MESS (Multi-purpose Exoplanet Simulation System): A Monte Carlo tool for the statistical analysis and prediction of exoplanets search results
M. Bonavita, G. Chauvin, S. Desidera, R.Gratton, M. Janson, J.L. Beuzit, M. Kasper, C. Mordasini
aa r X i v : . [ a s t r o - ph . S R ] O c t Astronomy&Astrophysicsmanuscript no. ms c (cid:13)
ESO 2018July 12, 2018
MESS (Multi-purpose Exoplanet Simulation System) ⋆ A Monte Carlo tool for the statistical analysisand prediction of exoplanets search results. ⋆⋆ M. Bonavita , , G. Chauvin , , S. Desidera , R.Gratton , M. Janson , J. L. Beuzit , M. Kasper , C. Mordasini INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy Department of Astronomy and Astrophysics - University of Toronto, 50 St. George Street M5S 3H4 Toronto ON Canada UJF-Grenoble 1 / CNRS-INSU, Institut de Planetologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble, F-38041,France Max Planck Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany European Southern Observatory (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching, GermanyReceived / Accepted
Abstract
Context.
The high number of planet discoveries made in the last years provides a good sample for statistical analysis, leading to someclues on the distributions of planet parameters, like masses and periods, at least in close proximity to the host star. We likely need towait for the extremely large telescopes (ELTs) to have an overall view of the extrasolar planetary systems. Those facilities will finallyensure an overlap of the discovery space of direct and indirect techniques, which is desirable to completely understand the nature ofthe discovered objects, obtaining both orbital parameters and physical characterization.
Aims.
In this context it would be useful to have a tool that can be used for the interpretation of the present results obtained with variousobserving techniques, and also to predict what the outcomes would be of the future instruments.
Methods.
For this reason we built MESS: a Monte Carlo simulation code which uses either the results of the statistical analysis of theproperties of discovered planets, or the results of the planet formation theories, to build synthetic planet populations fully describedin terms of frequency, orbital elements and physical properties. They can then be used to either test the consistency of their propertieswith the observed population of planets given di ff erent detection techniques (radial velocity, imaging and astrometry) or to actuallypredict the expected number of planets for future surveys, as well as to optimize the future multi-techniques observations for theircharacterization down to telluric masses. Results.
In addition to the code description, we present here some of its applications to actually probe the physical and orbitalproperties of a putative companion within the circumstellar disk of a given star and to test constrain the orbital distribution propertiesof a potential planet population around the members of the TW Hydrae association. Finally, using in its predictive mode, the synergyof future space and ground-based telescopes instrumentation has been investigated to identify the mass-period parameter space thatwill be probed in future surveys for giant and rocky planets
Key words.
Stars: brown dwarfs, planetary systems - Methods: data analysis, statistical
1. Introduction
Many statistical studies have been done using information com-ing from more than a decade of extensive searches for exoplan-ets, trying to answer questions either related to the properties ofthose objects, such as the mass, orbital period and eccentricity(Lineweaver & Grether 2003; Cumming et al. 2008), or aboutthe relevance of the host star characteristics (mass, metallicityand binarity) on the final frequency and distribution of plan-etary systems (see Fischer & Valenti 2005; Santos et al. 2004;Johnson et al. 2007). Since the most successful techniques (ra-dial velocity and transit) have focused on the inner ( ≤ AU )environment of main sequence solar-type stars, most of the avail-able information on the frequency of planets concern this classof stars.Recent discoveries of young distant planetary mass objectswith direct imaging (see e. g. Marois et al. 2008; Kalas et al.2008; Lagrange et al. 2008) are giving us a first hint on the po- ⋆ ⋆⋆ send o ff print requests to [email protected] tential of the direct detections in the exploration of the outer re-gion of the planetary systems, also raising many questions abouthow such objects could form (see Absil & Mawet 2009). Thisdefines the niche of the next generation high contrast imaging in-struments like the Gemini Planet Imager (GPI: Macintosh et al.2007) and VLT / SPHERE (Spectro-Polarimetric High-contrastExoplanet REsearch: Beuzit et al. 2008). These instruments willlikely allow us to extend such a systematic characterization tolarger scales ( ≥ AU ). Due to practical limitations (innerworking angle, best contrast achievable), these instrument willfocus on warm giant planets on orbits far away from their stars,paving the path for the ELTs facilities. A wide range of planetarymasses and separations, down to the rocky planets (and, in veryfavourable cases reaching the habitable zone), will be exploredwith 30-40 meter-class telescopes, finally allowing an overlapbetween the discovery spaces of direct and indirect techniques.In this context it is useful and crucial to predict the perfor-mances of the forthcoming instruments, not only in terms ofnumber of expected detections, but also trying to figure out what Bonavita et al.: MESS (Multi-purpose Exoplanet Simulation System) will be the explored parameter space and even the possible syn-ergies between di ff erent discovery techniques.Here we present our Monte Carlo simulation code MESS,whose aim is to provide a flexible and reliable tool for the statis-tical analysis and prediction of the results of planet searches.It produces synthetic planet populations, deriving all thephysical parameters of these planets together with the observ-ables that can be compared with the predicted capabilities of ex-isting or planned instruments. Such comparisons allow to derivesubsets of fully characterized detectable planets , as well as asnapshot of what the evolution of the sample of detected planetswould be in the next years.A detailed description of the code, and of all the assumptionswhich constitute its basis, is given in Sect. 2, while in Sect. 3we present the di ff erent operation modes of the code and theirapplications. Although the MESS has been built, and it has beenso far applied, only to analyze and / or predict the results of directimaging surveys, an extension of the code to di ff erent techniquesis planned. The first attempt in this direction are presented inSec. 3.3. Conclusions and suggestions for further work will befinally drawn in Sect. 4.
