Modelling high-order g-mode pulsators Nice 27/05/2008

A method for modelling high-order, g-mode pulsators: The case of γ Doradus stars.

A. MoyaInstituto de Astrofísica de Andalucía – CSIC, Granada, Spain

•Brief introduction

•The problem of mode identification

•Photometry (FRM and multicolour)

•Gamma Doradus Modelling Scheme

•Future prospects

•Rotational coupling

J.C. Suárez

S. Martín-Ruíz

P.J. Amado

R. Garrido

A. Grigacehene

M.A. Dupret

Modelling high-order g-mode pulsators Nice 27/05/2008

Brief introduction

γ Doradus stars

•High n

•Low ℓ

•Very low photometric amplitude

•Period close to 1 day

Space missions essential for improving our observational knowledge of these stars

Modelling high-order g-mode pulsators Nice 27/05/2008

Brief introduction

C CRadiative

rb rt

t

b

r

rnl drr

N

n

21

)1(Tassoul, 1980

•No Rotation

•No magnetic field

•Adiabatic approximation

t

b

r

rnl drr

N

n

31

)1( Smeyers & Moya, 2007

Modelling high-order g-mode pulsators Nice 27/05/2008

What is the meaning of mode identification?

In the approximation of the star to have spherical symetry, each mode can be asociated to a spherical armonic Yl

m(θ,φ)

Observed frequency (n,ℓ,m)

¿(n,ℓ,m)?¿(n,ℓ,m)?

The problem of mode identification

Modelling high-order g-mode pulsators Nice 27/05/2008

The problem of mode identification

There are two different observational techniques of modal identification:

1) Spectroscopy: This gives us part of the identification of the mode, that is (ℓ,m)

2) Photometry: We just have the periods of each mode and we have to connect with theoretical models to identify (n,ℓ,m)

Modelling high-order g-mode pulsators Nice 27/05/2008

The problem of mode identification

Possible tool: Asymptotic equidistance in period

12

1

2

1

2

)1(

2

)1(

221

)1(

221

1)1,(

IIn

In

nnP

This give information about ℓ and the Brunt-

Väisälä integral

1

1

)(

)(

11

22

2

1

P

P

Modelling high-order g-mode pulsators Nice 27/05/2008

The problem of mode identification

HD129019 Aurigae

Modelling high-order g-mode pulsators Nice 27/05/2008

Frequency Ratio Method (FRM)

Assumptions:

•Some knowledge of the spherical order ℓ (assume all modes having the same ℓ or we know each individual ℓ).

•No rotation, no magnetic field and adiabatic behaviour.

•The integral is almost constant for the different modes within a given model.

s

i

s

i

r

r

r

r

n

n

drrN

drrN

n

n

)5.0()1(

5.0)1(

nnn

n

n

n

,5.0

5.0

Moya et al., 2005, A&A 432, 189

Modelling high-order g-mode pulsators Nice 27/05/2008

Observations:

Physical parameters ≥ 3 frequencies

Frequency ratio method

Several sets of (n1,n2,n3,ℓ,Iobs)

Small set of possible theoretical models describing this star

FRM

(ν1, ν2,ν3)

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Teff Log g [Fe/H] Km/s

6996 4.04 -0.37

7079 4.47 -0.40

53

66

Freq c/d μHz

fI 1.216 14.069

fII 1.396 16.157

fIII 2.186 25.305

β1,2=0.871 β2,3=0.639 β1,3=0.556

±0.005 ±0.005 ±0.005

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Name n1 n2 n3 ℓ Iobs

t1 17 27 31 1 987.0

t2 21 33 38 1 1202.4

t3 21 33 38 2 694.2

t4 26 41 47 2 860.0

t5 30 47 54 2 984.3

t6 33 52 60 2 1087.9

Modelling high-order g-mode pulsators Nice 27/05/2008

Modelling high-order g-mode pulsators Nice 27/05/2008

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Modelling high-order g-mode pulsators Nice 27/05/2008

FRM with rotation

Suárez et al., 2005, A&A, 443, 271

The FRM still works for m=0 modes

There are not possible confusion between modes with different m

Two main conclusions:

Modelling high-order g-mode pulsators Nice 27/05/2008

)( cos ln ln

ln ln

) ( cos ln ln

ln ln

)( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tgg

gb

gF

tTT

Tb

TF

tll

bi Pm

l

Tl

l

m

l

Non-adiabaticcomputations

Surfacedistortion

Influence of the local effective temperature

variation

Influence of the localeffective gravity

variation

Equilibriumatmosphere models

(Kurucz 1993)g

rr/gg )(

2

e

e

Multicolor photometry

Modelling high-order g-mode pulsators Nice 27/05/2008

As a result of the numerical computations we can obtain

eff

eff

T

T

e

e

gg

RT

Tr

eff

effT

And the grow rate

M

r

M

r

dMWdt

WdMdt

0

0Where

FTT

W N

Multicolor photometry

Modelling high-order g-mode pulsators Nice 27/05/2008

Multicolor photometry

Current most evolved tool:

Time dependent convection

Modelling high-order g-mode pulsators Nice 27/05/2008

Observations giving physical parameters and

three frequencies Frequency ratio

method

Set of possible mode

identifications and

equilibrium models

Fix α in MLT and ℓ

Instability and non-adiabatic multicolor study with TDC (or

spectroscopy)Photometric multicolour

predictions (models, modes and free

parameters fixed)

