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 ● 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 ● 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 Photometric systems ● Large database being built – with filter sets of important surveys, and many contributions by TRILEGAL users ● Vega, AB, and ST systems, with zeropoint corrections when known ● Method for deriving bolometric corrections described in Girardi et al. (2002) ● Latest additions: GALEX, HST/WFC3, Megacam, Suprime-Cam Milky Way components ● Each galaxy component has its geometry ρ(r), ● star formation rate and age–metallicity relation, ψ (t) and Z (t), + the IMF (apparent binaries are included a posteriori) ● All functions may depend on r and t (but usually do not) ● Presently included: ● Triaxial bulge cf. Binney et al. (1997) ● Oblate spheroidal halo ● Thick disk (double exponential) ● Thin disk (double exponential with hz(t) ) ● Dust layer (exponential in z) Resulting volume-limited samples ● All disk stars (including white and brown dwarfs) within 80 pc Extinction ● Mean extinction coefficients for every filter computed with Cardelli et al. extinction curve with RV =3.1 ● Latest addition (v1.4): star-by-star extinction coefficients – Aʎ/AV varying with Teff, logg, and AV (Girardi et al. 2008) ● All extinction is therefore specified by the function AV (r), and its small-scale dispersion, σ(AV )(r) TRILEGAL initial calibration From the 7-passband CDFS stellar catalogue (Groenewegen et al. 2002): TRILEGAL initial calibration 2MASS towards the NGP (Girardi et al. 2005): TRILEGAL initial calibration 2MASS towards the galactic anticenter (Girardi et al. 2005): TRILEGAL initial calibration Hipparcos local sample (Girardi et al. 2005): all stars with V<7 and 1/π < 100 pc TRILEGAL initial calibration Hipparcos local sample (Girardi et al. 2005): all stars with V<8 and 1/π < 100 pc Tests to the initial calibration A very detailed modelling of UKIDSS data (Kerber et al. 2009). For a 0.21 sqrdeg area towards l=-220 deg, b=40 deg, first we simulate the images using TRILEGAL output (see poster by S. Rubele), then perform the photometry exactly as in the real UKIDSS image. Calibrating the Bulge ● Vanhollebeke et al. (2009) ● 7 parameters calibrated via a maximum-likelihood optimization method Calibrating the Bulge ● Results: (similar plots for 15 OGLE-II fields) ● Best-fitting parameters and (initial) error estimates Recalibrating disk(s) and halo ● Vanhollebeke et al. (2009) algorithm adapted to work with many more l.o.s. and input catalogues (2MASS, SDSS, OGLE, UKIDSS) ● Recalibration is running, but heavy computational work ● To have acceptable CPU times: ● presently limited to ~50 l.o.s., most coincide with SEGUE plate pointings, and ugrizJHKs filters ● thin disk is assumed to follow hz(t) α σW(t) – isothermal disks – with σW(t) taken from Geneva-Copenhagen survey (Nordstrom et al. 2007, Holberg et al. 2009) But With so many different ingredients, evolutionary tracks – spectra – filters – MW geometry – extinction maps – IMF – SFH some of which admitedly very uncertain (as the thin disk SFH) why should we trust the results ????? OK, THEY FIT THE PHOTOMETRY – STAR COUNTS IN COLOR-MAGNITUDE DIAGRAMS But why trusting the masses, radii, distances, ages ? The same code, near and far ● Underlying isochrones are being widely used and tested – the small bugs detected are usually corrected within 1 week ● Same code is being used to model star clusters and nearby galaxies from 40 pc to 4 Mpc – resulting stellar model improvements are immediately incorporated in the MW model e.g 180 sqrdeg of the LMC (+foreground) as seen by 2MASS Testing logg distributions The logg distribution from large spectroscopic surveys (Girardi et al. 2013), or equivalently, the giant/dwarf ratio. RAVE DR2 Testing logg distributions The logg distribution from large spectroscopic surveys (Girardi et al. 2013), or equivalently, the giant/dwarf ratio. RAVE DR2 Testing logg distributions The logg distribution from large spectroscopic surveys (Girardi et al. 2013), or equivalently, the giant/dwarf ratio. SEGUE Comparing with asteroseismic samples CoRoT LRs: ~ 3000 stars Mosser et al. 2010 Kepler data: ~ 10000 stars Hekker et al. 2011, Stello et al. uncertainties ~10-15% LRa01 Miglio et al. 2012, 2013 LRc01 Comparing with properties of Kepler dwarfs Chaplin et al. 2011 Discrepancies in stellar masses at ~1 Msun: – bad simulation of Kepler selection function? – problem with IMF and/or disk SFH ? – real problem with stellar models ? observed predicted by TRILEGAL Differential population studies with CoRoT observed zLRa01 < zLRc01 Miglio et al. 2013 MNRAS synthetic Summary ● Bayesian estimation of stellar parameters: now widely recognised as superior to isochrone fitting – well-tested versions using both parallax and asteroseismology cases ● Methods can also be used with photometry + spectroscopy alone – but with huge errors ● The next step: Bayesian inference using MW population models – useful as far as the planet-search surveys use well-defined (and easy to simulate) target selection criteria ● Such models becoming reliable → reproducing photometry + basic R, logg, and M distributions from large spectroscopic and asteroseismic surveys Where TRILEGAL v1.3 fails the most ● Low b in general, this is partially due to bad extinction model, partially because not calibrated there ● Rossetto et al. (2010): a large-area comparison between TRILEGAL v1.3 and 2MASS star counts What TRILEGAL does not ● No real dynamics is foreseen for the next few years, just simplified kinematics. Some good reasons for this: ● needs a mass distribution for the MW, which we do not know (e.g. what about a dark matter disk?) ● would increase a lot the CPU times ● less flexibility in playing with MW geometry ● Just smooth MW components (+ star clusters and background galaxies) ● No interacting binaries