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Accepted forpublicationin TheAstrophysical Journal PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 IMPROVED PARAMETERS FOR EXTRASOLAR TRANSITING PLANETS Guillermo Torres1, Joshua N. Winn2, and Matthew J. Holman1 Draft version2008 February 2 ABSTRACT We present refined values for the physical parameters of transiting exoplanets, based on a self- consistentanduniformanalysis oftransitlightcurves andthe observablepropertiesof the hoststars. Previouslyithasbeendifficulttointerprettheensemblepropertiesoftransitingexoplanets,becauseof the widely different methodologies that have been applied in individual cases. Furthermore, previous 8 studiesoftenignoredanimportantconstraintonthe meanstellardensitythatcanbe deriveddirectly 0 0 from the light curve. The main contributions of this work are i) a critical compilation and error 2 assessment of all reported values for the effective temperature and metallicity of the host stars; ii) the application of a consistent methodology and treatment of errors in modeling the transit light n curves; and iii) more accurate estimates of the stellar mass and radius based on stellar evolution a models, incorporating the photometric constraint on the stellar density. We use our results to revisit J somepreviouslyproposedpatternsandcorrelationswithintheensemble. Weconfirmthemass-period 6 correlation,and we find evidence for a new pattern within the scatter about this correlation: planets 1 aroundmetal-poorstarsaremoremassivethanthosearoundmetal-richstarsatagivenorbitalperiod. Likewise, we confirm the proposed dichotomy of planets according to their Safronov number, and we ] h find evidence that the systems with small Safronov numbers are more metal-rich on average. Finally, p we confirm the trend that led to the suggestion that higher-metallicity stars harbor planets with a - greater heavy-element content. o r Subject headings: methods: data analysis — planetary systems — stars: abundances — stars: fun- t damental parameters — techniques: spectroscopic s a [ 1. INTRODUCTION able diversity of planet characteristics that has been 3 found among the members of the transiting ensemble. The transiting exoplanets are only a small subset v The accuracy and precision with which the properties 1 of all the known planets orbiting other stars, but of the planet can be derived from transit data depend 4 they hold tremendous promise for deepening our un- stronglyonwhatevermeasurementsandassumptionsare 8 derstanding of planetary formation, structure, and evo- 1 lution. Observations of transits and occultations3 made regarding the host star. For example, for a given value of the transit depth, the inferred planetary radius . (along with the spectroscopic orbit of the host star) 1 scalesinproportiontotheassumedstellarradius;andfor not only allow one to measure the mass and ra- 0 a given spectroscopic orbit of the host star, the inferred dius of the planet, but also provide opportunities to 8 planetarymassscalesasthetwo-thirdspowerofthestel- 0 measure the stellar spin-orbit alignment (Queloz et al. lar mass. In the literature on transiting planets, a wide : 2000; Winn et al. 2007a), the planetary brightness v varietyofmethodshavebeenusedtoestimatetheradius temperature (Charbonneau et al. 2005; Deming et al. i andmassoftheparentstar,rangingfromsimplylooking X 2005), the planetary day-night temperature difference themupinatableofaveragestellarpropertiesasafunc- (Knutson et al.2007),andevenabsorptionlinesofplane- r tion of spectral type, all the way to fitting detailed stel- a taryatmosphericconstituents(Charbonneau et al.2002; lar evolutionary models constrained by the luminosity, Vidal-Madjar et al.2004;Tinetti et al.2007). Theseand effective temperature, and other observations that may otherobservationshavebeenaccompaniedbytheoretical be available for the star. As a result, the ensemble of progress in modeling the physical processes in the plan- planet properties at our disposal is inhomogeneous, and etary interiors and atmospheres, as well as the planets’ in many cases the uncertainty ofthose determinations is interactions with their parent stars. This rapid progress dominatedbysystematicerrorsinthestellarparameters has been stimulated in no small measure by the remark- that are treated differently by different investigators. Electronicaddress: [email protected] This situation is unfortunate because it hinders our 1Harvard-Smithsonian Center for Astrophysics, 60 Garden St., ability to gauge the reliability of any patterns that are Cambridge,MA02138, USA discerned among the ensemble properties of transiting 2Department of Physics, and Kavli Institute for Astrophysics exoplanets. With 23 systems that have been reported andSpaceResearch,Massachusetts InstituteofTechnology, Cam- bridge,MA02139, USA in the literature, this subfield should be poised for the 3Thewordtransitissometimesassumedinthefieldofexoplanet transitionfromahandfulofresultstoalargeanddiverse research to be synonymous with eclipse. In reality, it has a more enough sample for meaningful general conclusions to be restrictedmeaningandhaslongbeenusedinthe eclipsingbinary drawn,buttheheterogeneityofreportedresultsisclearly fieldtodescribeaneclipseofthelargerobjectbythesmallerone. Thetermoccultation isusedtorefertothepassageofthesmaller an obstacle. object (in this case, the planet) behind the larger one (the star) Itmaybe surprisingthatwearelimitedinmanycases (see, e.g., Popper 1976). To avoid confusion, we advocate that by our knowledge of the properties of the parent stars; occultation orsecondary eclipse arepreferabletoneologisms such one would think that stellar physics is a “solved prob- as“secondary transit”or“anti-transit”. 2 Torres et al. lem” in comparison to exoplanetary physics. However (inunitsofthemassofJupiter),whereiistheinclination it must be remembered that the host stars are usually angle of the orbit, P is the orbital period in days, e is isolated (and therefore no dynamicalmass measurement the eccentricity, and K the velocity semi-amplitude of ⋆ is possible), and that many of the host stars are dis- the star in ms−1. Even when sini is known precisely, tant enough that they do not even have measured par- the value of the planetary mass M that is derived from p allaxes. Of the 23 cases in the literature, five have Hip- the data will scale as M2/3, where M is the mass of the ⋆ ⋆ parcos parallaxes (Perryman et al. 1997). For those few star. Meanwhile, the light curve does not immediately it is fairly straightforward to estimate the stellar prop- yield R , the planetary radius; rather, the ratio of the p erties, but for the other systems, more indirect methods radii R /R is pinned down through the depth of the p ⋆ have been used. These indirect methods often rely on transit. the value of the stellar surface gravity that is derived The stellar mass and radius are usually inferred in- by measuring the depths and shapes of gravity-sensitive directly, from an analysis of high-resolution spectra of absorption lines in the stellar spectrum. This is a no- the star with the aid of model atmospheres, followed by toriously difficult measurement and the result is often acomparisonofthe atmospheric parameterswith stellar stronglycorrelatedwithotherparametersthataffectthe evolutionmodels. Thelatterstepisperformedsomewhat spectrum. Recently,however,Sozzetti et al.(2007),Hol- differently by different authors depending on the obser- man et al. (2007), and others demonstrated that it is vational constraints available. For this work we have possible to do better by using the informationabout the compiled and critically reviewed all of the available in- mean stellar density that is encoded in the transit light formation regarding the atmospheric properties of the curve. This information was typically overlooked prior host stars. This effort is described in 2.1. to these studies. § The approach adopted in this paper makes use of in- Thestudypresentedinthispaperwasmotivatedbythe formation from the light curves of transiting planets to desire for a homogeneous analysis, and by the desire to constrain the mean density of the star, ρ , following ⋆ takeadvantageofthephotometricestimatesofthestellar Sozzetti et al.(2007). Asdescribedtherein,thequantity meandensity. Wehaverevisitedthedeterminationofthe a/R (the planet-star separation a in units of the stel- ⋆ stellar parameters for all of the transiting planets that lar radius) is directly related to the mean stellar density have been reported in the literature. We have taken the (hereafter, simply the density), and can be derived from opportunity to merge all existing measurements of the the photometry without much knowledge about the star atmospheric parameters (mainly the effective tempera- (see below). The only dependence a/R has on the stel- ⋆ tureandmetallicity)withthegoalofpresentingthebest lar properties is through the limb-darkening coefficients, possible values. We have chosen a uniform method for which is typically a second-order effect (see 2.2). analyzing photometric data and have re-analyzed exist- § Many of the published light curve analyses do not re- inglightcurveswherenecessarytoprovidehomogeneity. port the value of a/R explicitly, and it is difficult or ⋆ Our hope was that by applying these procedures across impossible to reconstruct its value accurately from the the board,we and other investigatorscouldview the en- published information. It is especially difficult to obtain semble properties with greater clarity and uncover any a good measure of the true uncertainty in a/R from ⋆ interesting clues the transiting planets might provide us the published information. For this reason, and for the about the origin, structure, and evolution of exoplanets. sake of homogeneity, we have re-analyzed many of the This paper is organized as follows. 2 describes the high-quality photometric time-series available to us for § procedures by which we estimated the stellar proper- all transiting planets. We describe this effort in 2.2. ties, using the available spectroscopic and photometric § datasets,andaparticularsetoftheoreticalstellarevolu- 2.1. Atmospheric parameters tionmodels. As a checkonthe models, 3comparesthe § results of a subset of our calculations with those derived High-resolutionspectroscopicstudieshavebeenunder- from a different set of evolutionary models. 4 investi- taken for almost all of the parent stars of the known § gates alternate ways of estimating the stellar properties. transiting planets. The basic products of these studies 5dealswith GJ436,whichneeds specialtreatmentbe- are measurements of the effective temperature T , iron § eff causethe hoststarhassucha lowermassthanthe other abundance[Fe/H],andsurfacegravitylogg. Thequoted hoststars. 6presentsthe finalresults forthe planetary precision of these determinations varies widely, depend- § parameters. 