2. MESS (Multi-purpose Exoplanet SimulationSystem) / star contrast, degree of polarization, etc.3. given the detection capability relation of an instrument, ei-ther already available or planned, it selects a sub-sample of fully characterized detectable planets , which characteristicscan then be analyzed.The code then assumes a given star population, a planet popu-lation with associated physical and orbital properties based ona theoretical or semi-empirical approach, the corresponding ob-servables for di ff erent observing techniques, finally generate asynthetic population of planets to be compared with the instru-mental detection performances. Each step is described hereafter. The first input of the MESS is a sample of N Star stars, which havebeen targeted for planet searches or which are part of a samplefor future observations. Various stellar parameters are assumedto be known, such as the apparent magnitude, the distance, theluminosity, the spectral type, the mass, the age, the metallicity,etc.Fig. 1 shows the characteristics of a sample of 600 nearby( d <
20 pc) stars selected from the Hipparcos catalogue(Perryman & ESA 1997) and used to build the synthetic pop-ulation showed in Fig. 3.
Figure 1.
Principal characteristics of the sample of nearby stars usedto built the example synthetic population.
Upper Left:
ApparentMagnitude in the J Band vs distance in pc.
Upper Right:
Stellar Mass( M ⊙ ) vs distance in pc. Lower Left:
Histogram of stellar ages (Myrs).
Lower Right:
Histogram of stellar masses ( M ⊙ ). MESS also gives the possibility of taking into account the pres-ence of one (or more) additional stellar companions, in the anal-ysis. If a star in the sample is flagged as binary, the code uses theinformation about the binary orbit (if available) to compute thecritical semi-major axis for the dynamical stability of the system.This set the limiting value that the semi-major axis of a planetcan attain and still maintain its orbital stability, as a function ofthe mass-ratio and orbital elements of the binary, as shown byHolman & Wiegert (1999).Both the case of circumstellar (or Satellite S-type) and cir-cumbinary (or Planet P-type) orbit are considered, and the criti-cal semi-major axis is computed using Eq. 1 and 2 respectively,from Holman & Wiegert (1999). onavita et al.: MESS (Multi-purpose Exoplanet Simulation System) 3 a c / a b = . − . µ − . e b + . µ e b + . µ − . µ e b (1) a c / a b = . + . µ + . e b − . µ e b + − . µ − . e b + . e b µ (2)In both the equations, a c is critical semi-major axis , µ = M / ( M + M ), a b and e b are the semi-major axis and eccen-tricity of the binary, and M and M are the masses of the pri-mary and secondary stars, respectively. If not available fromliterature, the eccentricity is assumed to be e b = .
36, re-ported as mean value for the eccentricity of a binary sys-tem, by Duquennoy & Mayor (1991). In case the value ofthe semi-major axis is not available, then the code estimatesit as a b = . ρ ( arcsec ) d ( pc ) (see Fischer et al. 2002;Duquennoy & Mayor 1991) .Note that in the first case (S-type orbit), a c set the maxi-mum value that the semi-major axis of a planet can assume,before compromising the stability, while it represents the min-imum value of the semi-major axis of a stable planet, in the caseof a P-type orbit. The core of the code is the generation of the synthetic planets,that are fully characterized, both in terms of orbital parame-ter, and physical characteristics. Depending on the goal of thestudy one can choose between a
Semi-empirical approach ora
Theoretical Approach . These di ff erent approaches makes thecode suitable to constraint the planet properties under di ff erentassumptions, but also to test model predictions.If the Theoretical Approach is chosen, masses and period val-ues selected from a synthetic population provided by the out-put of the planetary formation models (see e.g. Mordasini et al.2009) are given as input. In this case all the orbital characteris-tics are also provided, together with the physical properties ofeach planet, so no random generation is needed, and the codeonly evaluates the observable and compares them to the provideddetectability relations. Di ff erent populations of planets obtainedassuming di ff erent stellar masses and metallicity values can beselected according with the characteristics of the real star in thesample, to take into account the e ff ects of the stellar characteris-tics on the planet formation processes. .The Semi-empirical Approach uses the power law distribu-tions in Eq. 3 and 4 for the mass and semi-major axis of theplanets as retrieved from the statistical analysis of the propertiesof the planets discovered so far to generate a seed population of N seed values of masses and periods (see Sec. 2.2.1). dNd (cid:16) M p (cid:17) ∝ (cid:16) M p (cid:17) α (3) dNdP ∝ P β (4) Note that Duquennoy & Mayor (1991) refers to solar-type star mul-tiplicity The results discussed in this paper has been obtained using mainlythe Semi-empirical Approach. An extensive use of the TheoreticalApproach, using as input the newest outcomes of the Bern formationmodels (Mordasini et al. 2011), will be the subject of a forthcomingpaper.
The user can also set a pre-determined grid of masses-periods and feed it to the code, without any assumption on thedistributions. This would be the case if, for example, bound-aries on the mass / semi-major axis space where planets can formare to be set using the outcomes of a formation model(see e.gMordasini et al. 2010), excluding from the sample the planetsnot compatible with the theory.If the Semi-empirical approach is used, mass, orbital param-eters, as well as temperature and radius of the planets, are ob-tained based on the assumption described in the next sections If the semi-empirical approach is chosen, the power law distribu-tions are fed to the Monte-Carlo core of the code, that randomlygenerates a fixed number of mass-period pairs. Both the plan-etary mass and period ranges can be given as inputs, togetherwith the power-law exponents. In a typical setup, the power-lawexponents are assumed to be α = − .
31 and β = − .
