Identifying Exo-Earth Candidates in Direct Imaging Data through Bayesian Classification
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Identifying Exo-Earth Candidates in Direct Imaging Data Through Bayesian Classification
Alex Bixel
1, 2 and Dániel Apai
1, 2, 3 Department of Astronomy/Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA Earths in Other Solar Systems Team, NASA Nexus for Exoplanet System Science Lunar and Planetary Laboratory, The University of Arizona, 1640 E. University Blvd, AZ 85721, USA
ABSTRACTFuture space telescopes may be able to directly image ∼
10 – 100 planets with sizes and orbitsconsistent with habitable surface conditions (“exo-Earth candidates” or EECs), but observers will facedifficulty in distinguishing these from the potentially hundreds of non-habitable “false positives” whichwill also be detected. To maximize the efficiency of follow-up observations, a prioritization schememust be developed to determine which planets are most likely to be EECs. In this paper, we presenta Bayesian method for estimating the likelihood that any directly imaged extrasolar planet is a trueexo-Earth candidate by interpreting the planet’s apparent magnitude and separation in light of existingexoplanet statistics. As a specific application of this general framework, we use published estimates ofthe discovery yield of future space-based direct imaging mission concepts to conduct “mock surveys”in which we compute the likelihood that each detected planet is an EEC. We find that it will bedifficult to determine which planets are EECs with > confidence using single-band photometryimmediately upon their detection. The best way to reduce this ambiguity would be to constrain theplanet’s orbit by revisiting the system multiple times or through a radial velocity precursor survey.Astrometric or radial velocity constraints on the planet’s mass would offer a lesser benefit. Finally, weshow that a Bayesian approach to prioritizing targets would improve the follow-up efficiency of a directimaging survey versus a blind approach using the same data. For example, the prioritized approachcould reduce the amount of integration time required for the spectral detection (or rejection) of waterabsorption in most EECs by a factor of two. Keywords: astrobiology, space telescopes, Bayesian statistics, astronomical techniques, habitable plan-ets, direct imaging INTRODUCTIONOne of the primary science goals driving the develop-ment of new space telescopes is the detection and char-acterization of Earth-like planets around nearby stars.Spectroscopy of the planets’ reflected light spectra wouldreveal whether they are potentially habitable – i.e., liq-uid water could exist on their surfaces. The presence ofbiosignature gasses such as O , O , and CH could beinterpreted as evidence for life beyond the Earth (e.g.,Seager et al. 2016; Fujii et al. 2018), although this inter-pretation would not be straightforward as many of thesegasses can be produced abiotically (e.g., Catling et al.2018; Meadows et al. 2018).Recently, final study reports have been publishedfor two space telescope design concepts with a pri-mary science goal of directly imaging and spectroscop-ically characterizing Earth-like planets around nearbystars. These are the Habitable Exoplanet Observa- tory (HabEx, The HabEx Study Team 2019) and theLarge UV/Optical/Infrared Surveyor (LUVOIR, TheLUVOIR Study Team 2019, hereafter L19). FollowingStark et al. (2014), both reports provide estimates forthe expected yield of planets across a range of sizes andinsolations. The HabEx report predicts the detectionand characterization of +18 − approximately Earth-sizedplanets in the habitable zone with a 4-meter aperture,while the LUVOIR report predicts +75 − with the 15-meter aperture “LUVOIR-A” design. These design con-cepts have been thoroughly investigated, and it is likelythat the design of any future direct imaging space tele-scope would enable it to detect comparable numbers ofpotentially habitable planets with moderate S/N pho-tometry.However, these dozens of “exo-Earth candidates”would be detected amidst hundreds of planets with at-mospheric compositions or equilibrium temperatures notconducive to Earth-like life, including planets outside a r X i v : . [ a s t r o - ph . E P ] F e b Bixel & Apai of the habitable zone, large mini-Neptunes with thickH/He envelopes, or Mars-sized planets which have beenstripped of their atmospheres. These non-habitableplanets often demonstrate the same observable param-eters (e.g., apparent magnitude and separation) as theexo-Earth candidates, but are far more common andmay therefore cause a significant number of false positivedetections. We demonstrate this problem in Figure 1.In fact, Guimond & Cowan (2018) show that separation-and contrast-based selection criteria could suffer from afalse discovery rate as high as 77% given just the detec-tion data in a single band, or 47% if prior constraints onthe orbit are available.While the “false positive” planets would be interest-ing to characterize in their own right, spectroscopyacross the full wavelength range could take weeks forthe faintest targets, diverting time and resources fromhigher priority targets. To separate the potentially hab-itable and non-habitable planets, a survey could usemulti-epoch broadband photometry to characterize theirorbits and spectroscopic observations of H O absorptionfeatures to confirm the presence of water vapor in hab-itable zone targets. However, even within the habitablezone, there exist many planets that are too small or largeto be habitable, and the identification of water absorp-tion features will require a significant investment of time(Kawashima & Rugheimer 2019). Additionally, warmerplanets within the runaway greenhouse limit could ex-hibit water absorption features. In order to maximizethe efficiency of an exo-Earth imaging mission, it is nec-essary to develop a method for identifying those planetsthat are most likely to be Earth analogs using the lim-ited data which will be available upon detection.We have previously advocated for a Bayesian approachto assessing the potential habitability of newly detectedexoplanets (Apai et al. 2018). The Bayesian approachallows one to probabilistically constrain the propertiesof the planet by leveraging knowledge from exoplanetstatistics on planet radii, masses, and orbital properties.It also allows one to fold in predictions from theoreticalmodels of planet formation and evolution as prior knowl-edge. As an example, in Bixel & Apai (2017) we usedthe Monte Carlo method to infer the likely compositionof Proxima Cen b (Anglada-Escudé et al. 2016) in lightof well-established statistical priors and the limited dataavailable about the system. We found that it is ∼ likely that the planet is small and rocky as opposed to a“mini-Neptune”. In this paper, we extend this approachto assess the likelihood that a directly imaged planet hasan appropriate composition and orbit to be potentiallyhabitable. In Section 2 we review our Bayesian framework andgive an example of how it can be applied to character-ize directly imaged planets. In Section 3 we discuss theprior assumptions upon which our framework is based,and how they might be improved in the coming decade.Using the planet yield estimates provided in LUVOIRfinal report as a baseline estimate for the yield of a hy-pothetical direct imaging mission, in Section 4 we con-duct mock surveys where we detect each target and es-timate the apparent likelihood that it has a potentiallyhabitable composition and orbit. In Section 5 we dis-cuss the results of these surveys, including what typesof non-habitable planets would be mistaken for poten-tially habitable planets, and which additional data couldhelp to resolve this ambiguity. Finally, we show that ourapproach to target prioritization could greatly enhancethe efficiency of follow-up observations after all of theplanets have been detected. A BAYESIAN FRAMEWORK FORCLASSIFYING DIRECTLY IMAGED PLANETS2.1.
Monte Carlo method
Here we review the Monte Carlo method for Bayesianinference. This method allows an observer to constrainthe unobservable properties of a planet based on limitedprecision measurements of its observable data values byassuming some understanding of the prior distributionof intrinsic properties and their relationship to the ob-servable data values.Suppose planets can be described by some set of in-trinsic properties θ and some resulting set of observabledata values x which can be calculated from θ . Given aprior probability distribution for the values of θ , P ( θ ) ,then the probability distribution for θ given x can becalculated using Bayes’ equation: P ( θ | x ) = P ( x | θ ) P ( θ ) (cid:82) θ P ( x | θ ) P ( θ ) The left term is commonly referred to as the posteriordistribution of θ , and P ( x | θ ) as the likelihood function.For most astrophysical applications there is no analyt-ical solution to this equation, so it must be solved nu-merically. Under the Monte Carlo method, we use theprior probability distribution P ( θ ) to simulate a set ofproperties θ sim , then calculate the simulated data values x sim directly. Next, we accept or reject this simulatedplanet based on the value of its likelihood function forsome observed set of data values x obs . Assuming thedata values are drawn from independent, normal prob-ability distributions with standard deviations (measure-ment uncertainties) σ obs , the likelihood function is thatof a multivariate Gaussian: dentifying Exo-Earth Candidates in Direct Imaging Data False positives for exo-Earth candidate detections
False positive False positive False positiveFalse positive 100 masData False positiveFalse positive False positive EEC
Size:Orbit: too smalltoo close Earth-sizedin habitable zone too largetoo far
Figure 1.
