Документ взят из кэша поисковой машины. Адрес оригинального документа : http://hea-www.harvard.edu/AstroStat/Stat310_1213/nms_20120904.pdf Дата изменения: Tue Sep 4 23:25:44 2012 Дата индексирования: Tue Oct 2 04:39:05 2012 Кодировка: Поисковые слова: п п п п п п р п р п р п р п р п р п п р п п р п

Combining Computer Models to Account for Mass Loss in Stellar Evolution
Nathan Stein
Statistics 310

September 4, 2012


Statistical Analysis of Stellar Evolution
Statistical analysis of stellar evolution relies on complicated models of the physical aging processes of stars Like a sampling distribution, these models predict observed quantities as a function of unknown parameters

456789/,) 4/(/#'&'(7)

!"#$%&'()*"+',)) -"() .&',,/()01",%2"3)

4('+89&'+) :;7'(1/2"37)

These models vary in complexity: some are solutions of coupled partial differential equations, some have simple analytic expressions We use tabulated versions of the more complex models, evaluated over a grid of parameter values


Opening the Black Box

Typically treat computer models as deterministic black box models We want to open the black box and see what the data can tell us about internal model components

456789/,) 4/(/#'&'(7)

!"#$%&'()*"+',)) -"() .&',,/()01",%2"3)

!"

4('+89&'+) :;7'(1/2"37)

What can we learn about the pro cesses of stellar evolution from observations of star clusters? We focus on the mass loss that stars experience along the way to their final stage as white dwarfs


Evolution of a Sun-like Star

1. Main sequence
p owered by hydrogen fusion

2. Red giant
no more hydrogen fuel, so the star co ols and swells and sheds its outer layers

3. Planetary nebula
remaining hot core ionizes the outer layers that have been ejected

4. White dwarf
once the outer layers are gone, the hot, dense core remains


Initial-Final Mass Relation (IFMR)

White dwarf mass < progenitor star mass The mapping between the progenitor mass and the white dwarf mass is called the initial-final mass relation (IFMR) Key ingredient in physics-based models of stellar evolution Interesting complication: relationship between two unobserved quantities (only one even observable)


Data

Observe stars through different photometric filters Focus on clusters of stars with the same age, chemical composition (metallicity), distance, absorption Stars have different initial masses Initial masses govern their rates of evolution see a snapshot of stars in different stages of evolution


q q q q q q q q q q q q qq q q q q q q q qq q q qq qq q qq q q q q q q q q q

q q

brighter

Red Giants
q q q q q q q

q q q qq q q q qq qq q q q q qq q qq q qq q q qq q q q q q q qq qq q q q q qq q q q qqq q q qq q q q q qq q q q q q q q q q q qq q qq q q q q qq q qqq q q q qq q q q q q q qq q qq q q q q q qq q q q qqq q qq qq qq qqq q q q qq q q q q q q q qq qqq q q q qq q q q q q q q qqqq q q

q q

Absolute Magnitude

Main Sequence

q q

q qq qq q q q q

q q

q q

q

q q q q q

q

fainter

White Dwarfs
qq q q q

q q

hotter Temperature

cooler


Basic Likelihood
>/%::6/3)) *'/:%('#'3&) 0(("() 45') *'&/,,676&8) 9362/,)*/::) ;6:&/37') 4<:"($2"3) !"#$%&'() *"+',)) -"() .&',,/() 01",%2"3)

=<:'(1'+) */536&%+':)

Yi = vector of observed magnitudes through different filters Mi = the mass of star i = vector of cluster parameters G
MS/RG

(Mi , ) = the stellar evolution model for main sequence stars

Observational uncertainties i assumed known Gaussian errors: Yi |Mi , , , i N (µi , i ), µi = GMS/RG (Mi , ) if star i is a main sequence star
indep


Basic Likelihood

f (Mi , ) = the initial-final mass relation = vector of IFMR parameters GWD (Mi , , ) = the stellar evolution model for white dwarfs Gaussian errors: Yi |Mi , , , i N (µi , i ), GMS/RG (Mi , ) GWD (Mi , , ) if star i is a main sequence star if star i is a white dwarf
indep

µi =


Binaries and Field Stars

Binary Systems
Between 1/3 and 1/2 of stars are binary systems that appear as one star Luminosities of component stars sum Magnitude = -2.5 log10 (luminosity) For main sequence-main sequence binaries, µij = -2.5 log
10