7 uses the new results to check on some ing on the signal-to-noise ratio and resolution of the ob- § of the previously proposedcorrelations among the prop- servations, the modeling techniques employed, and the erties of the transiting ensemble, and 8 provides final attitude taken toward systematic errors. § remarks. The Appendix lists the data sets and other For this work we have relied on published determina- issues that are particular to each system. tions, rather than any new spectroscopic data. In many 2. DETERMININGTHESTELLARPROPERTIES cases, a given transiting system has been described by more than one analysis, and they do not necessarily Fora transitingplanet, the basicdata arethe spectro- agree. In those cases, rather than arbitrarily choosing scopic orbit of the star (radial-velocity curve), and the themostrecentstudy,ortheonewiththesmallestuncer- photometric observations of transits (light curve). With tainties,wecriticallyexaminedalloftheavailablestudies such data, the planetary mass and radius cannot be de- and combined them. Our choices are documented in the termined independently of the stellar properties. The Appendix. We were guided by our own experience and radial-velocity curve can be used to determine increased the error estimates whenever they seemed op- Mpsini=4.919 10−3P1/3(1 e2)1/2K⋆[(M⋆+Mp)/M⊙]2/3 timistic. For example, many of the automated spectro- × − (1) scopic analysistools in common use today return formal Improved parameters 3 uncertaintiesfortheeffectivetemperaturesthatareonly face gravity, and metallicity. Thus, to some degree, the a few tens of degrees Kelvin; for one of the transiting determination of R /R , a/R , and i must be accompa- p ⋆ ⋆ systems the quoted error was only 13 K. There is ample nied by some assumptions about the stellar properties. literatureonthesubjectoftheabsoluteeffectivetemper- Our procedure was as follows. We modeled each sys- ature scale and the systematic errorsinherent in placing temusingatwo-bodyKeplerianorbit. Thestarhasmass any single system on such a scale. A recent investiga- M and radius R , and the planet has mass M and ra- ⋆ ⋆ p tionbyRam´ırez & Mel´endez(2004)hasshownthatthere diusR . TheorbithasperiodP,eccentricitye,argument p arestilldifferences oforder100Kbetweentemperatures of pericenter ω, and inclination i. For our purpose the derived from the spectroscopic condition of excitation uncertaintyinP wascompletelynegligible;wefixedP at equilibrium and from the Infrared Flux Method (IRFM, the most precisely determined value in the literature. In Blackwell & Shallis 1977; Blackwell et al. 1980). Other almost all cases, the radial-velocity data are consistent studieshaveindicateddiscrepanciesof50–100Kbetween with a circularorbit, andwe assumede=0 exactly. For different temperature scales (Ram´ırez & Mel´endez 2005; HAT-P-2 (HD 147506)we fixed the values of e and ω at Casagrande et al. 2006), and discussed at length possi- those that have been derived from radial-velocity data ble sources for these errors. In light of these findings, (and in the end we verified that changing these param- weconsideredit prudentto adopttemperature errorsno eters by 1σ does not significantly affect any of our final smaller than 50 K for our study, and then only when results). The initial condition is specified by a particu- there are several independent and consistent determina- lar midtransit time T . When the sky projections of the c tions or other evidence supporting that level of accu- star and planet do not overlap, the model flux is unity. racy. Similar concerns hold for the spectroscopic sur- When they do overlap we use the analytic formulas of face gravities, although we do not actually make use of Mandel & Agol(2002)tocompute the integralofthe in- them in this work, as described below. For the spec- tensity over the unobscured portion of the stellar disk, troscopic metallicities, we adopted a minimum uncer- assuming a quadratic limb-darkening law. tainty of 0.05 dex (for the best cases with multiple in- To arrive at a self-consistent solution including limb dependent measurements), even though smaller errors darkening, we began with initial values for M and M ⋆ p have occasionally been reported for individual analy- from the literature. We also chose values for the stel- ses. This is mainly because of the strong correlations lar T , logg, and metallicity, as described in the previ- eff presentamong[Fe/H],T ,andlogg(e.g.,Buzzoni et al. oussection. Thenweadoptedlimb-darkeningcoefficients eff 2001),aswellassomeevidenceforsystematicdifferences based on those stellar parameters, by interpolating the between different groups (see, e.g., Santos et al. 2004; tables of Claret (2000, 2004) for the appropriate band- Fischer & Valenti 2005; Gonzalez & Laws 2007). In the pass. At this point we estimated all of the remaining Appendixweprovideacompletelistingofthesourceswe parameters (using the procedure described in the next havedrawnfromineachcase. ThevaluesofT ,[Fe/H], paragraph) and used the result for a/R to update the eff ⋆ and logg finally adopted for all systems are listed in Ta- determinationofthestellarproperties. Then,anewpho- ble 1. Some degree of non-uniformity in these quantities tometric parameter estimation was performed, with re- is unavoidable due to the variety of procedures used by vised values of M , M and the limb-darkening coeffi- ⋆ p different authors, but we believe they represent the best cients, and so forth. This procedure convergedafter two availablesetfortheparentstarsbasedoncurrentknowl- or three iterations, in the sense that further iterations edge. changed none of the parameters by more than about a tenth of the statistical error. 2.2. Light curve fits The parameter estimations were carried out with a Foreachsystem,weexaminedthehighest-qualitytran- Markov Chain Monte Carlo algorithm (MCMC; see, sit photometry available to us in order to determine the e.g., Tegmark et al. (2004) for applications to cosmo- key parameters R /R , a/R , and i. Our methodol- logical data, Ford (2005) for radial-velocity data, and p ⋆ ⋆ ogy is described below, and the details of the data sets Holman et al. (2006) or Burke et al. (2007) for a similar that were used in each case are given in the Appendix. approachto transitfitting). It is basedonthe goodness- In some cases, the published determinations of R /R , of-fit statistic p ⋆ a/R , and i matched our own methodology very closely, and⋆we simply adopted the values from the literature; χ2 = Nf fj(obs)−fj(calc) 2, (2) these cases are also specified in the Appendix. σ j=1(cid:20) j (cid:21) Inthe absenceoflimbdarkening,the fourprimaryob- X servablesinatransitlightcurvearethe midtransittime, where f (obs) is the flux observedat time j and σ con- j j thedepth,thetotalduration,andthepartial-phasedura- trols the weights of the data points, and f (calc) is the j tion(ingressoregress). Asequenceofmeasuredmidtran- fluxcalculatedwithourmodel. IntheMCMCalgorithm, sittimes usuallyleads to a veryprecisedeterminationof astochasticprocessisusedtocreateasequenceofpoints theorbitalperiodP. Thedepthisequaltotheplanet-to- inparameterspacewhosedensityapproximatesthejoint starradiusratio,(R /R )2. AtfixedP,theparametersi a posteriori probability density for all parameters. One p ⋆ anda/R canbewrittenintermsofthetotalandpartial begins with an initial point and iterates a jump func- ⋆ durations through an application of Kepler’s Law (see, tion, which in our case was the addition of a Gaussian e.g., Seager & Mall´en-Ornelas2003). With limb darken- random number (“perturbation”) to a randomly chosen ing, however, there is no longer a well-defined depth or parameter. If the new point has a lower χ2, the jump is partial duration; an accurate light curve model must in- executed; if not, the jump is executed with probability clude a realistic intensity distribution across the stellar exp( ∆χ2/2). We adjusted the sizes of the perturba- − disk, which will depend on the stellar temperature, sur- tions until approximately 25% of jumps are executed ∼ 4 Torres et al. for each parameter. can be calculated from the models as Forthedataweightsσ ,weusedtheobservedstandard deviationof the out-of-tjransitdata (σ1), multiplied by a a = G 1/3 P2/3 (M +M )1/3 , (3) factor β 1 to account at least approximately for time- R 4π2 R ⋆ p ≥ ⋆ (cid:18) (cid:19) ⋆ correlatederrors(“red”noise)whichareoftensignificant for ground-based data. We chose β as follows. The key where G is the Newtonian gravitational constant, and timescale is the partial-phase duration, because the lim- the period P is well known from the photometry. The itingerrorinthedeterminationofa/R⋆andiisgenerally planetmassMp isnotknownapriori,butitsinfluenceis thefractionalerrorinthepartial-phaseduration. Weav- very small compared to the stellar mass M⋆, and even a eraged the out-of-transit data over this timescale, with roughvalueisusuallysufficientforthisapplication. Once each time bin consisting of N points depending on the the stellar mass is known,the process canbe repeatedif cadence of observations, and then calculated the stan- necessary with an improved value of Mp. dard deviation of the binned data, σ . Finally we set The stellar evolution models we use are those from N β = σ √N/σ . With white noise only, we would ob- the Yonsei-Yale (Y2) series by Yi et al. (2001) (see also N 1 ∗ Demarque et al. 2004), which are conveniently provided serve β = 1, but in practice β > 1 because the number with tools for interpolating isochrones in both age and ofeffectivelyindependentdatapointsissmallerthanthe metallicity.4 We explore the full range of metallicities actualnumber ofdata points. Forthis paper wedeliber- allowed by the observational errors in [Fe/H], sampling ately chose to analyze only those data for which β <2. 20 equally spaced values for each system. For each For each parameter, we took the mode of the MCMC ∼ metallicity we consider a range of ages from 0.1 to 14 distributionaftermarginalizingoverallotherparameters Gyr, in steps of 0.1 Gyr. These isochrones are inter- to be the “best value.” We defined the 68% confidence polated to a fine grid in mass, and compared point by limits p and p as the values for which the integral of lo hi point with the measuredvalues of T and a/R . All lo- the distribution between p and p is 0.68, and the in- eff ⋆ lo hi cations(“matches”)onthe isochronethat areconsistent tegrals from the minimum value to p and from p to lo hi with these quantities within the observationalerrors are themaximumvaluewereboth0.16=(1.00 0.68)/2. In − recorded. We also record the corresponding likelihood some cases the mode was all the way at one end of the given by exp( χ2/2), where probability distribution; in particular there were several − cases in which i = 90◦ was the mode of the inclination 2 2 2 distribution. In those cases we report phi as 90◦ and plo χ2 = ∆[Fe/H] + ∆Teff + ∆(a/R⋆) as the value for which the integral from zero to p was σ σ σ lo (cid:18) [Fe/H] (cid:19) (cid:18) Teff (cid:19) (cid:18) a/R⋆ (cid:19) (1.00 0.68) = 0.32. The final results are given in Ta- ble 2,−including the stellar density ρ computed directly and the ∆ quantities represent the difference between ⋆ from a/R and the period. the observed and model values at each point. Observa- ⋆ tionalerrorsareassumedto be Gaussian,andthe asym- metric error bars in a/R were taken into account. The ⋆ 2.3. Stellar masses, radii, luminosities, surface best-fit values for each stellar property are obtained by gravities, and ages computing the sum over all matches, weighted by their The fundamental parametersfor the hoststarsarede- corresponding likelihood. Additionally, we account for rivedhereusingstellarevolutionmodels. Werelyonthe the varyingdensity ofstarsoneachisochroneprescribed spectroscopicallydetermined T and[Fe/H], andwe re- by the Initial Mass Function (IMF), by multiplying the eff quire also an indicator of luminosity (L ) or some other weights by the number density of stars at each location ⋆ measure of evolution. Since most of these stars lack a as provided with the Y2 isochrones. The IMF adopted parallax measurement, the spectroscopic surface gravity is a power law with a Salpeter index. The effect of this has often been used in the past as a luminosity indi- latter weighting is generally small. cator. The effect of logg on the spectral lines is rela- TheresultsforeachsystemarelistedinTable3,where tively subtle, and strong correlations between logg and wegiveinadditiontothemassandradiusthetheoretical bothtemperatureandmetallicitymakethesedetermina- valuesoflogg⋆,luminosityL⋆,absolutevisualmagnitude tions challenging. Seager & Mall´en-Ornelas (2003) have MV, and evolutionary age. In several cases our results pointed out that an important property intrinsic to the differslightlyfromthosereportedinrecentdiscoverypa- star, the density, is encoded in the transit light curves pers that use the same atmospheric parameters adopted mainly through its dependence on the transit duration. here and apply the a/R⋆ constraint essentially in the As shown there, and more explicitly by Sozzetti et al. samewaywehave. Thepresentstudyrepresentsaslight (2007),thedensityisdirectlyrelatedtoa/R ,oneofthe improvementfor those systems due to the application of ⋆ parametersoften solvedfor in modeling the photometry. weights, as described above. Thisquantitycantypicallybedeterminedmoreprecisely The relative errors of the stellar masses and radii de- than logg and it is highly sensitive to the degree of evo- termined here have median values of about 6% and 4%, lution of the star, i.e., to its size. Thus, it serves as a better proxy for luminosity in many cases. We illustrate 4 In addition to the ironabundance, the enhancement of the α elements can also have a significant effect on the inferred stellar this below. properties. About halfofthe transitingsystems have atleastone To determine the stellar mass and radius, and other study in the literature reporting abundances for several of the α relevantpropertiesofthehoststars,wefollowtheproce- elements (Mg, Si, S, Ca, Ti, C, and O): HD 209458 (3 studies), dure described by Sozzetti et al. (2007) with minor im- TrES-1 (2 studies), WASP-1, XO-1, XO-2, and the five OGLE systems (one study each). In all cases the average enhancement provements,andcomparemodelisochronesdirectlywith is not significantly different from zero. We therefore assume here themeasuredvaluesofTeff,[Fe/H],anda/R⋆. Thelatter thatitiszeroforallsystems. Improved parameters 5 Fig. 1.