To illustrate the degeneracies which affect the interpretation of direct imaging data, we simulate the detection ofa planet orbiting a Solar-type star at 15 parsecs (center panel), as well as several planets of varying sizes, orbits, and albedoswhich have a similar projected separation and magnitude (surrounding panels). It is not clear whether this data point representsa true exo-Earth candidate, or one of many potential false positives. The color and size of each circle represents the planet’spotential radius; only green points are approximately Earth-sized ( ∼ . − . R ⊕ ). The color of the potential orbit representsits insolation; only green orbits are in the habitable zone. An ‘x’ marks the planet’s closest approach to the observer. Bixel & Apai P ( x | θ ) = m (cid:89) i =1 exp(( x obs ,i − x sim ,i ) / σ obs ,i ) If only an upper limit is available for a componentof x obs (e.g., magnitude), then the prior sample is firstpruned of simulated members exceeding that limit.This procedure is repeated in parallel for a large num-ber of planets until a statistically sufficient number areaccepted. The result of the likelihood-based selection isa sample of planets whose properties θ sim are distributedaccording to the posterior distribution. In other words,a histogram of the accepted values of θ sim representsthe probability distribution for θ obs , the properties ofthe observed planet.2.2. Constructing the prior sample
We use the priors in Table 1 to construct the priorsample according to the algorithm visualized in Figure 2,and we discuss the prior assumptions in detail in Section3. We consider several cases governing the amount ofdata available to the observer - these are listed in Table2. Before we simulate the properties of the directly im-aged planet, we first consider the properties of its hoststar, which the observer will only know with finite pre-cision. We represent this measurement uncertainty bydrawing a unique stellar radius and mass for each sys-tem from normal distributions with σ = 3% and 7%.We assume nearly exact measurements of the distanceand luminosity of the host star, and we discuss theseassumptions in Section 3.1.We generate the radius and period of the planet using Kepler occurrence rates as a prior probability distribu-tion, then use the radius to classify the planet as toosmall (“sub-terrestrial”), too large (“ice giant”), or of theproper size to maintain a habitable atmosphere againststellar irradiation (“terrestrial”). This requires us to ex-trapolate to planets smaller ( R (cid:46) . R ⊕ ) or with longerperiods ( P (cid:38) days) than those readily available inthe Kepler sample, as we discuss in Section 3.2. Wecalculate the mass from an empirical mass-radius rela-tionship, where we assume some intrinsic variance dueto stochastic planet formation histories and differencesbetween the host stars of the planets on which theserelationships are based. We draw eccentricities from abeta distribution and the remaining orbital elements areassumed to be isotropically distributed.The liquid water habitable zone (hereafter LWHZ) isthe range of orbital separations over which a broadlyEarth-like planet could feasibly host liquid water on itssurface. Kopparapu et al. (2014) find the zone’s bound-aries to be a function of the planet’s mass and the star’s effective temperature, hence we cannot infer a planet’smembership to the LWHZ based solely on its insola-tion. We calculate the LWHZ boundaries and determinewhether the planet lies in the runaway greenhouse, tem-perate, or maximum greenhouse regimes, interpolatingbetween the discrete planet masses modeled by Koppa-rapu et al. (2014) (0.1, 1.0, and 5.0 M ⊕ ), and taking theminimum or maximum mass values for planets outsideof this range. For planets with non-circular orbits weuse the mean flux approximation to determine whetherthey are in the habitable zone, which Bolmont et al.(2016) find to be valid for planets with low or moderateeccentricities receiving Earth-like insolations.Finally, we assign a geometric albedo to each planetusing one of two methods for each of the cases in Ta-ble 2. In Cases 1-5, we assign a monochromatic albedofrom a broad uniform prior with no dependence on theplanet’s class or orbital parameters. In Cases 6 and 7,we generate a spectrum for each planet by mapping itto a solar system analog with a comparable size and or-bit. These two cases allow us to test the usefulness ofcolor information for identifying potential exo-Earths.We dedicate more considerable discussion to the geo-metric albedos in Section 3.6.2.3. Calculating the observable data values
Once the full assortment of planet properties has beensimulated, we can proceed to calculate the observabledata values to compare against those of a newly detectedplanet. 2.3.1.
Apparent separation
The angular separation vector has two components,and can be calculated from the orbital elements and thedistance to the system d . We adopt the same referenceframe and notation as Murray & Correia (2010) , where i = 90 ◦ is an “edge-on” inclination and the observer is at z = ∞ , so the angular separation components orthogo-nal to the line of sight are: s x = ( r/d )[cos(Ω) cos( ω + f ) − sin(Ω) sin( ω + f ) cos( i )] s y = ( r/d )[sin(Ω) cos( ω + f ) + cos(Ω) sin( ω + f ) cos( i )] In most cases, the position angle has no meaningfullydefined zero point, so for simplicity we calculate only thenet separation | s | = (cid:113) s x + s y . The exception is Case4, where the coeval detection of a debris disk is used toconstrain the orbital plane of the planet; therefore theposition angle is meaningfully defined, and we compute s x and s y separately. Figure 4 and Equations 53 & 54 dentifying Exo-Earth Candidates in Direct Imaging Data Table 1.
A list of the prior assumptions which we use to build our sample in Case 1, along with relevant literaturereferences. The additional cases in Table 2 may modify these assumptions to reflect new prior knowledge or data.Parameter Description of prior ReferenceRadius and period
Kepler occurrence rates Mulders et al. (2018)Class “sub-terrestrial”, “terrestrial”, or “ice giant”based on radius, following Figure 4 Fulton et al. (2017); Zahnle &Catling (2017)Mass Empirical mass-radius relationship with in-trinsic spread Wolfgang et al. (2016)Habitable zone boundaries Planet mass-dependent LWHZ models (run-away and maximum greenhouse limits) Kopparapu et al. (2014)Albedo Uniform from 0.2 to 0.7Eccentricity Beta distribution ( α = 0 . , β = 3 . ) Kipping (2013) ω , Ω , M † Uniform from 0 to π cos( i ) Uniform from -1 to 1Exo-Earth Candidates(EECs) “terrestrial” class planets in the LWHZ † Argument of pericenter; longitude of the ascending node; mean anomaly
Table 2.
The different cases under which we conduct our mock surveys, representing the different data which theobserver might have.Case Description1 Each planet has a monochromatic geometric albedo drawn uniformly from 0.2 to 0.7. The planet’s monochro-matic magnitude and separation angle are observed in a single epoch with a signal-to-noise ratio of 7 andcentroid precision σ c = 3.8 mas.2 Prior radial velocity (RV) observations provide constraints of ± on the period, ± cm/s on the radialvelocity semi-amplitude, and ± ◦ on the mean anomaly, in addition to the data from Case 1.3 Simultaneous observations of a debris disk provide a ± ◦ constraint on the inclination and longitude of theascending node of the orbital plane. The planet’s orbital elements may be further offset from these by ± . in Ω and cos( i ) .4 Multiple epochs of direct imaging data permit constraints of ± ◦ on the phase angle, ± on the semi-major axis, and ± . on the eccentricity..5 Case 4 with an additional ± . µ as constraint on the semi-amplitude of the star’s motion due to the planet.6 We assume prior knowledge about the albedo distribution of small planets. Each planet’s spectral albedois determined by its class and location with respect to the habitable zone. Random monochromatic andpolychromatic offsets are also introduced. The planet’s magnitude in three bands and separation angle areobserved in a single epoch with a wavelength-integrated signal-to-noise ratio of 7.7 Combination of Cases 5 and 6: three-band photometry is available along with constraints on the planet’sorbit and the star’s astrometric motion from multi-epoch observations. Bixel & Apai
Simulating the prior sample of planets
Priors Properties
Isotropic priors Habitable zoneboundariesClassPeriodMassHost star propertiesMagnitude(s)
Observables
Uniform priorRadius-basedclassificationMass-radiusrelationshipLWHZ modelOccurrence ratesBeta prior Eccentricitycos( i) , Ω, ω, θGeometric albedoSpectral characterization Is it an exo-Earthcandidate (EEC)?K (cm/s) Case 2 θ (μas)
Cases 5 & 7
RadiusApparent separation
Figure 2.
