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  Astrometric exoplanet detection with Gaia Michael Perryman 1 , 2 , Joel Hartman, Gáspár Á. Bakos 3 , 4 Department of Astrophysical Sciences, Peyton Hall, Princeton, NJ 08544 and Lennart Lindegren Lund Observatory, Lund, Box 43, 22100 Sweden  ABSTRACT We provide a revised assessment of the number of exoplanets that should be discovered byGaia astrometry, extending previous studies to a broader range of spectral types, distances, andmagnitudes. Our assessment is based on a large representative sample of host stars from theTRILEGAL Galaxy population synthesis model, recent estimates of the exoplanet frequencydistributions as a function of stellar type, and detailed simulation of the Gaia observations usingthe updated instrument performance and scanning law. We use two approaches to estimatedetectable planetary systems: one based on the S/N of the astrometric signature per field crossing,easily reproducible and allowing comparisons with previous estimates, and a new and more robustmetric based on orbit fitting to the simulated satellite data.With some plausible assumptions on planet occurrences, we find that some 21000 ( ± 6000 )high-mass ( ∼ 1 − 15 M  J ) long-period planets should be discovered out to distances of  ∼ 500pc forthe nominal 5-yr mission (including at least 1000–1500 around M dwarfs out to 100pc), risingto some 70000 ( ± 20000 ) for a 10-yr mission. We indicate some of the expected features of this exoplanet population, amongst them  ∼ 25–50 intermediate-period ( P   ∼  2 − 3 yr) transitingsystems. Subject headings:  astrometry — space vehicles: instruments (Gaia) — planets and satellites: general 1. Introduction The current exoplanet census stands at around1800 1 , with some 600 discovered from radial ve-locity measures, and most of the others from pho-tometric transits. Only two (massive) astromet-ric discoveries have been claimed (Muterspaugh 1 Bohdan Paczyński Visiting Fellow 2 Adjunct Professor, School of Physics, University Col-lege Dublin, Ireland 3 Alfred P. Sloan Fellow 4 Packard Fellow 1 As of 2014 September 1,  lists 1821 confirmedplanets in 1135 systems, NASA’s  lists 1743 (along with  ∼ 4000 Kepler tran-sit candidates), while the more restrictive lists 1516 confirmed (with 1492 good orbits). et al. 2010; Sahlmann et al. 2014), while orbit con-straints for previously-known systems are providedby Hipparcos (e.g. Reffert & Quirrenbach 2011;Sahlmann et al. 2011) and HST–FGS astrometry(e.g. McArthur et al. 2010).The astrometric detectability and characteriza-tion of exoplanets should change quantitativelywith Gaia, which was launched on 2013 Decem-ber 19 and began routine operations in 2014 Au-gust. Previous work has estimated the potentiallydetectable numbers, both from periodic transitsearches in its high-accuracy multi-epoch photom-etry, and independently from the astrometric dis-placement of the host star.Photometric transit detection, demonstrated  a posteriori   in the comparable Hipparcos photom-1   a  r   X   i  v  :   1   4   1   1 .   1   1   7   3  v   1   [  a  s   t  r  o  -  p   h .   E   P   ]   5   N  o  v   2   0   1   4  etry for HD 209458b (Robichon & Arenou 2000)and HD 189733b (Bouchy et al. 2005), has beenconsidered both for Hipparcos (Castellano et al.2000; Laughlin 2000; Jenkins et al. 2002; Koen& Lombard 2002) and Gaia (Robichon 2002; Høg2002). Despite an accuracy of   ∼ 1mmag per tran-sit at  G  14 − 16  (Jordi et al. 2010), the low ca-dence makes the discovery of new transiting plan-ets non-trivial. Dzigan & Zucker (2012), who tookinto account the scanning law, Galactic structuremodels, and detection limits to   16mag, con-cluded that the low cadence and relatively smallnumber of measurements gives a limit on the de-tectable orbital period of   P     10 d, and a result-ing total number of expected discoveries from Gaia photometry   of between one thousand and severalthousand.The expected number of   astrometric   planet de-tections was superficially estimated at the time of the mission acceptance in 2000 at around 30000(Perryman et al. 2001), based on the limitedknowledge of exoplanet occurrences then available,and on the higher astrometric accuracies (by a fac-tor roughly 2) targeted at the time. Improvedstudies have since