Bayesian mass and age estimates for transiting exoplanet host stars
aa r X i v : . [ a s t r o - ph . I M ] D ec Astronomy & Astrophysicsmanuscript no. paper-arxiv c (cid:13)
ESO 2018September 11, 2018
Bayesian mass and age estimates for transiting exoplanet hoststars ⋆ P. F. L. Maxted , A. M. Serenelli , and J. Southworth Astrophysics Group, Keele University, Keele, Sta ff ordshire ST5 5BG, UKe-mail: [email protected] Instituto de Ciencias del Espacio (CSIC-IEEC), Facultad de Ciencias, Campus UAB, 08193, Bellaterra, SpainReceived ; accepted
ABSTRACT
Context.
The mean density of a star transited by a planet, brown dwarf or low mass star can be accurately measured from its lightcurve. This measurement can be combined with other observations to estimate its mass and age by comparison with stellar models.
Aims.
Our aim is to calculate the posterior probability distributions for the mass and age of a star given its density, e ff ective tempera-ture, metallicity and luminosity. Methods.
We computed a large grid of stellar models that densely sample the appropriate mass and metallicity range. The posteriorprobability distributions are calculated using a Markov-chain Monte-Carlo method. The method has been validated by comparison tothe results of other stellar models and by applying the method to stars in eclipsing binary systems with accurately measured massesand radii. We have explored the sensitivity of our results to the assumed values of the mixing-length parameter, α MLT , and initialhelium mass fraction, Y . Results.
For a star with a mass of 0.9 M ⊙ and an age of 4 Gyr our method recovers the mass of the star with a precision of 2% andthe age to within 25% based on the density, e ff ective temperature and metallicity predicted by a range of di ff erent stellar models. Themasses of stars in eclipsing binaries are recovered to within the calculated uncertainties (typically 5%) in about 90% of cases. Thereis a tendency for the masses to be underestimated by about 0.1 M ⊙ for some stars with rotation periods P rot < Conclusions.
Our method makes it straightforward to determine accurately the joint posterior probability distribution for the massand age of a star eclipsed by a planet or other dark body based on its observed properties and a state-of-the art set of stellar models.
Key words. stars: solar-type – binaries: eclipsing – planetary systems
1. Introduction
Studies of extrasolar planets rely on a good understanding of thestars that they orbit. To estimate the mass and radius of an ex-trasolar planet that transits its host star we require an estimatefor the mass of the star. The mass of the star will also stronglyinfluence the planet’s environment, e.g., the spectrum and inten-sity of the stellar flux intercepted by the planet and the natureand strength of the tidal interaction between the star and theplanet. The ages of planet host stars are used to investigate thelifetimes of planets and the time scales for tidal interactions be-tween the planet and the star (e.g., Matsumura et al. 2010; Lanza2010; Brown et al. 2011).The analysis of the light curve produced by a planetary tran-sit yields an accurate estimate for the radius of the star relative tothe semi-major axis of the planet’s orbit, R ⋆ / a , provided that theeccentricity of the orbit is known. This estimate can be combinedwith Kepler’s laws to estimate the density of the host star. Thedensity can be combined with estimates for the e ff ective tem-perature and metallicity of the star to infer a mass and age forthe star by comparison with stellar models or an empirical cal-ibration of stellar mass. In general, the comparison with stellarmodels is done using a maximum likelihood method, i.e., takingthe mass and age of the evolution track and isochrone that give ⋆ The source code and stellar model grids for our method are avail-able at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)or via http: // cdsweb.u-strasbg.fr / cgi-bin / qcat?J / A + A / . the best fit to the observed density (or R ⋆ / a ) and e ff ective tem-perature, estimated either by a least-squares fit to the observedproperties or “by-eye”.Maximum-likelihood estimates can be strongly biased incases where the mapping between the observed parameters andthe parameters of interest is non-linear. This is certainly thecase for stellar ages because the observed parameters changevery little during the main-sequence phase, but there are largechanges to the observed properties during the rapid evolutionof a star away from the main sequence. This can produce a“terminal age bias”, where the “best-fit” method applied to asample of stars produces a distribution of ages that is a pri-ori very unlikely, i.e., too few main-sequence stars and manystars in regions of the model paramater space correspondingto short-lived evolutionary phases. This problem is particularlyacute for cases where the uncertainties on the observed pa-rameters are large compared to the change in observed prop-erties during the main sequence phase. This is often the casefor the age estimates of single stars based mainly on sur-face gravity measured from the stellar spectrum or the abso-lute magnitude derived from the measured parallax and stellarflux. Bayesian methods that account for the a priori distributionof stellar ages have been developed to deal with this problemin spectroscopic stellar surveys (Jørgensen & Lindegren 2005;Pont & Eyer 2004; Serenelli et al. 2013; Schneider et al. 2014).Pont et al. (2009) applied a similar Bayesian approach in theirstudy of the HD 80606 planetary system, but there have been Article number, page 1 of 9 & Aproofs: manuscript no. paper-arxiv few other examples of this approach in exoplanet studies. Thismay be because there is currently no software available to theexoplanet research community that can be used to apply theseBayesian methods and that is straightforward to use.In general, the terminal age bias is expected to be less severefor planet host stars than for single stars because stellar den-sity measurements based on the analysis of a planetary transitare usually more precise than surface gravity estimates based onthe analysis of a stellar spectrum or the luminosity derived fromparallax and flux measurements. The mean stellar density is alsomore sensitive to the change in radius of a star as it evolves awayfrom the main sequence. However, precise stellar densities canalso cause problems because the broad sampling in mass, ageand