Flowchart illustrating our algorithm for simulating the prior sample of planets under Case 1. The red boxes representthe intrinsic planet properties which we simulate, and the red arrows indicate how they are used in the calculation of otherproperties. The blue boxes represent the priors which we use to simulate the properties, and the green boxes are the observablevalues which can be compared to the data. dentifying Exo-Earth Candidates in Direct Imaging Data
Apparent magnitude(s)
Following Madhusudhan & Burrows (2012) , wemodel the planet as a Lambertian sphere, in which casethe planet-to-star contrast ratio when observed at or-bital phase α is: L ( λ ) L ∗ ( λ ) = A g (cid:18) R P a (cid:19) (cid:20) sin( α ) + ( π − α ) cos( α ) π (cid:21) The phase angle is α = Cos − [sin( ω + f ) sin( i )] . Wedraw the geometric albedo from a prior distribution, andallow slightly super-Lambertian values ( A g > / ) asthese are observed in some wavelength ranges in the so-lar system.2.3.3. Radial velocity and astrometric semi-amplitudes andperiods
The semi-amplitude of the star’s periodic radial veloc-ity variation, assuming no other perturbers and M P (cid:28) M ∗ , is: K = (8 . cm/s ) ( M P /M ⊕ ) sin( i )( M ∗ /M (cid:12) ) / ( a/ AU ) / (1 − e ) / The semi-amplitude of the star’s astrometric motionis: θ = (3 . µ as ) ( M P /M ⊕ )( M ∗ /M (cid:12) ) ( a/ AU )( d/ pc ) In both cases the period of the stellar motion (andtherefore the planet’s orbit) is also measurable. How-ever the astrometric mass measurement requires thatthe system be observed for multiple epochs, in whichcase the planet’s orbital period can be derived from itsapparent motion about the host star.2.4.
Calculating the posterior probability distributions
We generate a posterior sample of simulated planetsfollowing the scheme in Section 2.1, where θ sim are thesimulated properties from Section 2.2, x sim are the simu-lated data values from Section 2.3, and x obs are the datavalues for the observed planet. This allows us to calcu-late a posterior probability distribution for each com-ponent of θ obs , the physical properties of the observedplanet. Based on just the planet’s apparent separationand magnitude, we are therefore able to place informa-tive constraints on its semi-major axis, radius, and mass.The posterior probability in which we are most in-terested is the probability that the planet is potentially Figure 1 and Equations 4 & 33 habitable - i.e., a “terrestrial”-class planet ( . (cid:46) R P (cid:46) . ) within the LWHZ. We designate these planets as“exo-Earth candidates” (EECs), following the terminol-ogy of The LUVOIR Study Team (2019) and The HabExStudy Team (2019), albeit including a slightly differentrange of sizes. Planets outside of this range we designateas “false positives”, with the false positive probability be-ing the likelihood that the observed planet - as judgedbased solely on its observed data values - is a false pos-itive instead of an EEC.2.5. Example: an exo-Earth candidate around a Solartwin
As a demonstration of our method, we use the pro-cedure in Section 2.2 to generate an Earth-sized planetin the center of the LWHZ of a Solar-type star at 15parsecs - an ideal exo-Earth candidate. We simulate thedetection of this planet by assuming a S/N = 7 mea-surement of its monochromatic magnitude, and a ± . mas measurement of its centroid.Acting as the observer - who has no prior knowledgeabout the planet’s true size and orbit - we use the pro-cedure outlined above to estimate the likelihood thatthat the planet is an exo-Earth candidate based on itsobserved magnitude and separation. The results of ouranalysis are summarized in Figure 3. We can confidentlysay that the planet is at least 1 AU from its star, and isunlikely to be farther than 3 AU - however, we cannotconstrain its orbit to the habitable zone with certainty.We can also tell that the planet is almost certainly largerthan . R ⊕ and smaller than R ⊕ - but this range in-cludes planets which are too small (sub-terrestrial) ortoo large (sub-Neptune) to be EECs.Most notably, we determine that it is only likelythat this planet is an EEC, as the majority of simu-lated planets which have a similar separation and mag-nitude are not habitable. In this specific example, eventhough the planet appears likely to have a size con-sistent with habitability, it also appears likely toorbit beyond the maximum greenhouse limit. This ex-ample shows that while it will be difficult to discrimi-nate between true EECs and false positives given justthe data available on detection, it will still be possi-ble to place meaningful probabilistic constraints on theplanet’s properties. PRIOR ASSUMPTIONSIn this section, we discuss our priors in more detailby reviewing the relevant literature and discussing howthey may be improved upon by future observations andmodeling efforts.3.1.
Stellar properties
Bixel & Apai
97 99 101 103 105
Separation (mas) M a g n i t u d e EECs (31% likely)False positives (69% likely) 1 2 3 a (AU) d P / d a R ( R ) d P / d R Figure 3.
We simulate the detection of an “ideal” exo-Earth candidate from 15 pc. (Left) The separation- and magnitude-phase space populated by a host of simulated EECs (green) and non-EECs (red). The data point with uncertainties is marked inblack. (Middle/right) Posterior distributions for the planet’s semi-major axis and radius, taking into account
Kepler statisticsand other priors. The true values are marked with dashed lines; this planet appears to the observer to have a wider orbit thanit actually does.
It is likely that much effort will be dedicated to char-acterizing the stellar targets of a direct imaging missionin advance of its launch. Still, the stellar properties willonly be constrained with finite precision - potentiallyseveral percent - so it is important that our prior sam-ple includes host stars spanning the range defined by therelevant uncertainties.
Gaia
DR2 (Gaia Collaboration et al. 2018) has al-ready provided high precision ( − µ as) parallaxmeasurements for nearby F-M spectral type stars, sowe treat this uncertainty as negligible. Optical/IR in-terferometry has allowed for the measurement of stellarradii to ∼ for targets at ∼ − pc (e.g. Ligiet al. 2016). Masses are more difficult to measure, sostellar atmosphere models are often used - as an ex-ample, Sharma et al. (2018) constrain model-dependentmasses for more than 10,000 stars using high resolutionspectroscopy, with a median precision of 7%. Follow-ing these examples, we draw the radius and mass of thehost star from normal distributions with widths of 3%and 7%, respectively.3.2. Radius and periodKepler allowed for the precise calculation of planet oc-currence for planets with periods shorter than 100 days.Mulders et al. (2018) find that the Kepler occurrencerates are well-described by independent broken powerlaws in both radius and period: dN pl d log P d log R ∝ f R ( R ) f P ( P ) where f P ( P ) = ( P/P break ) a P ( P < P break ) = ( P/P break ) b P ( P ≥ P break ) f R ( R ) = ( R/R break ) a R ( R < R break ) = ( R/R break ) b R ( R ≥ R break )The best-fit parameters for this model are ( P break , a P , b P ) = (12 , . , . and ( R break , a R , b R ) =(3 . , − . , − . The number of planets per system, N pl ,is found to be ∼ . Multiplicity will be an importantfactor for direct imaging surveys, as it could confusethe interpretation of the data or allow for simultane-ous follow-up observations of multiple planets - but thetopic is outside of the scope of this work. Here, we treateach detected planet independently, and normalize theabove power laws so that N pl = 1 .To properly simulate the abundance of planets in thehabitable zone of F, G, and K spectral type stars - as wellas smaller or cooler planets which might be mistakenfor them - requires us to extrapolate Kepler occurrencerates beyond the range of parameters within which theyare well-understood ( R (cid:38) . R ⊕ , P (cid:46) days). Thisextrapolation could be problematic; for example, the re-sults of Chen & Rogers (2016) suggest that planets onwider orbits can maintain thick volatile envelopes bet-ter against hydrodynamical escape, so we might find anover-abundance of large planets on wide orbits. We cangain some insight by studying the dependence of planetradii on insolation around low-mass stars, but these re-sults cannot necessarily be extrapolated to Solar-massregimes. dentifying Exo-Earth Candidates in Direct Imaging Data Kepler was generally not sensitiveto planets smaller than 0.5 R ⊕ , it is likely that somesuch planets will be found by direct imaging missionsand could masquerade as exo-Earth candidates. It istherefore necessary that we extrapolate the power lawof Mulders et al. (2018) down to 0.1 R ⊕ to ensure thatthe potentially large number of Mercury-sized objectsare represented in our simulations. However, since ourcutoff for exo-Earth candidates is R ≈ . R ⊕ , planetssmaller than 0.5 R ⊕ are less likely to be mistaken forEECs, so this extrapolation should not substantially in-fluence our results.3.3. Planet classes
We employ a radius-based classification scheme to sep-arate potentially habitable planets from those that aretoo small or too large to be habitable. This classifica-tion is motivated by two physical considerations affect-ing whether a planet can maintain a habitable atmo-sphere against irradiation over several Gyr.Empirical evidence suggests a change in planet com-positions between . − R ⊕ . Multiple authors find ev-idence for a split in planet densities in this range, withplanets larger than ∼ . R ⊕ mostly having densitiesmuch lower than the Earth’s (e.g., Weiss & Marcy 2014;Rogers 2015; Chen & Kipping 2017). Fulton et al. (2017)find a relative lack of Kepler planets with R ∼ . R ⊕ compared to smaller or larger radii; this “photoevapora-tion valley” was predicted by several authors who showthat smaller planets would lose thick envelopes due tohydrodynamic escape (e.g., Owen & Wu 2013). We in-terpret both results as evidence that planets larger than ∼ . − . R ⊕ have compositions more comparable toNeptune than the Earth, and are therefore not habitablein the traditional sense.Very small planets will also have trouble maintainingeven small and dense atmospheres against Earth-like in-solations. Zahnle & Catling (2017) find that a simplepower law relationship between a body’s escape veloc-ity and its effective insolation ( I ∝ v esc ) can predictwhether planets in the solar system (and some exoplan-ets) have atmospheres. According to this relation, aplanet with the same insolation and density as the Earthwould need to be larger than . R ⊕ to maintain a hab-itable atmosphere. However, we note that the Earthlies towards the inner edge of the LWHZ as calculatedby Kopparapu et al. (2014); it is possible that smallerplanets could maintain Earth-like atmospheres furtherout.Taking both of these considerations into account, weassign one of three classes to each planet based on its ra-dius: “sub-terrestrial” planets which are too small to be Radius ( R ) P r o b a b ili t y ( % ) sub-terrestrial terrestrial ice giant Figure 4.