74 respec-tively, from Cumming et al. (2008). The planetary masses spanthe range between 0.6 M Earth and 15 M Jup , and the periods ( P )are chosen between 2.5 days and 350 years (corresponding to 50AU for 1 M ⊙ star).A scaling of the planetary mass, and even of the period, withthe stellar mass can be also introduced, according to recent re-sults (e.g. Lovis & Mayor 2007). In addition, a dependence ofthe planet frequency on the stellar metallicity may also be con-sidered (see Fischer & Valenti 2005). For each mass-period pair in the seed generation, the code eval-uates the semi-major axis computed using Kepler’s third law,using the mass of each star in the input sample. Then, it gen-erates N gen values of all the orbital parameters: eccentricity ( e ),inclination ( i ), longitude of periastron ( ω ), longitude of ascend-ing node ( Ω ), and time of periastron passage ( T ). By default, allthese parameters are randomly generated following an uniformdistribution . The eccentricity distribution is cut at e = . ff ect the results of Direct Imagingsurveys. A full discussion of the impact of the eccentricity dis-tribution on the simulations results is held in Sec. 3.4.The date of observation is also required. If not available fromthe real data, an epoch of observation, t obs , is generated over atime-span chosen according with the considered instrument.The code also o ff ers the possibility to fix each orbital pa-rameter to known or predicted values, for all the planets in thepopulation.The coordinates, x and y , of the projected orbit on the planeperpendicular to the line of sight, are finally computed using theephemeris formulae of Heintz (1978), reported in Eq. 5 to 7. x = AX + FY (5) y = BX + GYX = cos E − e (6) Y = √ − e sin E ρ = q x + y (7) Note that in the case of the inclination, cos i and not i itself is uni-formly generated by the code Bonavita et al.: MESS (Multi-purpose Exoplanet Simulation System) where X and Y are the coordinates of the orbit (Eq. 6), ρ isthe projected separation, and A , B , F , G are the Thiele-Innes ele-ments, which can be obtained from the classical ones (the semi-major axis a , ω , Ω , and i ) using Eq. 8: A = a (cos ω cos Ω − sin ω sin Ω cos i ) B = a (cos ω sin Ω + sin ω cos Ω cos i ) (8) F = a ( − sin ω cos Ω − cos ω sin Ω cos i ) G = a ( − sin ω sin Ω + cos ω cos Ω cos i ) . In these equations, E is the eccentric anomaly (obtained from themean anomaly M ( Eq. 9) using Eq. 10) and ν the true anomaly(Eq. 11): M = t obs − T p ! π (9) E = M + e sin M + e MM = E − e sin E E = E + ( M − M )) / (1 − e cos E ) (10)tan ν/ = p (1 + e ) / (1 − e ) tan E / ρ (in arcsec), can be obtained ei-ther using Eq. 7 or Eq. 12 (which gives also an estimate of theradius vector: r ), then dividing for the star distance. ρ = r cos ( ν + ω ) sec ( θ − Ω ) (12) r = a (cid:16) − e (cid:17) / (1 + e cos ν ) Since we aim at consider both the thermal and reflected flux ofthe planets, we need two di ff erent estimates of the temperature.The first one is the internal temperature, T int , coming from theevolutionary models (see e.g. Bara ff e et al. 2003). The secondone is the equilibrium temperature, T eq , obtained through Eq. 13(from Sudarsky et al. 2003) T eq = " (1 − A B ) L ∗ πσ a , (13)where L ∗ is the star luminosity. The Bond albedo A B is assumedto be 0.35 in the J band (Jupiter value, see Hanel et al. 1981)and it is randomly generated between 0.3 and 0.52 in the visible(the latter being the Jupiter albedo in V band,see Sudarsky et al.2003)Our final assumed value for the e ff ective temperature of theplanet T e ff is given by: T ff = T + T . (14) MESS uses the approach developed by Fortney et al. (2007) toevaluate the planetary radius. Practically the radius is assumedto depend on the planet mass, with the following recipes:1. For Jupiter-like planets ( M ≥ M Earth ), an interpo-lation is performed within the published values given byFortney et al. (2007). Values of age and distance of eachstar are entered, yielding a value for R Gas . A core mass of10 M Earth is assumed.
Figure 2.
Summary of the planetary Mass - Radius relations adoptedfor the di ff erent mass ranges. All the model computation are made as-suming a host star of 1 M ⊙ , and the semi-major axis value is fixed to5 AU. Filled symbols corresponds to known transiting planets; opensymbols are for Solar System planets.
2. Equations 15 and 16 from Fortney et al. (2007) are used forthe smallest planets ( M ≤ M Earth ). These are either: R = (0 . im f + . M ) + (0 . im f + . M + (0 . im f + . R = (0 . rm f + . M ) + (0 . rm f + . M + (0 . rm f + . / rock and rock / iron planets, respectively. In theseequations, R is in R Earth and M is in M Earth , while im f isthe ice mass fraction (1.0 for pure ice and 0.0 for pure rock)and rm f is the rock mass fraction (1.0 for pure rock and 0.0for pure iron). In the typical MESS setup, the ice / rocky orrocky / iron fraction is set to 0.3 (50% of chance for a planetbeing mainly icy or rocky).3. Finally, predictions are uncertain for the Neptune-like plan-ets, where the transition between the two relations describedabove should occur. The most sensible approach seems to beto fit the mass-radius relation of the Solar System in the samemass-range (10 − M Earth ). This procedure provides a goodagreement with the radii of Uranus and Neptune and of thefew transiting Neptunes confirmed so far (as listed by TheExtrasolar Planet Encyclopaedia see Fig. 2).The resulting mass-radius relations are showed in Fig. 2,with over-plotted the data corresponding to the planets discov-ered with the transit technique and the planets from our SolarSystem, for comparison. Having in hands the full set of orbital and physical parametersof the planets, the code then provides an estimate of observable quantities such as the luminosity contrast or the degree of po-larization, needed for direct observations, but also quantifies theindirect e ff ects of the presence of the planet, providing a measureof the semi-amplitude of radial velocity (RV) and the astrometricsignal. MESS gives an estimate of both the intrinsic and reflected flux,in the selected band, for each planet. Throughout the paper wewill refer to the planets which luminosity is dominated by theintrinsic contribution as self-luminous or warm planets, as op-posed to the cold planets for which the reflected light providesmost of the contribution to the planet / star contrast.The intrinsic emission is estimated using the prediction ofevolutionary models at the age of the star (assumed to be alsothe age of the system). To this purpose two classes of modelscan be considered, based on di ff erent assumptions on the initialconditions: Hot Start models (Chabrier et al. 2000; Bara ff e et al.2003; Saumon & Marley 2008), which consider an initial spher-ical contracting state; and Core Accretion models (Marley et al.2007; Fortney et al. 2008), which couple planetary thermal evo-lution to the predicted core mass and thermal structure of a core-accretion planet formation model.In the following, we only consider the results obtained usingthe hot start models for the nearby sample. However, the problemof the initial condition and the uncertainties on the stellar agesare among the main limitations, in case of young stellar sam-ples, not only for our code, but also for any kind of study thatuses the same kind of approach (see e.g. Chauvin et al. 2010;Bonavita et al. 2010,for a detailed discussion). These limitationsalso apply to the theoretical approach, if the evolutionary mod-els are used to evaluate the planet intrinsic luminosity and radiiproduced by the models, as in Mordasini et al. (2010, 2011).For the evaluation of the reflected light, we scaled the Jupitervalue, according with the planet radius (expressed in Jupiterradii), semi-major axis, albedo and illuminated fraction of theplanet. This last contribution is computed through a phase de-pendent term, Φ ( β ), which is given by Eq. 17 (see Brown 2004),where β is the phase angle (angle at companion between star andthe observer) and z = r sin ( ν + ω ) is the radial coordinate of theradius vector. Φ ( β ) = (cid:2) sin β + ( π + β ) cos β (cid:3) /π (17)The Jupiter / Sun contrast is obtained using Eq. 18 whichgives an estimate of the fraction of stellar light captured by aplanet, depending on the values of the planet radius, semi-majoraxis and geometrical albedo, being Φ ( β ) = L Jup / L ∗ ) Ref = A Jup R a = . × − (18)Where A Jup = .
