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)