metallicity used for many grids of stellar models can pro-duce poor sampling of the observed parameter spacing, i.e., thetypical di ff erence in stellar density between adjacent model gridpoints can be much larger than the uncertainty on the observedvalue. This can produce systematic errors due to interpolation,particularly for stars near the end of their main-sequence evolu-tion where the evolution tracks have a complex behaviour thatis very sensitive to age, mass and composition. This also makesit di ffi cult to make reliable estimates of the uncertainties on themass and age. One method sometimes employed to estimate theuncertainties is to look for the mass and age range of all the mod-els that pass within the estimated errors on the observed values,e.g., all the models within the 1- σ error bars. This approach canbe misleading because the errors on the mass and age are oftenstrongly non-Gaussian and highly correlated . It also possible tomiss some combinations of mass, age and composition that pro-vide a reasonable match to the observed properties of the starbut that are not sampled by the stellar model grid or that fall justoutside the 1- σ error bars, particularly when the fitting is doneby-eye.The recently-launched European Space Agency missionGAIA (Perryman et al. 2001) will measure parallaxes and opti-cal fluxes for many stars that are transited by planets, brown-dwarfs and low-mass stars. The luminosity measurement de-rived from these observations can be an additional useful con-straint on the mass and age of the star. If the density, e ff ectivetemperature and composition of the star are also known thenfinding the best-fit mass and age of the star becomes an over-determined problem for stellar models with fixed helium abun-dance and mixing-length parameter, i.e., there are more observ-ables than unknowns. If the best-fit to the observed parameters ispoor then this opens up the possibility of using these stars to ex-plore whether the assumed values of the helium abundance andmixing-length parameter, or some other factor, can be adjustedto improve the agreement between the stellar models and realstars.To deal with these issues we have developed a Markov-chainMonte Carlo (MCMC) method that calculates the posterior prob-ability distribution for the mass and age of a star from its ob-served mean density and other observable quantities using a gridof stellar models that densely samples the relevant parameterspace. We have validated our method by applying it to data de-rived from di ff erent stellar models and by applying it to stars ineclipsing binary stars with precisely measured masses and radii.We have also quantified the systematic error in the estimatedmass and age due to the variations in the assumed helium abun-dance and convective mixing length parameter. The method hasbeen implemented as a program called bagemass that we havemade available for general use.
2. Method
Our method uses a grid of models for single stars produced withthe garstec stellar evolution code (Weiss & Schlattl 2008). Themethods used to calculate the stellar model grid are described inSerenelli et al. (2013) so we only summarise the main featuresof the models and some di ff erences to that description here. garstec uses the FreeEOS equation of state (Cassisi et al.2003) and standard mixing length theory for convection(Kippenhahn & Weigert 1990). The mixing length parameterused to calculate the model grid is α MLT = .
78. With thisvalue of α MLT garstec reproduces the observed properties ofthe present day Sun assuming that the composition is that givenby Grevesse & Sauval (1998), the overall initial solar metallic-ity is Z ⊙ = . Y ⊙ = . ff erent to the value inSerenelli et al. (2013) because we have included additional mix-ing below the convective zone in order reduce the e ff ect of grav-itational settling and so to better match the properties of metal-poor stars. Due to the e ff ects of microscopic di ff usion, the ini-tial solar composition corresponds to an initial iron abundance[Fe / H] i = + . ⊙ to2.0 M ⊙ in steps of 0.02 M ⊙ . The grid of initial metallicity valuescovers the range [Fe / H] i = − .
75 to − .
05 in steps of 0.1 dexand the range [Fe / H] i = − .
05 to + .
55 in 0.05 dex steps. Theinitial composition of the models is computed assuming a cosmichelium-to-metal enrichment ∆ Y / ∆ Z = ( Y ⊙ − Y BBN ) / Z ⊙ , where Y BBN = . ∆ Y / ∆ Z = . / H] i . For each value of initial mass and [Fe / H] i we extracted the output from garstec at 999 ages from the endof the pre-main-sequence phase up to an age of 17.5 Gyr or amaximum radius of 3 R ⊙ , whichever occurs first. We define thezero-age main sequence (ZAMS) to be the time at which the starreaches its minimum luminosity and measure all ages relative tothis time.To obtain the properties of a star from our model grid at ar-bitrary mass, [Fe / H] i and age we use the pspline implementationof the cubic spline interpolation algorithm. . We interpolated thestellar evolution tracks for stars that reach the terminal-age mainsequence (TAMS) on to two grids, one from the start of the evo-lution track to the TAMS, and one that covers the post-main se-quence evolution. The dividing line between the two is set by theage at which the central hydrogen abundance drops to 0. We use999 grid points evenly distributed in age for each model grid.For stars on the main sequence we interpolate between modelsas a function of age in units of the main sequence lifetime at thespecified values of mass and [Fe / H] i . The interpolating variablefor the post-main sequence properties is the age since the TAMSmeasured in units of the total time covered by the model grid.Splitting the grid of stellar models in this way improves the ac-curacy of the interpolation near the terminal-age main sequence.We used garstec to calculate some models for solar metal-licity at masses half-way between those used for our methodand then compared the interpolated values to those calculated by garstec . For the masses that we checked (1.01 M ⊙ and 1.29 M ⊙ )we find that the maximum error in the density is 0.01 ρ ⊙ , themaximum error in T e ff is 25 K and the maximum error in http://freeeos.sourceforge.net http://w3.pppl.gov/ntcc/PSPLINE Article number, page 2 of 9. F. L. Maxted et al.: Bayesian mass and age estimates for transiting exoplanet host stars log( L / L ⊙ ) is 0.015. The worst agreement between the calculatedand interpolated models occurs at very young ages ( < .
