Pre-Main Sequence evolutionary models

(low and intermediate mass)

Scilla Degl’InnocentiEmanuele Tognelli

Pier Giorgio Prada Moroni

(Physics Department, Pisa University)

Huge amount of observational data in young clusters and star forming regions in the Milky Way and Magellanic Clouds(Jeffries et al. 2014, Alcalà et al. 2014, Patience et al. 2014, Bouy et al. 2014, Sarro et al. 2014, Sacco et al. 2014, Robberto et al. 2013, Manara et al. 2013, Lopez-Marti et al. 2013, Spezzi et al. 2012, Alves De Oliveira et al. 2012, Bayo et al. 2012, Scandariato et al. 2012, Da Rio et al. 2012, Gougliermis et al. 2012, D’Orazi et al. 2009, Luhman et al. 2009, 2008, Enoch et al. 2009, Cignoni et al. 2010, Brandner et al. 2008, Reipurt et al. 2008 and references therein).

To infer the star formation rate and initial mass function updated Pre-MS evolutionary models are mandatory

Pisa Pre-MS models

Several Pre-Main Sequence database available in the literature(see e.g. Lagarde et al. 2012,Tognelli et al. 2011, Di Criscienzo et al. 2009, Dotter et al. 2008, Yi et al. 2001, Siess et al. 2000, Charbonnel et al. 1999, Palla & Stahler 1999, Baraffe et al. 1998, D’Antona & Mazzitelli 1997…)

The interpretation of a direct comparison among different evolutionary database is not so easy!(see e.g. Tognelli, Prada Moroni, Degl’Innocenti 2011, Hillebrand et al. 2008, Hillebrand and White 2004, etc..)

Quantitative analysis of the effects of uncertainties affecting both standard (no rotation, no magnetic fields, no mass loss etc..) and non standard PMS models

The adoption of a grey T(t) relationship is a too crude approximation for the integration of cold atmospheres (see e.g. discussions in Auman 1969, Dorman et al. 1989, Saumon et al. 1994, Baraffe et al. 1995, Allard et al. 1997, Chabrier & Baraffe 1997, Baraffe et al. 1998, 2000)

Thus all the evolutionary models are made with boundary conditions obtained from atmospheric models:P(tph,Teff, g, [Fe/H]) and T(tph,Teff, g, [Fe/H]) at tph provided by detailed, non-grey atmospheric models which solve the full radiative transport equation

- BH05 (default, Brott & Hauschildt 2005): 2000 K ≤ Teff ≤ 10 000 K;- CK03 (Castelli & Kurucz 2003): Teff > 10 000 K;- AHF11 (Allard et al. 2011): Teff < 2000

Boundary conditions at the bottom of the atmosphere

Main sources of uncertaintyon theoretical predictions

Chemical composition determination(metallicity, helium abundance, mixture..)

Adopted physical inputs

Physical mechanisms(diffusion, external convection efficiency,boundary conditions..)

(EOS, opacity, nuclear reaction rates..)

Main sources of uncertainty for the position of tracks and isochrones in the HR diagram

Chemical composition uncertaintiesUncertainties on [Fe/H], Z and Y

From [Fe/H] by adopting a solar scaled mixture one obtains the helium abundance Y and the total metallicityZ

By assuming:

- D[Fe/H] ≈ ±0.01, ±0.1 (we adopts ±0.05);- DYP ≈ 0.2485 ±0.008 (Cyburt 2004);- DY/DZ ≈ 2 ± 1 (Casagrande et al. 2007);- D(Z/X)sun ≈ +25/-10 % (see discussion in Tognelli et al. 2012)

Large differences in Teff in all the models both along the Hayashi track and in ZAMS

Luminosity is significantly affected only in not fully convective modelsReference values (Y=0.274, Z=0.01291, [Fe/H]=0)

(see e.g. Pagel & Portinari 1998 )

DTeff ≈ 50 – 100 K

DTeff ≈ 200 – 400 K

Dlog L/Lsun ≈ 0.01 – 0.1

(Tognelli et al. 2011)

DTeff ≈ 50 – 100 K

Warning

When comparing data with theoretical pre-MS tracks, one must use models with the same metallicity of the observed stars