been undertaken (Lattanzi et al.2000; Quist 2001; Sozzetti et al. 2001). The mostdetailed estimates have been made for subsets of the Gaia census by Casertano et al. (2008) forFGK dwarfs, and by Sozzetti et al. (2014) forM dwarfs.Casertano et al. (2008) derived an estimatednumerical yield for their sample based on starcounts from the Besançon Galaxy model, but con-strained to  V <  13  and  d <  200 pc to provideconstant astrometic precision and hence uniformGaia detectability thresholds for their orbit-fittingexperiments. They adopted an along-scan single-epoch measurement error of   ∼ 11 µ as ( ∼  8 µ as forsuccessive crossings of the two fields of view), tobe compared with the latest estimates of  ∼ 34 µ as,even for the brightest Gaia stars. They con-cluded that Gaia will detect  ∼ 8000 giant planets( M  p  >  1 − 3 M  J ) around FGK stars out to semi-major axes 3–4AU. Their comprehensive double-blind simulations also led to a number of conclu-sions on exoplanet detectability and orbit relia-bility (also for two-planet systems). For exam-ple, they showed that planets with astrometricS/N  >  3  per field crossing and period  P   ≤  5 yrcan be detected reliably and consistently, with avery small number of false positives. At twice thedetection limit, they found uncertainties in orbitalparameters and masses of typically 15–20%, whilefor favorable two-planet systems orbital elementswill be measured to better than 10% accuracy insome 90% of cases, with the mutual inclinationangle  ∆ i  determined with uncertainties   10 ◦ .Restricting their considerations to M dwarfs,Sozzetti et al. (2014) showed that Gaia should de-tect some 100 giant planets across the known sam-ple of M dwarf host stars within 30pc, and some2600 detections and  ∼ 500 accurate orbit determi-nations out to 100pc.Motivated by the start of the Gaia operations,we re-assess the number of exoplanets detectableby Gaia astrometry. Our main objectives are toextend the previous studies to a wider parameterrange (notably spectral type and distance), whiletaking account of recent estimates of exoplanet fre-quencies as a function of host star and planet prop-erties. We use a comprehensive host star Galaxypopulation model, the latest instrument perfor-mance estimates, and detailed simulations of thesatellite observations based on the scanning law.We quantify detection numbers both in terms of a simple S/N threshold per field crossing used inearlier work, as well as a more robust detectionstatistic based on orbit fitting.The paper is organized as follows. In Section 2we summarize the essential concepts and quan-tities relevant to astrometric exoplanet detectionwith Gaia. In Section 3.1 we describe the sam-ple of host stars used to quantify the numbers of planets detectable by Gaia astrometry drawn froma population synthesis Galaxy star count model,and in Section 3.2 we present the assumptions onthe exoplanet frequency estimates which we thenuse to simulate planets around each star. In Sec-tion 4 we derive preliminary detection statisticsbased on a simple consideration of the resultingastrometric signatures and the along-scan astro-metric error appropriate for that stellar magni-tude. In Section 5 we estimate planet discoverynumbers more rigorously by simulating the obser-vations that will be made by Gaia, and quanti-fying exoplanet detectability based on goodness-of-fit improvements from the orbit solutions. InSection 6 we focus on a statistically secure sub-set that is expected to transit, determining thedistribution of transit depths, and quantifying the2  number of transit events that will simultaneouslybe present in the Gaia epoch photometry. In Sec-tion 7 we discuss results for the subsets of FGKstars and M dwarfs, briefly comparing our pre-dicted yields with previous assessments, and weunderline some of the uncertainties on our latestpredicted numbers. 