Our probabilistic scheme for classifying planetsbased on their radii. “Sub-terrestrial” planets are so smallthat they will lose their atmospheres to thermal escape underLWHZ levels of irradiation. “Ice giants” are so massive thatthey will form and maintain thick volatile envelopes. Onlythe “terrestrial” planets are neither too small nor too largeto maintain a habitable atmosphere against irradiation. habitable, “terrestrial” planets which could have Earth-like atmospheres, and “ice giant” planets which are toolarge. There is likely some overlap between these cate-gories; for example, planets slightly larger than . R ⊕ or smaller than . R ⊕ might still have an Earth-like at-mosphere. To simulate this overlap we probabilisticallyassign each planet’s class from its radius using the prob-abilities in Figure 4. The “terrestrial” class includes allplanets with . < R < . R ⊕ and a fraction of planetswith . < R < . R ⊕ or . < R < . R ⊕ .3.4. Mass
To calculate each planet’s mass, we rely on the empir-ical mass-radius relationships of Wolfgang et al. (2016) (hereafter W16), which are calculated for smaller ( < . R ⊕ ) and larger ( < R ⊕ ) planets, reflecting the bi-modal split in planet compositions. They do not treatmass as a deterministic function of radius, but rathermodel a distribution of masses for each radius to cap-ture the intrinsic variability in planet compositions.We draw planet masses from truncated normal distri-butions defined by mean µ and variance σ , with mini-mum values of 0.01 µ and maximum values of M pure Fe -the mass of a pure iron composition as defined in W16.The parameters of the distributions are: µ, σ = . M ⊕ ( R/R ⊕ ) . , . M ⊕ ( R ≥ . R ⊕ )1 . M ⊕ ( R/R ⊕ ) . , . µ (0 . < R < . R ⊕ )1 . M ⊕ ( R/R ⊕ ) . , . µ ( R ≤ . R ⊕ ) These are the values of µ and σ fitted by W16, witha few caveats: Equation 2 and Table 1 Bixel & Apai
1. Their data did not allow the authors to determine σ M for the smaller planets; here, we arbitrarilychoose σ = 0 . µ (i.e., a 30% spread in density).2. Only a few uncertain data points and upper lim-its were available for planets with R < . R ⊕ , sowe instead assume approximately Earth-like den-sities.3. The large radius relationship was fitted for allplanets with R < R ⊕ , not just the ice giants;however, most of the precise data points neverthe-less had R > . R ⊕ .4. We model a few planets as large as R ⊕ , butthis mass-radius relationship is likely not valid be-yond R ⊕ ; indeed, it underestimates the mass ofJupiter ( ∼ R ⊕ ) by a factor of five. Ultimately,the overlap in masses and magnitudes betweenEarth-sized and Jupiter-sized planets is negligiblewhen considering potential false positives for exo-Earth candidate detections.Finally, we expect that the empirical mass-radius re-lationship will be improved upon in coming years by thediscovery of transiting rocky planets around low-massstars by TESS (Ricker et al. 2014), precision radius mea-surements from CHEOPS (Broeg et al. 2013), and massmeasurements through TTV or RV. By the time an exo-Earth direct imaging mission begins, observers shouldhave a better understanding of the relationship betweena planet’s size, composition, and mass with which tointerpret radial velocity or astrometric mass measure-ments. 3.5. Eccentricity
Planets in the solar system tend to have eccentric-ities smaller than 0.1, but multiple authors find evi-dence for a wider distribution of exoplanet eccentrici-ties in both transit (Kane et al. 2012) and radial veloc-ity data (Kipping 2013). Kipping (2013) determine thatthe eccentricity distribution of several radial velocity de-tected exoplanets is well-described by a beta function,with α = 0 . and β = 3 . - in which case > ofplanets have e > . .Eccentricity will have the effect of confusing the de-termination of a planet’s orbit from a single epoch ofimaging data, as a wider range of eccentric orbits couldbe consistent with the observed separation. Further-more, planets which orbit near the inner or outer edgeof the LWHZ may spend a fraction of their orbit outsideof the zone. To determine which of these planets areEECs we use the mean flux approximation - assumingthat a planet is habitable if the average insolation of its orbit is the same as a circular orbit in the habitablezone; Bolmont et al. (2016) find this to be an adequateapproximation for planets with low or modest eccentric-ities receiving a mean flux equal to the Earth’s, whilehighly eccentric planets tend to freeze out. However, itis possible that this approximation is not valid for ec-centric planets receiving a lower mean flux.We use the beta distribution of Kipping (2013) to drawplanet eccentricities and compute the projected separa-tion accordingly, though we truncate the distributionbeyond e > . (0.5% of planets) due to the additionalcomputing time required to solve Kepler’s equation forhighly eccentric orbits.We note that multi-planet systems such as our owntend to have more circularized orbits (Van Eylen & Al-brecht 2015). In principle, if a planet is detected with acompanion then it is a priori less likely to have an eccen-tric orbit, and it should be easier to determine whetherthe planet is an EEC. In the scope of this paper, how-ever, we treat all planets as the only member of theirsystem. 3.6. Albedo
We consider two different prescriptions for simulat-ing the geometric albedo: in Cases 1-5 we assume amonochromatic albedo drawn from a broad uniform dis-tribution, while in Cases 6 and 7 we assume a spectralalbedo model which depends on the planet’s class andposition with respect to the LWHZ.Constraining the actual distribution of planet surfaceand atmospheric properties is one of the goals of fu-ture imaging missions, so it might seem backwards tointerpret these observations by assuming the underlyingdistribution of geometric albedos as a prior. Neverthe-less, such an assumption is necessary in order to infer aplanet’s size from photometric data.A uniform prior represents a conservative approach tothe problem. We note that most planets in the solarsystem have geometric albedos ranging from 0.2 to 0.7across the UV to NIR wavelength range, with the excep-tion of Mercury’s very low albedo. Therefore for mostcases we draw a monochromatic geometric albedo foreach planet uniformly from 0.2 to 0.7.However, the proposed designs of direct imaging mis-sions allow for the simultaneous observations of a planetin 2-3 photometric bands, in which case color informa-tion would be available with no additional overhead.An efficient characterization strategy would make use ofthis color information to discriminate between EECs andtheir false positives, but to do so the observer must as-sume some prior knowledge about the diversity of plan-etary atmospheres and surfaces. As an example, we as- dentifying Exo-Earth Candidates in Direct Imaging Data for the Earth (Robinson et al. 2011,scaled from quadrature), Venus dayside, and Mars (nopublications listed). For Neptune we use the planet’sobserved geometric albedo as provided by Madden &Kaltenegger (2018) from 450-2500 nm, and set A g = 0 . from 300-450 nm (e.g., Mallama et al. 2017).To simulate both model uncertainty and physical di-versity among these solar system analogs, we allow thespectral albedos to vary by ± . monochromatically and ± . in each photometric bandpass. Additionally, weenforce a lower limit of 0.001 - to ensure that none of ourplanets are perfect blackbodies - and an upper limit of0.7, slightly more than the upper limit for the geometricalbedo of a Lambertian sphere. The range of spectralmodels for each category of planet is demonstrated inFigure 5.Finally, we consider the bandpasses in which the plan-ets are observed. For reference, LUVOIR’s proposedcoronagraphic instrument would be able to observe si-multaneously in bandpasses of each of its threechannels (L19). We choose wavelength ranges near thecenters of each channel: 335-390 nm, 715-830 nm, and1390-1610 nm.It is likely that a wide variety of terrestrial planetsexist; indeed, all four of the terrestrial planets in thesolar system are distinct in surface reflectance and at-mospheric absorption. Both of the ice giants, however,have similar albedo distributions. This suggests thatsome understanding of the albedo distribution of ice gi-ant analogues can be attained during the coming decade.Modeling efforts to understand the composition and ap-pearance of sub-Neptune type planets are already un-derway (e.g., Hu et al. 2015). New observatories suchas JWST, WFIRST, and ELTs could provide observa-tional tests of these models - the first through eclipse andtransit spectroscopy of warm and hot Neptunes, and thelatter two through direct imaging of ice giants at wideseparations. MOCK SURVEYS: METHODOLOGYThe LUVOIR team has released a final report (L19)which include estimates for the number of planets whichLUVOIR could detect as a function of planet radius,insolation, and host spectral type. In this section, we http://depts.washington.edu/naivpl/content/vpl-spectral-explorer Table 3.