An change in [Fe/H] between observations and models translates in a shift in Teff/L and hence in an error in the inferred mass and age

Δ[Fe/H]=0.2 dex leads to a shift in Teff of ≈100 K

ΔM=0.1 Mo Δt≈70 %

Chemical composition uncertaintiesSolar mixture

In PMS models solar mixture (at fixed Z) affects opacity:

Atmosphere. Molecules (CO, H2O, TiO) for logT(K) < 3.8 (Ferguson et al. 2005)

Interior. Hydrogen and metals: log T(K) = 5.2 – 5.8, e 6.4 – 6.8 iron group elements Fe (Cr, Fe, Ni) (Iglesias & Rogers 1996, Sestito et al. 2006)

Widely adopted solar mixtures

- GN93 (Grevesse & Noels 1993);

- GS98 (Grevesse & Sauval 1998);

- AS05 (Asplund et al. 2005);

- AS09 (Asplund et al. 2009);

(see also mixture evaluations by Caffau et al. 2010, Pereira et al. 2009,Grevesse et al. 2007)

Models with radiative cores are sensitive to a mixture variation (at fixed Z)(Models with Asplund et al. 2009 and Asplund et al. 2005 mixtures are very similar)

Chemical composition uncertaintiesSolar mixture

DTeff ≈ 90 – 100 K

DTeff ≈ 200 – 350 K

Physical inputs uncertaintiesEquation of state (EOS)

EOS plays a crucial role, in particular in the convective regions of low mass stars, which are almost adiabatic (see e.g. Mazzitelli 1989, D’Antona 1993 and references therein)

Teff and R of low-mass stars are determined by the adiabatic gradient, i.e. EOS

Effects of adopting different EOS*: For PMS stars with M < 1.0 Msun : maximum DTeff ≈ 50 - 70

K

*(PTEH95= Pols et al. 1995, FreeEOS08, see Irwin et al. 2004, SCVH95=Saumon et al. 1995, OPAL 2006,2001 see Rogers et al. 1996, Rogers & Nayfonov 2002)

(Tognelli et al. 2011)

If updated EOS are adopted the residual uncertainties do not affect in a relevant way the

evolution in the HR diagram

Physical input uncertainties14N(p,g)15O cross section

M≥ 1.5 Msun are affected starting from ZAMS

DTeff ≈ 40 K

DTeff ≈ 250 K

DTeff ≈ 350K

(Tognelli et al. 2011, see also Straniero et al. 2002, Imbriani et al. 2004, Degl’Innocenti et al. 2004, Weiss et al. 2005)

S(0)=1.57 ± 0.13 KeV-b (LUNA collaboration, see e.g. Formicola et al. 2003, Marta et al. 2008 and references therein)

Uncertainty on convection efficiency*

*Discussed by several authors, see e.g. Petersen 1990, Canuto & Mazzitelli 1992, D’Antona 1993, D’Antona & Mazzitelli 1994, 1997, Salaris & Cassisi 1996, Baraffe et al. 2000, D’Antona & Montàlban 2004, Montàlban et al. 2004, Landin et al. 2006

** Some evolutionary codes adopts the Full Spectrum Turbulent convection treatment (see e.g. Canuto & Mazzitelli 1991, Canuto et al. 1996)

A widely used approach is the mixing length theory** (Bohm-Vitense 1968) , in which the average convective efficiency depends on the free parameter a, to be calibrated with observations

The usual approach is the solar calibration which, however, does not rely on a physical argument, since there is no reason to expect that the efficiency of convection is the same for stars of different masses and different chemical compositions (see e.g. Canuto & Mazzitelli 1992, D’Antona & Mazzitelli 1994, 1998, Montalban et al. 2004)

Moreover there is no reason to assume a constant a value along the whole evolution of a star (see e.g. Siess & Livio 1997, Ludwig et al. 1999, Trampedach et al. 1999, 2007, Montalban & D’Antona 2006)

Uncertainty on convection efficiency

The assumed convection efficiency is related to the uncertainty on other inputs which affect the effective temperature

It is possible to obtain many solar models with significantly different pre-MS locations and shapes (see e.g. D’Antona & Mazzitelli 1994, Montalban et al. 2004)

(Tognelli, Prada Moroni, Degl’Innocenti 2011)

At present, a represents an uncertainty source

DTeff 50 KSSM

Uncertainty on convection efficiencyDependence on a:

- M < 0.2 – 0.1 Msun : mild dependence: almost adiabatic stars

- 0.2 Msun < M < 0.6 Msun : dependence only along the Hayashi track

.

