Detectability of Free Floating Planets in Open Clusters with JWST
AA CCEPTED FOR PUBLICATION IN A P J L
ETTERS ON
OVEMBER
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DETECTABILITY OF FREE FLOATING PLANETS IN OPEN CLUSTERS WITH JWST F ABIO P ACUCCI , A NDREA F ERRARA , E LENA
D’O
NGHIA , S CUOLA N ORMALE S UPERIORE , P
IAZZA DEI C AVALIERI , 7 56126 P
ISA (I TALY ) U NIVERSITY OF W ISCONSIN , 475 C
HARTER S T ., M ADISON , WI 53706 (USA) A LFRED
P. S
LOAN F ELLOW
Accepted for publication in ApJ Letters on 4 November 2013
ABSTRACTRecent observations have shown the presence of extra-solar planets in Galactic open stellar clusters, as in thePraesepe (M44). These systems provide a favorable environment for planetary formation due to the high heavy-element content exhibited by the majority of their population. The large stellar density, and corresponding highclose-encounter event rate, may induce strong perturbations of planetary orbits with large semimajor axes.Here we present a set of N -body simulations implementing a novel scheme to treat the tidal effects of externalstellar perturbers on planetary orbit eccentricity and inclination. By simulating five nearby open clusters wedetermine the rate of occurrence of bodies extracted from their parent stellar system by quasi-impulsive tidalinteractions. We find that the specific free-floating planet production rate ˙ N o (total number of free-floatingplanets per unit of time, normalized by the total number of stars) is proportional to the stellar density ρ (cid:63) of thecluster: ˙ N o = αρ (cid:63) , with α = (23 ± × − pc Myr − . For the Pleiades (M45) we predict that ∼ ofstars should have lost their planets. This raises the exciting possibility of directly observing these wanderingplanets with the James Webb Space Telescope in the NIR band. Assuming a surface temperature of the planetof ∼ K, a free-floating planet of Jupiter size inside the Pleiades would have a specific flux of F ν (4.4 µ m) ≈ × nJy, which would lead to a very clear detection ( S/N ∼ ) in only one hour of integration. Keywords: stars: kinematics and dynamics open clusters and associations: general planet-star interactionsplanetary systems methods: numerical INTRODUCTION
The first detection of an extra-solar planet orbiting a Sun-like star in a dense stellar environment dates back to 2012,when a NASA-funded team discovered Pr0201b and Pr0211binside the Praesepe open cluster (Quinn et al. 2012). Thiscluster contains more than 1000 stars, having a core radius of ≈ . pc and a distance from the Earth of ≈ pc (Kraus &Hillenbrand 2007). The newly discovered planet belongs tothe type of extra-solar planets termed Hot Jupiters (Mayor &Queloz 1995; Charbonneau et al. 2000), i.e., massive gas gi-ants that, unlike Jupiter, orbit very close to their parent stars.Hot Jupiters are the easiest planets to be detected with the ra-dial velocity or transit methods, due to their high mass andsmall semimajor axis (Moulds et al. 2013). However, the ori-gin of this extra-solar planet population is still uncertain. Sev-eral studies suggested that it is likely to result from a planetarymigration process that dramatically decreases the semimajoraxis of giant planets formed far away from the star, beyond theso-called ice line (Lin & Pringle 1976; Goldreich & Tremaine1980; Fabrycky & Tremaine 2007).Given the first confirmed planet detection in the Praesepeopen cluster, it is likely that several other planets are alsopresent in high-density environments. In particular, a studyby Meibom et al. (2013) shows that the frequency of plan-ets inside of clusters is the same as in the field. Indeed, thehigh metallicity found in most of the open clusters facilitatesthe process of planetary formation (Fischer & Valenti 2005;Chatterjee et al. 2012) and their relatively high stellar density,about 500 times larger than in the Solar neighborhood (Nilak-shi et al. 2002), makes stellar fly-by’s a very likely process.As a result, close encounters between stars in an open clusterare common and may dramatically affect the orbits of planetswith large semimajor axes (Adams et al. 2006). As planetary eccentricities grow, they may get captured into closer orbits,becoming Hot Jupiters; see, e.g., Wu & Lithwick (2012).Stellar fly-by’s and their effects on planets have been re-cently studied in various contexts. In the Solar System, forinstance, it is thought that the orbit of Sedna is a consequenceof stellar fly-by’s (Brasser et al. 2012), and recently Adams(2010) discussed the truncation of the Kuiper Belt by fly-by’s. In addition, stellar fly-by’s have also been investigatedin more recent works as a cause of planetary ejections (Ve-ras & Raymond 