Deriving the most likely stellar properties : Bayesian methods and star count models Léo Girardi INAF – Osservatorio Astronomico di Padova, Italy with contributions from A. Miglio, B. Chaplin, A. Bressan, P. Marigo, S. Rubele, L. Kerber, M. Groenewegen, L. da Silva, B. Rossetto, J. Johnson, et al. ”Know the star, know the planet” How to derive masses, ages, distances, radii, of candidate planet-hosting stars? What do we usually have: ● Multi-band photometry + 1. Parallaxes + spectroscopy for bright stars (V<8) 2. Asteroseismology + spectroscopy for Kepler+CoRoT targets ● Plenty of statistics about stars of similar colors & mags, in and outside clusters Where do stellar ages/masses come from 7 10 evolutionary tracks 100 − 0.1 M⊙ isochrones 10 − 10 yr L. Girardi, Marseille, May 13 – p. 3 Where do stellar ages/masses come from limitations we measure (mag,colour), • not (L, Teff ) tracks change with • metallicity (spectroscopy needed) at of close to the ZAMS: • gets only mass + upper limit to the age lower MS: only mass • isochrones 107 − 1010 yr L. Girardi, Marseille, May 13 – p. 3 Some more subtle limitations accurate tracks do not exist • they all use “fake physics” for convection: mixing length theory • and overshooting even the solar chemical composition is debated • what does really exist • updated tracks: with the best-ever microphysics (often • irrelevant) successful tracks: that reproduce more observations with less • change in parameters L. Girardi, Marseille, May 13 – p. 4 Ages (and masses) of star clusters and binaries Just choose the isochrone that passes above the most points L. Girardi, Marseille, May 13 – p. 5 M67 Ages (and masses) of isolated stars Observable parameters: , , [Fe/H] from spectrum • Teff log g (V ,parallax) MV L • → → no parallax? use (big errors however) • log g Gaussian errors for all parameters • Ages can come from: isochrone fitting (Edvardsson et al. 1993) • Bayesian approaches (Jørgensen & Lindegren 2005) • L. Girardi, Marseille, May 13 – p. 6 Isochrone fitting identify the isochrone that • passes above the star want error bars? • select isochrones at 1σ of star ± from Jørgensen & Lindegren (2005) L. Girardi, Marseille, May 13 – p. 7 Isochrone fitting Problems isochrones cross each • other not evenly spaced with age • behaviour changes with • [Fe/H] population density along • isochrones varies widely (evolutionary rate, IMF) from Jørgensen & Lindegren (2005) L. Girardi, Marseille, May 13 – p. 7 Bayesian estimation take into account all we know about the star (and similar stars), • and about the isochrones does not give an age, but an age distribution function • needs a prior for the ages • L. Girardi, Marseille, May 13 – p. 8 Bayesian estimation The Bayes theorem, the posterior probability P (τ,m,ξ) is P (τ,m,ξ) Pprior(τ,m,ξ) L(τ,m,ξ) ∝ × is what we expect a priori for • Pprior τ is the likelihood function • L For a set of n different data qi, with Gaussian errors n 1 L(τ,m,ξ) exp χ2/2 ∝ √ × − i=1 σ 2π ! Y n obs 2 qi qi(τ,m,ξ) χ2 = − σi i=1 X qi can be anything observed: Teff , log g, MV , B V , [M/H], ... − L. Girardi, Marseille, May 13 – p. 8 Bayesian estimation Separate into 3 indepedent probability functions for τ,m,ξ, and you have (Jørgensen & Lindegren 2005): f(τ) ψ(τ)G(τ) ∝ with ψ(τ) being the prior for the age, i.e. the star formation rate G(τ) L(τ,m,ξ) φ(m)dm dξ ∝ Z Z with φ(m) being the prior for the mass function, i.e. the IMF and the prior for metallicity ξ disappeared because assumed constant. G(τ) is what we want to determine. What about ψ(τ) ? A lot of freedom here. L. Girardi, Marseille, May 13 – p. 8 How does it work in practice (da Silva et al. 2006) given a star of MV σ, Teff σ, [Fe/H] σ • 0 ± 0 ± 0 ± take an isochrone of [Fe/H] and age • τ take a small section of 1 2 and mean (i.e. , , , etc.) • [Mi ,Mi ] x M g R the probability of star V belonging to this section is • [M 0, Teff ] 2 2 2 Mi (MV MV ) (T T ) P φ(M )dM exp 0 eff eff 0 12 1 i i −2 −2 ∝ M × − σ − σ Z i " MV Teff # add to cumulative histogram of • P12 P (x) integrate over entire isochrone • over all possible [Fe/H] values (Gaussian distribution) • over all possible values (flat distribution) • τ plot , compute median, mean, variance, etc. • P (x) x L. Girardi, Marseille, May 13 – p. 9 Sample in the CMD for 53 stars, da Silva et al. (2006), determined [Fe/H] 0.1 dex, • ± log g 0.2 dex, Teff 70 K ± ± L. Girardi, Marseille, May 13 – p. 10 Examples of PDFs L. Girardi, Marseille, May 13 – p. 11 Examples of PDFs L. Girardi, Marseille, May 13 – p. 12 Examples of PDFs L. Girardi, Marseille, May 13 – p. 13 Examples of PDFs L. Girardi, Marseille, May 13 – p. 14 Examples