A list of the solar system analogs which wesimulate each planet’s spectrum under Cases 6 and 7.The assumed model for the planet’s spectrum dependson its class/size and location with respect to the liquidwater habitable zone, as defined by the runaway andmaximum greenhouse limits.Class Location Assumed modelterrestrial in LWHZ Earthsub-terrestrial anywhere Marsterrestrial exterior to LWHZ Marsterrestrial interior to LWHZ Venusice giant anywhere Neptune
Wavelength (nm) G e o m e t r i c a l b e d o Figure 5. (Top) The set of base models which we use to sim-ulate geometric albedos under Cases 6 and 7 only, accordingto the scheme in Table 3. (Bottom) The actual range of sim-ulated spectra for each type of planet as observed in threephotometric bandpass (with wavelength offsets for visibility).We apply moderate differences to each planet’s spectrum tosimulate the underlying physical diversity, but broad differ-ences between the four groups can still be seen. produce a candidate sample based on these estimates,characterize each planet therein, and report on the effi-ciency of each strategy outlined in Table 2 for properlyidentifying exo-Earth candidates. While we rely on theyield estimates and stellar targets of L19 for our mocksurvey, we use these only as baseline estimates for theyield of a generic, hypothetical direct imaging survey; wedo not attempt to reproduce the results of the report orto assess the mission design.2
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Generating the candidate list
To properly generate a list of planets which could bedetected by a coronagraphic imaging mission requires acareful treatment of the instrument design, survey strat-egy, and additional sources of noise (e.g., dust) which isoutside of the scope of this work. L19 have performedsuch an analysis (following the methodology laid out byStark et al. (2014)) so we use their results to generateour candidate list .We acquired one of the simulated samples of host starsupon which the LUVOIR-A yield estimates are based,including masses, luminosities, and distances for 287stars (C. Stark, private correspondence). To assign a ra-dius to each star we use a simple scaling relation (Hansenet al. 2004): ( R ∗ /R (cid:12) ) = ( M ∗ /M (cid:12) ) . While this simulated list will not be the final targetlist of LUVOIR-A, it generally represents the diversityof host star properties which such a survey would en-counter. Next, we use the method outlined in Section2.2 to generate a large sample of planets around thesestars, and draw from that sample at random until theyield estimates for LUVOIR-A have been satisfied foreach bin in spectral type, planet size, and insolation.To ensure that the planets we simulate are detectable,we also enforce the same separation and brightness lim-its as L19. Namely, we only include targets whichare brighter than a planet-to-star contrast ratio of . × − and which are detected between 24–440mas - approximately the working angles of the proposedLUVOIR-A coronagraph at 500 nm.The yield estimates of the LUVOIR-A architectureproject the discovery of ∼ planets, ∼ of whichwould be exo-Earth candidates. For our purposes, theactual number of planets is not relevant - only their rela-tive abundance by size, distance, host spectral type, etc.- so we improve the accuracy of our results by inflatingthe yield estimates unilaterally by a factor of fifty.4.2. Survey cases
We run our mock surveys under seven cases repre-senting the different data which could available to theobserver. A brief description of each can be found inTable 2. 4.2.1.
Case 1: Detection data only
Under Case 1, the planet’s existence is entirely un-known before its direct imaging detection, and the see Figures 3.1 & 3.13 and Tables 8.7 for details on the targetlist, yield estimates, and instrument parameters observer only measures its apparent separation andmonochromatic magnitude. Following L19, we assumea signal-to-noise ratio of 7. If photon noise is dominant,the uncertainty on the planet’s centroid position is de-scribed by: σ c = FWHMSNRAssuming an effective wavelength of 500 nm, then σ c = 3.8, 1.9, and 1.0 mas for the 4-meter HabEx and 8- or15-meter LUVOIR architectures. This uncertainty willbe further affected by the pointing stability of the tele-scope during observations and the wavelengths at whichthe planet is observed; to be conservative, we choose σ c = 3.8 mas. Finally, we simulate the planet’s detectionby re-drawing its magnitude and apparent separationfrom normal distributions with widths defined by theseuncertainties.4.2.2. Case 2: Additional radial velocity detection
It has been emphasized that a radial velocity searchfor nearby Earth twins would be an important precursorto a space-based direct imaging mission (Dressing et al.2019), but the detection of Earth analogs around Solartwins is beyond the reach of current instrumentation.Doing so would require a significant investment of timeon major observing resources and new methods to cor-rect for systematic sources of noise such as stellar jitter(Plavchan et al. 2015).To investigate the potential benefits of a precursor ra-dial velocity search for interpreting planet detections,in Case 2 we simulate the direct imaging detection of aplanet (Case 1) along with a measurement of its orbitalparameters and radial velocity semi-amplitude. We seta baseline uncertainty of 5 cm/s on the measurementof the radial velocity semi-amplitude K - this value ischosen to be smaller than the value for an Earth twin( ∼ cm/s), but not negligible in comparison. Addi-tionally we allow a conservative 10% uncertainty on themeasurement of the orbital period, and a loose ± ◦ constraint on the planet’s mean anomaly.Several planets - including small planets in the LWHZwhich could masquerade as exo-Earths candidates - willhave K < cm/s. In these cases we assume the planetis undetected, so no constraints on its orbit are avail-able. An upper limit of cm/s is enforced so that thenon-detection also carries useful information about theplanet’s size.4.2.3. Case 3: Constraining the orbital plane using debrisdisks measurements
It is expected that several nearby systems contain ex-ozodiacal dust disks near their habitable zones (Ertel dentifying Exo-Earth Candidates in Direct Imaging Data ◦ ) constraints on the diskinclination and orientation (e.g., HL Tau, ALMA Part-nership et al. 2015). While the orientation of exozodia-cal disks may be harder to constrain if they are faint orexceed the outer working angle of the coronagraph - andwhile not all systems may have substantial disks at all -we here consider the “optimistic” case where every sys-tem has a disk, and the orientation parameters ( cos( i ) and Ω ) can be constrained to ± .In principle, if the orientation of the debris disk canbe tightly constrained and the planet shares exactly itsorbital plane with a circular orbit, then a single precisemeasurement of the planet’s apparent separation couldbe sufficient to determine its phase and semi-major axis.However, the orbital inclinations of solar system planetsdeviate by up to ◦ from the Solar spin axis - and largermisalignments could be possible in other systems - whileseveral of our simulated planets have eccentric orbits.We can still infer some information about the planet’sorbit given the orientation of the disk and some priorknowledge about how misaligned planetary systems tendto be. In Case 3, we use the orientation of a contem-poraneously detected disk as a prior constraint on themean orbital plane of the system, but allow for a dif-ference of ± . between cos( i ) and the longitude of theascending node ( Ω ) of the two components. We then in-terpret the observations using the same data as in Case1, but treating the two dimensions of the separation vec-tor separately.4.2.4. Cases 4 and 5: Multiple revisits and astrometricmass measurements
Constraining the planet’s orbital parameters will re-quire multiple revisits spaced over the orbital period, sorevisits will likely be folded into the observing strategyof future direct imaging missions. However, even withinthe habitable zone there will be numerous potential falsepositives for EECs, a fact which may limit the practicalbenefit of revisiting every system.Guimond & Cowan (2019) have shown that ∼ revis-its will be sufficient to constrain the orbital parameterswith better than precision. In Case 4 we assumethat the system has been revisited enough times for the planet’s orbital parameters to be constrained withcomparable precision. Specifically, we assume that thesemi-major axis is measured to ± , the eccentricityto ± . , and the orbital phase to ± ◦ .An ancillary benefit of revisiting targets would be theability to measure the star’s astrometric motion aboutthe system’s center of mass. Measuring the astrometricsemi-amplitude θ would allow observers to determine theplanet’s mass and potentially to identify which planetsare too small or large to be habitable. In Case 5 weassume that, in addition to the constraints provided inCase 4, the observer can measure θ with a precision of ± . µ as. This is the targeted astrometric precision ofthe High Definition Imager (HDI) instrument with theLUVOIR-A aperture (L19), and approximately / ofthe amplitude induced by an Earth twin at a distanceof 15 parsecs. If θ < . µ as, an upper limit of . µ asis enforced instead.4.2.5. Case 6: Color measurements
Future direct imaging missions could be able to ob-serve simultaneously in multiple bandpasses, in whichcase measurements of the planets’ colors would be avail-able as soon as they are detected. Color informationcould be used for preliminary planet classification; forexample, Batalha et al. (2018) predict that it will bepossible to differentiate between cloudy and cloud-freeJovian planets with reflected light imaging in three fil-ters using WFIRST or ELTs.However, color information is only useful for inferringthe planet’s properties if we make a prior assumptionabout the spectral albedos of extrasolar planets. Someobservational constraints are currently available througheclipse observations of close-in giant planets, but forsmaller planets we can only rely on planets in the so-lar system and theoretical models of the surfaces andatmospheres of known transiting and RV-detected plan-ets. It is therefore worthwhile to determine what effectan approximate prior understanding of spectral albedoswould have on the interpretation of direct imaging data.In Case 6 we simulate planets with spectral albedosreflective of (though not identical to) solar system ana-logues; our detailed assumptions are described in Section3.6. Each planet is observed in a 10% bandpass at thecenter of three wavelength channels, with a signal-to-noise ratio weighted by the square root of the bandpass-integrated flux (i.e., photon noise). The signal to noiseintegrated across all three bandpasses is 7, as in Case 1.If S/N < 2 in a given bandpass (typically in the UV orinfrared), then a 2 σ upper limit is enforced instead.4.2.6. Case 7: Maximum information Bixel & Apai
In the final case we consider all of the informationwhich a larger telescope would be able to acquire ona target after several revisits. These include measure-ments of the orbital parameters and astrometric semi-amplitude as well as brightness measurements in threebands. We assume the same measurement precision andsensitivity limits as in Cases 4 and 5, and we interpretthe color information following the method of Case 6.4.3.