2. Measurement and detection principles2.1. Gaia astrometry Gaia, as for its predecessor Hipparcos, uti-lizes a small number of key measurement prin-ciples (observations above the atmosphere, twowidely-separated viewing directions, and a uni-form ‘revolving scanning’ of the celestial sphere)to create a catalog of star positions, proper mo-tions, and parallaxes of state-of-the-art accuracies(Perryman et al. 2001). Crucially, both missionsprovide  absolute   trigonometric parallaxes, ratherthan the relative parallaxes accessible to narrow-field astrometry from the ground. The observa-tions are reduced to an internally consistent andextremely ‘rigid’ catalog of positions and propermotions, but whose  system   orientation and angu-lar rate of change are essentially arbitrary, sincethe measured arc lengths between objects are in-variant to frame rotation. Placing both positionsand proper motions on an inertial system corre-sponds to determining these 6 degrees of freedom(3 orientation and 3 spin components). For Gaia,they will be derived using the large numbers of ob-served quasars (Claeskens et al. 2006; Perrymanet al. 2014).On-board detection ensures that objects brighterthan  G  ∼  20  at that measurement epoch   will bedetected and observed astrometrically and pho-tometrically, the latter through low-resolutionspectrophotometry at the trailing edge of the as-trometric field (see Jordi et al. 2010, Figs 1–2).The highest photometric accuracy will come fromthe unfiltered  G  band astrometric field photom-etry, which will range (per field crossing) from1mmag or better for  G <  14 mag to  ∼ 0 . 2 mag at G  = 20 mag (Jordi et al. 2010, Figure 19).Final astrometric accuracies (in positions, par-allaxes, and annual proper motions) should beroughly constant at  ∼  10 µ as (micro-arcsec) be-tween  V   ∼  7 − 12 , degrading according to pho-ton statistics to  ∼ 20–25 µ as at  V   = 15 , and to ∼ 300 µ as at  V   = 20  (precise values depend onphotometric passband, star color, and astromet-ric parameter). These final accuracies result fromthe combination of the one-dimensional measure-ments throughout the mission, assembled using aglobal iterative adjustment (Perryman et al. 2001;O’Mullane et al. 2011; Lindegren et al. 2012).A single star at finite distance and with recti-linear space motion can be described by just 5 as-trometric parameters, representing its position ( α , δ  ), proper motion ( µ α ,  µ δ ), and parallax ( ̟ ). Anyorbiting companions, including those of planetarymass, will perturb the stellar motion and resultin deviations of the individual (‘intermediate’) as-trometric data from a simple 5-parameter model.Detectability will depend on the amplitude of thedeviations (Section 2.2), and the number and cov-erage of the individual measurements.The number of individual field of view cross-ings,  N  fov , along with the final mission accuraciesfrom which they are constructed, are primarily de-pendent on ecliptic latitude,  β  . This results fromthe satellite ‘scanning law’, which is optimized tomaintain a constant (solar) thermal payload il-lumination, while maximizing separability of theastrometric parameters.  N  fov  is independent of magnitude, and ranges between about  N  fov  ∼  60 at  β <  10 ◦ to about  N  fov  ∼  80  at  β >  80 ◦ , witha maximum of about  N  fov  ∼  150  at intermedi-ate ecliptic latitudes,  β   ∼ 45 ◦ , where the scanningdensity is highest (Table 1). The high values of  N  fov  around  β   ∼± 45 ◦ do not necessarily improveplanet detection substantively, adding little to thenumber of distinct epochs and projection geome-tries.Our simulations of detection and orbit recon-struction require estimates of   σ fov , the along-scanaccuracy per field of view crossing as a function of  G  magnitude (Table 2). In terms of the centroid-ing accuracy for each of the 9 astrometric CCDs, σ η , we adopt σ fov  = ( σ 2 η 9 + σ 2att  + σ 2cal ) 0 . 5 ,  (1)where  σ att  is the contribution from (both ran-dom and systematic modelling) attitude errors,and  σ cal  is that from calibration errors. Both areassumed constant over the field crossing; we adopt σ att  = 20 µ as (Risquez et al. 2013), and similarlyfor  σ cal  (Lindegren et al. 2012). Evidently, both3  are provisional pending results from the global it-erative solution.For bright stars,  G    12 , signal saturation isavoided by CCD ‘gating’, activated according tothe star’s measured brightness, allowing a reducednumber of active TDI lines, and designed to resultin a more-or-less constant measurement precisionover the range  G ∼ 3 − 12 mag.In terms of the inverse relative number of pho-tons in the image z  = 10 0 . 4(max[ G, 12] − 15) ,  (2)normalized to  z  = 1  at  G  = 15 , we use σ η  = (53000 z  + 310 z 2 ) 0 . 