DTeff until 500 K

- 0.6 Msun < M < 1.2 Msun :dependence along the whole pre-MS

- M > 1.2 Msun: dependence only along the Hayashi track

The uncertainty on aML also affects both mass and age estimates of young stars if obtained through the comparison of the stellar position in the HR diagram

Uncertainty on convection efficiency

(Tognelli, Prada Moroni, Degl’Innocenti 2011)

The greatest difference occurs near the Hayashi track

Uncertaintes: matching point between interior and atmosphere

The value of tph , the optical depth at which interior and atmosphere matches must be chosen in a way that:

- enough large to guarantee photon diffusion approximation; - enough small to reduce discrepancies due to inconsistencies between atmospheric and interior models .

generally: tph : 2/3 ÷ 100

(see e. g Montàlban et al. 2004, Tognelli et al. 2011)

Main effects on cold objects: low mass stars

and Hayashi track

DTeff ≈ 90 K

Reference value:tph = 10

(see e. g. Morel et al. 1994)

Model uncertainties: conclusions

PMS models (extremely) sensitive to:

original chemical composition, EOS, convection efficiency, atmospheric treatment

The adoption of updated evolutionary models for the comparison with observations is mandatory

Pisa pre-MS database

Very fine grid of masses and chemical compositions

11653 tracks and 10080 isochrones.

- 43 stellar mass values, 0.2 ÷ 7.0 Msun

- 20 metallicities (Z);

- 3 original helium abundances (Y);

- 2 original deuterium abundances (Xd);

-3 mixing length values (aML).

-Isochrone ages: 1 – 100 Myr for each set

Standard solar model: Y=0.253, Z=0.0138, a=1.68 Xd=2 . 10-5

(Tognelli, Prada Moroni, Degl’Innocenti 2011)

The database is available at the URL: http://astro.df.unipi.it/stellar-models/

Uncertainties on the initial conditions

The very common procedure of starting PMS evolution from a model on the Hayashi track with a large radius and luminosity defining it as “the zero age model” is not, in principle, a realistic procedure because it neglects the protostellar phase (see e.g. Stahler 1980, 1983, Hartmann et al. 1997)

However… IF we believe that once the main accretion phase is finished, the evolution quickly converges to that of standard hydrostatic models (see e.g. Stahler 1983, Palla & Stahler 1999)…..in this case one can analyse the effects of starting from different points along the Hayashi track.

Models should start from the birth line defined as the locus in the HR diagram where the star ends its accretion phase (Stahler 1983)

Under these hypothesis birthline is located in the region of the HR diagram corresponding to large and bright structures

The structure loses memory of this initial model and when this occurs?

That is ages inferred for very young stars are reliable?

“Zero Age Model” : expanse structure with Tc = 1÷2 105 K no deuterium burning

If Tc ≤ 3.5 105 K the tracks converge to an unique solution before 1 Myr (See also discussions in Baraffe et al. 2000, D’Antona & Montalban 2006, Landin et al. 2006, Scandariato et al. 2012)

Does the accretion phase affects PMS evolution?

Stars form from the fragmentations of molecular clouds, which provide the seeds of future stars (protostars). Then the remaining mass is accreted during constant and/or time dependent accretion episodes.

Despite of the huge observational effort* the accretion geometry (disk or spherical), the accretion rate and its time dependence, the mass and radius of the initial protostellar seed are still uncertain

Regarding protostellar evolution**: 1) spherical approach (proposed by Stahler et al. 1980, see also 1981, 1986, Palla & Stahler 1991) 2) disk accretion model (proposed by Hartmann et al. 1997 and Siess & Livio 1997)

The evolution of the protostar is strongly dependent on the accretion parameters, mainly on the fraction of the accretion energy deposed inside the star. *See e.g. Hartigan et al. 1991, Hartmann et al. 1998, 2011, Hillenbrand et al. 1998, Lada et al. 2000, 2001, Haisch et al. 2001, Calvet et al. 2004, Allers et al. 2006, Lada et al. 2006, Luhman et al. 2008,2012, Flaherty & Muzerolle 2008, Enoch et al. 2009, Hernandez et al. 2010, Rigliaco et al. 2011, Spezzi et al. 2012, Manara et al. 2012, Ercolano et al. 2014, Alcalà et al. 2014, Dunham et al. 2014 and references therein.