— Evolutionary tracks for all host stars except GJ 436 fromtheYonsei-YalemodelsofYietal.(2001). Thenumberingof Fig. 2.—Evolutionaryageversusmassfortransitingplanethost thesystemsfollowsthatinthetables. Themassesandmetallicities starsbasedonthemodels byYietal.(2001). GJ436isexcluded adopted for the tracks, as well as the location of each star on the (see text). The numbering of the systems is the same as in the diagram,arefromthebestfittotheobservations. tables. respectively,butwithwide rangesdepending onthe pre- cision of the observables (2% to 13% for σ /M , and M⋆ ⋆ 1.3% to 12% for σ /R ). While all of the stars in the R⋆ ⋆ currentsamplearehydrogen-burningstars,thedegreeof evolution within the main sequence varies considerably (see Figure 1). Some systems such as TrES-3 (#11) and XO-1(#15) arenear the zero-agemainsequence; others likeHAT-P-4(#20)havealreadylivedfor 90%oftheir ∼ main-sequence lifetime. The age is a critical ingredient for the theoretical modeling of the structure and evolu- tion of exoplanets (see, e.g., Burrows et al. 2007), yet it is among the most difficult properties to determine for isolated main-sequence stars. For the host stars of tran- siting planets the evolutionary ages span the full range, as seen in Figure 2. The difficulty mentioned above is most evident for the less evolved objects (M less than ⋆ about 0.9 M⊙), which are all seen to have large error barsthatcanreachthe upper limitof14Gyrconsidered here. For those cases the nominal ages reported in this work should be used with caution. The surface gravities inferred from the models (logg ; ⋆ Table 3) have formal uncertainties that are typically Fig. 3.— Surface gravities for the planet host stars inferred about5timessmallerthanthosemeasuredspectroscopi- from stellar evolution models by Yietal. (2001), compared with those measured spectroscopically. The dotted line represents the cally(logg ;Table1). Thisreflectsthestrengthofthe spec one-to-one relation, and the systems are numbered as inprevious constraintprovidedbya/R⋆. Thevalues ofloggspec and figures. logg are compared against each other in Figure 3. On ⋆ averagethey agreequite well (the mean O C difference ing double-lined eclipsing binaries with very accurately − is 0.027 dex), and the rms scatter of the differences measured masses and radii have shown that the agree- − is 0.15 dex although three of the systems present differ- ment with theory is in general very good, and is within ences larger than 0.2 dex. An illustration of the supe- a few percent for solar-type stars (see, e.g., Andersen riorconstraintaffordedbya/R is showninFigure 4 for ⋆ 1991; Pols et al. 1997; Lastennet & Valls-Gabaud2002). OGLE-TR-132 and WASP-2, two of the more dramatic Asasimpletest,weconsideredasecondsetofmodelsby examples. In neither case is the parallax known. Girardi et al.(2000)thathasoftenbeenusedbyotherin- vestigatorsinthefieldoftransitingplanets. Weusedthe 3. COMPARISONWITHOTHERMODELS isochroneinterpolationtools providedonthe website at Thereisundoubtedlysomesystematicerrorintroduced the Osservatorio Astronomico di Padova5 to explore the by imperfections in the stellar evolution models them- agreement with the observed values of T , [Fe/H], and eff selves, but this type of error is difficult to evaluate. Ex- tensivecomparisonsbetweenmodelsandobservationsus- 5 http://stev.oapd.inaf.it/∼lgirardi/cgi-bin/cmd. 6 Torres et al. Fig. 4.—Measuredpropertiesoftwoextrasolarplanethoststars displayedon plots analogous tothe H-R diagram. The constraint onthelocationofthestarsbasedonthesurfacegravities(loggspec) andtemperaturesisshowninthetoppanels,andthesamefora/R⋆ versusTeff isshowninthebottompanels. Isochronesarefromthe seriesofevolutionary models byYietal.(2001)forthemeasured metallicity in each case (Table 1), and are shown for ages of 1 to 13Gyr(lefttoright)insteps of1Gyr. Thea/R⋆ valuesareseen to provide a much better handle on the stellar parameters (mass, radius,etc.). a/R , and to infer the stellar mass and other properties ⋆ in the same way as above. There are small differences in the physical assumptions between these models and thosefromtheY2 series,butoveralltheyarerathersim- ilar. ForthisapplicationthePadovamodelsareavailable for metallicities smaller than Z = 0.030 (corresponding to [Fe/H] = +0.20), which allows us to compare results for nine of the transiting systems.6 Figure 5 displays this comparisonforthe stellarmassesandradii,showing that in all cases the results from both models agree to well within their uncertainties. Similar tests were carried out using the models of Fig. 5.— Comparison of the stellar masses and radii de- Baraffe et al.(1998). Thesemodelsenjoywidespreaduse rivedfromthe Y2 models (Yietal.2001)andthe Padova models (Girardietal.2000)fortransitingplanethosts,showingtheexcel- forlower-massstarsandbrowndwarfs,butthey arealso lentagreement. Thedottedlinesrepresenttheone-to-onerelation, computedformassesaslargeas1.4M⊙. They areavail- andthenumberingofthesystemsisthesameasinpreviousfigures. ableforthreedifferentvaluesofthemixinglengthparam- eter α over restricted ranges in mass and metallicity. ML The value that fits the observed properties of the Sun of solar. Of the six systems in this range, one (OGLE- (against which all models are calibrated) is α = 1.9. TR-113) gave no solution consistent with the observed ML Forcoolerobjects near the bottom ofthe main sequence values of Teff and a/R⋆. This is because the Baraffe the value used almost universally is α =1.0. Here we models are limited to ages less than about 10 Gyr for ML adoptthe formervalue,since ourstarsareallmoremas- this mass range, whereas both the Yi et al. (2001) and sive than about 0.8 M⊙ with the exception of GJ 436, theGirardi et al.(2000)modelsindicateanageofabout which we discuss in more detail in 5. For the mass 13 Gyr for this star. Whether OGLE-TR-113 is truly range of interest and for α = 1.9§the Baraffe et al. thisoldisunknown. Itseemsatleastaslikelythatsome ML (1998)modelsarepubliclyavailableonlyforsolarmetal- of the other observational constraints are in error. The licity, so the comparison was limited to transiting sys- remaining 5 systems show excellent agreement with the tems with [Fe/H] within about 0.1 dex of the Sun, and resultsfromboththeY2 modelsandthePadovamodels, with [Fe/H] uncertainties that keep them within 0.2 dex within the formal uncertainties. Thesetestsofthestellarevolutionarymodelsareobvi- 6 Higher-metallicity Padova models have been published by ouslynotexhaustive,anditmaywellbethecasethatall Salasnichetal.(2000),butthoseemployphysicalassumptionsdif- ofthesetsofmodelsweconsideredhavesomedeficiencies ferent enough that they cannot be merged with the Girardietal. in common. However, the general pattern of agreement (2000) models we use here. In addition, from a practical point of does lend some degree of confidence to the results. We view, the use of these higher-metallicity models is not yet imple- mentedintheonlineinterpolationroutineasofthiswriting. proceedundertheassumptionthatthesystematicerrors Improved parameters 7 inthesecalculationsarenotthedominantsourceoferror inthestellarparametersderivedhere(exceptperhapsfor OGLE-TR-113,as noted above). 4. ADDITIONALOBSERVATIONALCONSTRAINTS As a further test of the accuracy of our stellar radius determinations, in this section we consider the consis- tency check provided by the near infrared (NIR) surface brightness(SB)relations,whichyieldtheangulardiame- terφofastardirectlyintermsofitsapparentmagnitude and color. Unlike the parallax, which would in principle yield the stellar luminosity but is known for only five of the brighter systems in the sample, φ can be computed for all host stars from existing photometry. We use the empirical relation logφ =c (V K)+c 0.2K (4) SB 1 2 − − derivedbyKervella et al.(2004),inwhichthecoefficients arec =0.0755 0.0008andc =0.5170 0.0017,theK- 1 2 ± ± bandmagnitudeisintheJohnsonsystem,andφ isthe SB limb-darkened value of the angular diameter expressed Fig. 6.— Angular diameters φSB computed from the near in- frared surface brightness relation of Kervellaetal. (2004) com- in milli-arc seconds. The relation represented by eq.(4) pared against the values derived from our modeling, making use is extremely tight, with a scatter well under 1%. Near of the apparent V magnitudes for all host stars and ignoring ex- infrared magnitudes are available for all stars from the tinction. The dotted line represents the one-to-one relation, and 2MASS catalog and were transformed to the Johnson thesystemsarenumberedasinpreviousfigures. system following Carpenter (2001). The best available the argument around and use the φ values as further SB V magnitudes collected from the literature are listed in constraints in our modeling. By trial and error we find Table 1. thatauniformreddeningvaluebetweenE(B V)=0.02 − Angular diameters from our stellar evolution model- and0.03issufficienttoreducetheaveragediscrepancyin ing in 2.3 can be derived for comparison with φ by the angular diameters for the fainter stars to zero. This SB § making use of our theoretical radii and absolute visual modest amount of reddening is consistent with expecta- magnitudes in Table 3, along with the apparent V mag- tions for objects that are between 100 and 500 pc away, nitudes, using as are these. In addition to the angular diameters, Table 4 lists the φ =9.3047R /100.2(V−MV+5). (5) mod ⋆ parallaxes π and corresponding distances we derive mod With the stellar radius expressed in solar units, the nu- from V and MV, ignoring extinction. Hipparcos par- merical constant is such that φmod is in mas. Neither allaxes πHIP are given for comparison as well, for the this equation nor the previous one take into account in- objects that have them. terstellarextinction,althougheq.(4)isactuallyquitein- sensitive to extinction. We discuss this below. 5. THECASEOFGJ436 The comparison between the angular diameters from As the only M dwarf among the currently known eq.(4) and eq.(5) is shown in Table 4 for all systems transiting planet host stars, GJ 436 presents a special except for GJ 436, for reasons to be described in the challenge. The Y2 stellar evolution models we used next section, and the OGLE stars, which are likely to in all other cases are not intended for the lower main be significantly affected by extinction since they lie sev- sequence, as they lack the proper non-grey model at- eralkpcawayneartheGalacticplane. Theuncertainties mosphere boundary conditions to the interior equations listedinclude allcontributions fromthe photometryand that have been shown to be critical for cool objects model-derivedquantities,aswellastheerrorsinthecoef- (see, e.g., Chabrier & Baraffe 1997). The Padova mod- ficientsofeq.(4). Theprecisionofφ istypicallyseveral els of Girardi et al. (2000) have the same shortcoming, SB times better than that of φ . A graphical comparison although both of these models are perfectly adequate mod is shown in Figure 6, with the one-to-one relation repre- for hotter stars. On the other hand, the calculations sentedwithadiagonalline. Thegoodagreementoverthe by Baraffe et al. (1998) with α = 1.0 are specifically ML full range of an order of magnitude in φ is an indication designed for low-mass stars, and have in fact been in- that our model radii from 2.3 are accurate. voked by other authors for this system as a rough con- § One effect of extinction in this analysis would be to sistency check. For the most part, the previous mass make the values based on eq.