of PDFs L. Girardi, Marseille, May 13 – p. 15 Examples of PDFs L. Girardi, Marseille, May 13 – p. 16 Examples of PDFs L. Girardi, Marseille, May 13 – p. 17 Examples of PDFs L. Girardi, Marseille, May 13 – p. 18 Examples of PDFs L. Girardi, Marseille, May 13 – p. 19 Examples of PDFs de Medeiros et al. (2009) L. Girardi, Marseille, May 13 – p. 20 Sanity checks observed vs. derived diameters L. Girardi, Marseille, May 13 – p. 21 Sanity checks “colour excess vs. distance” “colour excess vs. Teff ” L. Girardi, Marseille, May 13 – p. 22 Sanity checks the mass–metallicity relation L. Girardi, Marseille, May 13 – p. 23 Sanity checks 2 Hyades giants in the sample • HD27371: τ = 0.53 0.09 Gyr, M = 2.70 0.13 M⊙ • ± ± HD27697: τ = 0.67 0.13 Gyr, M = 2.54 0.14 M⊙ • ± ± best turn-off age is 0.625 0.05 Gyr (Perryman et al. 1998) • ∼ ± 3 Hya binaries indicate 0.63 Gyr (Lastennet et al. 1999) • ∼ L. Girardi, Marseille, May 13 – p. 24 SOLAR-LIKE OSCILLATIONS KIC8006 Sun 161 SOHO KIC8379 KIC6603 927 624 KIC6106 KIC6116 415 048 Chaplin et al. 2011 Seismology of planet-host e.g. stars Kepler-37 Barclay et al. 2013 Kepler-68 Gilliland et al. 2013 Kepler-56 Huber et al. 2013 Ensemble of 77 KOI with seismic constraints Solar-like oscillations: average parameters Δν: large frequency separation BiSON data Solar-like oscillations: average parameters νmax : frequency of maximum power Mosser et al. 2010 Brown et al. 1991 Kjeldsen & Bedding 1995 Solar-like oscillations: average parameters average seismic parameters: Mass and radius estimate: EVOLUTIONARY STATE OF GIANTS Distinguishing between RGB and RC stars evolutionary track 1.4 Msun Composite YC population RC vs RGB EVOLUTIONARY STATE OF GIANTS RGB RC vs RGB period spacing of high- RC order g modes where EVOLUTIONARY STATE OF GIANTS Observations with Kepler and CoRoT: He-burning Kepler He-burning RGB CoRoT RGB Bedding et al. 2011, Nature Mosser et al. 2011, A&A ”Grid-based methods” in asteroseismology Basic observables: Δγ, γmax, ΔP, Teff , and their errors → Weighted likelihood over all isochrone sections That's the same as the classical Bayesian estimation, but for minor details: ● Some authors do not weight on the occupation probability along the isochrone ● ΔP, when used, is just a on/off flag for the evolutionary stage (RGB or RC) PARAM v1.3 http://stev.oapd.inaf.it/param ● http://stev.oapd.inaf.it/param This part works Almost works... Going further: the role of star count models ● Johnson, Morton & Wright 2013: using TRILEGAL Galaxy model as a prior for the stellar mass Small correction in this case, but no reason for not taking it into account in large surveys with uniform & simple selection criteria Going further: the role of star count models ● Gaidos & Mann 2013: bias and selection effects in transiting planet surveys ● Gaidos 2013: identifying most likely Earth-size Kepler transiting planets in the habitable zone ● Batalha et al. 2013: simulating Kepler false-alarm probabilities ● ... Given the increased use of such models in planet-host characterization, how reliable are their predicted mass, radius, and age distributions? (Predicted CMD distributions are not in question) TRILEGAL ● TRIdimensional modeL of thE GALaxy (or ”very nice” in Southern Brazil) ● A population synthesis code to simulate resolved stellar populations in general – star clusters, background galaxies, and the Milky Way ● Main particularities: ● stands on well-tested stellar models / isochrones ● (m)any photometric system(s) TRILEGAL scheme (v1.3) TRILEGAL web interface ● http://stev.oapd.inaf.it/trilegal TRILEGAL web interface ● http://stev.oapd.inaf.it/trilegal Evolutionary tracks / isochrones ● Girardi et al. (2000) for most stars – 0.2 to 7 Mꙩ ● Bertelli et al. (1994) for massive stars ● Marigo & Girardi (2007) for AGB stars ● Vassiliadis & Wood for PNN + Benvenuto & Althaus for white dwarfs (under revision; Zabot et al. 2010) ● Chabrier et al. (2000) for very-low mass and brown dwarfs – down to 0.01Mꙩ Stellar atmosphere models ● Mainly ATLAS9 ODFNEW (Castelli & Kurucz 2004), for -2.5<[M/H]<+0.5], 3500<Teff/K<50000, ● Blackbody for OB stars with Teff>50000 K (replacement planned) ● Koester et al. for DA white dwarfs (being updated, Zabot et al. 2010) ● Fluks et al. (1994) for M giants (under revision, Aringer et al. 2010) ● Either Loidl et al. (2003, v1.3) or Aringer et al. (2009, v1.4+) for C-type giants ● BDUSTY1999 (Allard et al. 2000) for very-low mass and brown dwarfs – down to 500 K Stellar atmosphere models ● Mainly ATLAS9 ODFNEW (Castelli & Kurucz 2004), for 3500<Teff/K<50000 ● Blackbody for OB stars with Teff>50000 K (replacement planned) ● Koester et al.