** For models of accreting stars see also Behernd & Maeder 2001, Yorke & Sonnhalter 2002, Baraffe et al. 2009, 2012, Baraffe & Chabrier 2010, Hosokawa et al. 2010, 2011, Haemmerlé et al. 2013

For hydrodinamical simulations of collapsing clouds, see Masunaga et al. 1998, Masunaga & Inutsuka 2000, Boyod Whitwort 2005, Vorobyov & Basu 2005,2006, 2010, Machida et al. 2010, Tomida et al. 2010, 2013, Dunham & Vorobyov 2012 and references therein

Non standard PMS models

The ’’classical theory’’ of stellar evolution (spherical symmetry, no effects of rotation, mass loss, magnetic fields and so on) fails to explain nearly all observed Li and Be surface abundance patterns

Several non standard models was proposed* (rotational induced mixing, traveling internal gravity waves, tachocline, a combination of gravity waves and rotation, internal magnetic fields, mass loss etc ..)

However the effects on L, Teff and evolutionary time of PMS models is quite negligible (see e.g. Charbonnel et al. 2013, Eggenberget et al. 2012, Landin et al. 2006), but for models which assume a variation of Teff and radius due to the presence of strong magnetic fields

*Reviews by Pinsonneault 1997, Delyannis 2000, Jeffries 2000, 2006, Pasquini 2000, Charbonnel 2000, Siess 2000, Randich 2006.See also Zahn (1984), 1992, Michaud (1986), Pinsonneault et al. 1989, 1992, Charbonneau & Michaud 1991, Garcia Lopez & Spruit 1991, Chaboyer & Zahn 1991, Charbonneau 1992, Spergel & Zahn 1992,Schatzman 1993, Deliyannis & Pinsonneault 1997, Mendez et al. 1997, 1999, Zahn et al. 1997, Kumar & Quataert 1997, Gough & Mac Intire 1998, Maeder & Zahn 1998, Talon & Charbonnel 1999, Charbonnel & Talon 1999, 2005, Brun et al. 1999, Montalban & Schatzmann 2000, Talon et al. 2002, Piau et al. 2003, Michaud et al. 2004, Landin et al. 2006, Eggenber et al. 2008, Pace et al. 2012, Lagarde et al. 2012, Eggenberger et al. 2012, Charbonnel et al. 2013

Observational test: binary systems*

Few (27) PMS stars with direct mass measurements (see e.g. Mathieu et al. 2007, Stempels et al. 2008, Cusano et al. 2010)

Comparison made by means of a Bayesian method (by Jorgensen & Lindegren 2005)

*see e.g. Simon et al. 2000, Steffen et al. 2001, Hillebrand & White 2004, Stassun et al. 2004, Mathieu et al. 2007, Baden et al. 2007, Alecian et al. 2007, Stassun 2008, Jackson et al. 2009, Torres et al. 2010, Feiden & Chaboyer 2012 and references therein

(Gennaro, Prada Moroni, Tognelli 2010)

fuori 2-sigmafuori 2-sigma

Observational test: binary systemsIn agreement with previous results:

For standard models masses tend to underestimated, however (Gennaro, Prada Moroni, Tognelli 2010)in the case of double line eclipsingbinary systems the agreement isquite good (maximum relative differences15% - 20% and in some cases differences as small as 5%, but V1174 Ori)

Regarding the age the situation is slightly worse (30% of the systems result not coeval)

Suggestions for the preference for colder models (low external convection efficiency, a=1 ?) in agreement with previous results (see e.g. D’Antona et al. 2000, Simon et al. 2001, Steffen et al. 2001, Baraffe et al. 2002, D’Antona & Montalban 2003, Stassun et al. 2004, Covino et al. 2004, Claret 2006, Alves de Oliveira 2013)