35 is the value of the Jupiter albedo in the theJ-Band, (see Hanel et al. 1981).Then we end with a final value of the contrast in reflectedlight given by Eq.19.( L p / L ∗ ) Ref = (cid:16) L Jup / L ∗ (cid:17) Ref Φ ( β ) ( R p / R Jup ) ( a / a Jup ) (19)As a consequence of Eq. 19, the results of MESS will be sensi-tive to the choice of A λ , especially for the cold planets, in whichthe contribution of the reflected light is dominant. Following the outcomes of Jupiter observations and theoretical models (Seee.g. Burrows 2004), we decided to uniformly generate the val-ues of the albedo between 0.2 and 0.7. The code anyways o ff ersthe option to fix the value of the albedo to a chosen value, for allthe planets in the generation. A test of the impact of the choiceof the albedo value on the redults of the simulations is presentedin Sec. 3.4. The indirect e ff ects of the presence of the planet, such as thesemi-amplitude of radial velocity (RV) variations and the astro-metric signal can be inferred, knowing all the orbital character-istics for each planet. The degree of polarization Π is assumed to be of the form (seee.g. Stenflo 2005): Π = Π max × (1 − cos β ) / (1 + cos β ) (20)where Π max is the maximum polarization value (which is as-sumed to be randomly generated between 0.1 and 0.3), and β is the same as in Eq. 17. Then the contrast due to the polarizedlight of the planets is Π times the contribution in reflected lightevaluated with Eq. 19. Depending on the purpose of the analysis, the code can generatethe planet population in two di ff erent ways:a) Full population : the value of N seed sets the spacing of themass-period grid, and for each point on it N gen planets aregenerated, ending with N seed × N gen planets per each star. Thepopulation for each star is saved in an independent file. Thisapproach is useful for the statistical analysis of existing data,since in this case MESS provides the fraction of detectableplanets per star, which can be used to derive the global prob-ability of finding a planet over the whole target list. This canbe then compared with the real results.b) Reduced population : only one orbit is generated for eachpoint in the mass-period grid. N gen in this case sets the num-ber of planet in a planetary system associated with each star .The final population is then composed by N star × N seed plan-ets, and all the planets are saved together in one file. Then thepredicted detection performances of a given instrument canbe used, to derive the population of objects that are expectedto be detected around each star, if the whole input sample isobserved.As an example, we generated a reduced population (assum-ing 5 planets per star) of planets around the stars of the nearbysample described in Sec. 2.1. We choose the semi-empirical ap-proach, and used the typical setup we discussed in Sec. 2.2.1 .Fig. 3 shows the position of the planets in the mass vs semi-major axis plane. note that no consideration on the planet stability is made, and to thepurpose of the analysis each planet is considered separately Note that the whole calculation of the physical characteristics andobservables described in Sec. 2.2.3 to 2.3.2 can be skipped (with con-siderable gain in computing speed), the code providing in this case onlythe orbital elements Bonavita et al.: MESS (Multi-purpose Exoplanet Simulation System)
Figure 3.
Mass semi-major axis distribution of the synthetic planets inthe populations generated by MESS using the semi-empirical approach.The di ff erent classes of planets (see text) are plotted using di ff erentcolours: red / orange for the warm / cold Jupiters, green for the Neptunelike planets, blue for the rocky planets. Figure 4.
Distribution of radial velocity vs. period of the syntheticplanets for the population showed in Fig. 3.
The planets are separated into the three classes, using di ff er-ent colours: – Giant (or Jupiter-like) planets ( M planet > M Earth ). Adistinction between
Cold Jupiters (orange dots) and
WarmJupiters (red dots), as defined in Sec. 2.3.1, is also made. – Neptune-like planets (10 M Earth ≤ M planet ≤ M Earth : greendots) – Rocky planets ( M planet < M Earth : blue dots)The distribution of the observable quantities for the planetshowed in Fig. 3 are summarized in Fig. 4, 5 and 6.
The last step is the comparison of the observables of the gen-erated synthetic planets with the detection limits of di ff erentobserving techniques, with the possibility to actually combinethem. It is important, especially in case of comparative studies, Figure 5.
Distribution of the astrometric signal vs. period of the syn-thetic planets for the population showed in Fig. 3.
Figure 6.