5 ,  (3)which is a fit to the values 92, 230, 590, 960, 1600,2900 µ as quoted for  G  = 13 ,  15 ,  17 ,  18 ,  19 ,  20  byLindegren et al. (2012, Table 1).We complete these forms with an expression forthe sky-averaged parallax accuracy, which can beapproximated by σ ̟  = 1 . 2 × 2 . 15  σ fov / √  68 . 9 = 0 . 311 σ fov  ,  (4)where 2.15 is the geometric factor linking the (sky-averaged) parallax accuracy with the error perfield crossing, 68.9 is the (sky-averaged) numberof field crossings per star over the nominal 5-yrmission including dead time (Table 1), and 1.2 isa margin (Lindegren et al. 2012).Values of   z ,  σ η ,  σ fov , and  σ ̟ , as a function of  G  magnitude, are given in Table 2. 2.2. The astrometric signature As a planet detection and characterization tech-nique, astrometry aims at measuring the influenceof an orbiting planet in addition to the two otherclassical astrometric effects: the linear path of the system’s barycenter projected on the sky (thestar’s proper motion), and the reflex motion (thestar’s parallax) resulting from the Earth’s orbitalmotion around the Sun. Both star and planet or-bit the star–planet barycenter and, after account-ing for the parallax and proper motion terms, theorbit of the primary therefore appears projectedon the plane of the sky as an ellipse with semi-major axis given by a ⋆  =  M  p M  ⋆  a p  ,  (5)where  M  p  and  M  ⋆  are the planet and star massrespectively, and  a p  is the semi-major axis of theplanet orbit with respect to the barycenter.The observable for astrometric planet detectionis the corresponding quantity in angular measure,generally referred to as the  astrometric signature  ,given by α  =  M  p M  ⋆   a p 1 AU   d 1 pc  − 1 arcsec  ,  (6)where  d  is the distance, and  M  p  and  M  ⋆  arein common units. The definition may also beadopted for  e   = 0 , but with detectability depen-dent on orbital phase (Section 4). The effect islinearly proportional to  a p  and, importantly, ap-plies equally to hot or rapidly-rotating stars. Butwhile the technique is most sensitive to massiveplanets at large  a p , measurement timescales mustbe proportionally long (of order of the orbital pe-riod).The size of the effect calculated for all con-firmed exoplanets to date (2014 September 1) isshown in Figure 1a as a function of orbit period.Vertical lines illustrate the period limits betweenwhich Gaia will be most efficient in its discoveryspace ( 0 . 2    P     6 yr, see Section 5). On theassumption that an astrometric signature of   ∼ 1–3 times the parallax standard error could be de-tected (see Section 4), a sizeable fraction of knownsystems will have their exoplanet-induced photo-centric motion determined at some level by Gaia.Figure 1b restricts the plot to the known  tran-siting   planets, and demonstrates that Gaia as-trometry will provide little orbital information forthe majority of known transiting planets. No tran-siting planets have  α >  30 µ as, and the great ma- jority have  α  ≪  1 µ as. Indeed, known exoplanetswith large  α  are almost exclusively those discov-ered by radial velocity measurements. Even thenearest hot Jupiters will be undetected astromet-rically, and the same applies to the  P    6 d plan-ets which might be discovered in the Gaia pho-tometric data (Dzigan & Zucker 2012). Gaia as-trometry may nonetheless clarify the existence of massive outer companions of hot Jupiters, whichhave been invoked to explain their inward migra-tion (e.g. Bakos et al. 2009; Neveu et al. 2013;Knutson et al. 2014).Various astrophysical noise sources will con-tribute to the accuracy of astrometric measure-4  ments in principle, but appear to lie below rel-evant limits in practice, and have been ignoredhere. These include the effects of variable stellarsurface structure (star spots, plages, granulation,and non-radial oscillations) on the observed photo-centre (e.g. Reffert et al. 2005; Ludwig 2006; Eriks-son & Lindegren 2007; Lanza et al. 2008; Makarovet al. 2009), relativistic modeling at  ∼  1 µ as (e.g.Anglada-Escudé et al. 2007), and possible effectsat optical wavelengths of interstellar and inter-planetary scintillation and stochastic gravitationalwave noise (Perryman et al. 2001). 