Link (?) with the underestimate of radii in PMS binary systems noticed by several authors (see e.g. Stassun et al. 2006, 2007, Mathieu et al. 2007, Jackson et al. 2009, Torres et al. 2010, Feiden & Chaboyer 2012, Somers & Pinsonneault 2014)

- It is a fragile element destroyed through proton capture at relatively low temperature, i.e. T ≈ 2.5 million degrees, easily reached even during the early pre-MS evolution;

- It is only destroyed in PMS stars: lithium surface abundance depends only on the initial amount of lithium present in the star and on the efficiency of the depletion processes;

-Lithium is a very good tracer of both the efficiency of the mixing processes present in the star and the temperature stratification. It gives precious information about the structure of the convective envelope

( see e.g. reviews in Delyannis et al. 2000, Jeffries 2000, Pinsonneault et al. 2000, Charbonnel et al. 2000, Talon 2008, Talon & Charbonnel 2010, Jeffries 2006…)

Lithium abundance in PMS starsWhy lithium is so important?

General features of Lithium burning in pre-MS

1 - Onset of lithium burning: Fully convective star.

2 - Formation of a radiative core: Lithium burning still possible at the base of the convective envelope if TCE ≥ TLi .

3 - The extension of the convective envelope reduces: TCE decreases, lithium burning is halted.

1

2

3

3

PMS 7Li surface depletion strongly depends on the mass

and on the metallicity of the star

Lithium pre-MS depletion is extremely sensitive to the convection efficiency

General features of Lithium burning in pre-MS

- Mixing Length Theory (Bohm-Vitense 1968) Introduction of the free parameter α (lc = α HP)

- Calibration of α: Sun, CMDs for main- sequence stars.

Standard models fail in reproducing the observed 7Li abundances in young clusters if a solar/MS convection efficiency is assumed in PMS hints for a reduced convection efficiency(see e.g. D’Antona & Mazzitelli 1994, 1997, Ventura et al. 1998, Schlatt & Weiss 1999, Piau & Turck-Chièze 2002, D’Antona & Montalban 2003, Landin et al. 2006, Eggenberger et al. 2012, Tognelli et al. 2012, Sommers & Pinsonneualt 2014 and references therein )

Uncertainties on predicted surface lithium abundance*

- Chemical composition

Uncertainties on YP, (Z/X), [Fe/H], ΔY/ΔZ, solar mixture- Physical Inputs

Opacity coefficients (radiative opacity): uncertainty of about ± 5% from differences between OPAL 2005 (see e.g. Iglesias & Rogers 1996) and OP (Seaton et al. 1994,

Badnell et al. 2005), see also Neuforge-Verheecke et al. 2001, Bahcall et al. 2005, Badnell et al. 2005, Valle et al. 2012).

7Li +p reaction Rates: uncertainty about ± 5% (Lattuada et al. 2001; Pizzone et al. 2003).

EOS: uncertainties evaluated by comparing the results obtained from several EOS widely used in the literature (OPAL2001, SCVH95, PTEH)

Initial D and 7Li values : XD= 2 . 10-5 (see e.g. Geiss & Gloeckler 1998, Linsky et al. 2006, Steigmann et al. 2007)A(Li) = log NLi/NH + 12 = 3.2 ± 0.2 (see e.g. Jeffries 2006, Lodders et al. 2009)

* See also Mazzitelli 1989, D’antona 1993, Swenson et al. 1994, D’Antona & Mazzitelli 1994, Chabrier & Baraffe 1997, Piau & Turck-Chièze 2002, D’Antona & Montalban 2003, Landin et al. 2006, Sestito et al. 2006, Tognelli et al. 2011, Somers & Pinsonneault 2012 and references therein

Uncertainties on predicted surface lithium abundance

Error bars: we quadratically added the differences in surface lithium abundances and effective temperature obtained taking into account the quoted error sources

The uncertainties on the chemical composition, physical inputs have a

strong effect on the predictions of surface lithium abundance in particular

for low-mass starschemical composition

physical inputs

Tognelli, Degl’Innocenti, Prada Moroni 2012

Open clusters: Ic 2602, α Per, Blanco 1, Pleiades, Ngc 2516

30-40 Myr < age < 130-150 Myr

-0.10 ≤ [Fe/H] ≤ +0.07

Colour-Magnitude Diagram : high quality photometric observations (HIPPARCOS when available)