Planet / star contrast vs. projected separation of the same plan-ets showed in Fig. 3. and since the MESS does not produce the detection limits, tomake sure that the detection performances that are fed as inputto the code have been estimated by correctly taking into accounteach instrumental biases, specific to each technique, and the stel-lar characteristics. In the context of the MESS applications, thecode has been extensively used considering two possible inputsfor the detection performances: – The
1D mode , which selects the detectable planets using athreshold or a curve giving the lower detection limits (RV, as-trometric precisions or contrast performances) as a functionof the period, the semi-major axis, the angular separationsetc., defined by the instrumental capabilities – The
2D mode , which is especially built for the analysis of theperformances of the Deep Imaging instruments. This modetakes advantage from the knowledge of all the orbital ele-ments of the planets, to place them on a two dimensionaldetection map. This mode allows using all the spatial infor-mation stored in the images. Using the whole 2D map notonly allows to take into account possible peculiar character-istics of the circumstellar environment, such as the presence onavita et al.: MESS (Multi-purpose Exoplanet Simulation System) 7 of disks, but also prevent to under / overestimate the contrastcurve depending on the method chosen for the extraction it-self (see Bonavita et al. 2010).
3. Applications
Once the synthetic population of planets has been created, thenext step is to compare the characteristics of the generated plan-ets with the detection limits appropriate for the instrument underconsideration.MESS o ff ers three di ff erent operation modes (OM), depend-ing on which kind of analysis is needed.1. The Hybrid Mode (MESS HM) which is the most flexibleone, and an be used to probe the physical and orbital proper-ties of a putative companion around one given system basedon the combination of di ff erent techniques, a priori informa-tion on the possible orbit given the presence of other planetsor circumstellar disk.2. The Statistical Analysis Mode (MESS SAM) , which isbuilt for the analysis of real data and uses the full popula-tion defined in Sect. 2.4. It enables to test di ff erent set ofplanet populations or constrain the maximum occurrence ofplanets for a given population that would be consistent withthe results of detection and / or null-detection of a completesurvey of a large target sample.3. The Predictive Mode (MESS PM) , which starts from the reduced population (see Sect. 2.4), and given the predictedperformances of a planned instruments, can be used to selectthe most suitable targets given the science goals of the instru-ment itself, to test the results of di ff erent observing strategiesand finally to foresee possible synergies with other instru-ments. The first and more versatile MESS mode is the so-called Hybridmode. This mode can be used for the study of particularly inter-esting targets, or to test specific hypothesis. It allows for exampleto take into account all available informations about the orbit of aplanet already discovered around the target, in order to put con-straints on the planet generation. A preliminary version of thismode has been used to put constraint on the presence of a plan-etary companion embedded in the disk surrounding the T-Tauristar LkCa15 (see Bonavita et al. 2010).We present here an analogous analysis made for TWA 11.This star has been found to be surrounded by a debris disk bySchneider et al. (2009). Using STIS, Schneider et al. (2009) pro-vided a full characterization of the disk geometry, and suggesteda possible unseen companion responsible for some of the ob-served properties. We then decided to use MESS HM to verifywhich kind of constraints can be put using the VLT-NACO ob-servations of this star.A pixel-to-pixel 2D noise map was estimated from the re-duced NACO images, using a sliding box of 5 × σ threshold to build the finaldetection limit maps to be used for the statistical analysis. Thesemaps were also converted in terms of minimum mass map us-ing the evolutionary model predictions at the age of the system.Fig. 7 shows an example of the resulting sensitivity map . Note that the decreasing values of the non-detection probability atseparations lower than 30-50 AU are due to systematic errors. In factthe detection limit drops to unrealistic low values really close to the star
Figure 7.
2D map giving the values of the minimum mass of de-tectable companions (6 σ ) as a function of projected separation,around TWA 11We considered only circular orbits coplanar with the disk,with an inclination and a longitude of the ascending node fixedby the disk properties reported in Schneider et al. (2009): i Disk = . ± . Ω = PA ± = (27 . ± ρ = . ′′ (Jura et al. 1995). As pointed out by Schneider et al. (2009), thevalue of the outer boundary of the disk is consistent with thepresence of the companion. Using Eq. 1 we in fact obtaineda value for the critical semi-major axis for the planet stability( a crit ) of about 165 AU. Taking into account these constraints,we set the range of explored semi-major axes to 35-160 AU.The results of our simulations, in terms of non detection proba-bility maps as a function of the companion mass and semi-majoraxes, are shown in Fig.8. The disk boundaries are also shown, asreported by Schneider et al. (2009).Is it clear that with the NACO images we are not able to putstrong constraint on planetary-mass objects, but surely low-starcompanions and brown dwarfs more massive than 30 M Jup canbe excluded at a > AU and 20 M Jup ones for a > AU . The MESS SAM operation mode allows to test the consistencyof various sets of (mass, eccentricity, semi-major axes) paramet-ric distributions of a planet population with observational data.Given the detection performances of a survey, the frequency ofdetected simulated planets (over the complete sample) enablesderivation of the probability of non-detection of a given planetpopulation associated with a normalized distribution set. Thenthe comparison with the survey results tests directly the disagree-ment with observations at an appropriate level of confidence.As an example of the use of SAM@MES, statistical analysismode, we present the analysis of a small sample of young neigh-bourhood stars, part of the TWA association, and observed withNACO@VLT. These stars are part of a bigger sample for whichthe observations and statistical analysis, done with a preliminary
Bonavita et al.: MESS (Multi-purpose Exoplanet Simulation System)
Figure 8.
Non detection probability map of a faint companionaround TWA11 as a function of its mass and semi-major axis,in the case of a circular orbit. Inclination and longitude of theascending node have been fixed using the disk properties i Disk = . ± . Ω = PA ± = (27 . ± P D ) of companions of various massesand orbital parameters (semi-major axis a , eccentricities e , incli-nation i , longitude of the ascending node Ω , longitude of perias-tron ω and time of periastron passage T p ). We used the empiricalapproach, generating a full population of 10.000 planets for eachtarget, with a mass range spanning between 0.3 and 30 M Jup anda cut-o ff in semi-major axis of 100 AU.Each simulated companion was placed on the 2D minimummass map according to its position on the projected orbit to testits detectability, comparing its mass with the minimum valueachievable at the same position in the FoV.Only circumbinary planets were considered around TWA 22,adopting the total mass of the system as M S tar . In fact the binarybeing so close ( ρ = . ′′ see Bonnefoy et al. 2009) leads to avalue of the critical semi-major axis for circumstellar planets a CS of only 0.456 AU and of 8.395 AU for the circumbinary ones.Two sets of indices for the power-law distribution weretested:1. The ones derived by Cumming et al. (2008) (CM08): α = − . β = − . α = − . β = − . α and β to the CM08 values, we also intro-duced di ff erent values for the scaling of the planetary mass withthe primary mass.The results of all these simulations are summarized in Fig. 9. A second more general use is to constrain the exoplanet fraction f within the physical separation and mass probed by a survey, Figure 9.