10-G

MS/RG,j

(Mi 1 , )/2.5

+ 10

-GMS/RG,j (Mi 2 , )/2.5

All main sequence stars are modeled as binaries

Field Stars
Appear in observational field of view, but not part of cluster Mixture model, with field stars assumed uniformly distributed in magnitude space


Age Metallicity Initial Mass Distance Absorption Initial Mass

Computer Model for Stellar Evolution
Main Sequence Comp Model

Gaussian Measurement Error

IFMR WD mass Age on MS

Observed Magnitudes

Age Distance Absorption

White Dwarf Comp Model

+
Field Star Contamination


Component Computer Models

If star is a white dwarf, the MS/RG computer mo del returns how long it lived as a main sequence and red giant star (the progenitor age).
prog age

= FMS/RG ([Fe/H] , M )

The white dwarf co oling mo del computes the effective temperature and radius of the star as a function of its cooling age (total age minus progenitor age) and its current mass. (
Teff

,

radius

)=F

co oling

(

age

-

prog age

,M

WD

)


Component Computer Models

The log of the gravitational force experienced at the surface of the white dwarf is computed using Newton's law:
log g

= log10 (G M

WD

/radius 2 )

The white dwarf atmosphere mo del uses the surface gravity and the effective temperature to derive the emergent spectrum of the star's atmosphere as a function of wavelength. The model then integrates the emergent spectrum over the filter response to calculate the modeled magnitudes. µ = Fatmosphere (Teff , log g )


Component Computer Models

(
Teff

prog age radius

= FMS

/RG

([Fe/H] , M )
prog age 2 radius

,

) = Fco

oling

(age -
WD

,M )

WD

)



log g

= log10 (G M

/

)

µ = Fatmosphere (

Teff

,

log g


Component Computer Models

(
Teff

prog age radius

= FMS

/RG

([Fe/H] , M )
prog age 2 radius

,

) = Fco

oling

(age -
WD

,M )

WD

)



log g

= log10 (G M

/

)

µ = Fatmosphere (

Teff

,

log g


Component Computer Models

(
Teff

prog age radius

= FMS

/RG

([Fe/H] , M )
prog age 2 radius

,

) = Fco

oling

(age -
WD

,M )

WD

)



log g

= log10 (G M

/

)

µ = Fatmosphere (

Teff

,

log g


Parameterizing the IFMR

We let the IFMR be a deterministic function of the initial mass M and parameters : MWD = f (M , ) We primarily consider a linear IFMR Simple functional forms are reasonable because visible white dwarfs in any particular cluster will typically span a relatively narrow range of initial masses


4

2

0

V

10

8

6

!"#$%&'()'$*'%+),$-./% 01$23,"#$2%!"#$%&'()'$*'%45'%

67#3'%89",:%.;2',<"=1$2% 01$23,"#$%68%011>#$5%45'2%
0.0 0.5 B-V 1.0

?68%@,'*),21,%45'2A%B%% ?!&%45'A%C%?68%011>#$5%45'2A%

12

68%.;2',<"=1$2%01$23,"#$% 68%0),,'$3%!"22'2%

68%@,'*),21,%45'2%01$23,"#$% 68%D$#=">%!"22'2%


Prior Distributions

Primary mass: log10 (mass) N (-1.02, 0.6772 ) , based on population distribution Uniform on the ratio of smaller to larger mass Uniform on log10 (age) between limits of stellar evolution models Cluster membership prior probabilities come from external information (when available) Informative priors on metallicity, distance, and absorption Prior distribution on the IFMR parameters is uniform on the region corresponding to monotonically increasing IFMRs 0.1M < mass < 8.0M


Statistical Computation

At least 3N + 3 parameters for cluster with N stars Local modes based on choices of cluster members vs field stars Joint posterior
N

p ( , , M, R, Z | Y) p ( , )
i =1

[i pc (Yi | Mi , Ri , , )pc (Mi , Ri )]Zi в [(1 - i )pf (Yi )pf (Mi , Ri )]1
-Zi

Z i R pc pf

= vector of cluster memb ership indicators = prior probability of cluster memb ership for star i = vector of ratios of secondary to primary mass = cluster star likeliho od or prior = field star likeliho od or prior


