Probabilistic Approach

Spectroscopists usually quote on their analysis the best-fit parameters on their quantities plus some error margin. Yet this is only a subset of the information that can actually drawn from their analysis - it would only be if the errors on the stellar parameters were not correlated, i.e. if

Cov(Teff,[Fe/H]) =0. This is by far not true - the errors are highly correlated by the underlying

physics. The stellar effective surface temperature Teffand also surface gravity log(g)do affect

the ionisation equilibria, the atomic level populations, etc., and hence spectral line strengths and inferred elemental abundances. Hence the metallicity estimate will be a function on those two variables, while it affects their values by itself. The strong correlation between estimated temperature and metallicity is found everywhere in literature and can e.g. be guessed from figs. 6 and 7 in Zwitter et al. (2008), or fig. 12 in Siebert et al. (2011). So the quantity handling all the available information are not the single best-fit values, but the multidimensional probability distribution of the stellar parameter estimates on the given spectrum:

Psp([Fe/H],Teff,log(g),[α/Fe],Xsp) (5.4)

where Xsp sums up all the spectral parameters not listed in this course, e.g. micro-turbulence

- if not a 3D model atmosphere is run (see a discussion of advanced spectroscopic analysis in Asplund, 2005b), that does not require this parameter - , rotational velocity of the star, reddening, etc.

Yet, as discussed by Asplund (2005a) in view of the IRFM, spectroscopy is not the only thing we know: We also have photometric information about the examined objects, with Hipparcos

evolution models give very firm constraints on the possible loci of stars in parameter space: In simple models their parameters are a simple function of their initial mass, metallicity and age:

(Teff,log(g),C) = f′(Mi,[Fe/H],[α/Fe],τ), (5.5)

where C denotes the full information by the absolute magnitude and colours of stars. This relation provides in theory a very accurate mean of discarding major parts of the parameter space of spectroscopic results.

As has been shown by various sources (e.g. Chieffi et al., 1991; Chaboyer et al., 1992), the dif- ferent metallicity indicators can moreover be replaced by an effective [Me/H] without significant accuracy. In an ideal world stellar evolution would hence give us a three-dimensional region in parameter space, where stars can reside, which before the isochrone turnoff with its age de- pendence even collapses into a two-dimensional sheet. In reality those predictions are affected uncertainties in atmosphere models, in the modelling of the stellar interior (see e.g. Magic et al., 2010) and in a couple of initial parameters (Charlot et al., 1996), particularly the initial helium problem particularly on the metal poor side (Casagrande et al., 2007). This should be accounted for by broadening the above relation. In a simple way speaking in the formalism of Burnett & Binney (2010) this can be achieved by augmenting the observable errors a bit. In this case it will be advisable to work with variable errors: While stellar models are very precise on the upper main sequence, their reliability deteriorates towards the coolest and evolved objects, i.e. M dwarf and the giant branches.

Using the mapping given by stellar evolution models, we can write down the probability distri- bution: P(Teff,log(g),C) =G′(σ)◦ ◦ Z f′(Mi,[Fe/H],[α/Fe],τ) ·Pp(τ,Mi,[Fe/H],[α/Fe],X)dMid[Fe/H]d[α/Fe]dX (5.6)

where Pp(τ,Mi,[Fe/H],[α/Fe],X)sums up our a priori information on these quantities and the

result gets convolved with G′(σ), taken as described above to be some Gaussian broadening according to the error vectorσ= (σTeff,σlog(g),σC). In case we want to accommodate a Galactic

model, one would need to further work in the distance distribution and the distance modulus for the sample.

On the left hand side we must now attain the parameter space that we need for compatibility with the spectroscopic results (i.e. we need a probability distribution in the same dimensions):

Replace the function f′by an extension of f′that is equivalent with f′on(Teff,log(g),C)space

and is an identity mapping on the other parameters (and do so in parallel with G′):

P([Fe/H],[α/Fe],Teff,log(g),Mi,τ,C,X) =G(σ)◦

◦

Z

f(Mi,[Fe/H],[α/Fe],τ)

·Pp(τ,Mi,[Fe/H],[α/Fe],X)dMid[Fe/H]d[α/Fe]dX (5.7)

This entity looks evil, but it is not. We can gain it by a simple weighted (via our priors) integral over the isochrones and any condition we impose upon our parameter set can be gained by multiplying the corresponding probability function onto P and then (like in Burnett & Binney, 2010) normalising the sum to one, after we have integrated out all dimensions we do not like. So we can get the probability distribution in our favourite parameter space:

Pi([Fe/H],[α/Fe],Teff,log(g)) =

Z

P([Fe/H],[α/Fe],Teff,log(g),Mi,τ,C,X)dMidτdCdX (5.8)

Once this is established, we can combine the spectroscopic information with the stellar evolution and a priori knowledge by a simple multiplication:

P′=PspPi (5.9)

Isochrone sets and priors

To get some rudimentary grip on the uncertainties in stellar models we make use of two indepen- dent data sets: A dense grid drawn from the web interface of the Padova isochrones (Bertelli et al., 2008, 2009) and a dense grid of BASTI isochrones (Pietrinferni et al., 2004, 2006, 2009) that was kindly provided by S. Cassisi for our own probabilistic age determinations in Casagrande et al. (2011). The Padova sample consists of isochrone data sets at 56 metallicities ranging from

Z =0.0001 to Z =0.07. Their solar metallicity is at the Grevesse & Sauval (1998) estimate of Z_⊙ =0.017, while the price they have to pay for the closer proximity to recent estimates (Asplund et al., 2009) is a rather low solar ( Y_⊙ =0.26) and hence at a standard ∆Y/∆Z too

low primordial helium abundance compared to standard Big Bang Nucleosynthesis (Steigman, 2010) (another point might be their lack of diffusion, which acts in the same direction). In the query we hence had to limit Y >0.23, where their grid ends. We applied a dense age spacing of 0.01 dex or respectively 2.3%. The BASTI isochrones still use the (Grevesse & Noels, 1993) solar metallicity and we apply their standard helium abundances. We have 20 metallicity bins from Z=0.0001 to Z=0.04 and an age spacing of at maximum 100 Myr.

As initial mass function we apply the Salpeter (1955) IMF, with the usual exponentα =2.35, any other IMF of desire can be readily implemented and tested for.

It would be tempting to use a more specific age prior or even age-metallicity relation, but this would raise concerns about that results might be influenced by prejudices on the age of stellar components. Many studies make firm conclusions that the oldest populations in the Galaxy exceed 10 Gyr in age (cf. Aumer & Binney, 2009; Sch¨onrich & Binney, 2009a). Hence we use a simple flat age distribution between 0 and 14 Gyr as a prior.

These are delivered by the infrared-flux method (IRFM, Blackwell & Shallis, 1977; Casagrande et al., 2010). In addition the remaining colour information can be scanned for information on reddening, metallicity, alpha enhancement or in some cases gravity. Photometric calibrations are a vastly used tool and if parameters apart from Teffare required, narrow band filter systems like

Str¨omgren or Washington photometry contain the best information. Yet some tentative estimates can be derived from broad band photometry as well. Overall, photometric methods deliver a third probability distribution Pph(Teff,log(g),[Fe/H],[α/Fe]). Again this can be combined with

the two other sources to the final probability distribution

Pf =PspPiPph, (5.10)

from which our improved parameter estimates can be drawn. The photometrically derived tem- peratures are generally of a quality excelling that of spectroscopic ones in their scatter and their zero points: It was shown in previous studies (Casagrande et al., 2011) that photometric tem- peratures from the IRFM are (apart from the most metal poor objects) in sound agreement with stellar evolution models.