01 Gyr)and, for the 1.29 M ⊙ star, near the “blue hook” as the star evolveso ff the main sequence. Away from these evoutionary phases theerror in the interpolated values is at least an order-of-magnitudesmaller that these “worst case” values. It is much easier to calculate the probability distribution func-tions of a star’s mass and age if the observed quantities can beassumed to be independent. To enable us to make this assump-tion we define the data to be a vector of observed quantities d = (T e ff , L ⋆ , [Fe / H] s , ρ ⋆ ), where T e ff is the e ff ective tempera-ture of the star, L ⋆ its observed luminosity, [Fe / H] s characterisesthe surface metal abundance and ρ ⋆ is the stellar density.The density of stars that host a transiting extrasolar planetcan be determined directly from the analysis of the light curve ifthe eccentricity is known. From Kepler’s third law it follows that ρ ⋆ = ⋆ π R ⋆ = π GP (1 + q ) a R ⋆ ! , where P and a are the period and semi-major axis of the Keple-rian orbit and q = M c / M ⋆ is the mass ratio for a companion withmass M c . The quantity ( a / R ⋆ ) can be determined with good pre-cision from a high quality light curve alone if the orbit is knownto be circular since it depends only on the depth, width and shapeof the transit (Seager & Mallén-Ornelas 2003). For non-circularorbits the same argument applies but in this case the transit yields( d t / R ⋆ ), where d t is the separation of the stars during the tran-sit, so the eccentricity and the longitude of periastron must bemeasured from the spectroscopic orbit or some other method sothat the ratio d t / a can be determined. The error in the mass ratiowill give a negligible contribution to the uncertainty on ρ ⋆ forany star where the presence of an extrasolar planet has been con-firmed by radial velocity observations. The same argument ap-plies to brown-dwarf or low mass stellar companions, but somecare is needed to make an accurate estimate of the mass ratio,e.g., via the mass function using an initial estimate of the stellarmass, and the additional uncertainty in ρ ⋆ should be accountedfor. Most of the stars currently known to host exoplanets do nothave accurately measured trigonometrical parallaxes. For thesestars we set log(L ⋆ / L ⊙ ) = ± ⋆ = π d f ⊕ where f ⊕ , theflux from the star at the top of the Earth’s atmosphere correctedfor reddening, and d , the distance to the star measured fromits trigonometrical parallax. The values of T e ff and [Fe / H] s canbe derived from the analysis of a high quality stellar spectrum.These may be less accurate than the quoted precision becauseof the approximations used in the stellar model atmospheres andthe uncertainties in atomic data used in the analysis. For that rea-son we set lower limits of 80 K on the standard error for T e ff and0.07 dex for the standard error on [Fe / H] s (Bruntt et al. 2010).For stars similar to the Sun a change of 80 K in the assumedvalue of T e ff results in a change of about 0.02 dex in the value of[Fe / H] s derived from the analysis of the spectrum (Doyle et al.2013). This is at least a factor of 3 lower than the minimum stan-dard error that we have assumed for [Fe / H] s so we ignore thisweak correlation between T e ff and [Fe / H] s . For a few stars thee ff ective temperature can be determined directly from f ⊕ and thedirectly measured angular diameter. The quoted uncertainties on these directly-measured T e ff values should not be used if theyare much smaller than about 80 K because our method does notaccount for the strong correlation between T e ff and L ⋆ in thesecases and some allowance must be made for the uncertainties inthe e ff ective temperature scale of the stellar models. The vector of model parameters that are used to predict the ob-served data is m = ( τ ⋆ , M ⋆ , [Fe / H] i ), where τ ⋆ , M ⋆ and [Fe / H] i are the age, mass and initial metal abundance of the star, respec-tively. The observed surface metal abundance [Fe / H] s di ff ersfrom the initial metal abundance [Fe / H] i because of di ff usionand mixing processes in the star during its evolution.We use the MCMC method to determine the probability dis-tribution function p ( m | d ) ∝ L ( d | m ) p ( m ). To estimate the like-lihood of observing the data d for a given model m we use L ( d | m ) = exp( − χ / χ = ( T e ff − T e ff , obs ) σ T + ( log( L ⋆ ) − log( L ⋆, obs ) ) σ L + ( [Fe / H] s − [Fe / H] s , obs ) σ / H] + ( ρ ⋆ − ρ ⋆, obs ) σ ρ . In this expression for χ observed quantities are denoted bythe subscript ‘obs’, their standard errors are σ T , σ log L , etc.,and other quantities are derived from the model, as describedabove. In cases where asymmetric error bars are quoted on val-ues we use either the upper or lower error bar, as appropriate,depending on whether the model value is greater than or lessthan the observed value. The probability distribution function p ( m ) = p ( τ ⋆ ) p (M ⋆ ) p ([Fe / H] i ) is the product of the individualpriors on each of the model parameters. The assumed prior on[Fe / H] i normally has little e ff ect because this parameter is wellconstrained by the observed value of [Fe / H] s so we generallyuse a ‘flat’ prior on [Fe / H] i , i.e., a uniform distribution over themodel grid range. It is possible in our method to set a prior onM ⋆ of the form p ( M ⋆ ) = e α M ⋆ . The prior on M ⋆ is the product ofthe present day mass function for planet host stars, the mass dis-tribution of the target stars in the surveys that discovered theseplanets and the sensitivity of these surveys as a function of stellarmass, but since none of these factors is well determined we nor-mally use a flat prior for M ⋆ , i.e, α =
0. We also use a flat prioron τ ⋆ over the range 0 – 17.5 Gyr. In general, and for all the re-sults below unless stated otherwise, we simply set set p ( m ) = / H] or τ ⋆ of the form p ( x ) = e − ( x − x lo ) / σ , x < x lo , x lo < x < x hi e − ( x − x hi ) / σ , x > x hi . We generate a set of points m i (a Markov chain ) with theprobability distribution p ( m | d ) using a jump probability dis-tribution f ( ∆ m ) that specifies how to generate a trial point m ′ = m i + ∆ m . The trial point is always rejected if any ofthe model parameters are outside their valid range. Otherwise,a point is always included in the chain if L ( m ′ | d ) > L ( m i | d ) andmay be included in the chain with probability L ( m ′ | d ) / L ( m i | d )if L ( m ′ | d ) < L ( m i | d ) (Metropolis-Hastings algorithm). If thetrial point is accepted in the chain then m i + = m ′ , otherwise m i + = m i (Tegmark et al. 2004). Article number, page 3 of 9 & Aproofs: manuscript no. paper-arxiv
We randomly sample 65,536 points uniformly distributedover the model grid range and take the point with the lowestvalue of χ as the first point in the chain. From this startingpoint, m , we find the step size for each parameter such that | ln( L ( m | d ) − ln( L ( m ( j )0 | d ) | ≈ .