2.3. Orbit constraints from astrometricdata A 3d Keplerian orbit is described by 7 pa-rameters, for example the classical elements a,e,P,t p ,i, Ω ,ω . The semi-major axis  a  and ec-centricity  e  specify the size and shape of the orbit.The period  P   is related to  a  and the componentmasses through Kepler’s third law, while  t p  spec-ifies the position of the object along its orbit atsome reference time, generally with respect to aspecified pericenter passage. The three angles ( i ,the orbit inclination to the plane tangent to thecelestial sphere;  Ω , the longitude of the ascend-ing node; and  ω , the argument of pericenter) givethe projection of the true orbit into the observed(apparent) orbit; they depend solely on the orien-tation of the observer with respect to the orbit.Both radial velocity and astrometry measurethe host star’s barycentric motion rather than thatof the planet directly, and somewhat different in-formation is provided by each measurement tech-nique. 2 2 We recall that the sizes of the three related orbits – thestellar orbit around the barycenter, the planet orbit aroundthe barycenter, and the relative orbit of the planet aroundthe star – are in proportion  a ⋆  :  a p  :  a rel  =  M  p  : M  ⋆  : ( M  ⋆  +  M  p ) , with  a rel  =  a ⋆  +  a p . Furthermore, e rel  =  e ⋆  =  e p ,  P  rel  =  P  ⋆  =  P  p , the three orbits arecoplanar, and the orientations of the two barycentric or-bits ( ω ) differ by  180 ◦ . From the line-of-sight (radial) ve-locity variations alone, not all 7 Keplerian elements areaccessible. Specifically: (i)  Ω  is undetermined; (ii) onlythe combination  a ⋆  sin i  is determined, with neither  a ⋆ nor  sin i  individually; (iii) measurements provide a valuefor the ‘mass function’, which for  M  p  ≪  M  ⋆  reduces to M ≃  ( M  3p  sin 3 i ) /M  2 ⋆ . It follows that if   M  ⋆  can be esti-mated from its spectral type (or otherwise), then  M  p  sin i can be determined, although the planet mass remains un-certain by the unknown factor  sin i . From astrometry, all 7 Keplerian elements areaccessible in some form, irrespective of the orbitinclination to the line-of-sight. Specifically, theorbit solution (including the planet location alongthe orbit as a function of time) gives  i  and  α . FromEquation 6,  a ⋆  can be determined from  α  if   d  isknown, with  a ⋆ + a p  (and hence  a p ) obtained fromKepler’s third law assuming that  ( M  ⋆ + M  p ) ≃ M  ⋆ can be estimated from the star’s spectral type orfrom evolutionary models. Then  M  p  is determinedfrom Equation 5. If the planet is invisible (the casefor all but a few more massive long-period plan-ets which have been imaged) the orbital motion of the star around the system barycenter is correctlydetermined by astrometry only if the star positionis measured with respect to an ‘absolute’ refer-ence frame, which is the case for Gaia (Perrymanet al. 2014). Astrometric measurements alone are,however, unable to identify which of the nodes isascending, i.e. where the planet moves away fromthe observer through the reference plane, an am-biguity resolved by radial velocity observations.For multiple exoplanet systems, and if the or-bital contributions from each can be separated, as-trometry can also establish the relative inclinationbetween pairs of orbits (e.g. van de Kamp 1981,Equation 16.5; Casertano et al. 2008; McArthuret al. 2010).Four orbit elements ( a ⋆ ,e,ω,t  p ) are in commonbetween astrometric and spectroscopic orbit so-lutions. Combined observations therefore furtherconstrain and improve the 3d orbit, as well asthe individual component masses (e.g. Wright &Howard 2009). 3. Host star and planet distributions3.1. Star counts As an input to a new set of simulations, weused the population synthesis Galaxy star countmodel TRILEGAL (TRIdimensional modeL of thE GALaxy, Girardi et al. 2005, 2012). This isbased on a theoretical stellar luminosity function φ ( M,rrr,λ )  [i.e., as a function of absolute mag-nitude  M  , Galactic position  rrr  = ( ℓ,b,r ) , andphotometric passband  λ ], derived from a set of evolutionary tracks, together with suitable distri-butions of stellar masses, ages, and metallicities.TRILEGAL (version 1.6) includes five distinctGalaxy components: the thin and thick disks, the5
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