2012; de Juan Ovelar et al. 2012; Craig &Krumholz 2013; Parker & Quanz 2012). Most of the theo-ries of planet formation suggest that planets should reside inresonances (Masset & Snellgrove 2001), but these are foundto be seldom (Wright et al. 2011a). One possibility is thatstellar fly-by perturbations lead to the break-up of multireso-nant states (Morbidelli et al. 2007; Batygin & Brown 2010),triggering large-scale instabilities that could explain the ran-dom a - e distribution of exoplanets. Additional complicationsarise from the fact that solar systems can eject planets ontheir own through instabilities (Rasio & Ford 1996; Nesvorn´y2011; Batygin et al. 2012; Nesvorn´y & Morbidelli 2012; Bo-ley et al. 2012).In what follows we present the results of numerical simu-lations following the impulsive perturbation of planetary or-bits with orbital distances from the parent star in the range (5 − AU , roughly from the orbital distance of Jupiter tothe farthest extra-solar planets with a clear semimajor axis de-tection discovered so far (see exoplanets.org, Wright et al.2011b). Several planets with much larger semimajor axeshave been discovered (Luhman et al. 2011), but the associateduncertainty is also much larger. Our aim is to show that a siz-able fraction of planets within this distance range are ejectedfrom their planetary systems by tidal interactions with nearby a r X i v : . [ a s t r o - ph . E P ] N ov stars. METHODOLOGY
We have developed a numerical code to accurately describethe tidal effects of a gravitational encounter between a Jupiter-like planet orbiting a central star, and a perturbing star movingon a straight line or parabolic trajectory (see D’Onghia et al.2010 for the geometry of the problem). The close encounterhas been studied in a four-parameter space, namely (a) thedistance between the parent star and the planet, d p , (b) the in-terstellar distance at pericenter, D , (c) the inclination betweenthe perturber and planet orbital planes, i , and (d) the relativevelocity of the perturber, v rel . Each parameter interval hasbeen sampled with five points, for a total of 625 different val-ues. For any given point in the parameter space, the resultingvariations of the orbital eccentricity and inclination are savedin the “interaction matrix”. This matrix is used as a sub-gridmodel for a N -body simulation of an open cluster. The use ofa sub-grid approach presents some limitations with respect toa fully self-consistent model of N particles. Nonetheless, thesub-grid model is less computationally expensive than othermethods, allowing us to build a better statistics on the escap-ing planets. Building the Interaction Matrix
The following Table 1 summarizes the values of the inter-action parameters:
Table 1
Interaction Parameters D (AU) v rel (AU/yr) i ( ◦ ) d p (AU)30 0.01 0 5100 0.05 10 151000 0.1 30 3010000 0.5 45 45100000 1.0 60 60A gravitational encounter between two stars yields a modi-fication of the planetary orbit, caused by an energy transferbetween the perturbing star and the planet. The interactionmatrix provides information about the values of ∆ e and ∆ i for all the evaluation points.Denoting by M the mass of both stars and m the mass ofthe planet, we fix the system of reference on the parent starand suppose that the perturber follows a straight line trajec-tory. The parabolic trajectory case has also been implementedas in D’Onghia et al. (2010), but the total number of free-floating planets differs in the two cases by less than . forall the five clusters. The initial orbital plane of the planet co-incides with the x − y plane, with the x axis pointing in thedirection of the pericenter. In this reference frame, the forceacting on the planet can be decomposed into the centripetalforce exerted from the parent star and the tidal force on theplanet, caused by the passage of the perturber. The equationsof motion are solved via a Runge-Kutta fourth-order method.The orbital eccentricity and the inclination are evaluated dur-ing the integration with the following expressions: i = arctan (cid:32) z (cid:112) x + y (cid:33) (1) e = (cid:115)(cid:18) (cid:15)λ µ (cid:19) (2) where (cid:15) = E/m is the specific orbital energy, λ = L/m isthe specific angular momentum and µ = G ( M + m ) . Theparent star is displaced at the origin of the reference frameand remains at rest during the entire evolution of the system.A single planet is assigned to each star and it is displaced on acircular orbit at a distance specified by the interaction matrix.The perturbing star has the following initial conditions for theposition: [ x, y, z ] = [ D cos( i ) , y shift , D sin( i )] (3)The value of y shift is calculated assuming that the perturberreaches the pericenter at half of the integration time t end . Wechoose to integrate up to a time when the ratio between themagnitude of the perturber-planet force and the magnitude ofthe parent star-planet force is equal to a constant, T int : t end = 1 v × (cid:115) M d p mT int − ( D − d p ) (4)The time evolution (for a single point in the parameter space)yields the total variations in eccentricity and inclination. The Open Cluster Simulation