Non-detection probability for the stars listed in Tab. 1,based on various sets of period and mass distribution. Mass andperiod distribution are extrapolated and normalized from RVstudies.
Top:
Variation of the non detection probability usingtwo di ff erent sets of power-law distributions (see text). Bottom: variation of the non-detection probability fixing α = − .
31 and β = − .
74 (Cumming et al. 2008) and di ff erent scaling the massof the planet with the primary mass.in the case of null or positive detections. Contrary to what wasassumed before, f becomes an output of the simulation, whichactually depends on the assumed (mass, period, eccentricity) dis-tributions of the giant planet population. This statistical analysisaims at determining f , within a confidence range, as a functionof mass and semi-major axis, given a set of individual detectionprobabilities p j directly linked to the detection limits of each starobserved during the survey and to the considered giant planetdistributions.The probability of planet detection for a survey of N starscan in fact be described by a binomial distribution, given a suc-cess probability f p j , with f being the fraction of stars with plan-ets, and p j the individual detection probabilities of detecting aplanet if present around the star j . Each individual p j can bereplaced by h p j i , the mean survey detection probability of de-tecting a planet if present. Finally, assuming that the number ofexpected detected planets is small compared to the number ofstars observed ( f h p j i << f max for agiven level of confidence CL: f max = − ln (1 − CL) N h p j i (21)Fig. 10 shows the results obtained applying this module atthe sample of stars listed in Tab. 1.Although the significance of our results is not really high,given the small size of the sample, they are still in agreementwith the results of the whole analysis presented by Chauvin et al.(2010) and with the results of the other deep imaging surveys(see e.g. Nielsen & Close 2010; Lafreni`ere et al. 2007). An extensive analysis, with MESS SAM, of the results of the majordeep imaging surveys published in the last decade is ongoing, and willbe presented in a forthcoming dedicated paper.onavita et al.: MESS (Multi-purpose Exoplanet Simulation System) 9
Table 1.
Sample of TWA stars considered in our analysis. In addition to name, coordinates, galactic latitude ( b ), spectral type,distance, V and K photometry, the observing mode direct imaging (DI) or coronagraphy (COR) and the status of the primary(single, binary Bin, triple) are also listed Name α δ
SpT Mass d Age V K Mode Notes[J2000] [J2000] M ⊙ (pc) (Myr) (mag) (mag)TWA22 10 17 26.9 -53 54 28 M5 0.15 18 8 13.2 7.69 DI / CI, S27, Ks Bin ( ρ = . ′′ )TWA14 11 13 26.3 -45 23 43 M0 0.55 63 8 13.8 8.50 DI / CI, S27, KsTWA12 11 21 05.6 -38 45 16 M2 0.30 32 8 13.6 8.05 DI / CI, S27, KsTWA19 11 47 24.6 -49 53 03 G5 1.50 104 8 9.1 7.51 DI / CI, S13, HTWA23 12 07 27.4 -32 47 0 M1 0.40 37 8 12.7 7.75 DI / CI, S13, HTwa25 12 15 30.7 -39 48 42 M5 0.15 44 8 11.4 7.31 DI / CI, S27, KsTWA11 12 36 01.0 -39 52 10 A0 2.10 67 8 5.8 5.77 DI / CI, S27, H Bin( ρ = . ′′ ), Star with diskTwa17 13 20 45.4 -46 11 38 K5 1.00 133 8 12.6 9.01 DI / CI, S13, H
Figure 10.
Top:
Survey mean detection probability derived asa function of the semi-major axis, assuming parametric massand period distributions derived by Cumming et al. (2008). Theresults are reported for individual masses: 1.5,3,6,9,12,15,30 M Jup . The integrated probability for the planetary mass regimeis shown with the thick green line.
Bottom:
Planet fraction upperlimit derived as a function of semi-major axis, given the samemass and period distributions.
The MESS SAM can also be used to test the predictions of spe-cific planet formation theories. An extensive use of this OM hasbeen made to analyze a sample of massive stars (B-type andearly A-type) observed with NIRI, to test the applicability ofplanet formation by disk instability in those systems. Startingfrom a uniform mass versus semi-major axis grid with a sam-pling of 5 AU in semi-major axis and 1 M jup in mass, 10 or-bits were generated for each grid point. Models of disc insta-bility (Bell et al. 1997; Mordasini et al. 2011,and Klahr et al., inprep.) were then used to provide boundaries in the mass ver-sus semi-major axis space, within which sub-stellar companionscan form by this mechanism. These boundaries were dependenton the stellar properties, and so appropriate values should beused for each target in the sample. The planets falling withinthe allowed range were subsequently evaluated against the 1Ddetection limits from the high-contrast images of the survey. Inthis way, by testing a range of planet distributions within the setboundaries, meaningful limits could be placed on the frequencyof planet and brown dwarf formation by disk instability in mas- Semi−major axis (AU) M a ss ( M j up )
50 100 150 200 250 300102030405060708090100 00.10.20.30.40.50.60.70.80.91
Detection limit41 AridetectionprobabilityFormation limits
Figure 11.
Non detection probability map, for 41 Ari (HIP13209).sive disks. The full analysis is presented in detail in the surveypaper (Janson et al. 2011). An example of a detection probabilitymap in mass versus semi-major axis space is shown in Fig. 11.
