . − .
949 0 . − . . − .
58 0 . − .
949 0 . − . . − .
86 1 . − .
949 0 . − . . − .
09 1 . − .
949 0 . − . . .
20 1 . − . − .
659 0 . − . .
94 1 . − . − .
660 0 . − . − .
22 1 . − . − .
658 0 . − . − .
18 1 . − . − .
661 0 . − . − .
01 1 . − .
780 0 .
650 0 . . − .
39 0 . − .
780 0 .
645 0 . . − .
61 1 . − .
780 0 .
645 0 . . .
29 0 . − .
780 0 .
650 0 . . .
52 1 . − .
780 0 .
649 0 . . − .
72 1 . − .
780 0 .
648 0 . . − .
27 0 . − . − .
653 0 . − . .
84 1 . − . − .
655 0 . − . .
01 1 . − . − .
658 0 . − . − .
16 1 . − . − .
655 0 . − . .
79 1 . − . − .
658 0 . − . − .
69 0 . − . − . − . − . − .
24 0 . − . − . − . − . .
05 1 . − . − . − . − . − .
61 1 . − . − . − . − . − .
12 1 . − . − . − . − . .
24 1 . − . − . − . − . .
45 1 . − .
464 0 .
675 0 . − . .
26 0 . − .
464 0 .
675 0 . − . − .
51 1 . − .
463 0 .
679 0 . − . − .
10 1 . − .
463 0 .
678 0 . − . − .
10 1 . − . − . − . . − .
56 1 . − . − . − . . − .
08 2 . − . − . − . . − .
06 1 . − . − . − . . .
20 2 . − .
318 0 . − . − . − .
68 1 . − .
318 0 . − . − . − .
92 0 . − .
318 0 . − . − . .
33 1 . − .
318 0 . − . − . − .
85 1 . − . − . − . . − .
81 1 . − . − . − . . − .
04 0 . − . − . − . . .
15 1 . Article number, page 21 of 24 &A proofs: manuscript no. main
Table 6. continued. t − .
25 Π cos ψ sin ψ δ Λ σ Λ yr mas mas − . − . − . . − .
37 0 . − .
171 0 . − . . − .
36 1 . − .
171 0 . − . . .
33 2 . − . − .
674 0 . . − .
91 1 . − . − .
670 0 . . .
14 0 . − . − .
670 0 . . − .
12 1 . − .
014 0 . − . . .
48 2 . − .
014 0 . − . . − .
66 2 . − .
014 0 . − . . .
40 1 . − .
014 0 . − . . .
29 2 . . − .
674 0 . . .
12 1 . . − .
674 0 . . .
79 1 . . − .
673 0 . . − .
46 4 . .
152 0 .
654 0 . . .
63 1 . .
152 0 .
651 0 . . .
30 1 . .
152 0 .
657 0 . . − .
30 2 . .
152 0 .
654 0 . . .
52 1 . . − .
650 0 . − . − .
21 1 . . − .
649 0 . − . − .
49 0 . . − .
647 0 . − . − .
64 1 . . − .
645 0 . − . − .
22 1 . . − .
644 0 . − . − .
38 1 . . − .
647 0 . − . − .
60 1 . .
320 0 .
663 0 . . − .
51 1 . .
320 0 .
663 0 . . .
61 1 . .
320 0 .
660 1 . . − .
17 1 . .
320 0 .
659 1 . . − .
46 1 . . − . − . − . .
18 2 . . − . − . − . .
48 2 . . − . − . − . − .
25 2 . . − . − . − . − .
88 1 . . − . − . − . .
49 2 . . − . − . − . .
18 1 . .
476 0 .
687 0 . − . .
94 1 . .
476 0 .
685 0 . − . .
77 2 . .
476 0 .
686 0 . − . .
88 1 . .
476 0 .
687 0 . − . .
87 1 . . − . − . − . − .
03 1 . . − . − . − . .
52 2 . . − . − . − . − .
87 1 . .
625 0 . − . − . .
52 2 . .
625 0 . − . − . − .
32 1 . . − . − . . − .
26 2 . . − . − . . .
79 2 . . − . − . . − .
77 1 . . − . − . . − .
75 2 . .
769 0 . − . − . − .
13 2 . .
769 0 . − . − . − .
92 2 . .
769 0 . − . − . .
20 2 . . − .
658 0 . . − .
42 1 . . − .
656 0 . . − .
17 2 . . − .
656 0 . . − .
88 1 . . − .
656 0 . . .
60 1 . . − .
657 0 . . − .
63 1 . .
921 0 . − . . .
37 1 . .
921 0 . − . . .
79 1 . . − .
679 0 . . .
29 2 . . − .
681 0 . . .
50 1 . . − .
681 0 . . − .
54 1 . . − .
678 0 . . .
01 1 . Article number, page 22 of 24obert J. De Rosa et al.: A significant mutual inclination between the planets within the π Mensae system
Table 6. continued. t − .
25 Π cos ψ sin ψ δ Λ σ Λ yr mas mas . − .
681 0 . . − .
56 2 . .
085 0 .
670 0 . . − .
52 1 . .
085 0 .
670 0 . . .
35 2 . .
085 0 .
668 0 . . − .
12 1 . .
085 0 .
668 0 . . .
75 2 . . − .
651 0 . − . .
50 2 . . − .
650 0 . − . .
08 2 . . − .
647 0 . − . − .
54 1 . .
254 0 .
646 0 . . .
74 2 . .
254 0 .
646 0 . . .
69 2 . .
254 0 .
652 0 . . .
53 1 . .
254 0 .
651 0 . . .
18 2 . .
254 0 .
648 0 . . − .
83 1 . .
254 0 .
648 0 . . − .
81 1 . .
714 0 . − . − . .
38 1 . .
714 0 . − . − . .
70 1 . .
828 0 . − . . .
70 1 . .
828 0 . − . . .
61 1 . Article number, page 23 of 24 &A proofs: manuscript no. main
Table 7.
Predicted
Gaia scan timings, orientations and parallax factors for the π Mensae system. t − . MJD
Π sin θ cos θ yr d − .
849 56895 .
50 0 . . − . − .
739 56935 .
80 0 . . − . − .
739 56935 .
87 0 . . − . − .
578 56994 .
51 0 . − . − . − .
477 57031 . − . − . . − .
477 57031 . − . − . . − .
417 57053 .
47 0 . − . . − .
416 57053 .
54 0 . − . . − .
319 57089 . − . . . − .
318 57089 . − . . . − .
151 57150 . − . . − . − .
056 57184 .
89 0 . . . .
032 57217 . − . . − . .
032 57217 . − . . − . .
219 57285 . − . − . − . .
219 57285 . − . − . − . .
295 57313 .
09 0 . . − . .
394 57349 . − . − . . .
454 57371 .
31 0 . − . − . .
454 57371 .
38 0 . − . − . .
555 57408 . − . − . . .
555 57408 . − . − . . .
618 57431 .
01 0 . − . . .
714 57466 . − . . . .
794 57495 .
47 0 . − . . .
884 57528 . − . . − .31269