Classifying the targets
Once we have constructed a sample of targets andsimulated their detection, we classify them using theinference framework described in Section 2. This yieldsfor each planet the likelihood, according to the observer,that the planet is an EEC or a false positive. In Figure 6we plot the likelihood that the planet is an EEC for eachsimulated EEC in the sample. Since the observer doesnot know a priori that these planets are EECs, we seethat they cannot make a confident identification uponthe planets’ initial detections, but given additional dataor multiple revisits they can identify several EECs withconfidence.We can break down the false positive probability bysize and orbit, determining for each observed EEC theinferred likelihood that it is, for example, a sub-Neptuneon an orbit exterior to the habitable zone. In Figure 7we plot this probability for each combination of class andorbit averaged over the sample of observed EECs. Thisplot illuminates the key sources of ambiguity in classi-fying EECs. For example, we see that it is difficult todistinguish between an EEC and a sub-terrestrial planetwith a temperate orbit, or a planet which has the propersize but lies just interior or exterior to the habitablezone. MOCK SURVEYS: RESULTS5.1.
Which planets are “false positives”?
Figure 7 demonstrates that true EECs share the ob-servable parameter space with a wide range of planetsboth within and outside of the LWHZ. These can bebroadly separated into: (i) planets which are too smallor large to be habitable, but are yet in the habitablezone, (ii) planets which are of the proper size to be hab-itable, but are not in the habitable zone and (iii) planetswhich are both of the wrong size and not in the habit-able zone. Each of these categories are approximatelyequal in their potential to masquerade as EECs.As we demonstrate in Figure 6, the observer will typ-ically be unable to distinguish between true EECs andtheir many potential false positives given just the datanecessary for the planet’s detection. Even when a planetwhich is in fact an EEC is detected, the observer will only be able to make this determination with < confidence.5.2. Do constraints on the orbit help to identify EECs?
Since most of the “false positives” are planets outsideof the habitable zone, it stands to reason that measure-ments which constrain the planet’s orbit would be use-ful for identifying EECs. Indeed, if the planet can beindependently detected through RV, and its period con-strained with precision, then it can typically be con-strained to the habitable zone with > confidence.However, approximately 25% of EECs remain below our5 cm/s detection limit, in which case the orbit cannotbe established.Observing the planet multiple times over an orbitalperiod will help to rule out planets outside of the habit-able zone with similar confidence, assuming 10% uncer-tainties on the orbital parameters. Nevertheless, evenif a planet can be constrained to the habitable zone, itmay yet be too large or small to be habitable. In gen-eral, constraints on the orbit and phase will only allowthe observer to distinguish EECs from temperate sub-terrestrial planets or ice giants with ∼ confidence.5.3. Do constraints on the mass help to identify EECs?
We find that measurements of the astrometric semi-amplitude θ , when combined with magnitude measure-ments, can modestly increase the observer’s ability toidentify EECs. For example, our average confidencefor identifying EECs given multiple revisits to deter-mine the orbit (Case 4) improves by about 18% if weinclude an astrometric measurement or upper limit onthe planet’s mass (Case 5), and several individual EECscan be identified with very high confidence. These couldbe the highest priority targets for deeper spectroscopicfollow-up.Under Case 2, if both the measured period and ra-dial velocity semi-amplitude K are used to constrain theplanet’s properties then EECs can be identified with anaverage confidence of 52%. If the measured value of K is ignored, however, then this confidence drops to .In other words, the measurement of the planet’s massaffords an extra 12% confidence that the planet is anEEC.Constraining the orbit generally reduces the potentialfor false positives more than constraining the mass, inpart because constraints on the planet’s size are avail-able based on its brightness alone. On the other hand,the relationship between radius, mass, and compositionis more well-understood than the prior distribution ofplanet albedos (which almost certainly is not uniform).An inference about the planet’s composition made from dentifying Exo-Earth Candidates in Direct Imaging Data How confidently can we identify exo-Earth candidates in mock observations?
0% 20% 40% 60% 80% 100%0%10%20%30%40% (average)
Case 1:initial detection
Confidence that the planet is an EEC % o f d e t e c t e d EE C s
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 2:initial detection+ radial velocity
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 3:initial detection+ debris disk
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 4:initial detection+ revisits
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 5:initial detection+ revisits+ astrometry
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 6:initial detection+ color
0% 20% 40% 60% 80% 100%0%10%20%30%40%50%
Case 7: (74%) initial detection+ revisits+ color+ astrometry
Confidence that the planet is an EEC % o f d e t e c t e d EE C s Figure 6.
We simulate the detection of ∼ EECs around nearby stars, then infer their properties from the mock dataunder each of the cases in Table 2. Above we plot the probability - as inferred by an uninformed observer - that each detectedplanet is an EEC, as well as the average value (dashed line). In the ideal case, this value would be 100% for all EEC targets,but typically it is smaller because of the limited data available to the observer. With additional data (Cases 2–7), the observercan be more confident that the detected planet is an EEC. Bixel & Apai
What kinds of planets could be mistaken for exo-Earth candidates? sub-terrestrialR 0.8 R terrestrial0.8 R 1.6 R ice giantR 1.6 R initial detection Planet class F a l s e p o s i t i v e p r o b a b ili t y sub-terrestrial terrestrial ice giant0%5%10%15%20% Case 2:initial detection+ radial velocity sub-terrestrial terrestrial ice giant0%5%10%15%20%
Case 3:initial detection+ debris disk sub-terrestrial terrestrial ice giant0%5%10%15%20%
Case 4:initial detection+ revisits sub-terrestrial terrestrial ice giant0%5%10%15%20%
Case 5:initial detection+ revisits+ astrometry sub-terrestrial terrestrial ice giant0%5%10%15%20%
Case 6:initial detection+ color sub-terrestrial terrestrial ice giant0%5%10%15%20%
Case 7:initial detection+ revisits+ color+ astrometry
Planet class F a l s e p o s i t i v e p r o b a b ili t y Figure 7.
We simulate the detection of ∼ EECs around nearby stars, then infer their properties from the mock data undereach of the cases in Table 2. Above we plot the sample-averaged probability - as inferred by an uninformed observer - that theplanet is instead a false positive with a non-habitable class or orbit. Additional data will suppress the false positive probability- for example, fitting the orbit with finite precision will reduce the likelihood that the planet is outside of the habitable zone. dentifying Exo-Earth Candidates in Direct Imaging Data
Do color measurements help to identify EECs?