(5) appear too small. Ex- determinations for GJ 436 have relied upon empirical amination of the differences shows a hint of this, partic- mass-luminosity relations, but even those relations have ularly among the fainter (more distant) stars: the av- their problems. Previous radius estimates for GJ 436 erage difference (φ φ )/φ is +0.5 1.8% for have often rested on the assumption of numerical equal- mod SB SB h − i ± the 6 objects brighter than V = 11, and 3.9 1.6% ity between M and R for M stars (Gillon et al. 2007a; ⋆ ⋆ − ± for the others. Since reddening values for individualsys- Deming et al. 2007). tems are presently unknown, and cannot be determined The importance of the GJ 436 system is undeniable, accurately enough from the information at hand, we are asitharborsthesmallesttransitingplanetthathasbeen unable to correctforthis. We areequallyunable to turn found to date (comparable in size to Neptune). Fur- 8 Torres et al. thermore, it is the nearest transiting exoplanet, at a tioned above. We do not believe so, because the mass distance of only 10 pc. Given these facts, it is some- regime of GJ 436 is very different. The host stars of the what surprising that until recently the mass of the star other transiting planets are typically more than twice as ( 0.4M⊙)wasonly knownto about10%(Maness et al. massive as GJ 436. ∼ 2007;Gillon et al.2007b),onaparwiththeworstofthe determinations in Table 3 (that of WASP-2,with poorly 6. PLANETARYPARAMETERS determined spectroscopic parameters and more than 10 The combination of the stellar properties in Table 1 timesfartheraway). Similarlimitationsholdforthestel- withthelightcurveparametersinTable2,alongwiththe lar radius. As a result, our knowledge of the planetary measured orbital periods and velocity semi-amplitudes, parameters has suffered. Torres (2007) showed that the yields the homogeneous set of planet parameters pre- Baraffe et al.(1998)modelsintheiroriginalformdonot sented in Table 5. In addition to the planetary mass yield satisfactory results for GJ 436 since different an- and radius, we have computed and tabulated the plan- swers for M⋆ and R⋆ are obtained depending on which etary surface gravity and mean density, as well as the constraints are used (including the absolute magnitude, orbital semimajor axis and other useful characteristics. since a reliable parallax measurement is available from As pointed out by Southworth et al. (2007), Winn et Hipparcos). Moreover,someoftheotherpredictedquan- al. (2007), Beatty et al. (2007), and others, the surface tities are at odds with empirical determinations. These gravity of a transiting planet can be derived without difficulties had not been previously emphasized. knowledge of the stellar mass or radius, using only the However, as described by Torres (2007), reliable pa- radial-velocitycurveandthetransitlightcurve. Wetake rameters can still be obtained from the models by mod- this opportunity to correct the general expression for ifying the theoretical radii and temperatures in such a logg presented by Sozzetti et al. (2007), valid also for p wayastopreservethebolometricluminosities. Thisrec- the case of eccentric orbits, which neglected to account ognizes the fact that R⋆ and Teff as predicted by theory fortheprojectionfactorimplicitintheimpactparameter havebeenshowntodisagreewithaccuratemeasurements b as derived from the light-curve fits: for M dwarfs in double-lined eclipsing binaries (see, e.g., logg = 2.1383 logP +logK (6) Popper 1997; Clausen et al. 1999; Torres & Ribas 2002; p ⋆ − − − Ribas 2003; Lo´pez-Morales & Ribas 2005), whereas the 1 b 1 e2 2 luminosities appear to be unaffected (Delfosse et al. log 1 − + 2000; Torres & Ribas 2002; Ribas 2006; Torres et al. −2 −(cid:20)a/R⋆1+esinω(cid:21) ! 2006). Torres(2007)appliedsimultaneouslytheobserva- a/R 1 tionalconstraintsonT ,a/R ,thecolorindexJ K,and +2log ⋆ + log(1 e2) . theabsoluteK-bandmeffagnitud⋆eM ,andallowe−dthera- (cid:18)Rp/R⋆(cid:19) 2 − K dius/temperatureadjustmentfactortobeafreeparame- The numerical constantis such that the gravity is in cgs ter. Inthiswayaself-consistentsolutionwasachievedfor units when P and K are expressed in their customary ⋆ all parameters, and in addition the radius/temperature units of days and m s−1. factor showed excellent agreement with previous esti- Thefirstpropertiesonegenerallywantstoknowabout mates for other M dwarfs. The stellar mass and radius a transiting planet are its mass and radius. The large are M⋆ =0.452+−00..001142 M⊙ and R⋆ =0.464+−00..000191 R⊙. We size measured for the first transiting planet discovered, adopttheseresultshere,thusplacingGJ436onasimilar HD 209458b,has been widely debated and still presents footing as the other transiting systems in that the prop- somewhat of a challenge to theory. It is now accom- erties of the host stars are all based on current stellar panied by several other inflated planets, underlying our evolution models. incompleteknowledgeofthephysicsoftheseobjects. An The results for GJ 436arelistedin Table 3 alongwith updated version of the now classical diagram of M ver- p those of the other 22 stars. A few words of caution are susR isshowninFigure7,inwhichfiveotherexamples p in order. As pointed out by Torres(2007), the predicted are seen to be at least as large as HD 209458b (#3), or absolute visual magnitude for this star is unreliable due perhaps even larger,given the uncertainties. to missing molecular opacity sources shortward of 1 µm The fundamental physical properties of the known in the model atmospheres that are used as boundary transitingplanetscoveraconsiderablerange—morethan conditions in the stellar evolution calculations (see, e.g., two orders of magnitude in mass, a factor of nearly 5 Baraffe et al. 1998; Delfosse et al. 2000). Also, because in radius, and a factor of 3 in orbital separation—and of the unevolved status of the star, the nominal age is these are discussed in the next section. But the geomet- probablymeaninglesssincetheobservationalconstraints ric properties that determine the light-curve shapes are allow any age within the range of 1–10 Gyr explored in also varied. In Figure 8 we present a “portrait gallery” the modeling effort of Torres (2007). Finally, this mod- of all known transiting planets. Stars and planets are eling effort assumed that the stellar metallicity is solar. rendered to scale, emphasizing the wide range of stellar ForGJ436thisisavalidassumptionsincethemeasured types probed by the photometric searches. The orbital composition is [Fe/H] = 0.03 0.20 (see Appendix), geometries are indicated with solid horizontal lines rep- − ± but the uncertainty in [Fe/H] was not accountedfor due resenting an edge-on orientation, and the path of each to the lack of proper models, and therefore the errors of planet across the stellar disk shown with dotted lines the stellar propertiesin Table 3 may be underestimated. at the actual impact parameter. (Of course, the data do A careful reader may wonder whether our use of the notdistinguishbetweenpositiveandnegativeimpactpa- Baraffe et al. (1998) models in 3 as a check on the rameters.) The resulting light curves calculated for the results from the Y2 isochrones i§s contradicted by our V band are all shown with the same vertical (flux) and remarks on the radius/temperature discrepancies men- horizontal(time) scale, to facilitate comparison. Depths Improved parameters 9 residuals. We seem to have found such a third vari- able: the metallicity of the host star. Figure 9b displays the O C residuals from the top panel as a function − of [Fe/H], indicating a rather clear correlation (dashed line): ∆M =(+0.152 0.050) (1.17 0.23) [Fe/H]. p ± − ± × Afterremovalofthistrend,therelationbetweenM and p period becomes tighter (Figure 9c). The scatter in the mass-period relation is reduced from 0.26 M in the Jup toppanelto0.17M inthebottompanel. Itseemsun- Jup likelythatthisisastatisticalfluke. However,thescatter isstilllargerthantheformalobservationaluncertainties, suggesting that these three variables are not completely determinative. What might be the implications of this metallicity de- pendence? It has been proposed that the mass-period relation is related to the process by which close-in ex- oplanets migrated inward from their formation sites, or morespecifically, to the mechanismthat halts migration at orbital periods of a few days. The trend of larger masses at shorter orbital periods could suggest that the halting mechanism depends on mass, and larger planets Fig. 7.—Mass-radiusdiagramforalltransitingplanets(HAT-P- areabletomigratefurtherin. Thedependenceonmetal- 2isoffthescale,withMp =8.7MJup). Linesofconstant density licitymaythenbeinterpretedtoindicatethatplanetsin areshown. metal-poor systems need to be more massive in order to vary quite significantly (3 mmag in V for HD 149026b, migrateinwardtothesameorbitalperiodasmoremetal- 26 mmag for OGLE-TR-113b; see Table 2), as do the rich planets. In this context, the trend in Figure 9b overallshapesofthetransitevents,whichinseveralcases could be interpreted as evidence that the efficiency of are grazing enough to depart from the canonicalprofiles themigration(orhalting)mechanismisaffectedtosome often depicted in the literature. degree by the chemical composition. A dependence of migrationonmetallicityisinfactpredictedbysomethe- 7. DISCUSSION ories, and could arise, as pointed out by Sozzetti et al. Many investigators have sought and claimed possible (2006), either from slower migration rates in metal-poor correlations between various stellar and planetary pa- protoplanetary disks (Livio & Pringle 2003; Boss 2005) rametersoftransitingsystems. Inprinciplesuchcorrela- or through longer timescales for giant planet forma- tionscouldleadtoimportantinsightsintotheformation, tion around metal-poor stars, which would effectively structure, and evolution of exoplanets. A primary mo- reduce the efficiency of migration before the disk dis- tivation for presenting a more complete, accurate, and sipates (Ida & Lin 2004; Alibert et al. 2005). However, homogeneous set of these parameters in this work was the above processes are more aimed at addressing the to facilitate such studies. The relatively large array of apparentlackofshort-periodplanets amongverymetal- properties now available offers the opportunity to find poor stars claimed by some authors (see Sozzetti et al. new correlations,orto revisitold ones incorporatingad- 2006),whereasamongthetransitingplanetsitisnotthe ditional variables. While it is beyond the scope of the lackofmoremetal-poorexamplesweareconcernedwith, present work to investigate all possible correlations with but rathertheir different properties (suchas mass) com- statistical rigor, in this section we check on three of the pared to metal-rich planets at the same orbital period. most intriguing and potentially important relations that Togiveaquantitativeexample,wefindthatforaperiod have been proposed. of 2.5 days (near the average for known transiting sys- tems), the mass of a planet with [Fe/H] = 0.2 is 40% 7.1. Planetary mass versus orbital period − ∼ largerthanonewithaveragemetallicity([Fe/H]=+0.13, Mazeh et al. (2005) were the first to point out the ap- excluding HAT-P-2b and GJ 436b), while the mass of a parentcorrelationbetweenM andP fortransitingplan- planet with [Fe/H] = +0.4 is about 30% smaller. This p ets (see also Gaudi et al. 2005). The original suggestion rangeofmetallicities,from[Fe/H]= 0.2to+0.4(cover- − was based on only 6 systems, but additional discoveries ingafactorof4inmetalenhancement),isapproximately have generally supported the trend of decreasing mass thefullrangeobserved. Wenotethatthestrengthofthe with longer periods, although the scatter has also be- metallicity effect is modest rather than overwhelming, come larger. This is shown in Figure 9a. HAT-P-2b and this too is in agreement with theoretical expecta- would be an extreme outlier in this plot; we have ex- tions (e.g., Livio & Pringle 2003). The massive planet cluded it because it is so much more massive than the HAT-P-2b obviously does not conform to this trend of other planets and may belong to a different category of Mp versus period, which may indicate some fundamen- planet (see 7.2). Similarly, we have excluded GJ 436b tal difference either in its formation or migration. because it is§so much less massive than the others, and An alternative interpretation, also proposed by maybearockyorrock-iceplanetratherthanagasgiant. Mazeh et al. (2005), is that the Mp versus P relation is A simple linear fit is shown for reference (dashed line). more a reflection of survival requirements in close prox- We also investigated the scatter in this relation, seek- imity to the star, due to thermal evaporation from the ing any “third variable” that might correlate with the extreme UV flux. Close-in planets must be more mas- 10 Torres et al. Fig. 8.— “Portraitgallery”of transiting extrasolar planets. Star and planet sizes areshownto scale. The vertical and horizontal axes ofalllightcurvesarealsoonthesamescale. Planettrajectoriesareshownwiththeirmeasuredimpactparameters(dottedlines). Orbital periodsareindicatedineachpanel.