5, where m ( j )0 denotes a set ofmodel parameters that di ff ers from m only in the value of oneparameter, j . We then produce a Markov chain with 10,000 stepsusing a multi-variate Gaussian distribution for f ( ∆ m ) with thestandard deviation of each parameter set to this step size. Thisfirst Markov chain is used to find an improved set of model pa-rameters and the second half of this chain is used to calculate thecovariance matrix of the model parameters. We then calculatethe eigenvalues and eigenvectors of the covariance matrix, i.e.,the principal components of the chain. This enables us to deter-mine a set of transformed parameters, q = ( q , q , q ), that areuncorrelated and where each of the q i have unit variance, as wellas the transformation from q to m . We then produce a Markovchain with 50,000 steps using a multi-variate Gaussian distribu-tion for f ( ∆ q ) with unit standard deviation for each of the trans-formed parameters. The first point of this second Markov chainis the set of model parameters with the highest value of L ( m i | d )from the first Markov chain. This second Markov chain is the oneused to estimate the probability distribution function p ( m | d ). Weuse visual inspection of the chains and the Gelman-Rubin statis-tic (Gelman & Rubin 1992) to ensure that the chains are wellmixed, i.e., that they properly sample the parameter space. Thenumber of steps used in the two Markov chains can be varied,e.g., longer chains can be used to ensure that the parameter spaceis properly sampled in di ffi cult cases.Our algorithm is implemented as a fortran program called bagemass that accompanies the on-line version of this article andthat is also available as an open source software project.
3. Validation of the method
In Table 1 we compare the predicted values of T e ff and ρ ⋆ fromour garstec models for a star at an age of 4 Gyr with an initialmass of 0.9 M ⊙ and solar composition to those of three othergrids of stellar models. These are all grids of “standard stellarmodels’ in the sense that they assume a linear relation betweenhelium enrichment and metallicity, and the mixing length pa-rameter is calibrated using a model of the Sun. The DartmouthStellar Evolution Program (DSEP) model grid is described inDotter et al. (2008). The “VRSS 2006” model grid is describedin VandenBerg et al. (2006) and the Geneva 2012 models are de-scribed by Mowlavi et al. (2012). The VRSS 2006 models do notinclude di ff usion or gravitational settling of elements.The range in T e ff values is 65 K, with the garstec modelsbeing at the top end of this range. The values of ρ ⋆ vary byabout 3.5%, with the garstec models predicting the lowest den-sity while the VRSS 2006 models predict the highest density.The di ff erences are mainly due to di ff erences in the assumed so-lar metallicity that is used for the zero-point of the [Fe / H] scaleand the assumed value of α MLT in each model grid.We used bagemass to find the best-fitting (maximum-likelihood) mass and age estimates for these model stars basedon the values given in Table 1. We assigned standard errors of80 K to T e ff and 0.07 dex to [Fe / H] i , i.e., the minimum standarderrors on these values that we use for the analysis of real stars.We assumed that the error on ρ ⋆ is ± . ρ ⊙ . The results are http://sourceforge.net/projects/bagemass Table 1.
Maximum-likelihood mass and age estimates from bagemass for model stars with masses of 0.9M ⊙ [Fe / H] i = . Model T e ff ρ ⋆ [Fe / H] s Mass Age[K] [ ρ ⊙ ] [M ⊙ ] [Gyr] garstec − .
035 0.900 4.00DSEP 2008 5372 1.514 − .
044 0.884 4.59VRSS 2006 5388 1.539 0 0.903 3.15Geneva 2012 5370 1.517 − .
019 0.891 4.18 : : : : ¯ )0 : : : : P r e d i c t e d m a ss ( M ¯ ) Fig. 1.
Bayesian mass estimates for stars in detached eclipsing bi-nary star systems. Symbols / colours denote the following orbital periodranges: P orb < / green; 6 < P <
12 d – open cir-cles / black; P >
12 d – crosses / blue. also shown in Table 1. The results for the garstec models showthat our method is self-consistent to better than the 1-per centlevel in recovering the stellar mass and age of the star. Compar-ing our results to those of other models, we see that the system-atic error due to di ff erences in the stellar models in the recoveredstellar mass is less than about 2%. The systematic error in the re-covered age is larger ( ≈ ff erent masses or ages, e.g.,di ff erences in the treatment of convective overshooting can leadto large di ff erences in the predicted age for more massive stars. We used DEBCat to identify 39 stars in 24 detached eclips-ing binary systems that are suitable for testing the accuracy ofthe mass estimated using our method when applied in the massrange typical for planet-host stars. The masses and radii of thestars in this catalogue have been measured directly to a preci-sion of better than 2%. We have restricted our comparison tostars in binary systems with orbital periods P orb > ⊙ , stars without a measured valueof [Fe / H] s that include an estimate of the standard error, andstars with [Fe / H] s < − .
4. The limits on R ⋆ and [Fe / H] s werechosen so that the stars are not close to the edge of the of thestellar model grid. For the majority of planet host stars currently Article number, page 4 of 9. F. L. Maxted et al.: Bayesian mass and age estimates for transiting exoplanet host stars ¡ : ¡ : : : P r e d i c t e d m a ss ¡ o b s e r v e d m a ss ( M ¯ ) Fig. 2.