Beside the analysis of the real data, MESS can be also used topredict the output of forthcoming searches, the goal being to pro-vide information about the capabilities of future planet searchinstruments. With this mode, the flexibility of the code reachesits maximum, providing a wide range of possible applications.Once the synthetic planet population has been created, andassuming the characteristics of a given instrument, MESS PMallows predicting the number of detections expected from a fu-ture facility. This provides informations on :1. the expected frequency of planets2. the properties of these objects3. the kind of constraints that their observation can put on theplanet formation theories.Furthermore, it also allows to test di ff erent instrumental config-urations and observational strategies that can be adopted, thusproviding a tool to tune the instrument characteristics, in orderto fulfil the requirements needed to access a certain domain inthe parameter space, and reach the proposed science goals. Table 2.
Instruments for direct imaging of exoplanets considered in ouranalysis. References:
B10
Beuzit et al. (2010);
G05
Green et al. (2005);
G07
Graham et al. (2007);
K10
Kasper et al. (2010);
R10
Rieke et al.(2010);
S10
Stuik et al. (2010);
T10
Trauger & Moody (2010)Instrument Contrast Wavel. IWA Year Ref.( µ m) (”)8 m ground-based telescopesVLT-SPHERE 10 − − − − − − − − − > − As an example of the application of MESS PM, we report theresults of a comparison of the capability of a set of instrumentsfor the direct imaging of exoplanets, planned for the next decade,which are briefly described in Tab. 2.Since the purpose of the presented analysis is purely illustra-tive, we adopted for each instrument an averaged detectabilityrelation, taken from the reference indicated in Tab. 2, then us-ing only the 1D approach. The sample of stars used is the onedescribed in Sec. 2.2.1 and whose properties are summarizedin Fig. 1. This sample was originally selected as a preliminarysample for the planet search survey to be done with SPHERE,the next generation planet finder of VLT (Beuzit et al. 2008), andit’s therefore optimized for this kind of instruments, which possi-bly introduces some biases against some of the other instrumentanalyzed. The analysis was made using the reduced population,assuming five planet per star.The results of the analysis, showed in Fig. 12 and also sum-marized in Tab. 3, foresee that enormous progress that can beexpected in the next decade. The available measurements are al-ready giving us indirect information on far away planets aroundyoung stars, but passing through the intermediate step of nextgeneration image and finally with the advantage of ELT instru-ments we will have a wide view on planetary systems at di ff erentstages of their evolution. Table 3.
Summary of expected detections from imagers in the nextdecadeInstrument Year Young Old Nept. RockyGiants GiantsGr. based 8m 2011 tens fewJWST 2014 tens few1.5m Space Coro. ? tens tens tens fewELT’s > Once the RV and astrometric modules will be completed,MESS PM will provide an estimate of both the direct and in-direct signatures of the presence of the planets, and thus be usedto compare the outcomes of imaging with dynamical methods.These are interesting, because the latter allow determining theplanet masses, thus eliminating the degeneracy with age, whichis currently one of the major problems a ff ecting direct detec-tions. Moreover, possible synergies between di ff erent discoverymethods are becoming more and more likely, ELT’s instrumentsrepresenting the ideal link between direct and indirect detec-tions, covering both young, nearby systems discovered by nextgeneration imagers and also meant to provide the first images ofplanets already detected by RV.Fig. 13 and 14 summarize the results of the preliminary ver-sion of the RV and astrometric modules of MESS PM. The plan-ets showed are the same as in the lower right panel of Fig. 12.If confirmed, these results would suggest that the discoveryspace for EPICS at E-ELT overlaps well with those from ra-dial velocity (RV) instruments (HARPS at ESO 3.6m telescope,ESPRESSO at VLT, and especially CODEX at E-ELT) as wellas with that of GAIA (Casertano et al. 2008).The RV module being still under test, and without havingenough data to perform a consistent and accurate analysis ofthe performances and comparison between the instruments un-der scrutiny, this analysis is not meant to tell which instrumentis going to provide the highest number of detection, but just atshowing the potential of the further versions of the code. Figure 13.
Planets expected to be detected by EPICS (nearbysample) in the RV signal vs. period plane, compared with de-tection limits for RV instruments (HARPS, ESPRESSO andCODEX). The colour code is the same as in Fig. 1.
In this last section we present the results of some tests which goalwas to show how MESS can be used to investigate the influenceof the various physical parameters considered as inputs for theplanet generation. In particular we focused on the eccentricitydistribution and on the value of the planetary albedo. onavita et al.: MESS (Multi-purpose Exoplanet Simulation System) 11
Figure 12.
Planets expected to be discovered by SPHERE (representative of planet finders on 8m class ground-based telescopes),JWST-MIRI, 1.5m Space Coronagraphs, and EPICS / E-ELT (representative of 30-40m class telescopes) in the mass vs separationplane. Di ff erent colours are used for warm giant (orange), cold giant (red), Neptune-like (green), and rocky planets (blue) respec-tively. Eccentricity distribution
Direct Imaging surveys are, by definition, mostly sensitive toplanets in wide orbits. Also, planets on highly eccentric orbitscould also be preferred targets, since they are more likely tobe found farther out with respect to planets on a circular orbitwith the same semi-major axis. This could led to a bias towardshigh eccentricity planets in our results. As mentioned in Sec. 2.2,the eccentricity distribution of the planets generated by MESS isuniform, and cut at e = . standard setup , thus with theuniform eccentricity distribution cut at e = .
6. The red, green,blue, purple and light blue lines show the outcomes of the sim-ulations done by fixing e = e = . e = . e = . e = .
8, respectively. As expected, the higher eccentricity val-ues can lead to an higher fraction of detected planets, for a givensemi-major axis value.This simple exercise shows not only that the standard setup of the MESS does not introduce any systematic bias towardshigh eccentricity, but also that the code allows us to easily take this kind of biases into account, if they are proven to be real, bychanging the simulation parameters.As a final remark, it has to be said that the e ff ect of the eccen-tricity is important only in the case of warm planets, as the onesthat could be found around our TWA targets. As the age of thestars increases, the reflected light contribution to the planet con-trast becomes more and more important, thus counterbalancingthe e ff ect of the eccentricity. Albedo distribution
As mentioned in Sec. 2.3, the Albedo of the planets in the syn-thetic population is randomly generated between 0.2 and 0.7.Especially for the cold planets, the value of the albedo can bea critical parameter for the planet detection. We then decided toperform a check to see how big is the impact on the simulationresults. With an approach similar to the one used to test the ec-centricity e ff ect (see Sec. 3.3.3), we performed di ff erent sets ofsimulation, A λ being the only free parameter. We used an hypo-thetical G2V star ( J = , age = . Gyrs ) at 20 parsecs as target,and the detection limits of EPICS (see Tab. 2).Fig. 16 shows the results of the standard setup ( A λ randomlygenerated between 0.2 and 0.7, black solid line), together withthe ones obtained by fixing A λ to 0 .