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Preview Improved parameters for extrasolar transiting planets

Accepted forpublicationin TheAstrophysical Journal PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 IMPROVED PARAMETERS FOR EXTRASOLAR TRANSITING PLANETS Guillermo Torres1, Joshua N. Winn2, and Matthew J. Holman1 Draft version2008 February 2 ABSTRACT We present refined values for the physical parameters of transiting exoplanets, based on a self- consistentanduniformanalysis oftransitlightcurves andthe observablepropertiesof the hoststars. Previouslyithasbeendifficulttointerprettheensemblepropertiesoftransitingexoplanets,becauseof the widely different methodologies that have been applied in individual cases. Furthermore, previous 8 studiesoftenignoredanimportantconstraintonthe meanstellardensitythatcanbe deriveddirectly 0 0 from the light curve. The main contributions of this work are i) a critical compilation and error 2 assessment of all reported values for the effective temperature and metallicity of the host stars; ii) the application of a consistent methodology and treatment of errors in modeling the transit light n curves; and iii) more accurate estimates of the stellar mass and radius based on stellar evolution a models, incorporating the photometric constraint on the stellar density. We use our results to revisit J somepreviouslyproposedpatternsandcorrelationswithintheensemble. Weconfirmthemass-period 6 correlation,and we find evidence for a new pattern within the scatter about this correlation: planets 1 aroundmetal-poorstarsaremoremassivethanthosearoundmetal-richstarsatagivenorbitalperiod. Likewise, we confirm the proposed dichotomy of planets according to their Safronov number, and we ] h find evidence that the systems with small Safronov numbers are more metal-rich on average. Finally, p we confirm the trend that led to the suggestion that higher-metallicity stars harbor planets with a - greater heavy-element content. o r Subject headings: methods: data analysis — planetary systems — stars: abundances — stars: fun- t damental parameters — techniques: spectroscopic s a [ 1. INTRODUCTION able diversity of planet characteristics that has been 3 found among the members of the transiting ensemble. The transiting exoplanets are only a small subset v The accuracy and precision with which the properties 1 of all the known planets orbiting other stars, but of the planet can be derived from transit data depend 4 they hold tremendous promise for deepening our un- stronglyonwhatevermeasurementsandassumptionsare 8 derstanding of planetary formation, structure, and evo- 1 lution. Observations of transits and occultations3 made regarding the host star. For example, for a given value of the transit depth, the inferred planetary radius . (along with the spectroscopic orbit of the host star) 1 scalesinproportiontotheassumedstellarradius;andfor not only allow one to measure the mass and ra- 0 a given spectroscopic orbit of the host star, the inferred dius of the planet, but also provide opportunities to 8 planetarymassscalesasthetwo-thirdspowerofthestel- 0 measure the stellar spin-orbit alignment (Queloz et al. lar mass. In the literature on transiting planets, a wide : 2000; Winn et al. 2007a), the planetary brightness v varietyofmethodshavebeenusedtoestimatetheradius temperature (Charbonneau et al. 2005; Deming et al. i andmassoftheparentstar,rangingfromsimplylooking X 2005), the planetary day-night temperature difference themupinatableofaveragestellarpropertiesasafunc- (Knutson et al.2007),andevenabsorptionlinesofplane- r tion of spectral type, all the way to fitting detailed stel- a taryatmosphericconstituents(Charbonneau et al.2002; lar evolutionary models constrained by the luminosity, Vidal-Madjar et al.2004;Tinetti et al.2007). Theseand effective temperature, and other observations that may otherobservationshavebeenaccompaniedbytheoretical be available for the star. As a result, the ensemble of progress in modeling the physical processes in the plan- planet properties at our disposal is inhomogeneous, and etary interiors and atmospheres, as well as the planets’ in many cases the uncertainty ofthose determinations is interactions with their parent stars. This rapid progress dominatedbysystematicerrorsinthestellarparameters has been stimulated in no small measure by the remark- that are treated differently by different investigators. Electronicaddress: [email protected] This situation is unfortunate because it hinders our 1Harvard-Smithsonian Center for Astrophysics, 60 Garden St., ability to gauge the reliability of any patterns that are Cambridge,MA02138, USA discerned among the ensemble properties of transiting 2Department of Physics, and Kavli Institute for Astrophysics exoplanets. With 23 systems that have been reported andSpaceResearch,Massachusetts InstituteofTechnology, Cam- bridge,MA02139, USA in the literature, this subfield should be poised for the 3Thewordtransitissometimesassumedinthefieldofexoplanet transitionfromahandfulofresultstoalargeanddiverse research to be synonymous with eclipse. In reality, it has a more enough sample for meaningful general conclusions to be restrictedmeaningandhaslongbeenusedinthe eclipsingbinary drawn,buttheheterogeneityofreportedresultsisclearly fieldtodescribeaneclipseofthelargerobjectbythesmallerone. Thetermoccultation isusedtorefertothepassageofthesmaller an obstacle. object (in this case, the planet) behind the larger one (the star) Itmaybe surprisingthatwearelimitedinmanycases (see, e.g., Popper 1976). To avoid confusion, we advocate that by our knowledge of the properties of the parent stars; occultation orsecondary eclipse arepreferabletoneologisms such one would think that stellar physics is a “solved prob- as“secondary transit”or“anti-transit”. 2 Torres et al. lem” in comparison to exoplanetary physics. However (inunitsofthemassofJupiter),whereiistheinclination it must be remembered that the host stars are usually angle of the orbit, P is the orbital period in days, e is isolated (and therefore no dynamicalmass measurement the eccentricity, and K the velocity semi-amplitude of ⋆ is possible), and that many of the host stars are dis- the star in ms−1. Even when sini is known precisely, tant enough that they do not even have measured par- the value of the planetary mass M that is derived from p allaxes. Of the 23 cases in the literature, five have Hip- the data will scale as M2/3, where M is the mass of the ⋆ ⋆ parcos parallaxes (Perryman et al. 1997). For those few star. Meanwhile, the light curve does not immediately it is fairly straightforward to estimate the stellar prop- yield R , the planetary radius; rather, the ratio of the p erties, but for the other systems, more indirect methods radii R /R is pinned down through the depth of the p ⋆ have been used. These indirect methods often rely on transit. the value of the stellar surface gravity that is derived The stellar mass and radius are usually inferred in- by measuring the depths and shapes of gravity-sensitive directly, from an analysis of high-resolution spectra of absorption lines in the stellar spectrum. This is a no- the star with the aid of model atmospheres, followed by toriously difficult measurement and the result is often acomparisonofthe atmospheric parameterswith stellar stronglycorrelatedwithotherparametersthataffectthe evolutionmodels. Thelatterstepisperformedsomewhat spectrum. Recently,however,Sozzetti et al.(2007),Hol- differently by different authors depending on the obser- man et al. (2007), and others demonstrated that it is vational constraints available. For this work we have possible to do better by using the informationabout the compiled and critically reviewed all of the available in- mean stellar density that is encoded in the transit light formation regarding the atmospheric properties of the curve. This information was typically overlooked prior host stars. This effort is described in 2.1. to these studies. § The approach adopted in this paper makes use of in- Thestudypresentedinthispaperwasmotivatedbythe formation from the light curves of transiting planets to desire for a homogeneous analysis, and by the desire to constrain the mean density of the star, ρ , following ⋆ takeadvantageofthephotometricestimatesofthestellar Sozzetti et al.(2007). Asdescribedtherein,thequantity meandensity. Wehaverevisitedthedeterminationofthe a/R (the planet-star separation a in units of the stel- ⋆ stellar parameters for all of the transiting planets that lar radius) is directly related to the mean stellar density have been reported in the literature. We have taken the (hereafter, simply the density), and can be derived from opportunity to merge all existing measurements of the the photometry without much knowledge about the star atmospheric parameters (mainly the effective tempera- (see below). The only dependence a/R has on the stel- ⋆ tureandmetallicity)withthegoalofpresentingthebest lar properties is through the limb-darkening coefficients, possible values. We have chosen a uniform method for which is typically a second-order effect (see 2.2). analyzing photometric data and have re-analyzed exist- § Many of the published light curve analyses do not re- inglightcurveswherenecessarytoprovidehomogeneity. port the value of a/R explicitly, and it is difficult or ⋆ Our hope was that by applying these procedures across impossible to reconstruct its value accurately from the the board,we and other investigatorscouldview the en- published information. It is especially difficult to obtain semble properties with greater clarity and uncover any a good measure of the true uncertainty in a/R from ⋆ interesting clues the transiting planets might provide us the published information. For this reason, and for the about the origin, structure, and evolution of exoplanets. sake of homogeneity, we have re-analyzed many of the This paper is organized as follows. 2 describes the high-quality photometric time-series available to us for § procedures by which we estimated the stellar proper- all transiting planets. We describe this effort in 2.2. ties, using the available spectroscopic and photometric § datasets,andaparticularsetoftheoreticalstellarevolu- 2.1. Atmospheric parameters tionmodels. As a checkonthe models, 3comparesthe § results of a subset of our calculations with those derived High-resolutionspectroscopicstudieshavebeenunder- from a different set of evolutionary models. 4 investi- taken for almost all of the parent stars of the known § gates alternate ways of estimating the stellar properties. transiting planets. The basic products of these studies 5dealswith GJ436,whichneeds specialtreatmentbe- are measurements of the effective temperature T , iron § eff causethe hoststarhassucha lowermassthanthe other abundance[Fe/H],andsurfacegravitylogg. Thequoted hoststars. 6presentsthe finalresults forthe planetary precision of these determinations varies widely, depend- § parameters. 