Lithium data by Sestito & Randich (2005)

Large homogeneous database of observational data for surface 7Li abundance in open clusters of different ages and chemical compositions

Comparison with selected young open clusters

Isochrone fitting: determine the “best value” of the age and of the mixing length parameter for MS stars (αMS) with the related errors

Uncertainty on age determination: 10-20 Myr, mainly from the lack of stars near the overall contraction region

Not negligible for mid- and low-mass stars in cluster younger than about 60-70 Myr, completely negligible in the other cases

Consistency: the models adopted for surface lithium predictions are the same as the ones adopted for the isochrone fitting

Comparison with selected open clusters

The method

Comparison with selected open clusters

Lithium predictions

- one chooses the best value of αPMS to reproduce the observed surface 7Li abundance in each cluster;

- αMS is constrained by the comparison with the Colour-Magnitude Diagram.

Standard models + models in which we allow different values of α during pre-MS and MS evolution

- Additional physical mechanisms not taken into account in standard PMS models

- Hints of non-constancy of α for stars in different masses and evolutionary phases from observations (see e.g Chieffi et al. 1995, Morel et al. 2000, Ferraro et al. 2006, Yildiz 2007, Gennaro et al. 2012, Piau et al. 2011, Bonaca et al. 2012) and detailed hydrodinamical simulations (e.g. Ludwig et al. 1999; Trampedach 2000)

Ages are in quite good agreement with previous

determinations

40 Myr 60 Myr

110 Myr120 Myr

130 Myr

Tognelli et al. 2012

Previous indications for a low PMS convection efficiency are confirmed*

-It’s possible to reproduce the observed 7Li(Teff) profile by adopting the same αPMS for all the selected clusters (independence of pre-MS mixing length parameter on the chemical composition and age for clusters younger than about 150-200 Myr)

Clearly it’s not possible to reproduce the spread in Li abundance for cool stars (Teff ≤ 5500K) observed in some young clusters (e.g. Pleiades)

αPMS = 1.0

*See e.g. Ventura et al. 1998; Simon et al. 2000; Steffen et al. 2001; D’Antona & Montalban 2003; Stassun et al. 2004, Landin et al. 2006, Somers & Pinsonneault 2014

0.65 M

1.00 M1.30 M

1.20 M 1.00 M

0.60 M

0.70 M

1.00 M

0.70 M

1.00 M1.00 M1.40 M

0.65 M

(Tognelli et al. 2012)

Rotating models (with or without gravity waves) provide (slightly) higher Li depletion especially for low mass stars rotation alone cannot solve the problem (see e.g. Pinsonneault et al. 1989, Martin & Claret 1996, Mendes et al. 1999, Landin et al. 2006, Eggenberger et al. 2012, Chaboyer et al. 2013)

Several possible solutions of the problem(s):-Rotation + star disk coupling + low convection efficiency ( see e.g. Eggenberger et al. 2012)

-inhibition of convection by magnetic activity ( see e.g. Ventura et al. 1998, Mullan & Mac Donald 2001, D’Antona & Montalban 2003, Landin et al. 2006, Chabrier et al. 2007, Morales et al. 2008, 2010, Mac Donald & Mullan 2012, Feiden & Chaboyer 2013, Somers & Pinsonneault 2014)

-Effects of accretion history (see e.g. Baraffe & Chabrier 2010)

-Li dispersion is (at least in part) spurious : stellar surface activity, wrong temperature scale, inhomogeneous reddening, rotational broadening of Li absorption lines in rapid rotators ( see e.g. Luhman et al. 1997, King et al. 2000, Margheim et al. 2002, Hillenbrand & White 2004, Pace et al. 2012 )

Comparisons with binary stars with dinamical mass and 7Li abundance values (ASAS J052821, EK Cep, RXJ 0529, V1174 Ori) cannot help to discriminate between low and high convection efficiency due to the still large observational errors (Tognelli et al. 2012, see also D’Antona & Montalban for RXJ 0529)

????