The code NBODY6 (Aarseth 1999) has been modified, withthe addition of a separate routine to manage the planetary con-figurations, which is called once at each time step. A sim-ulation of N stars stars has been initialized with a Plummersteady-state distribution (Ernst et al. 2011) in the phase-space.All these objects are one solar mass stars, without any stellarevolution, and the cluster is supposed to be isolated in space.This physical system has been simulated up to the age τ ofthe stellar cluster. Our simple model consists of only singlestars, although the binary star fractions in open clusters areobserved to be high. This is a limitation of our model sincethe encounters with binary stars would likely be quite com-mon, but we do believe that our simple model provides a ro-bust lower limit on the number of free-floating planets in thesehigh-density stellar environment.The choice of the Plummer model instead of more accuratemodels (e. g. the King ones) is justified by the fact that, unlikethe globular clusters, open clusters are not completely relaxedobjects, so that the difference between the two models is notsignificant. The cluster is subject to a natural evaporation withtime due to high-speed encounters which eject stars from thecluster. As a consequence, the number of interacting stars isalso variable with time.Most of the empirical radial distributions of exoplanetsfound in literature are related to inner planets, with semimajoraxis smaller than a few AUs (see e.g. Bovaird & Lineweaver2013). Unfortunately, planets in exactly the range we areprimarily interested in ( ∼ AU ) are the most difficult todetect. Given this observational bias and focusing on therange (5 − AU of semimajor axis, it is possible to fit theobserved distribution of planets (retrieved from the databaseexoplanets.org) with an exponential decay function, namely N ( r ) = A × e ( − r/a ) , where A is a normalization constant, a is the e-folding length and N ( r ) is the number of planets witha semi-major axis distance r from the central star. The initialdistribution of eccentricities is flat, with e = 0 . .Every gravitational encounter between a pair of stars causesa tidally-driven modification of the planetary orbit of their twoplanets. For every star in the simulation, the routine loopsover all the other stars and computes D , i and v rel , while d p has already been assigned. The corresponding variations ofthe planetary inclination and eccentricities are calculated byinterpolation from the sub-grid model and applied only whenthe interstellar distance calculated between the pair of starshas increased with respect to the same quantity calculated atthe last time step. This approximation is supported by thefact that the variations of eccentricities and inclinations in thetested encounters are very focused around the time of peri-center. The accuracy in the determination of the pericenter isrelated to the magnitude of the time step used by NBODY6. Ifthe pericenter falls between two time steps, the values calcu-lated from the interaction matrix for the orbital modificationsare always underestimated. A partial solution for this problemis to increase the values of ∆ e and ∆ i by a fraction of theirvalues which is proportional to the following quantity: Rv rel ∆ T D (5)where ∆ T is the numerical time step (when it goes to zero,the correction vanishes) and R is a random number with uni-form probability between 0 and 1, which expresses our lackof knowledge of the real position of the pericenter. We sup-pose that the orbit of each planet remains circular through-out the simulation. Firstly, we assumed that the radius re-mains constant: this situation is physically unrealistic, but of-fers a very secure lower-limit for the number of free-floatingplanets. Then, we assumed that the radius is modified pro-portionally to the eccentricity, with the simple scaling law: d new ∼ d old × (1 + e ) . In this second case, the total numberof free-floating planets increases by roughly 9%.Several checks on the overall consistency of the simulationhave been performed, as the one concerning the effect of themean field, i.e. the cumulative gravitational effects of the starsthat are very far from