7 uses the new results to check on some ing on the signal-to-noise ratio and resolution of the ob- § of the previously proposedcorrelations among the prop- servations, the modeling techniques employed, and the erties of the transiting ensemble, and 8 provides final attitude taken toward systematic errors. § remarks. The Appendix lists the data sets and other For this work we have relied on published determina- issues that are particular to each system. tions, rather than any new spectroscopic data. In many 2. DETERMININGTHESTELLARPROPERTIES cases, a given transiting system has been described by more than one analysis, and they do not necessarily Fora transitingplanet, the basicdata arethe spectro- agree. In those cases, rather than arbitrarily choosing scopic orbit of the star (radial-velocity curve), and the themostrecentstudy,ortheonewiththesmallestuncer- photometric observations of transits (light curve). With tainties,wecriticallyexaminedalloftheavailablestudies such data, the planetary mass and radius cannot be de- and combined them. Our choices are documented in the termined independently of the stellar properties. The Appendix. We were guided by our own experience and radial-velocity curve can be used to determine increased the error estimates whenever they seemed op- Mpsini=4.919 10−3P1/3(1 e2)1/2K⋆[(M⋆+Mp)/M⊙]2/3 timistic. For example, many of the automated spectro- × − (1) scopic analysistools in common use today return formal Improved parameters 3 uncertaintiesfortheeffectivetemperaturesthatareonly face gravity, and metallicity. Thus, to some degree, the a few tens of degrees Kelvin; for one of the transiting determination of R /R , a/R , and i must be accompa- p ⋆ ⋆ systems the quoted error was only 13 K. There is ample nied by some assumptions about the stellar properties. literatureonthesubjectoftheabsoluteeffectivetemper- Our procedure was as follows. We modeled each sys- ature scale and the systematic errorsinherent in placing temusingatwo-bodyKeplerianorbit. Thestarhasmass any single system on such a scale. A recent investiga- M and radius R , and the planet has mass M and ra- ⋆ ⋆ p tionbyRam´ırez & Mel´endez(2004)hasshownthatthere diusR . TheorbithasperiodP,eccentricitye,argument p arestilldifferences oforder100Kbetweentemperatures of pericenter ω, and inclination i. For our purpose the derived from the spectroscopic condition of excitation uncertaintyinP wascompletelynegligible;wefixedP at equilibrium and from the Infrared Flux Method (IRFM, the most precisely determined value in the literature. In Blackwell & Shallis 1977; Blackwell et al. 1980). Other almost all cases, the radial-velocity data are consistent studieshaveindicateddiscrepanciesof50–100Kbetween with a circularorbit, andwe assumede=0 exactly. For different temperature scales (Ram´ırez & Mel´endez 2005; HAT-P-2 (HD 147506)we fixed the values of e and ω at Casagrande et al. 2006), and discussed at length possi- those that have been derived from radial-velocity data ble sources for these errors. In light of these findings, (and in the end we verified that changing these param- weconsideredit prudentto adopttemperature errorsno eters by 1σ does not significantly affect any of our final smaller than 50 K for our study, and then only when results). The initial condition is specified by a particu- there are several independent and consistent determina- lar midtransit time T . When the sky projections of the c tions or other evidence supporting that level of accu- star and planet do not overlap, the model flux is unity. racy. Similar concerns hold for the spectroscopic sur- When they do overlap we use the analytic formulas of face gravities, although we do not actually make use of Mandel & Agol(2002)tocompute the integralofthe in- them in this work, as described below. For the spec- tensity over the unobscured portion of the stellar disk, troscopic metallicities, we adopted a minimum uncer- assuming a quadratic limb-darkening law. tainty of 0.05 dex (for the best cases with multiple in- To arrive at a self-consistent solution including limb dependent measurements), even though smaller errors darkening, we began with initial values for M and M ⋆ p have occasionally been reported for individual analy- from the literature. We also chose values for the stel- ses. This is mainly because of the strong correlations lar T , logg, and metallicity, as described in the previ- eff presentamong[Fe/H],T ,andlogg(e.g.,Buzzoni et al. oussection. Thenweadoptedlimb-darkeningcoefficients eff 2001),aswellassomeevidenceforsystematicdifferences based on those stellar parameters, by interpolating the between different groups (see, e.g., Santos et al. 2004; tables of Claret (2000, 2004) for the appropriate band- Fischer & Valenti 2005; Gonzalez & Laws 2007). In the pass. At this point we estimated all of the remaining Appendixweprovideacompletelistingofthesourceswe parameters (using the procedure described in the next havedrawnfromineachcase. ThevaluesofT ,[Fe/H], paragraph) and used the result for a/R to update the eff ⋆ and logg finally adopted for all systems are listed in Ta- determinationofthestellarproperties. Then,anewpho- ble 1. Some degree of non-uniformity in these quantities tometric parameter estimation was performed, with re- is unavoidable due to the variety of procedures used by vised values of M , M and the limb-darkening coeffi- ⋆ p different authors, but we believe they represent the best cients, and so forth. This procedure convergedafter two availablesetfortheparentstarsbasedoncurrentknowl- or three iterations, in the sense that further iterations edge. changed none of the parameters by more than about a tenth of the statistical error. 2.2. Light curve fits The parameter estimations were carried out with a Foreachsystem,weexaminedthehighest-qualitytran- Markov Chain Monte Carlo algorithm (MCMC; see, sit photometry available to us in order to determine the e.g., Tegmark et al. (2004) for applications to cosmo- key parameters R /R , a/R , and i. Our methodol- logical data, Ford (2005) for radial-velocity data, and p ⋆ ⋆ ogy is described below, and the details of the data sets Holman et al. (2006) or Burke et al. (2007) for a similar that were used in each case are given in the Appendix. approachto transitfitting). It is basedonthe goodness- In some cases, the published determinations of R /R , of-fit statistic p ⋆ a/R , and i matched our own methodology very closely, and⋆we simply adopted the values from the literature; χ2 = Nf fj(obs)−fj(calc) 2, (2) these cases are also specified in the Appendix. σ j=1(cid:20) j (cid:21) Inthe absenceoflimbdarkening,the fourprimaryob- X servablesinatransitlightcurvearethe midtransittime, where f (obs) is the flux observedat time j and σ con- j j thedepth,thetotalduration,andthepartial-phasedura- trols the weights of the data points, and f (calc) is the j tion(ingressoregress). Asequenceofmeasuredmidtran- fluxcalculatedwithourmodel. IntheMCMCalgorithm, sittimes usuallyleads to a veryprecisedeterminationof astochasticprocessisusedtocreateasequenceofpoints theorbitalperiodP. Thedepthisequaltotheplanet-to- inparameterspacewhosedensityapproximatesthejoint starradiusratio,(R /R )2. AtfixedP,theparametersi a posteriori probability density for all parameters. One p ⋆ anda/R canbewrittenintermsofthetotalandpartial begins with an initial point and iterates a jump func- ⋆ durations through an application of Kepler’s Law (see, tion, which in our case was the addition of a Gaussian e.g., Seager & Mall´en-Ornelas2003). With limb darken- random number (“perturbation”) to a randomly chosen ing, however, there is no longer a well-defined depth or parameter. If the new point has a lower χ2, the jump is partial duration; an accurate light curve model must in- executed; if not, the jump is executed with probability clude a realistic intensity distribution across the stellar exp( ∆χ2/2). We adjusted the sizes of the perturba- − disk, which will depend on the stellar temperature, sur- tions until approximately 25% of jumps are executed ∼ 4 Torres et al. for each parameter. can be calculated from the models as Forthedataweightsσ ,weusedtheobservedstandard deviationof the out-of-tjransitdata (σ1), multiplied by a a = G 1/3 P2/3 (M +M )1/3 , (3) factor β 1 to account at least approximately for time- R 4π2 R ⋆ p ≥ ⋆ (cid:18) (cid:19) ⋆ correlatederrors(“red”noise)whichareoftensignificant for ground-based data. We chose β as follows. The key where G is the Newtonian gravitational constant, and timescale is the partial-phase duration, because the lim- the period P is well known from the photometry. The itingerrorinthedeterminationofa/R⋆andiisgenerally planetmassMp isnotknownapriori,butitsinfluenceis thefractionalerrorinthepartial-phaseduration. Weav- very small compared to the stellar mass M⋆, and even a eraged the out-of-transit data over this timescale, with roughvalueisusuallysufficientforthisapplication. Once each time bin consisting of N points depending on the the stellar mass is known,the process canbe repeatedif cadence of observations, and then calculated the stan- necessary with an improved value of Mp. dard deviation of the binned data, σ . Finally we set The stellar evolution models we use are those from N β = σ √N/σ . With white noise only, we would ob- the Yonsei-Yale (Y2) series by Yi et al. (2001) (see also N 1 ∗ Demarque et al. 2004), which are conveniently provided serve β = 1, but in practice β > 1 because the number with tools for interpolating isochrones in both age and ofeffectivelyindependentdatapointsissmallerthanthe metallicity.4 We explore the full range of metallicities actualnumber ofdata points. Forthis paper wedeliber- allowed by the observational errors in [Fe/H], sampling ately chose to analyze only those data for which β <2. 20 equally spaced values for each system. For each For each parameter, we took the mode of the MCMC ∼ metallicity we consider a range of ages from 0.1 to 14 distributionaftermarginalizingoverallotherparameters Gyr, in steps of 0.1 Gyr. These isochrones are inter- to be the “best value.” We defined the 68% confidence polated to a fine grid in mass, and compared point by limits p and p as the values for which the integral of lo hi point with the measuredvalues of T and a/R . All lo- the distribution between p and p is 0.68, and the in- eff ⋆ lo hi cations(“matches”)onthe isochronethat areconsistent tegrals from the minimum value to p and from p to lo hi with these quantities within the observationalerrors are themaximumvaluewereboth0.16=(1.00 0.68)/2. In − recorded. We also record the corresponding likelihood some cases the mode was all the way at one end of the given by exp( χ2/2), where probability distribution; in particular there were several − cases in which i = 90◦ was the mode of the inclination 2 2 2 distribution. In those cases we report phi as 90◦ and plo χ2 = ∆[Fe/H] + ∆Teff + ∆(a/R⋆) as the value for which the integral from zero to p was σ σ σ lo (cid:18) [Fe/H] (cid:19) (cid:18) Teff (cid:19) (cid:18) a/R⋆ (cid:19) (1.00 0.68) = 0.32. The final results are given in Ta- ble 2,−including the stellar density ρ computed directly and the ∆ quantities represent the difference between ⋆ from a/R and the period. the observed and model values at each point. Observa- ⋆ tionalerrorsareassumedto be Gaussian,andthe asym- metric error bars in a/R were taken into account. The ⋆ 2.3. Stellar masses, radii, luminosities, surface best-fit values for each stellar property are obtained by gravities, and ages computing the sum over all matches, weighted by their The fundamental parametersfor the hoststarsarede- corresponding likelihood. Additionally, we account for rivedhereusingstellarevolutionmodels. Werelyonthe the varyingdensity ofstarsoneachisochroneprescribed spectroscopicallydetermined T and[Fe/H], andwe re- by the Initial Mass Function (IMF), by multiplying the eff quire also an indicator of luminosity (L ) or some other weights by the number density of stars at each location ⋆ measure of evolution. Since most of these stars lack a as provided with the Y2 isochrones. The IMF adopted parallax measurement, the spectroscopic surface gravity is a power law with a Salpeter index. The effect of this has often been used in the past as a luminosity indi- latter weighting is generally small. cator. The effect of logg on the spectral lines is rela- TheresultsforeachsystemarelistedinTable3,where tively subtle, and strong correlations between logg and wegiveinadditiontothemassandradiusthetheoretical bothtemperatureandmetallicitymakethesedetermina- valuesoflogg⋆,luminosityL⋆,absolutevisualmagnitude tions challenging. Seager & Mall´en-Ornelas (2003) have MV, and evolutionary age. In several cases our results pointed out that an important property intrinsic to the differslightlyfromthosereportedinrecentdiscoverypa- star, the density, is encoded in the transit light curves pers that use the same atmospheric parameters adopted mainly through its dependence on the transit duration. here and apply the a/R⋆ constraint essentially in the As shown there, and more explicitly by Sozzetti et al. samewaywehave. Thepresentstudyrepresentsaslight (2007),thedensityisdirectlyrelatedtoa/R ,oneofthe improvementfor those systems due to the application of ⋆ parametersoften solvedfor in modeling the photometry. weights, as described above. Thisquantitycantypicallybedeterminedmoreprecisely The relative errors of the stellar masses and radii de- than logg and it is highly sensitive to the degree of evo- termined here have median values of about 6% and 4%, lution of the star, i.e., to its size. Thus, it serves as a better proxy for luminosity in many cases. We illustrate 4 In addition to the ironabundance, the enhancement of the α elements can also have a significant effect on the inferred stellar this below. properties. About halfofthe transitingsystems have atleastone To determine the stellar mass and radius, and other study in the literature reporting abundances for several of the α relevantpropertiesofthehoststars,wefollowtheproce- elements (Mg, Si, S, Ca, Ti, C, and O): HD 209458 (3 studies), dure described by Sozzetti et al. (2007) with minor im- TrES-1 (2 studies), WASP-1, XO-1, XO-2, and the five OGLE systems (one study each). In all cases the average enhancement provements,andcomparemodelisochronesdirectlywith is not significantly different from zero. We therefore assume here themeasuredvaluesofTeff,[Fe/H],anda/R⋆. Thelatter thatitiszeroforallsystems. Improved parameters 5 Fig. 1.