Even more precise observational data are needed:

- masses, radii, rotation velocities, lithium abundances, magnetic activity for PMS binary stars

- effective temperatures, rotation velocities, magnetic activity, accretion rates, lithium abundances for young clusters and star forming regions

……

…in a way to constrain the large amount of models that we, as theoreticians, have fun to produce!

…Backup…

- Evolutionary Code: FRANEC Updated physical inputs (see Tognelli et al. 2011)

Equation of State: OPAL EOS 2006 (Rogers & Nayfonov 2002);

Radiative Opacity: OPAL 2005 (Iglesias & Rogers 1996) for log T[K] > 4.5, Ferguson et al. (2005) for log T[K] ≤ 4.5;

Heavy Elements Solar Distribution: Asplund et al. (2005);

Boundary Conditions: Brott & Hauschidt (2005) for T < 10 000 K and Castelli & Kurucz (2003) for T ≥ 10 000 K. Match between atmosphere and interior computations at the optical depth τph = 10;

Uncertainties on theoretical predictions - Chemical composition of the star.

From the observations we obtained [Fe/H] (spectroscopic). Helium and metals abundance (Y, Z) are compute using the relation:

for solar-scaled metal distribution plus a linear relation between helium and metal abundance. (YP : primordial helium abundance. (Z/X) : photospheric metal-to-hydrogen ratio abundance in the Sun. ΔY/ΔZ : helium-to-metal enrichment ratio.)

Uncertainties on YP, (Z/X), [Fe/H] and ΔY/ΔZ propagate into uncertainties on Y and Z.

Typical uncertainties on such quantities:

- Δ[Fe/H] ≈ ± 0.05 dex

- Δ(Z/X)/(Z/X) ≈ 15% (Bahcall et al. 2005)

- YP = 0.2485 ± 0.0008 (Steigman 2006)

- 1 ≤ ΔY/ΔZ ≤ 5 (see e.g. Gennaro et al. 2010)

- εLi-7 = 3.2 ± 0.2 (see e.g. Lodders 2009)

A change in Y and Z affects the structure of the

star hence lithium depletion

plus

For PMS stars the temporal evolution of the luminosity is slightly dependent on the EOS with a maximum difference for the ZAMS age of ≈ 20% while, except for M≤ 0.1 Msun , Dlog L/Lsun ≤ 0.02

Physical inputs uncertaintiesEOS

PMS isochrones

If updated EOS are adopted the residual uncertainties does not affect in a relevant way the evolution in the HR diagram (Tognelli et al. 2011)

The cross section update influences only the age determination for old clusters (see Straniero et al. 2002, Imbriani et al. 2004, Degl’Innocenti et al. 2004, Weiss et al. 2005)

(14N+p) LTO McHe LCNO

HB L3αHB

* For (14N+p)/2 ΔLogLHB~0.01 (Z=0.0002)

* For (14N+p)/2 ΔLogLHB~ -0.01 (Z=0.001)

Maximum age variation by adopting the vertical method ~ 1 Gyr

CNO cross sections and cluster ages

Degl’Innocenti et al. 2004

Pisa Models: D-burning• Cosmological D abundance: 3.8x10-5 ≤ XD ≤ 4.5x10-5

(Cyburt et al. 2004, Steigman et al. 2007, Pettini et al. 2008)

• As stellar generations follow each other, D is astrated, since stars are net destroyers of D

• Solar neighbourhood D abundance: XD ≈ 2.5-3x10-5

(Geiss & Gloeckler 1998, Vidal-Madjar et al. 1998, Linsky 1998, Linsky et al. 2007, Steigman et al. 2007)

D is destroyed at about 106 K (early pre-MS).

p(d, 3He)g Q = 5.493 MeV (Qpp ≈ 26 MeV)

Temporarily halts gravitational constraction

Contrazione gravitazionale:riparte quando

Xd ≈ 1/100 valore iniziale

Deuterium abundance:

Pisa Models: D-burning• The value of XD affects

only the very early evolution

• After a few Myr the isochrones converge

• XD= 4x10-5 for all stars • XD= 2x10-5 for Z>0.007

Pisa Solar ModelYini Zini Ysup Zsup α Rcz

KS66 0.2532 0.0137 0.2222 0.0126 1.97 0.7294

CK03 0.2532 0.0137 0.2222 0.0126 1.75 0.7295

BH05 0.2533 0.0137 0.2221 0.0126 1.68 0.7295

Yini Zini Ysup Zsup α Rcz

BS05 0.2614 0.0140 0.2300 0.0125 1.96 0.7289

GZ06 0.2570 0.0135 0.2273 0.0124 1.99 0.7306

S09 0.2593 0.0139 0.2292 0.0126 2.10 0.7280

Other standard solar models

Atmosfere stellari implementate:Atmosfere non-grigie (tabelle modelli teorici dettagliati, ideali per M < 1 Msun).