the star under investigation. Its effect onthe orbital parameters of a planet is negligible when comparedto the gravitational effects of a strong encounter. We performa fit of the ever-increasing eccentricity of the planet before astrong encounter and extrapolate the fit up to the final timeof the simulation, noticing that the corresponding variation ofthe eccentricity ( ∆ e ∼ − ) is negligible for our purposes.For the very same reason, also the N -body encounters, with N > , have negligible effects on the extraction rate of theplanets.If the total energy of a planet becomes positive, it escapesfrom its planetary system. Such free-floating planet is treatedas a separate particle, with velocity magnitude equal to the es-cape velocity and random direction. Wandering planets inter-act with all the other stars, but not with other wandering plan-ets and their dynamics is followed by another routine addedto NBODY6. If a free-floating planet approaches another starwithin a distance of 250 AU, its total energy, E , with respectto that star is computed and, if negative, the planet is con-sidered as re-captured and is eliminated from the list of free-floating planets (Perets & Kouwenhoven 2012). RESULTS
A summary of our results for five different galactic openclusters (NGC188, NGC6530, M16, M44, M45) is reportedin Figure 1, where we plot the specific free-floating planetproduction rate (total number of free-floating planets per unitof time, normalized by the total number of stars) as a functionof the central stellar density of the open cluster.For our simulations, we have used the parameters shown inTable 2 for the five galactic open stellar clusters: −3 ]0.00000.00050.00100.00150.00200.00250.0030 Sp e c i f i c F r ee - F l o a t i n g P r o d u c t i o n R a t e [ M y r − ] M16 M44 M45NGC188NGC6530
Figure 1.
Specific free-floating planet production rate as a function of theopen cluster central stellar density, for five different galactic open clusters.
Table 2
Open Clusters DataNumber of stars Core radius (pc) Age (Myr)M44 1391 1.3 700M45 3151 1.4 100M16 1560 1.4 2.5NGC6530 365 1.4 2NGC188 5388 2.9 6000The number of stars is derived using the astronomicaldatabase SIMBAD, which also allows to set the membershipprobability. In any case, the five different galactic open clus-ters present in the paper have been chosen in order to providea wide range of stellar densities in the cluster, as long as dif-ferent ages.The relation between the specific free-floating planet pro-duction rate ˙ N o and the stellar density ρ (cid:63) is given by ˙ N o = αρ (cid:63) where α = (23 ± × − pc Myr − . See also Spurzemet al. (2009) where similar results are obtained. This strik-ingly simple relation indicates that the presence and abun-dance of free-floating planets is clearly related to the envi-ronmental stellar density. In addition, these clusters formedmore massive than they are today and with very different cen-tral densities and density structures, so that our predictionsfor the free-floating planets formation rate are again strictlylower limit estimates. The evolution of the eccentricity of theplanetary population of the Pleiades is shown in Figure 2.The contour plot describes the overall increase with time ofthe eccentricity of the planetary population. A direct compar-ison between our results, obtained with a sub-grid approach,and a fully self-consistent integration of a system of N par-ticles has been made. In the case of the Pleiades cluster, thetotal number of escapers is higher by less than 10% with theuse of a self-consistent integration. OBSERVABILITY OF FREE-FLOATING PLANETS
As free-floating planets do not orbit around a star, their di-rect detection, usually hampered by the unmanageably highcontrast with the star for normal planets (Sumi et al. 2011;Dong et al. 2013; Delorme et al. 2012; Beichman et al.2013; Gould & Yee 2013), might be possible in the infraredwith JWST (Burrows et al. 2003). Figure 3 shows a sim-
Figure 2.
Time evolution of the eccentricities of the planetary populationof the Pleiades, up to 100 Myrs of evolution. The contour plot shows thedistribution of eccentricities as a function of time, along with the distributionof eccentricities for the population of bounded planets at the final simulationtime. The lowermost orange bar refer to (unperturbed) planets that are stillin the initial eccentricity bin ( e ≈ . ). This population is the dominantone and it has not been included in the contour plot for the sake of clarity.Finally, the top panel reports the cumulative number of free-floating planetsas a function of time, reaching a level ∼ at the end of the simulation. Free-floating planet @ 500 K
Figure 3.