In Figure 8 we plot the average inferred probabilitythat an EEC is similar to the Earth, Neptune, Venus, orMars for Cases 1 (no color) and 6 (color). We find thatcolor information is useful for distinguishing betweenspectra with positive versus negative slopes between theUV and visible channels. Specifically, adding a colormeasurement allows the observer to distinguish betweenEarth and Martian analogs particularly well - if smallplanets in the habitable zone tend to look like Mars,color information will be a valuable discriminant. Ac-cording to Figure 6, observing the color and constrain-ing the planet’s orbit could allow the observer to identifymost EECs with > confidence.We choose solar system planets as our templates asthey cover a relatively broad range of insolations andplanet sizes, and with the exception of very hot exoplan-ets they remain the only planets for which albedo mea-surements are presently available. In our set of modelsshown in Figure 5, the Earth stands out due to its mod-est scattering slope in the optical versus a much strongerfeature in Neptune’s spectrum, or opposite features forVenus and Mars. Yet a habitable planet does not needto look like the Earth, and indeed there is evidence thatduring the Archean the Earth had a substantially red-der appearance due to organic haze particles (e.g., Arneyet al. 2016, 2017). Similarly, the solar system providesno examples of an Earth-sized planet beyond the habit-able zone. Here we assume such worlds have a Martianappearance, but this is likely inaccurate for icy worldsor planets with dense atmospheres.We compare our range of simulated spectra to themodels of Krissansen-Totton et al. (2016), who com-pute optimal photometric bandpasses to distinguish be-tween several different examples of potential exoplanetreflectance spectra. Their optimized bandpasses are431-531 nm ("blue"), 569-693 nm ("green"), and 770-894 nm ("red") - different than those used in this work.In Figure 9 we place our models on a color-color plotsimilar to Figure 2 in the cited work, along with a subsetof the models considered therein. We find that our solarsystem analogs with simulated physical diversity covera comparable range in color space, so we believe that weadequately represent a diversity of planet appearanceseven though our range of base models is limited.We stress that this result is sensitive both to our priorassumptions and to the bandpasses in which we chooseto observe our targets. More work must be done to un-derstand the potential diversity of terrestrial planets and to determine which photometric bandpasses are optimalfor distinguishing them from EECs (e.g., Krissansen-Totton et al. 2016). In Section 5.9 we discuss how obser-vational constraints on the albedos of potential false pos-itives could be derived within the coming decade. Thesenew discoveries can then be folded into our Bayesianframework, and the results of Case 6 suggest that doingso could allow for the confident distinction between falsepositives and true EECs on the basis of color informa-tion.5.5. Can a debris disk be used to constrain the orbitalplane?
We find that measuring the orientation of the de-bris disk provides relatively little information about theplanet’s orbit. Specifically, we are on average only ∼ more confident that the observed EECs lie within thehabitable zone when we have measured the orientationof the disk. The benefits are slightly greater in caseswhere the system is observed from a “pole-on” orienta-tion ( | cos( i ) | > . ), in which case the uncertainty inthe centroid measurement translates to a smaller uncer-tainty in the semi-major axis versus the “edge-on” cases( | cos( i ) | < . ). Nevertheless, unless the centroid pre-cision is much better than . mas and the observedsystems are at least as well-aligned as the planets in thesolar system, measuring the disk orientation will gener-ally not allow an observer to constrain the orbit withoutrevisits.5.6. Can EECs be identified given maximalphotometric information?
In Case 7 we assume that the observer has revisitedthe system multiple times to constrain the orbit withthree-band photometry and has additionally measured(or placed upper limits on) the astrometric motion ofthe star due to the planet. This is the most informationwhich could be obtained for the typical system withoutsubstantial follow-up or precursor observations, and wefind that it would allow the observer to confidently iden-tify most EECs, with an average confidence of 87%. Thissuggests a promising roadmap toward selecting targetsfor follow-up, but as in Case 6 this result is dependenton the observer’s prior assumptions about exoplanet re-flectance spectra.5.7.
Would Bayesian prioritization improve follow-upefficiency?
A practical way to frame our results is in terms offollow-up efficiency. A logical next step after detectinga directly imaged planet (and optionally constraining itsorbit) would be to search for water absorption in a nar-row part of the spectrum to further test its habitability.8
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Does color information help to identify exo-Earth candidates?
Neptune Venus Mars0%10%20%30%40%
Case 1:initial detection
Neptune Venus Mars0%10%20%30%40%
Case 6:initial detection+ color
Type of planet F a l s e p o s i t i v e p r o b a b ili t y Figure 8.
We test whether low-to-moderate S/N color measurements would help to identify EECs, assuming that all planetslook approximately like a solar system analog depending on their class and orbit (Table 3 and Figure 5). Plotted is the typicalinferred probability that the EEC is actually a false positive with a Neptune, Venus, or Mars-type surface and atmosphere.Without color information (left) it is difficult to distinguish Earth analogs from smaller or cooler Mars-like planets, or largerNeptune-like planets. With color information (right), the slope of the optical spectrum provides a useful discriminant betweenthe Earth and Mars, but does little to reduce the ambiguity due to larger Neptunes. "Red" / "Green" albedo ratio " B l u e " / " G r ee n " a l b e d o r a t i o Blue ice sub-NeptuneTitan MoonEuropa
Earth Neptune Venus Mars
Figure 9.
We plot a set of simulated Earth, Neptune, Venus,and Mars analogs in color-color space - each planet’s spec-trum is modified from the solar system model to simulatephysical diversity. We also plot data points from Krissansen-Totton et al. (2016) (Figure 2) who calculate these color ra-tios for a large diversity of solar system bodies, materials,and exoplanet models. While we consider fewer base mod-els than these authors, our simulated planets cover a similarrange in color space.
However, unless a strategy is employed for prioritizingor pruning the target list, a significant amount of timewill be spent “following-up” non-habitable planets. In-deed, Kawashima & Rugheimer (2019) have shown that3-10 hours may be required to confidently detect H Oabsorption in an exo-Earth atmosphere from 5 parsecsusing a 10-meter telescope. For the typical target ob-served by the 15-meter LUVOIR-A from 15 parsecs, ap-proximately ∼ hours may be required.Our inference framework allows us to prioritize targetsbased on the likelihood that they are, in fact, exo-Earthcandidates. After probabilistically classifying each ob- served target - including both EECs and non-EECs -we prioritize them by the probability that they are trueEECs and submit them for follow-up observations withthe most likely candidates first. In Figure 10 we sum-marize the efficiency of this prioritization strategy forCases 1 (initial detection only) and 4 (multiple revisits).We compare our Bayesian approach to two non-prioritized follow-up strategies: first, removing all tar-gets whose projected separations are wider than themaximum greenhouse limit, and second, removing alltargets that are found to be outside of the habitablezone after multiple revisits. In both of these cases wealso remove targets with a contrast ratio brighter than − . These separation- and magnitude-based cuts ex-clude (cid:46) of bright or eccentric EECs but severallarger or cooler false positives. The remaining targetsare then followed-up blindly. Finally, we plot a line rep-resenting a perfectly efficient follow-up strategy with nofalse positives.We find that prioritizing the targets using ourBayesian framework allows us to re-observe them witha much greater efficiency - using the same data - thana blind approach. In fact, a prioritized approach usingjust the detection data is initially as efficient as takinga blind approach after each planet’s orbit has been char-acterized. We can discuss these efficiency gains in termsof integration time. Assuming 450 targets have beendiscovered in the initial census and 25 hours of integra-tion time are required to search each planet for water,the results in Figure 10 suggest that probabilistic targetprioritization could reduce the required amount of inte-gration time to follow-up 50% of EECs by 28 days (ifthe orbits have been precisely constrained) to 95 days(if only the detection data are available). Since this pri-oritization scheme does not require additional data on dentifying Exo-Earth Candidates in Direct Imaging Data Efficiently surveying planets for spectroscopic H O absorption % of targets followed up % o f EE C s f o ll o w e d u p initial detection, prioritizedinitial detection+ revisits, prioritized separation-based cut, fainter than 10 LWHZ targets only, fainter than 10 ideal; no false positives0 100 200 300 400 Estimated integration time (days)
Figure 10.
The efficiency of different post-detection follow-up strategies (e.g., to search for water absorption), quantified asthe percentage of all targets which must be followed up before a given percentage of EECs have been covered; the ordinateis proportional to the amount of time required for follow-up. The gray line represents ideal survey efficiency, where no falsepositives are re-observed. The colored lines represent strategies which make use of the detection data only (blue) or multiplerevisits to establish the orbit (red). The dashed lines are non-prioritized (blind) approaches which first remove very bright orwidely-separated planets, or planets whose orbits are constrained to be outside of the LWHZ. The solid lines are prioritizedapproaches in which we observe the planets which are most likely to be EECs first. The upper axis estimates the amount oftime required to search each planet’s atmosphere for water absorption, assuming 25 hours of integration time per target. the system, it could be naturally folded into the surveystrategy of any direct imaging mission.5.8.
Is the Bayesian approach always appropriate?