— Evolutionary tracks for all host stars except GJ 436 fromtheYonsei-YalemodelsofYietal.(2001). Thenumberingof Fig. 2.—Evolutionaryageversusmassfortransitingplanethost thesystemsfollowsthatinthetables. Themassesandmetallicities starsbasedonthemodels byYietal.(2001). GJ436isexcluded adopted for the tracks, as well as the location of each star on the (see text). The numbering of the systems is the same as in the diagram,arefromthebestfittotheobservations. tables. respectively,butwithwide rangesdepending onthe pre- cision of the observables (2% to 13% for σ /M , and M⋆ ⋆ 1.3% to 12% for σ /R ). While all of the stars in the R⋆ ⋆ currentsamplearehydrogen-burningstars,thedegreeof evolution within the main sequence varies considerably (see Figure 1). Some systems such as TrES-3 (#11) and XO-1(#15) arenear the zero-agemainsequence; others likeHAT-P-4(#20)havealreadylivedfor 90%oftheir ∼ main-sequence lifetime. The age is a critical ingredient for the theoretical modeling of the structure and evolu- tion of exoplanets (see, e.g., Burrows et al. 2007), yet it is among the most difficult properties to determine for isolated main-sequence stars. For the host stars of tran- siting planets the evolutionary ages span the full range, as seen in Figure 2. The difficulty mentioned above is most evident for the less evolved objects (M less than ⋆ about 0.9 M⊙), which are all seen to have large error barsthatcanreachthe upper limitof14Gyrconsidered here. For those cases the nominal ages reported in this work should be used with caution. The surface gravities inferred from the models (logg ; ⋆ Table 3) have formal uncertainties that are typically Fig. 3.— Surface gravities for the planet host stars inferred about5timessmallerthanthosemeasuredspectroscopi- from stellar evolution models by Yietal. (2001), compared with those measured spectroscopically. The dotted line represents the cally(logg ;Table1). Thisreflectsthestrengthofthe spec one-to-one relation, and the systems are numbered as inprevious constraintprovidedbya/R⋆. Thevalues ofloggspec and figures. logg are compared against each other in Figure 3. On ⋆ averagethey agreequite well (the mean O C difference ing double-lined eclipsing binaries with very accurately − is 0.027 dex), and the rms scatter of the differences measured masses and radii have shown that the agree- − is 0.15 dex although three of the systems present differ- ment with theory is in general very good, and is within ences larger than 0.2 dex. An illustration of the supe- a few percent for solar-type stars (see, e.g., Andersen riorconstraintaffordedbya/R is showninFigure 4 for ⋆ 1991; Pols et al. 1997; Lastennet & Valls-Gabaud2002). OGLE-TR-132 and WASP-2, two of the more dramatic Asasimpletest,weconsideredasecondsetofmodelsby examples. In neither case is the parallax known. Girardi et al.(2000)thathasoftenbeenusedbyotherin- vestigatorsinthefieldoftransitingplanets. Weusedthe 3. COMPARISONWITHOTHERMODELS isochroneinterpolationtools providedonthe website at Thereisundoubtedlysomesystematicerrorintroduced the Osservatorio Astronomico di Padova5 to explore the by imperfections in the stellar evolution models them- agreement with the observed values of T , [Fe/H], and eff selves, but this type of error is difficult to evaluate. Ex- tensivecomparisonsbetweenmodelsandobservationsus- 5 http://stev.oapd.inaf.it/∼lgirardi/cgi-bin/cmd. 6 Torres et al. Fig. 4.—Measuredpropertiesoftwoextrasolarplanethoststars displayedon plots analogous tothe H-R diagram. The constraint onthelocationofthestarsbasedonthesurfacegravities(loggspec) andtemperaturesisshowninthetoppanels,andthesamefora/R⋆ versusTeff isshowninthebottompanels. Isochronesarefromthe seriesofevolutionary models byYietal.(2001)forthemeasured metallicity in each case (Table 1), and are shown for ages of 1 to 13Gyr(lefttoright)insteps of1Gyr. Thea/R⋆ valuesareseen to provide a much better handle on the stellar parameters (mass, radius,etc.). a/R , and to infer the stellar mass and other properties ⋆ in the same way as above. There are small differences in the physical assumptions between these models and thosefromtheY2 series,butoveralltheyarerathersim- ilar. ForthisapplicationthePadovamodelsareavailable for metallicities smaller than Z = 0.030 (corresponding to [Fe/H] = +0.20), which allows us to compare results for nine of the transiting systems.6 Figure 5 displays this comparisonforthe stellarmassesandradii,showing that in all cases the results from both models agree to well within their uncertainties. Similar tests were carried out using the models of Fig. 5.— Comparison of the stellar masses and radii de- Baraffe et al.(1998). Thesemodelsenjoywidespreaduse rivedfromthe Y2 models (Yietal.2001)andthe Padova models (Girardietal.2000)fortransitingplanethosts,showingtheexcel- forlower-massstarsandbrowndwarfs,butthey arealso lentagreement. Thedottedlinesrepresenttheone-to-onerelation, computedformassesaslargeas1.4M⊙. They areavail- andthenumberingofthesystemsisthesameasinpreviousfigures. ableforthreedifferentvaluesofthemixinglengthparam- eter α over restricted ranges in mass and metallicity. ML The value that fits the observed properties of the Sun of solar. Of the six systems in this range, one (OGLE- (against which all models are calibrated) is α = 1.9. TR-113) gave no solution consistent with the observed ML Forcoolerobjects near the bottom ofthe main sequence values of Teff and a/R⋆. This is because the Baraffe the value used almost universally is α =1.0. Here we models are limited to ages less than about 10 Gyr for ML adoptthe formervalue,since ourstarsareallmoremas- this mass range, whereas both the Yi et al. (2001) and sive than about 0.8 M⊙ with the exception of GJ 436, theGirardi et al.(2000)modelsindicateanageofabout which we discuss in more detail in 5. For the mass 13 Gyr for this star. Whether OGLE-TR-113 is truly range of interest and for α = 1.9§the Baraffe et al. thisoldisunknown. Itseemsatleastaslikelythatsome ML (1998)modelsarepubliclyavailableonlyforsolarmetal- of the other observational constraints are in error. The licity, so the comparison was limited to transiting sys- remaining 5 systems show excellent agreement with the tems with [Fe/H] within about 0.1 dex of the Sun, and resultsfromboththeY2 modelsandthePadovamodels, with [Fe/H] uncertainties that keep them within 0.2 dex within the formal uncertainties. Thesetestsofthestellarevolutionarymodelsareobvi- 6 Higher-metallicity Padova models have been published by ouslynotexhaustive,anditmaywellbethecasethatall Salasnichetal.(2000),butthoseemployphysicalassumptionsdif- ofthesetsofmodelsweconsideredhavesomedeficiencies ferent enough that they cannot be merged with the Girardietal. in common. However, the general pattern of agreement (2000) models we use here. In addition, from a practical point of does lend some degree of confidence to the results. We view, the use of these higher-metallicity models is not yet imple- mentedintheonlineinterpolationroutineasofthiswriting. proceedundertheassumptionthatthesystematicerrors Improved parameters 7 inthesecalculationsarenotthedominantsourceoferror inthestellarparametersderivedhere(exceptperhapsfor OGLE-TR-113,as noted above). 4. ADDITIONALOBSERVATIONALCONSTRAINTS As a further test of the accuracy of our stellar radius determinations, in this section we consider the consis- tency check provided by the near infrared (NIR) surface brightness(SB)relations,whichyieldtheangulardiame- terφofastardirectlyintermsofitsapparentmagnitude and color. Unlike the parallax, which would in principle yield the stellar luminosity but is known for only five of the brighter systems in the sample, φ can be computed for all host stars from existing photometry. We use the empirical relation logφ =c (V K)+c 0.2K (4) SB 1 2 − − derivedbyKervella et al.(2004),inwhichthecoefficients arec =0.0755 0.0008andc =0.5170 0.0017,theK- 1 2 ± ± bandmagnitudeisintheJohnsonsystem,andφ isthe SB limb-darkened value of the angular diameter expressed Fig. 6.— Angular diameters φSB computed from the near in- frared surface brightness relation of Kervellaetal. (2004) com- in milli-arc seconds. The relation represented by eq.(4) pared against the values derived from our modeling, making use is extremely tight, with a scatter well under 1%. Near of the apparent V magnitudes for all host stars and ignoring ex- infrared magnitudes are available for all stars from the tinction. The dotted line represents the one-to-one relation, and 2MASS catalog and were transformed to the Johnson thesystemsarenumberedasinpreviousfigures. system following Carpenter (2001). The best available the argument around and use the φ values as further SB V magnitudes collected from the literature are listed in constraints in our modeling. By trial and error we find Table 1. thatauniformreddeningvaluebetweenE(B V)=0.02 − Angular diameters from our stellar evolution model- and0.03issufficienttoreducetheaveragediscrepancyin ing in 2.3 can be derived for comparison with φ by the angular diameters for the fainter stars to zero. This SB § making use of our theoretical radii and absolute visual modest amount of reddening is consistent with expecta- magnitudes in Table 3, along with the apparent V mag- tions for objects that are between 100 and 500 pc away, nitudes, using as are these. In addition to the angular diameters, Table 4 lists the φ =9.3047R /100.2(V−MV+5). (5) mod ⋆ parallaxes π and corresponding distances we derive mod With the stellar radius expressed in solar units, the nu- from V and MV, ignoring extinction. Hipparcos par- merical constant is such that φmod is in mas. Neither allaxes πHIP are given for comparison as well, for the this equation nor the previous one take into account in- objects that have them. terstellarextinction,althougheq.(4)isactuallyquitein- sensitive to extinction. We discuss this below. 5. THECASEOFGJ436 The comparison between the angular diameters from As the only M dwarf among the currently known eq.(4) and eq.(5) is shown in Table 4 for all systems transiting planet host stars, GJ 436 presents a special except for GJ 436, for reasons to be described in the challenge. The Y2 stellar evolution models we used next section, and the OGLE stars, which are likely to in all other cases are not intended for the lower main be significantly affected by extinction since they lie sev- sequence, as they lack the proper non-grey model at- eralkpcawayneartheGalacticplane. Theuncertainties mosphere boundary conditions to the interior equations listedinclude allcontributions fromthe photometryand that have been shown to be critical for cool objects model-derivedquantities,aswellastheerrorsinthecoef- (see, e.g., Chabrier & Baraffe 1997). The Padova mod- ficientsofeq.(4). Theprecisionofφ istypicallyseveral els of Girardi et al. (2000) have the same shortcoming, SB times better than that of φ . A graphical comparison although both of these models are perfectly adequate mod is shown in Figure 6, with the one-to-one relation repre- for hotter stars. On the other hand, the calculations sentedwithadiagonalline. Thegoodagreementoverthe by Baraffe et al. (1998) with α = 1.0 are specifically ML full range of an order of magnitude in φ is an indication designed for low-mass stars, and have in fact been in- that our model radii from 2.3 are accurate. voked by other authors for this system as a rough con- § One effect of extinction in this analysis would be to sistency check. For the most part, the previous mass make the values based on eq.