Right panel: Simulated image of the central region of the Pleiades,after 10 Myrs of evolution showing the expected flux at λ = 4 . µm fromstars and free-floating planets. Left panel: a typical field of the NIRCamcamera onboard the future JWST observatory. The arrow shows the locationof a free-floating planet with a surface temperature of ∼ K. ulated image (for the typical 2.2 arcmin field of view ofthe JWST/NIRCam instrument) of the central region of thePleiades cluster, evolved up to 10 Myrs.The values of the flux (in nJy, at 4.4 µ m) are calculated as-suming the appropriate spectrum for both the stars (assumedto be Sun-like) and the free-floating planets (assumed to havea surface temperature T eff = 500 K). The free-floating planetinside the field is indicated with a white arrow; its flux is F ν (4.4 µ m) ≈ × nJy and it is detectable in s of in-tegration time with a S/N ∼ (see JWST Exposure TimeCalculator for reference).Free-floating planets can then be detected by the NIRCamif T eff > ∼ K, because for lower surface temperatures thespecific flux would be too faint. For example, with T s = 300 K, the corresponding specific flux would be F ν (4 . µm ) ≈ nJy, which needs ∼ hr of integration to reach S/N ∼ level. The actual observation of free-floating planets implies thepossibility of distinguishing them from very far away stars(which are redshifted to the IR) and brown dwarfs alreadyformed in isolation inside the open cluster. The fact thatbackground stars are dynamically detached should serve as amean to distinguish them from our free-floating planets pop-ulation, thanks to their higher redshift and possibly differentproper motion. The distinction between free-floating planetsextracted from bounded planetary systems and isolated browndwarfs is more difficult. The rate of formation of isolatedbrown dwarfs inside open clusters may be deduced from thelow-mass end of the IMF. Comparing the theoretical predic-tion with the observed population of free-floating planets, theexcess probability over the formation rate of isolated browndwarfs could be accounted as a measure of the occurrence ofthe free-floating planets, predicted by our model. CONCLUSIONS
In this work we have shown that tidal interactions betweenstars in relatively high-density environments, such as openstellar clusters, may well cause the ejection of external plan-ets from their systems. By means of using an accurate sub-grid model setup for a large parameter space, we have per-formed a purely N -body simulation of different Galactic openstellar clusters, with different morphological properties andages. Assigning to each star a planetary body of Jupiter-massand variable distance from the parent star, we have studiedthe occurrence of free-floating planets over the entire evo-lution of each cluster, also accounting for the possibility ofa planet being recaptured by another star. In particular, wehave shown that the specific free-floating planet productionrate ˙ N o is linearly related with the stellar density of the clus-ter ρ (cid:63) : ˙ N o = αρ (cid:63) , with α = (23 ± × − pc Myr − .Specifically, for the Pleiades we predict that ∼ of thestars should have lost their planet during the evolution of thecluster.The high contrast with the star for bounded planets posesvery strict, and usually unmanageable, limits on their directobservation. On the contrary, the large population of free-floating planets predicted in this work might be detected withthe next generation of space telescopes, given that their sur-face temperature is sufficiently high. The study on observabil-ity shows that even with a relatively low-temperature planetat K, a detection may be feasible in the infrared band,using the NIRcam instrument onboard the future JWST ob-servatory. A clear detection becomes much more feasible ifthe free-floating planets (or at least a fraction of them) reachtemperatures of at least ∼ K. The coolest brown dwarfcandidates have very similar surface temperatures (Luhmanet al. 2012), so the lower bound of ∼ K might be reachedby a large fraction of the population of free-floating planets.The detection of free-floating planets would open a new path-way to exoplanetary and stellar-cluster studies, allowing us totest the survival and evolution of planets under extreme inter-stellar conditions.We are grateful to Kostantin Batygin for suggestions anda careful reading of the manuscript. ED gratefully acknowl-edges the support of the Alfred P. Sloan Foundation.REFERENCES