Error in the Bayesian mass estimate for stars in detached eclips-ing binary star systems as a function of orbital period. Errors in excessof 2 standard deviation are plotted with filled circles. known the luminosity of the star has not been measured directly,so we do not include the luminosity of the eclipsing binary starsas a constraint in this analysis. The properties of the stars in thissample and the masses we derive using our method are given inTable 2. Kepler-34 and Kepler-35 are also planet host stars, withplanets on circumbinary orbits.The observed and predicted masses are shown in Fig. 1.There is a clear tendency for our grid of standard stellar mod-els to underpredict the mass of some stars by about 0.15M ⊙ . Weimposed an arbitrary limit on the orbital period of the binary sys-tems we have used so we have investigated whether the orbitalperiod is a factor in this analysis. The approximate orbital periodof each star is indicated by the plotting symbol / colour in Fig. 1.It is clear that all the stars with discrepant masses have short or-bital periods ( < ∼ P orb < P rot > ∼
15 d and in the mass range 0.8 –1.3 M ⊙ . There are no suitable data for stars in long-period bina-ries to test whether this conclusion holds in the mass range 1.3 –1.6 M ⊙ . There are also no suitable data to test our method in themass range 0.6 – 0.8 M ⊙ , or for stars with masses 0.8 – 1.3 M ⊙ and rotation periods 7 d < P rot <
14 d.The binary star WOCS 40007 (2MASSJ19413393 + ff ries et al. (2013) find the age of NGC 6819 to be about2.4 Gyr from color-magnitude diagram isochrone fitting and anage estimate for the binary system of 3 . ± . :
22 0 :
24 0 :
26 0 :
28 0 :
30 0 :
32 0 :
34 0 : : : : : ® M L T Fig. 3.
Helium abundance Y and mixing length parameter α MLT for 42solar-like stars from Metcalfe et al. (2014). stellar model fits to their mass and radius measurements of thetwo stars. We obtain ages of 3 . ± . . ± . ff ries et al. The results derived using our grid of stellar models with fixedvalues of α MLT and Y calibrated on the Sun are subject to somelevel of systematic error due to the variations in these valuesbetween di ff erent stars. Fortunately, observed constraints on thevariation in both of these factors are now available thanks to Ke-pler light curves of su ffi cient quality to enable the study of solar-like oscillations for a large sample of late-type stars. The Keplerdata for 42 solar-type stars have been analysed by Metcalfe et al.(2014) including α MLT and Y as free parameters in the fit of thestellar models to observed frequency spectrum. The values of Y and α MLT derived from the asteroseismology are shown in Fig. 3.It can be seen that there is no strong correlation between Y and α MLT , and that there is additional scatter in these values beyondthe quoted standard errors. If we add 0.02 in quadrature to thestandard errors on Y then a least-squares fit of a constant to thevalues of Y gives a reduced chi-squared value χ ≈
1, so weuse 0.02 as an estimate of the scatter in Y . Similarly, for α MLT we assume that the scatter around the solar-calibrated value is0.2. It is debatable whether these derived values of Y are reli-able measurements of the actual helium abundance of these starsor whether the derived values of α MLT are an accurate reflectionof the properties of convection in their atmospheres. However,the frequency spectrum of solar-type stars is very sensitive to themean density of the star so it is reasonable to use the observedscatter in Y and α MLT from the study of Metcalfe et al. as a wayto estimate the systematic error due to the uncertainties in thesevalues in the masses and ages derived using our method.In principle it may be possible to estimate an appropriatevalue of α MLT to use for a given star based on an empirical ortheoretical calibration of α MLT against properties such as mass,surface gravity, etc. We have decided not to attempt this and in-stead to treat the scatter in α MLT and Y as a sources of possiblesystematic error in the masses and ages. The systematic errorin the mass due to the uncertainty on Y is given by the quan- Article number, page 5 of 9 & Aproofs: manuscript no. paper-arxiv
Table 2.
Bayesian mass and age estimates for stars in detached eclipsing binary star systems ( h M ⋆ i ) compared to the mass of the stars directlymeasured from observations ( M obs ). Star P orb T e ff [Fe / H] s ρ ⋆ M obs h M ⋆ i [d] [K] [ ρ ⊙ ] [M ⊙ ] [M ⊙ ] Ref.WOCS 40007 A 3.18 6240 ±
90 0 . ± .
03 0 . ± .
010 1 . ± .
020 1 . ± .
038 1WOCS 40007 B 5960 ±
150 0 . ± .
016 1 . ± .
018 1 . ± .
056 1CD Tau A 3.43 6190 ±
60 0 . ± .
15 0 . ± .
008 1 . ± .
016 1 . ± .
079 2CD Tau B 6190 ±
60 0 . ± .
014 1 . ± .
016 1 . ± .
072 2CO And A 3.65 6140 ±
130 0 . ± .
15 0 . ± .
009 1 . ± .
007 1 . ± .
090 3CO And B 6160 ±
130 0 . ± .
008 1 . ± .
007 1 . ± .
092 3CoRoT 105906206 B 3.69 6150 ±
160 0 . ± . . ± .
017 1 . ± .
03 1 . ± .
071 4GX Gem B 4.04 6160 ± − . ± .
10 0 . ± .
002 1 . ± .
010 1 . ± .
041 5UX Men A 4.18 6190 ±
100 0 . ± .
10 0 . ± .
015 1 . ± .
001 1 . ± .
056 6,7UX Men B 6150 ±
100 0 . ± .
018 1 . ± .
001 1 . ± .
057 6,7WZ Oph A 4.18 6160 ± − . ± .
07 0 . ± .
012 1 . ± .
007 1 . ± .
052 8,9WZ Oph B 6110 ±
100 0 . ± .
011 1 . ± .
006 1 . ± .
053 8,9V636 Cen A 4.28 5900 ± − . ± .
08 0 . ± .
013 1 . ± .
005 0 . ± .
045 10,11V636 Cen B 5000 ±
100 1 . ± .
022 0 . ± .
003 0 . ± .
024 10,11CoRoT 102918586 B 4.39 7100 ±
120 0 . ± .