In principle, a Bayesian prioritization scheme shouldalways be superior to a blind follow-up strategy becauseit leverages additional information about exoplanet de-mographics. In practice, this prior knowledge is al-ways incomplete and potentially inaccurate, which couldmake Bayesian prioritization risky in the sense that itmight fall prey to unexpected types of false positives.For example, to produce planets for our mock surveys weuse the yield estimates of L19 which include more sophis-ticated treatments of the mission design and a differentestimation and extrapolation of
Kepler occurrence rates- but then we use our own algorithm to probabilisticallyclassify each planet. Therefore, the prior assumptionswe make when characterizing these simulated planetsare not exactly representative of their true parent dis-tribution, which makes our prioritization less efficient.Indeed, we see in Figure 10 that if the planets’ or-bits are known, then the blind approach to prioritizingtargets is more efficient than the Bayesian prioritized approach for the last ∼ of EECs. However we arguethat this is realistic - the observer’s prior assumptionswill always differ from reality to a degree - and we notethat our approach is still efficient despite this mismatch.For this reason we believe Bayesian prioritization willyield overall better results, but observers might prefer ablind approach for low priority targets.5.9. What priors can be improved in the comingdecades?
The accuracy of our probabilistic classification de-pends on the extent and accuracy of existing knowledgeof exoplanet statistics. Considering the state of the field ∼
20 years ago, it is reasonable to assume that an ob-server using this method to interpret observations ∼ years from now will base their judgment on better priorassumptions than are currently available. Here we spec-ulate on ways in which the prior inputs to our methodcould be refined in the coming decades.5.9.1. What is the largest potentially habitable planet?
In our work we assume that planets larger than ∼ . R ⊕ will form and maintain volatile envelopes over0 Bixel & Apai
Gyr timescales, thereby resembling “mini-Neptunes”more than “super-Earths”. The most compelling evi-dence for this comes from density measurements for alimited sample of small planets (Weiss & Marcy 2014;Rogers 2015; Chen & Kipping 2017) and the gap in
Ke-pler radius occurrence rates near ∼ . R ⊕ (Owen &Wu 2013; Fulton et al. 2017), but the exact value ofthis transition radius and its dependence on parameterssuch as insolation and spectral type requires further re-search (e.g., Fulton & Petigura 2018; MacDonald 2019;Martinez et al. 2019). In the near future, TESS (Rickeret al. 2014) will detect hundreds of small planets orbitingbright stars (Sullivan et al. 2015; Bouma et al. 2017; Bar-clay et al. 2018; Ballard 2019). In combination with pre-cise radii measurements by CHEOPS (Broeg et al. 2013)and ground-based radial velocity measurements, thesediscoveries will enhance the sample of − R ⊕ plan-ets with measured densities. Later on, PLATO (ESA2017) will detect several hundred planets smaller than R ⊕ on orbits as wide as the Earth’s, providing statis-tics for planet radii over a broader range of insolationsthan Kepler . Models of atmospheric evolution can beused to combine these lines of evidence into a more com-prehensive understanding of which planets should havenon-habitable volatile envelopes (e.g., Owen & Wu 2013;Lopez & Fortney 2014; Gupta & Schlichting 2019).5.9.2.
What is the period distribution for planets on wideorbits?
This work relies on extrapolation from
Kepler occur-rence rates for shorter period planets, but ice giants inthe habitable zone and beyond could be a significantsource of false positives for EEC detection. WFIRSTcould detect hundreds of wide-orbit planets through itsmicrolensing survey, some with masses lower than theEarth (Barry et al. 2011; Penny et al. 2019). PLATOwill also detect a number of transiting planets on orbitsas wide as the Earth’s (ESA 2017). Even if these datado not fully probe the relevant range of planet radii andperiods, they may provide enough points for interpola-tion to accurately predict the occurrence of terrestrialplanets and ice giants within the habitable zone and be-yond.5.9.3.
What do the spectra of false positives look like?
Our assumed distribution of planet albedos is the leastwell-vetted prior assumption in this work, but in Cases6 and 7 we show that prior knowledge of planet albe-dos could greatly enhance survey efficiency. The mostpromising avenue for constraining planet albedos in thenext decade is through direct imaging of super-Earthsand sub-Neptunes in the habitable zone and on wider or-bits with ELTs and WFIRST (e.g., Kasper et al. 2010; Traub et al. 2016; Savransky & Garrett 2016; Artigauet al. 2018; Weinberger et al. 2018; Akeson et al. 2019).These observations would be valuable for determiningthe optimum wavelength ranges for discriminating be-tween ice giants and other types of planets.Several authors have shown that post-runaway atmo-spheres like Venus’ could develop on extrasolar planetscomparable in size to the Earth within the inner edgeof the habitable zone, which itself depends on the spec-tral type of the host star and planetary factors such asmass, rotation rate, and ocean coverage (e.g., Koppa-rapu et al. 2013, 2014; Yang et al. 2014; Kodama et al.2018). Transit spectroscopy with JWST could revealthe atmospheric composition of a small number of ter-restrial planets within the runaway greenhouse limit oflow-mass stars (Lustig-Yaeger et al. 2019). While notdirectly measuring the albedo, these observations couldreveal whether Venus analogs are common, and model-ing efforts could reconstruct their likely appearance inreflected light.5.9.4.
Testing priors against new data
Finally, it will be possible for observers to validatetheir priors during the course of the direct imaging sur-vey. As a simple example, if more faint planets are dis-covered than predicted under the assumed priors, thenit is likely that either the occurrence rates under-predictat small radii or there are more low-albedo planets thanare present in the solar system. This information couldthen be forwarded into a probabilistic analysis of whichplanets are most likely to be Earth-like before substan-tial time is committed for follow-up observations. CONCLUSIONSWe have developed a Bayesian framework with whichto infer the properties of a directly imaged planet on thebasis of limited photometric data, with the primary goalof identifying exo-Earth candidates for deeper spectro-scopic follow-up. This framework is dependent on a mul-titude of priors drawn from observed exoplanet statisticsand a few theoretical models. We use it to characterizethe ability of future direct imaging missions to identifypotentially habitable planets upon their initial detectionusing only photometry. We determine the key ambigui-ties involved in this determination and explore potentialsolutions, such as constraining the orbit through multi-ple revisits.Assuming a uniform prior on a monochromatic albedo,we have found that the detection data alone is not suffi-cient to determine whether the planet has a potentiallyhabitable size or orbit. In the best cases, a few exo-Earth candidates could be identified with ∼ con-fidence, but the average EEC would only be identified dentifying Exo-Earth Candidates in Direct Imaging Data ∼ confidence. This translates to a potentialfalse discovery rate of ∼ , consistent with previousresults (Guimond & Cowan 2018).Constraints on the planet’s orbit could be achievedthrough a precursor RV survey or by revisiting the sys-tem multiple times. This would allow the observer toconstrain the planet to the habitable zone with confi-dence, but would not eliminate the problem of false pos-itives posed by very small or large planets in the hab-itable zone. A mass measurement could be somewhatuseful for ruling these false positives out, but would byno means be definitive. By revisiting a system multi-ple times to establish its orbit and measuring the massastrometrically, an observer could still only distinguishEECs from false positives with a typical confidence of ∼ - but could also identify several individual EECswith high confidence (>90%).The use of color information to discriminate betweenEECs and false positives could dramatically reduce theseambiguities, but requires that prior assumptions bemade about the possible appearance of planets (e.g., asa function of their size and insolation). While currentdata and models do not allow such prior assumptions,in principle this could change over the next decade. Inour example, we find that by revisiting a planet to es-tablish its orbit and measuring its brightness in threebands, the strong majority of EECs could be identifiedwith confidence.Even though we are not always able to confidentlyidentify EECs using our method, we show that a tar-get prioritization strategy which leverages a Bayesianapproach will be more efficient than a non-probabilisticapproach with the same data. Such an approach could reduce the time required for preliminary spectroscopicfollow-up by a factor of two.Our confidence in these results is dependent on ourconfidence in the priors we have chosen to use, and someof these - such as the radius and semi-major axis dis-tribution of temperate planets around G dwarfs - arebased on extrapolation. Many could be improved in thecoming decade - such as the relationship between planetradius, mass, and bulk composition through transit andradial velocity observations and advances in planet for-mation theory, or the reflected spectra of hot and coldsub-Neptunes with JWST, WFIRST, and ELTs. Near-term efforts to improve our prior knowledge will enablefuture observers to more efficiently survey nearby sys-tems in search of potentially habitable worlds.ACKNOWLEDGMENTSThe authors are grateful to Christopher Stark for pro-viding a sample of host stars from the LUVOIR yieldestimates, and Ilaria Pascucci for offering feedback onthe manuscript. A.B. acknowledges support from theNASA Earth and Space Science Fellowship Program un-der grant No. 80NSSC17K0470. The results reportedherein benefited from collaborations and/or informationexchange within NASA’s Nexus for Exoplanet SystemScience (NExSS) research coordination network spon-sored by NASA’s Science Mission Directorate. This re-search has made use of NASA’s Astrophysics Data Sys-tem. Software:
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