Statistical Computation

Marginal posterior p ( , | Y ) = ···
Z1 N

···
ZN

p ( , , M, R, Z | Y) d Md R

p ( , )
i =1

i

pc (Yi | Mi , Ri , , )pc (Mi , Ri )dMi dRi +

(1 - i )pf (Yi ) Because of conditional independence, marginalizing out nuisance parameters involves lots of 1- and 2-dimensional integrals over compact regions (e.g. Mi [0.15, 8.0], Ri [0, 1]), which can be numerically approximated in parallel Use MCMC on the lower dimensional ( , )


Sensitivity to Misspecification

Deterministic models GMS/RG are assumed known, but there are uncertainties, different implementations, etc. Performed a simulation to test the sensitivity of inferences to misspecification of GMS/RG Used the models of Yi et al. (2001) and Dotter et al. (2008) Simulated eight clusters under both sets of models at different ages using the linear IFMR of Williams et al. (2009): M
WD

= 0.339 + 0.129M

Fit the clusters using both sets of models and a linear IFMR


ages 95% Posterior Intervals (M

Sun

Simulated: Yi Fit: Yi
8

Simulated: Dotter Fit: Yi

Simulated: Yi Fit: Dotter

) 6

logAge

4

8.4 8.6 8.8 9.0

9.1 9.2 9.4 9.6

2

2

4

6

8
Sun

2

4

6

8
Sun

2

4

6

8
Sun

True Initial Mass (M

)

True Initial Mass (M

)

True Initial Mass (M

)

95% Posterior Intervals (M

Sun

) 0.5 1.0 1.5

0.5

1.0

1.5
Sun

0.5

1.0

1.5
Sun

0.5

1.0

1.5
Sun

True Final Mass (M

)

True Final Mass (M

)

True Final Mass (M

)


Data Analysis

Analyzed three clusters: NGC 2477, the Hyades, and M35 Results for NGC 2477 and the Hyades under both the Yi et al. (2001) and Dotter et al. (2008) models Results for M35 under the Yi et al. (2001) models (M35 is too young for the Dotter et al. (2008) models)


Initial and Final Mass Inferences

Yi et al. models NGC 2477 Hyades M35 (NGC 2168) 1.2

(i) 1.2 (ii) (iii) (iv) White Dwarf Mass (MSun)

Dotter et al. models NGC 2477 Hyades

White Dwarf Mass (MSun)

1.0

0.8

(i) Williams et al 2009 (ii) Weidemann 2000 (iii) Salaris et al 2009 (linear) (iv) Salaris et al 2009 (piecewise) 2 4 Initial Mass (MSun) 6 8

0.6

0.6

0.8

1.0

2

4 Initial Mass (MSun)

6

8


Sensitivity of NGC 2477 Inferences to IFMR Model

Yi et al. models 1.2 1.2 Linear IFMR Piecewise linear IFMR White Dwarf Mass (MSun)

Dotter et al. models Linear IFMR Piecewise linear IFMR

White Dwarf Mass (MSun)

1.0

0.8

0.6

2

4 Initial Mass (MSun)

6

8

0.6

0.8

1.0

2

4 Initial Mass (MSun)

6

8


Sensitivity of NGC 2477 Inferences to IFMR Model

Yi et al. models 1.2 1.2 Linear IFMR Quadratic IFMR White Dwarf Mass (MSun)

Dotter et al. models Linear IFMR Quadratic IFMR

White Dwarf Mass (MSun)

1.0

0.8

0.6

2

4 Initial Mass (MSun)

6

8

0.6

0.8

1.0

2

4 Initial Mass (MSun)

6

8


Sensitivity of Hyades Inferences to Error Model

Yi et al. models 1.2 1.2 Gaussian likelihood t likelihood White Dwarf Mass (MSun)

Dotter et al. models Gaussian likelihood t likelihood

White Dwarf Mass (MSun)

1.0

0.8

0.6

2

4 Initial Mass (MSun)

6

8

0.6

0.8

1.0

2

4 Initial Mass (MSun)

6

8


Acknowledgments

David A. van Dyk (Imperial College London) Ted von Hippel (Embry-Riddle Aeronautical University) Steven DeGennaro (UT Austin) Elizabeth Jeffery (James Madison University) William H. Jefferys (UT Austin, U Vermont)