05 0 . ± .
013 1 . ± .
03 1 . ± .
043 12EK Cep B 4.43 5700 ±
200 0 . ± .
05 0 . ± .
009 1 . ± .
012 1 . ± .
069 13YZ Cas B 4.47 6890 ±
240 0 . ± .
06 0 . ± .
008 1 . ± .
007 1 . ± .
040 14BK Peg A 5.49 6270 ± − . ± .
07 0 . ± .
002 1 . ± .
007 1 . ± .
033 15BK Peg B 6320 ±
90 0 . ± .
013 1 . ± .
005 1 . ± .
047 15V785 Cep A 6.50 5900 ± − . ± .
06 0 . ± .
015 1 . ± .
007 1 . ± .
059 16V785 Cep B 5870 ±
100 0 . ± .
017 1 . ± .
007 1 . ± .
057 16BW Aqr B 6.72 6220 ± − . ± .
11 0 . ± .
018 1 . ± .
021 1 . ± .
086 17EW Ori A 6.94 6070 ±
100 0 . ± .
09 0 . ± .
012 1 . ± .
011 1 . ± .
051 18EW Ori B 5900 ±
100 0 . ± .
014 1 . ± .
009 1 . ± .
051 18V568 Lyr A 14.47 5650 ±
90 0 . ± .
05 0 . ± .
011 1 . ± .
004 1 . ± .
058 19V568 Lyr B 4820 ±
150 1 . ± .
034 0 . ± .
002 0 . ± .
038 19KIC 6131659 A 17.53 5790 ± − . ± .
20 1 . ± .
016 0 . ± .
007 0 . ± .
053 20LV Her A 18.44 6210 ±
160 0 . ± .
21 0 . ± .
013 1 . ± .
010 1 . ± .
104 21LV Her B 6020 ±
160 0 . ± .
013 1 . ± .
008 1 . ± .
092 21V565 Lyr A 18.80 5600 ±
90 0 . ± .
05 0 . ± .
014 0 . ± .
003 1 . ± .
042 22V565 Lyr B 5430 ±
130 1 . ± .
017 0 . ± .
003 0 . ± .
051 22Kepler-35 A 20.73 5610 ± − . ± .
20 0 . ± .
007 0 . ± .
005 0 . ± .
041 23Kepler-35 B 5200 ±
100 1 . ± .
016 0 . ± .
005 0 . ± .
043 23AI Phe B 24.59 6310 ± − . ± .
10 0 . ± .
008 1 . ± .
004 1 . ± .
096 24Kepler-34 A 27.80 5920 ± − . ± .
15 0 . ± .
006 1 . ± .
003 1 . ± .
074 25Kepler-34 B 5860 ±
140 0 . ± .
007 1 . ± .
002 1 . ± .
061 25KX Cnc A 31.22 5900 ±
100 0 . ± .
10 0 . ± .
006 1 . ± .
003 1 . ± .
057 26KX Cnc B 5850 ±
100 0 . ± .
006 1 . ± .
003 1 . ± .
048 26KIC 8410637 B 408.3 6490 ±
170 0 . ± .
11 0 . ± .
021 1 . ± .
017 1 . ± .
067 27
References. (1) Je ff ries et al. (2013); (2) Ribas et al. (1999); (3) Sandberg Lacy et al. (2010); (4) da Silva et al. (2014); (5) Lacy et al. (2008);(6) Andersen et al. (1989) (7) Hełminiak et al. (2009);(8) Clausen et al. (2008b); (9) Clausen et al. (2008a); (10) Clausen et al. (2008b);(11) Clausen et al. (2009); (12) Maceroni et al. (2013) (13) Martin & Rebolo (1993); (14) Pavlovski et al. (2014); (15) Clausen et al. (2010b);(16) Meibom et al. (2009); (17) Clausen et al. (2010b); (18) Clausen et al. (2010a); (19) Brogaard et al. (2011) ; (20) Bass et al. (2012);(21) Torres et al. (2009); (22) Brogaard et al. (2011) ; (23) Welsh et al. (2012); (24) Andersen et al. (1988); (25) Welsh et al. (2012);(26) Sowell et al. (2012); (27) Frandsen et al. (2013) tity σ M , Y , which is the change in in the best-fit mass, M b , pro-duced by applying our method using the grid of stellar models inwhich the helium abundance is increased by + .
02 compared tothe value used in our grid of standard stellar models. Similarly, σ τ, Y is the systematic error in the age due to the uncertainty on Y calculated in the same way. For the systematic errors in themass and age due to the uncertainty in α MLT we calculate thebest-fitting mass and age for a grid of stellar models calculatedwith garstec for α MLT = .
50 and multiply the resulting changein mass or age by the factor 0 . / (1 . − . σ M ,α is the Article number, page 6 of 9. F. L. Maxted et al.: Bayesian mass and age estimates for transiting exoplanet host stars change in the best-fitting mass due to an increase of 0.2 in α MLT and similarly for the age and σ τ,α . The two grids of stellar mod-els necessary for these calculations are included in the versionof bagemass that accompanies this article so the user is free todecide whether they should account for this potential source ofadditional uncertainty for the star they are analysing.
