AMLA
Seminar:


Mario
Gennaro,
MPIA
‐
Heidelberg


Wednesday,
24th
June
2009


Es#ma#ng
parameters
of
stellar
popula#ons


Outline


PART I : description (and some applications) of a Bayesian method for the determination of stellar ages of single stars; based on Jørgensen and Lindegren (2005); A&A, 436, 127

Both methods aim to give a statistical reliability to the parameters estimated from data using stellar models; i.e. a robust estimator of the parameter as well as meaningful Confidence Interval.

PART II : a maximum-likelihood method for fitting colour-magnitude diagrams; based on Naylor and Jeffries (2006), MNRAS, 373, 1251

PART
I:
problem
formula#on


Model parameters: p = (τ, ζ, m) ζ is some form for the metallicity Y, mixture functions of ζ

Observational data: q = ([Me/H], log Teff, MV) But any color may + uncertainties be also used

Models provide a mapping p q(τ, ζ, m) Age determination is a particular case of the inverse problem, i.e. finding τ(q); this is possible if dim(p) ≤ dim(q) and if the mapping is non-degenerate

The problem can be solved by numerical inversion (isochrone fitting) or using a probabilistic approach.

PART
I:
Bayesian
es#ma#on
of
stellar
ages


Maximum likelihood (or min chi-sq) estimate of age: is it good?

NOT ALWAYS

Bayes theorem (+ norm.):

where:

and:

PART
I:
Bayesian
es#ma#on
of
stellar
ages


Which is the ‘’best’’ isochrone to fit the point on the right?

It is clear that we need some additional information to

understand which is the most probable age

Experience can guide us (guess what is the most probable age and I’ll buy you a PIZZA), but in order to have a quantitative estimate we

need to specify the form of the prior probability distribution

PART
I:
Bayesian
es#ma#on
of
stellar
ages


The ‘’arrow-step’’ is reasonable if one wants to use his data to investigate any relation among the parameters, i.e. an age-metallicity relation

•  SFR is chosen to be flat (for the same reason)

•  Metallicity distribution is also chosen to be flat (it is good if there are good observations of [Me/H])

•  IMF is chosen to be a power law

Then, with a flat SFR, age determination is entirely based on the study of G(τ)

PART
I:
Bayesian
es#ma#on
of
stellar
ages


G function for the star of the previous example (for three values of the nominal error)

The “speed of evolution” is automatically taken into

account here via the integration on the mass

(not shown): the behavior of G is not sensitive to the exponent in the IMF and the choice of the metallicity prior (as far as obs.

errors are not too big)

PART
I:
Bayesian
es#ma#on
of
stellar
ages


Best value for the age: mode (by MC simulation it has been shown that this is the less biased estimator when G is not “well behaved”)

Confidence interval is defined to be the shortest interval outside which the G function is below a limiting value Glim . The 68% confidence level corresponds to a value Glim = 0.61 (no proof here!). This is also validated via MC simuation

PART
I:
Some
applica#ons
of
the
method


Ages and masses in the Geneva – Copenhagen survey of solar neighborhood (Nordström et al., 2004) have been estimated with this technique.

It is shown that age-metallicity / age-position-metallicity / age-velocity relations may be well studied for a big sample of stars with homogeneously derived parameters (but always pay attention to observational biases !)

Application to stellar clusters: assuming that the observed stars in a cluster are all single, member, coeval and that the obs. data are independent, then:

PART
I:
Some
applica#ons
of
the
method


The G function provides a robust estimate of the cluster age even though data may show many kind of complications

The open cluster M67

Non Members ?

BS ?

Bad Phot ?

PART
II:
overview


•  Stellar cluster: population of stars that share the same global properties: age, distance, chemical composition

•  Often these properties are derived by some kind of by-eye comparison

•  Is there an objective way to do the comparison? •  Fitting a curve to data with 1D uncertainties -> chi-sq •  Can be extended to 2D uncertainties (linear and non linear models)

But no population really lies on a single curve in the CMD (binarity/multiplicity, IMF)

•  Is it possible to define some goodness-of-fit parameter to choose among isochrones?

•  Binning data (Dolphin, 2002)? Not always possible/desirable •  Introducing a new method that takes into account the 2D distribution of points in the CMD

PART
II:
the
τ2
distribu#on



Probability for a single point:

DM = 0; t = 40 Myr

Where:

(uncorrelated Gaussian errors)

Definition (similar to χ2):

An example of ρ, with BF=0.5

PART
II:
some
facts
(no
proof!)


1)  If we consider no error on the color, i.e. ci =c always then there is no dependence on c and ρ(c,m) δ(m) this implies τ2 χ2

2)  Even with 2D errors τ2 minimization yields the correct result when fitting a straight line

3)  The latter can be generalized and applied to the fit of curves with small curvature (and still works fine!)

The fact that τ2 works fine in these special cases is an important test and gives a good “feeling” that it can work fine also in its full 2D (or more?) implementation

PART
II:
cumula#ve
distribu#on


Once a minimum value for τ2 has been found, the question is what is the probability of obtaining an higher value? How significant is the result?

The Pr(τ2) for single points may be combined to give a cumulative distribution of τ2 for many data points.

The model that gives the minimum τ2 and the σ of the single data can be used to calculate the predicted τ2 value for the model: Pr(τ).

The expected τ2 distribution and the observed one can be compared to understand if the fit is a good fit or not

σ = 0.01 σ = 0.1

PART
II:
an
example
(NGC
2547)


Best fitting isochrone, with a binary fraction of 0.5

The corresponding τ2 grid

PART
II:
conclusion


This method is quite powerful and general; it has been used also in the color-color space to fit a mean extinction to stars in the same cluster

It has been applied to several young clusters, providing consistent and directly comparable values of their ages

Of course the results depend on the adopted evolutionary models, and (not strongly) by the assumption on binary fraction and IMF…but this may as well be an advantage if on wants to recover and not simply assume these two quantities.

Thank
you
and
….be
careful,
machines
are
learning!


Thank
you
and
….be
careful,
machines
are
learning!


Thank
you
and
….be
careful,
machines
are
learning!


Thank
you
and
….be
careful,
machines
are
learning!


Thank
you
and
….be
careful,
machines
are
learning!