2, 0 .
35 (the Jupiter value,
Figure 14.
Planets expected to be detected by EPICS (nearbysample) in the astrometric signal vs. period plane, comparedwith detection limits for astrometric satellites GAIA. Colourcode is the same as in Fig. 1.
Figure 15.
Fraction of detected planet as a function of the semi-major axis value (AU) for di ff erent values of the planet eccen-tricity. The black solid line shows the results obtained with the standard setup .see Sec. 2.3.1), 0 . .
4. Summary and conclusions
In this paper, we presented MESS (Multi-purpose ExoplanetSimulation System), a Monte Carlo tool for the statistical analy-sis and prediction of survey results for exoplanets.Our aim was to build an extremely versatile code, that couldbe used to test the outcomes of any instrument / technique for thedetection of planets. We consider several assumptions on: Figure 16.
Fraction of detected planet as a function of the semi-major axis value (AU) for di ff erent values of the planet albedo( A λ ). The black solid line shows the results obtained with the standard setup . – The star population, and how to take into account the proper-ties of each star and their e ff ect on either the characteristicsof the planets or the instrument capabilities. The binarity as-pects is also included to take into account the possible e ff ectsof a stellar companion to the planet formation. – The planet population, providing the complete set of orbitalelements and a large number of physical parameters of theplanets (radius, temperature, luminosity, etc.), either gener-ated using the information coming from the analysis of theplanets confirmed up to now (semi-empirical approach) orusing the results of the planet formation theories (theoreticalapproach). – The predicted observables (luminosity and polarimetric con-trast, RV semi-amplitude, astrometric signal) – The synthesis of a planet population, that can be easilyadapted to the purpose of the investigation – The final comparison with the detection limits, with the pos-sibility to combine the informations coming from di ff erentobserving techniques, to select a sub-sample of detectable planets whose characteristics can then be investigated.The code is such that each and every one of these assumptionscan be released and / or changed. This not only provides a toolwhich is independent from the models (e.g. the planet formationtheory chosen if the theoretical approach is used, or the evolu-tionary models used to estimate the planet luminosity and radius)but also makes it relevant to test model prediction, as well as toconstraint the properties of the known planets under di ff erentinitial conditions.So far only the Direct Imaging module of the code has beenextensively used, but the combination of various techniques isunder test and will o ff er rich perspective for future combinedstudies of exoplanets.Three main applications of the MESS code have been shown:1. The Hybrid mode, built for the analysis of single objects,is presented in Sec.3.1. It can be used to probe the physicaland orbital properties of a putative companion around a givensystem based on the combination of di ff erent techniques, andpossibly a priori information on the orbit given the presenceof other planets or of a circumstellar disk. onavita et al.: MESS (Multi-purpose Exoplanet Simulation System) 13
2. The SAM mode (Sec.3.2), optimized for the analysis of alarge sample of stars, shows its full potential in Sec. 3.2,by providing a detailed statistical analysis of a sample ofstars observed with direct imaging. Both the agreement ofthe observations with the observed parameter distributions(Sec. 3.2.1) and the planet formation theories (Sec. 3.2.3) aretested, using the semi-empirical and theoretical approach, re-spectively.3. The PM mode finally aims at the prediction of the outcomesof future searches, and can be used to tune not only the maininstrument parameters, but even the observing strategy.However, an extensive use of the code requires a completeknowledge of the instrument under test, of all the error sourcesand of the detection capabilities. Then, to really extend the useof MESS to other facilities one should first properly set all theneeded parameters. As already mentioned before, both the RVand the astrometric part are currently included in a very simplis-tic way. A better treatment of the dependence of the detectabilitywith astrometry from the orbital parameters should be included.A rigorous treatment of the stellar jitter evaluation must be im-plemented to allow a better comparison between the imaging andradial velocity capabilities. Especially in the case of E-ELT in-struments, this would allow to better define the synergies be-tween the various channels, for a more focused observing strat-egy.Moreover, a precise measure of the stellar characteristics isalso needed, in order to minimize the e ff ects that errors on theseparameters, such as the age of the system or the presence of stel-lar companions, can have on the analysis.Finally, the inclusion of an analysis of the planet stability incase of multiple objects is planned, together with an extensiveuse of the theoretical approach, using the outcomes of the mostrecent Bern models (Mordasini et al. 2010).Each technique performances vary with the star properties(age, mass, distance...), have di ff erent observables (luminosity,minimum mass, radius...), di ff erent observing strategies. It istherefore extremely important to take this into account to ringthe maximum constrains, first on the properties of giant plan-ets (physical, orbital parameter space) that will actually entirelyshape the planetary system architecture, then possibly on thetelluric planets. A better characterization of the giant and tel-luric planet orbital and physical properties, including their de-pendency with the host properties, is critical for a better un-derstanding of their formation processes as various mechanismsmay be at play (Boley 2009; Dodson-Robinson et al. 2009;Stamatellos & Whitworth 2009), but also of their architectureand dynamical evolution. At the end, one additional and impor-tant issue is to understand the required physical conditions thatwill lead to the formation of telluric planets in habitable zonewithin planetary systems shaped by giant planets, and that willpossibly lead to the formation of Life. Acknowledgements.
This work was done as part of M. Bonavita PhD thesis,which was founded by the Italian Institute for Astrophysics (INAF).The authors would like to thank Prof. W. Benz and Dr. Y. Alibert for pro-viding essential inputs for the development of the theoretical approach of theMESS.We also acknowledge support from the French National Research Agency(ANR) through project grant ANR10-BLANC0504-01
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