4. Results
We have applied our method to over 200 stars that host transitingplanet or brown dwarf companions and have found that the soft-ware runs without problems in all these cases and that our resultsare generally in good agreement with published results. Here weonly report the results for a selection of stars to illustrate themain features of our method and to highlight some interestingresults.The input data used for our analysis is given in Table 3. Forstars that have a trigonometrical parallax in van Leeuwen (2007)with precision σ π /π < ∼ . ⋆ using a valueof f ⊕ estimated by integrating a synthetic stellar spectrum fitby least-squares to the observed fluxes of the star. Optical pho-tometry is obtained from the Naval Observatory Merged Astro-metric Dataset (NOMAD) catalogue (Zacharias et al. 2004), theTycho-2 catalogue (Høg et al. 2000) and the Carlsberg MeridianCatalog 14 (Copenhagen University et al. 2006). Near-infraredphotometry was obtained from the Two Micron All Sky Sur-vey (2MASS) and Deep Near Infrared Survey of the South-ern Sky (DENIS) catalogues (The DENIS Consortium 2005;Skrutskie et al. 2006). The synthetic stellar spectra used for thenumerical integration of the fluxes are from Kurucz (1993). Red-dening can be neglected for these nearby stars given the accu-racy of the measured fluxes and parallaxes. Standard errors areestimated using a simple Monte Carlo method in which we gen-erate 65,536 pairs of π and f ⊕ values from Gaussian distribu-tions and then find the 68.3% confidence interval of the resultinglog( L / L ⊙ ) values.The results of the analysis for our selection of planet hoststars are given in Table 4, where we provide the values of themass, age and initial metallicity that provide the best fit to theobserved properties of the star, M b , τ b and [Fe / H] b , respectively,the value of χ for this solution, and the mean and standard devi-ation for each of the posterior distributions of age and mass. Thelikely evolutionary state of the star is indicated by the quantity p MS , which is the fraction of points in the chain for which thecentral hydrogen abundance is not 0, i.e., p MS is the probabilitythat the star is still on the main-sequence.
5. Discussion
An example of applying bagemass to one star (HAT-P-13) isshown in Fig. 4. We have chosen this star as an example becauseit shows the di ffi culties that can arise in the analysis of stars closeto the end of their main sequence evolution. The complexity ofthe evolution tracks and isochrones in the Hertzsprung-Russelldiagram is clear, even though we have restricted the models plot-ted to one value of [Fe / H] i (the best-fit value). The joint poste-rior distribution for the mass and age is clearly bimodal, withpeaks in the distribution near ( M , τ ⋆ ) = (1 .
20 M ⊙ , . ⊙ , 4.1 Gyr). It can be di ffi cult to ensure that the Markovchain has converged for distributions of this type where there are http://cdsweb.u-strasbg.fr/denis.html two solutions separated in the parameter space by a region of lowlikelihood. We used another MCMC analysis for this star with500,000 steps in the chain to verify that the default chain lengthof 50,000 is su ffi cient even in this di ffi cult case to produce anaccurate estimate of the posterior probability distribution. Themaximum-likelihood solution in this case has a mass and age( M b , τ b ) = (1 .
21 M ⊙ , . h M ⋆ i , h τ ⋆ i ) = (1 . ± .
06 M ⊙ , . ± . ρ ⋆ . In logarithmic terms the standard error on ρ ⋆ for this star is0.023 dex. This is much lower than the typical uncertainty for es-timates of log g based on the spectroscopic analysis of a late-typestar ( ≈ . g . Even so, the full Bayesian analysis is worthwhile toobtain an objective estimate of the true range of possible massesand ages for a planet host star, particularly for stars near themain sequence turn o ff or if the error on ρ ⋆ is large. We havealso found that the joint probability distribution for mass and agefrom the full Bayesian analysis can also be useful for improvingthe power of statistical tests based on these quantities, e.g., com-paring the ages derived from stellar models to gyrochronologicalages (Maxted et al., in prep.).HD 209458 was the first transiting exoplanet discovered(Charbonneau et al. 2000) and is also one of the brightest andbest-studied. The properties of this planetary system are also typ-ical for hot Jupiter exoplanets. The fit to the observed propertiesof this star including the luminosity constraint is good ( χ = . h τ ⋆ i and h M ⋆ i for these stars in Table 4 we see that the adding paral-lax measurement with the best precision currently available doesnot lead to a significant improvement in the precision of the massand age estimates – the slightly better precision of the age esti-mate for HD 209458 is mostly due to the higher precision ofthe density estimate. A similar argument applies in the case ofthe stars HD 189733 and WASP-52. These stars are less massivethan HD 209458 and close to the limit at which age estimatesfrom stellar models become impossible because there is no sig-nificant evolution of the star during the lifetime of the Galaxy.For stars without a parallax measurement the number of ob-servables is the same as the number of model parameters, butit is still possible that no models in the standard model gridgive a good fit to the observed properties of the star. This isthe case for Qatar 2, which is a good example of a star that ap-pears to be older than the Galactic disc (10 Gyr, Cojocaru et al.2014). We used the Markov chain calculated for this star to esti-mate an upper limit to the probability that the age of this starderived from our stellar models is less than 10 Gyr and find P ( τ ⋆ <
10 Gyr) < . τ b = . σ τ, Y and σ τ,α are not reliable in this case.It has long been known that some K-dwarfs appear tobe larger by 5% or more than the radius predicted by stan-dard stellar models (Hoxie 1973; Popper 1997). This “radiusanomaly” is correlated with the rotation rate of the star, but alsoshows some dependence on the mass and metallicity of the star(López-Morales 2007; Spada et al. 2013). The dependence on Article number, page 7 of 9 & Aproofs: manuscript no. paper-arxiv
Fig. 4.
Left panel: Results of our MCMC analysis for HAT-P-13 in the Hertzsprung-Russell diagram. Black dots are individual steps in the Markovchain. The dotted line (blue) is the ZAMS. Solid lines (red) are isochrones for ages 6 . ± . . ± .
06 M ⊙ . All isochrones and tracks are for [Fe / H] i = .
47. Right panel: Joint posterior distribution for the mass and age andHAT-P-13 from our MCMC analysis.
Table 3.
Observed properties of a selection of stars that host transiting extrasolar planets.
Star P [d] T e ff [K] [Fe / H] ρ/ρ ⊙ f ⊕ [pW m − ] log( L ⋆ / L ⊙ ) Ref.HAT-P-13 = TYC 3416-543-1 2.92 5653 ± + . ± .
08 0 . ± .
013 1HD 209458 3.52 6117 ± + . ± .
05 0 . ± .