(5) appear too small. Ex- determinations for GJ 436 have relied upon empirical amination of the differences shows a hint of this, partic- mass-luminosity relations, but even those relations have ularly among the fainter (more distant) stars: the av- their problems. Previous radius estimates for GJ 436 erage difference (φ φ )/φ is +0.5 1.8% for have often rested on the assumption of numerical equal- mod SB SB h − i ± the 6 objects brighter than V = 11, and 3.9 1.6% ity between M and R for M stars (Gillon et al. 2007a; ⋆ ⋆ − ± for the others. Since reddening values for individualsys- Deming et al. 2007). tems are presently unknown, and cannot be determined The importance of the GJ 436 system is undeniable, accurately enough from the information at hand, we are asitharborsthesmallesttransitingplanetthathasbeen unable to correctforthis. We areequallyunable to turn found to date (comparable in size to Neptune). Fur- 8 Torres et al. thermore, it is the nearest transiting exoplanet, at a tioned above. We do not believe so, because the mass distance of only 10 pc. Given these facts, it is some- regime of GJ 436 is very different. The host stars of the what surprising that until recently the mass of the star other transiting planets are typically more than twice as ( 0.4M⊙)wasonly knownto about10%(Maness et al. massive as GJ 436. ∼ 2007;Gillon et al.2007b),onaparwiththeworstofthe determinations in Table 3 (that of WASP-2,with poorly 6. PLANETARYPARAMETERS determined spectroscopic parameters and more than 10 The combination of the stellar properties in Table 1 timesfartheraway). Similarlimitationsholdforthestel- withthelightcurveparametersinTable2,alongwiththe lar radius. As a result, our knowledge of the planetary measured orbital periods and velocity semi-amplitudes, parameters has suffered. Torres (2007) showed that the yields the homogeneous set of planet parameters pre- Baraffe et al.(1998)modelsintheiroriginalformdonot sented in Table 5. In addition to the planetary mass yield satisfactory results for GJ 436 since different an- and radius, we have computed and tabulated the plan- swers for M⋆ and R⋆ are obtained depending on which etary surface gravity and mean density, as well as the constraints are used (including the absolute magnitude, orbital semimajor axis and other useful characteristics. since a reliable parallax measurement is available from As pointed out by Southworth et al. (2007), Winn et Hipparcos). Moreover,someoftheotherpredictedquan- al. (2007), Beatty et al. (2007), and others, the surface tities are at odds with empirical determinations. These gravity of a transiting planet can be derived without difficulties had not been previously emphasized. knowledge of the stellar mass or radius, using only the However, as described by Torres (2007), reliable pa- radial-velocitycurveandthetransitlightcurve. Wetake rameters can still be obtained from the models by mod- this opportunity to correct the general expression for ifying the theoretical radii and temperatures in such a logg presented by Sozzetti et al. (2007), valid also for p wayastopreservethebolometricluminosities. Thisrec- the case of eccentric orbits, which neglected to account ognizes the fact that R⋆ and Teff as predicted by theory fortheprojectionfactorimplicitintheimpactparameter havebeenshowntodisagreewithaccuratemeasurements b as derived from the light-curve fits: for M dwarfs in double-lined eclipsing binaries (see, e.g., logg = 2.1383 logP +logK (6) Popper 1997; Clausen et al. 1999; Torres & Ribas 2002; p ⋆ − − − Ribas 2003; Lo´pez-Morales & Ribas 2005), whereas the 1 b 1 e2 2 luminosities appear to be unaffected (Delfosse et al. log 1 − + 2000; Torres & Ribas 2002; Ribas 2006; Torres et al. −2 −(cid:20)a/R⋆1+esinω(cid:21) ! 2006). Torres(2007)appliedsimultaneouslytheobserva- a/R 1 tionalconstraintsonT ,a/R ,thecolorindexJ K,and +2log ⋆ + log(1 e2) . theabsoluteK-bandmeffagnitud⋆eM ,andallowe−dthera- (cid:18)Rp/R⋆(cid:19) 2 − K dius/temperatureadjustmentfactortobeafreeparame- The numerical constantis such that the gravity is in cgs ter. Inthiswayaself-consistentsolutionwasachievedfor units when P and K are expressed in their customary ⋆ all parameters, and in addition the radius/temperature units of days and m s−1. factor showed excellent agreement with previous esti- Thefirstpropertiesonegenerallywantstoknowabout mates for other M dwarfs. The stellar mass and radius a transiting planet are its mass and radius. The large are M⋆ =0.452+−00..001142 M⊙ and R⋆ =0.464+−00..000191 R⊙. We size measured for the first transiting planet discovered, adopttheseresultshere,thusplacingGJ436onasimilar HD 209458b,has been widely debated and still presents footing as the other transiting systems in that the prop- somewhat of a challenge to theory. It is now accom- erties of the host stars are all based on current stellar panied by several other inflated planets, underlying our evolution models. incompleteknowledgeofthephysicsoftheseobjects. An The results for GJ 436arelistedin Table 3 alongwith updated version of the now classical diagram of M ver- p those of the other 22 stars. A few words of caution are susR isshowninFigure7,inwhichfiveotherexamples p in order. As pointed out by Torres(2007), the predicted are seen to be at least as large as HD 209458b (#3), or absolute visual magnitude for this star is unreliable due perhaps even larger,given the uncertainties. to missing molecular opacity sources shortward of 1 µm The fundamental physical properties of the known in the model atmospheres that are used as boundary transitingplanetscoveraconsiderablerange—morethan conditions in the stellar evolution calculations (see, e.g., two orders of magnitude in mass, a factor of nearly 5 Baraffe et al. 1998; Delfosse et al. 2000). Also, because in radius, and a factor of 3 in orbital separation—and of the unevolved status of the star, the nominal age is these are discussed in the next section. But the geomet- probablymeaninglesssincetheobservationalconstraints ric properties that determine the light-curve shapes are allow any age within the range of 1–10 Gyr explored in also varied. In Figure 8 we present a “portrait gallery” the modeling effort of Torres (2007). Finally, this mod- of all known transiting planets. Stars and planets are eling effort assumed that the stellar metallicity is solar. rendered to scale, emphasizing the wide range of stellar ForGJ436thisisavalidassumptionsincethemeasured types probed by the photometric searches. The orbital composition is [Fe/H] = 0.03 0.20 (see Appendix), geometries are indicated with solid horizontal lines rep- − ± but the uncertainty in [Fe/H] was not accountedfor due resenting an edge-on orientation, and the path of each to the lack of proper models, and therefore the errors of planet across the stellar disk shown with dotted lines the stellar propertiesin Table 3 may be underestimated. at the actual impact parameter. (Of course, the data do A careful reader may wonder whether our use of the notdistinguishbetweenpositiveandnegativeimpactpa- Baraffe et al. (1998) models in 3 as a check on the rameters.) The resulting light curves calculated for the results from the Y2 isochrones i§s contradicted by our V band are all shown with the same vertical (flux) and remarks on the radius/temperature discrepancies men- horizontal(time) scale, to facilitate comparison. Depths Improved parameters 9 residuals. We seem to have found such a third vari- able: the metallicity of the host star. Figure 9b displays the O C residuals from the top panel as a function − of [Fe/H], indicating a rather clear correlation (dashed line): ∆M =(+0.152 0.050) (1.17 0.23) [Fe/H]. p ± − ± × Afterremovalofthistrend,therelationbetweenM and p period becomes tighter (Figure 9c). The scatter in the mass-period relation is reduced from 0.26 M in the Jup toppanelto0.17M inthebottompanel. Itseemsun- Jup likelythatthisisastatisticalfluke. However,thescatter isstilllargerthantheformalobservationaluncertainties, suggesting that these three variables are not completely determinative. What might be the implications of this metallicity de- pendence? It has been proposed that the mass-period relation is related to the process by which close-in ex- oplanets migrated inward from their formation sites, or morespecifically, to the mechanismthat halts migration at orbital periods of a few days. The trend of larger masses at shorter orbital periods could suggest that the halting mechanism depends on mass, and larger planets Fig. 7.—Mass-radiusdiagramforalltransitingplanets(HAT-P- areabletomigratefurtherin. Thedependenceonmetal- 2isoffthescale,withMp =8.7MJup). Linesofconstant density licitymaythenbeinterpretedtoindicatethatplanetsin areshown. metal-poor systems need to be more massive in order to vary quite significantly (3 mmag in V for HD 149026b, migrateinwardtothesameorbitalperiodasmoremetal- 26 mmag for OGLE-TR-113b; see Table 2), as do the rich planets. In this context, the trend in Figure 9b overallshapesofthetransitevents,whichinseveralcases could be interpreted as evidence that the efficiency of are grazing enough to depart from the canonicalprofiles themigration(orhalting)mechanismisaffectedtosome often depicted in the literature. degree by the chemical composition. A dependence of migrationonmetallicityisinfactpredictedbysomethe- 7. DISCUSSION ories, and could arise, as pointed out by Sozzetti et al. Many investigators have sought and claimed possible (2006), either from slower migration rates in metal-poor correlations between various stellar and planetary pa- protoplanetary disks (Livio & Pringle 2003; Boss 2005) rametersoftransitingsystems. Inprinciplesuchcorrela- or through longer timescales for giant planet forma- tionscouldleadtoimportantinsightsintotheformation, tion around metal-poor stars, which would effectively structure, and evolution of exoplanets. A primary mo- reduce the efficiency of migration before the disk dis- tivation for presenting a more complete, accurate, and sipates (Ida & Lin 2004; Alibert et al. 2005). However, homogeneous set of these parameters in this work was the above processes are more aimed at addressing the to facilitate such studies. The relatively large array of apparentlackofshort-periodplanets amongverymetal- properties now available offers the opportunity to find poor stars claimed by some authors (see Sozzetti et al. new correlations,orto revisitold ones incorporatingad- 2006),whereasamongthetransitingplanetsitisnotthe ditional variables. While it is beyond the scope of the lackofmoremetal-poorexamplesweareconcernedwith, present work to investigate all possible correlations with but rathertheir different properties (suchas mass) com- statistical rigor, in this section we check on three of the pared to metal-rich planets at the same orbital period. most intriguing and potentially important relations that Togiveaquantitativeexample,wefindthatforaperiod have been proposed. of 2.5 days (near the average for known transiting sys- tems), the mass of a planet with [Fe/H] = 0.2 is 40% 7.1. Planetary mass versus orbital period − ∼ largerthanonewithaveragemetallicity([Fe/H]=+0.13, Mazeh et al. (2005) were the first to point out the ap- excluding HAT-P-2b and GJ 436b), while the mass of a parentcorrelationbetweenM andP fortransitingplan- planet with [Fe/H] = +0.4 is about 30% smaller. This p ets (see also Gaudi et al. 2005). The original suggestion rangeofmetallicities,from[Fe/H]= 0.2to+0.4(cover- − was based on only 6 systems, but additional discoveries ingafactorof4inmetalenhancement),isapproximately have generally supported the trend of decreasing mass thefullrangeobserved. Wenotethatthestrengthofthe with longer periods, although the scatter has also be- metallicity effect is modest rather than overwhelming, come larger. This is shown in Figure 9a. HAT-P-2b and this too is in agreement with theoretical expecta- would be an extreme outlier in this plot; we have ex- tions (e.g., Livio & Pringle 2003). The massive planet cluded it because it is so much more massive than the HAT-P-2b obviously does not conform to this trend of other planets and may belong to a different category of Mp versus period, which may indicate some fundamen- planet (see 7.2). Similarly, we have excluded GJ 436b tal difference either in its formation or migration. because it is§so much less massive than the others, and An alternative interpretation, also proposed by maybearockyorrock-iceplanetratherthanagasgiant. Mazeh et al. (2005), is that the Mp versus P relation is A simple linear fit is shown for reference (dashed line). more a reflection of survival requirements in close prox- We also investigated the scatter in this relation, seek- imity to the star, due to thermal evaporation from the ing any “third variable” that might correlate with the extreme UV flux. Close-in planets must be more mas- 10 Torres et al. Fig. 8.— “Portraitgallery”of transiting extrasolar planets. Star and planet sizes areshownto scale. The vertical and horizontal axes ofalllightcurvesarealsoonthesamescale. Planettrajectoriesareshownwiththeirmeasuredimpactparameters(dottedlines). Orbital periodsareindicatedineachpanel.

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