Multicolour photometric observations

Gamma Doradus Modeling Scheme

Modelling high-order g-mode pulsators Nice 27/05/2008

Teff Log g [Fe/H] Km/s

6990 4.17 -0.18 18

Freq c/d

f1 0.7948

f2 0.7679

f3 0.3429

=0.966 ±0.010

=0.447 ±0.010

=0.431 ±0.010

1

2

f

f

2

3

f

f

1

3

f

f

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Name n1 n2 n3 ℓ Iobs

t1 33 34 77 1 681.14

t2 57 59 133 2 678.24

And lower Iobs

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Mass Teff Log g Log L/LΘ R/RΘ Age Ith [Fe/H] αov

1.4 7006 4.28 0.63 1.41 600 681.5 -0.1 0.3

Model fulfilling

FRM constraints

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Multicolor analysis with TDC (Dupret et al. and Grigahcene et al.) for the model coming from FRM

Different αMLT and atmospheric models

Strömgren filters

ℓ=2

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Stability analysis with TDC for different αMLT

αMLT

1.6

2.0

1.4

1.8αMLT=1.6

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Physical parameters

Mass Teff Log g Log L/LΘ R/RΘ Age [Fe/H]

1.4 7006 4.28 0.63 1.41 600 -0.1

Theoretical parameters

αMLT Ith αov

1.6 681.5 0.3

Modal identification

n1 n2 n3 ℓ

57 59 133 2

Freq c/d

f1 0.7948

f2 0.7679

f3 0.3429

GDMS (9 Aurigae)

Modelling high-order g-mode pulsators Nice 27/05/2008

Rotational coupling

δmλ(coup)= β·δmλ(1)+(1-β) δmλ(2)

)( cos ln ln

ln ln

) ( cos ln ln

ln ln

)( cos )2)(1(

)(cos 10ln

2.5

e

e

eff

eff

eff

eff

tgg

gb

gF

tTT

Tb

TF

tll

bi Pm

l

Tl

l

m

l

Modelling high-order g-mode pulsators Nice 27/05/2008

Rotational coupling

Modelling high-order g-mode pulsators Nice 27/05/2008

Rotational coupling

Modelling high-order g-mode pulsators Nice 27/05/2008

Rotational coupling

How to obtain information in this case

Modelling high-order g-mode pulsators Nice 27/05/2008

Frequency ratio method gives a set of possible

models fitting the physical parameters and the

observed frequencies, fixing the parameters directly related with the Brunt-Väisälä frequency as

metallicity, overshooting, etc.

+

Time dependent convection-pulsation interaction can give a

range for α by studying the instability regions,

estimating also the multicolour photometric

observables for those theoretical models

GDMS

Physical source of information

Modelling high-order g-mode pulsators Nice 27/05/2008

Future prospects

Test these methods with different γ Doradus stars

a) With more than 3 frequencies

b) Belonging a cluster

c) Include most evolved tools with rotation and develop the rest.

Modelling high-order g-mode pulsators Nice 27/05/2008

Future prospects

Extent to other g-mode pulsators as SPB, some SdB, etc.

Through a statistical extension of the asymptotic expression

Modelling high-order g-mode pulsators Nice 27/05/2008

t

b

r

rn drr

N

n

21

)1(

Future prospects

t

b

r

re

n drr

Nn

n

61

2

)1(

t

b

r

re

n drr

Nn

n

41

2

)1(

Fully radiative

star

Convective core- radiative

envelope

Convective core- radiative envelope –convective envelope

Modelling high-order g-mode pulsators Nice 27/05/2008

Future prospects

t

b

r

rn drr

N

An

)1(

A is obtained by fitting this expression with the numerical spectrum of the

differential equations for different stars

Modelling high-order g-mode pulsators Nice 27/05/2008

THANK YOU

MERCI

GRACIAS

OBRIGADO

DANKE

GRAZIE

DEKUJI

DZIĘKUJĘ

Modelling high-order g-mode pulsators Nice 27/05/2008

The star HD12901

Name [Fe/H] M/M Teff L/L Log g Xc Age ρ/ρ L

A1 -0.4 1.2 3.83 0.52 4.28 0.50 1990 9.19 2

A2 -0.4 1.3 3.85 0.72 4.17 0.40 2100 6.10 2

A3 -0.4 1.4 3.84 0.90 4.02 0.26 2090 3.47 2

B1,C1 -0.6 1.2 3.83 0.67 4.12 0.27 3120 5.36 1,2

B2,C2 -0.6 1.3 3.83 0.84 3.98 0.17 2720 3.20 1,2

B3,C3 -0.6 1.4 3.83 0.98 3.88 0.10 2290 2.20 1,2

Modelling high-order g-mode pulsators Nice 27/05/2008

White and/or multi-colourphotometric

Observations

Frequencies and/or amplitude ratios

& Phase differences

Equilibrium models(evolution code)

Adiabatic and/or non-adiabaticcomputations

Mode identification

Mixing length (

Improving the fit

Stellar parameters

Scuti

Doradus

Cephei

SPB

Convection

ChemicalComposition (Z)

Atmosphere models - Limb darkening

Hydrodynamic

Overshooting

, ...

Photometry

Modelling high-order g-mode pulsators Nice 27/05/2008

Brief introduction

What is the astroseismology?

Is to infer properties of the stellar interiors by observing, identifying and fitting the proper modes some stars pulse with the equilibrium and pulsating

stellar models

One of the main problems is the modal identification, that is, to label each observed mode

with its frequency and the numbers (n,l,m)