008 23 . ± . + . ± .
04 2WASP-32 2.72 6042 ± − . ± .
09 0 . ± .
050 3,4HD 189733 2.22 5050 ± − . ± .
05 1 . ± .
170 27 . ± . − . ± .
025 2WASP-52 1.75 5000 ± + . ± .
12 1 . ± .
080 5Qatar 2 1.34 4645 ± − . ± .
08 1 . ± .
016 6
References. (1) Southworth et al. (2012); (2) Southworth (2010); (3) Brown et al. (2012); (4) Teske et al. (2014); (5) Hébrard et al. (2013);(6) Mancini et al. (2014).
Table 4.
Bayesian mass and age estimates for a selection host stars of transiting extrasolar planets.
Star τ b [Gyr] M b [M ⊙ ] [Fe / H] i , b χ h τ ⋆ i [Gyr] h M ⋆ i [M ⊙ ] p MS σ τ, Y σ τ,α σ M , Y σ M ,α HAT-P-13 6.59 1.206 + .
473 0.02 6 . ± . . ± .
08 0.29 − .
05 1 . − . − . + .
068 0.19 2 . ± . . ± .
04 1.00 0 .
20 1 . − . − . − .
025 0.01 3 . ± . . ± .
05 1.00 0 .
47 1 . − . − . − .
008 0.02 4 . ± . . ± .
02 1.00 2 .
32 1 . − . − . + .
078 0.01 7 . ± . . ± .
04 1.00 2 .
53 3 . − . − . + .
149 3.58 15 . ± . . ± .
01 1.00 − .
04 0 . − .
018 0 . ff ectthe structure of a star by producing a high coverage of starspots,which changes the boundary conditions at the surface of the star,or by reducing the e ffi ciency of energy transport by convection.Whatever the cause of the radius anomaly in K-dwarfs, the exis-tence of inflated K-dwarfs is likely to be the reason why Qatar 2has an apparent age >
10 Gyr. One method that has been pro-posed to deal with the radius anomaly is to simulate the magneticinhibition of convection by reducing the mixing length parame-ter (Chabrier et al. 2007). For Qatar 2, we found that models with α MLT < . α MLT to use for a given star a ff ected by the radius anomaly and so no wayto estimate the ages of such stars.Clausen et al. (2010b) found a reasonable match to the ob-served masses and radii of both stars in the eclipsing binarysystem BK Peg using both the VRSS stellar evolution mod-els (VandenBerg et al. 2006) and the Yale-Yonsei “ Y ” models(Demarque et al. 2004), so the di ff erence between the observedmass of BK Peg A and the predicted mass with our grid ofstandard stellar models needs some explanation. Both modelgrids used by Clausen et al. (2010b) use old estimates for thereaction rate N(p , γ ) O. This reaction is the bottle-neck inthe CNO cycle and the relevant reaction rate has been redeter-mined experimentally and theoretically in the last decade, re-sulting in a reduction by a factor of 2 compared to previous val-ues (Adelberger et al. 2011). We are able to reproduce this goodfit to the mass and radius of BK Peg A with garstec if we usethe old (inaccurate) reaction rate. With the new reaction rate and
Article number, page 8 of 9. F. L. Maxted et al.: Bayesian mass and age estimates for transiting exoplanet host stars the correct value of the star’s mass the “red-hook” in the stel-lar evolution track as the star leaves the main sequence does notreach values of T e ff low enough to match the observed T e ff value.We also experimented with variations in the assumed convectiveovershooting parameter, α ov , but this does not help to produce agood fit.Decreasing Y or, to a lesser extent, α MLT leads to an increasein the estimated mass for stars with mass > ≈ . M ⊙ . This sug-gests that a comprehensive analysis similar to the one presentedhere but applied to eclipsing binary stars will yield useful in-sights into stellar models for such stars. Indeed, Fernandes et al.(2012) have undertaken just such an analysis using a maximumlikelihood method to estimate the helium abundance and mixinglength parameter for 14 stars in seven eclipsing binary stars withmasses from 0.85 M ⊙ to 1.27 M ⊙ . They find a weak trend forstars with large rotation velocities to be fit better by models withlower values of α MLT , in qualitative agreement with the resultsthat we have found here. We have assumed that rotation is alsothe reason why the mass esimated using our method is too lowby about 0.1 M ⊙ for some stars with masses > ∼ M ⊙ in detachedeclipsing binaries. This issue deserves further investigation, ide-ally using Bayesian methods similar to those developed here.Given the number of model parameters and other factors such asrotation and magnetic activity that should be considered, thereis a clear need for accurate mass, radius, T e ff and [Fe / H] mea-surements for more eclipsing binary stars. In particular, there isa lack of data for stars in long period eclipsing binaries withmasses 1.3 – 1.6 M ⊙ , i.e, at the upper end of the mass distribu-tion for planet host stars.
6. Conclusion
The software used to produce the results in this paper is writtenin standard fortran , is freely available and the installation re-quires only one, widely-used additional software library ( fitsio ,Pence 1998). The method has been validated against other mod-els and tested using the observed properties of detached eclipsingbinary stars. The method and its limitations are fully describedherein. As a result, it is now straightforward to determine accu-rately the joint posterior probability distribution for the mass andage of a star eclipsed by a planet or other dark body based on itsobserved properties and a state-of-the art set of stellar models.
Acknowledgements.
JS acknowledges financial support from the Science andTechnology Facilities Council (STFC) in the form of an Advanced Fellowship.AMS is supported by the MICINN ESP2013-41268-R grant and the General-itat of Catalunya program SGR-1458. The pspline package is provided by theNational Transport Code Collaboration (NTCC), a U.S. Department of Energy(DOE) supported activity designed to facilitate research towards development offusion energy as a potential energy source for the future. The NTCC websiteis supported by personnel at the Princeton Plasma Physics Laboratory (PPPL),under grants and contracts funded by the US Department of Energy.
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