- BH05 (default, Brott & Haushildt 2005): 2000 K ≤ Teff ≤ 10 000 K; - CK03 (Castelli & Kurucz 2003): Teff > 10 000 K; - AHF11 (Allard et al. 2011): per stelle di piccola massa Teff < 2000 K;

Atmosfere grigie (modelli semplificati, M ≥ 1 Msun).

- KS66 (Krishna-Swamy 1966)

Effetto su strutture

con inviluppi convettivi!

DTeff ≈ 100 - 200 K DTeff ≈ 100 - 150 K DTeff ≈ 100,(max > 200 K)

Comparison with other authors

Tognelli, Prada Moroni & Degl’Innocenti 2011

Test osservativo: BINARIE (2/6)

Metodo Bayesiano: Formalismo presentato da Jorgensen & Lindegren (2005) testato su oggetti di MS e post-MS. Generalizzabile per sistemi binari.

La probabilità a posteriori di ottenere le osservabili { q } (= {L, Teff,R, g}) dai modelli teorici che dipendono dai parametri { p } (= {t, M, Z}) e da dei meta-parametri { Ξ } (= {Yp, aML,DY/DZ}) e’ definita come,

- L(p, Ξ | q) = likelihood.- f0 = prior.- h = normalizzazione.

La potenza del metodo Bayesiano sta nell’utilizzo del prior!!

OPAL-OP radiative opacities

Test osservativo: BINARIE (2/6)

Metodo Bayesiano: Formalismo presentato da Jorgensen & Lindegren (2005) testato su oggetti di MS e post-MS. Generalizzabile per sistemi binari!

La probabilità a posteriori di ottenere le osservabili { q } (= {L, Teff,R, g}) dai modelli teorici che dipendono dai parametri { p } (= {t, M, Z}) e da dei meta-parametri { Ξ } (= {Yp, aML,DY/DZ}) e’ definita come,

- L(p, Ξ | q) = likelihood.- f0 = prior.- h = normalizzazione.

La potenza del metodo Bayesiano sta nell’utilizzo del prior!!

Test osservativo: BINARIE (4/6)

Procedura (Jorgensen & Lindegren 2005, Gennaro et al. 2012)

-Per ogni sistema binario si calcola distribuzione di probabilità al variare dei parametri {p} per le diverse combinazioni dei meta-parametri {Ξ} (con/senza prior sulla massa)

-Si calcolano le likelihood relative per i parametri incogniti (es. età, massa, metallicità): marginalizzazione sui modelli assumendo indipendenza dei parametri.

-Il valore più probabile è dato dalla moda della likelihood relativa (distribuzioni asimmetriche).

-Intervallo di confidenza [xmin, xmax]: ottenuto eliminando il 16% dell’area totale dalle code.

-Bayes Factor (BF): permette di confrontare direttamente modelli ottenuti usando divers valori dei meta-parametri.

Metodo quantitativo per identificare il set di parametri con maggiore “evidenza”

Distance (pc)

148

158

238

131

385422

Main properties of the selected clusters

Sestito et al. (2005)

Typical uncertainties:

Δ log NLi/NH ≈ 0.1 - 0.2 dex

Δ Teff ≈ 100 - 150 K

Temperature scale (Bessel 1979):

- independent on the metallicity:,

- no activity correction.

Teff (K) = 1808(B-V)02 – 6103(B-V)0 + 8899

Results: Lithium surface abundances 4 (4) - Binary Systems: HR Diagram.

Results: Lithium surface abundances 4 (4) - Binary Systems: Surface Lithium Abundance.

- Lithium data still uncertain (ΔALi ≈ 0.3 - 0.5 dex)