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```A tool to simulate COROT light-curves
1:
LESIA, Observatory of Paris/Meudon
2: IAS, Orsay
Purpose :
to provide a tool to simulate CoRoT light-curves of the seismology
channel.
Interests :
● To help the preparation of the scientific analyses
● To test some analysis techniques (e.g. Hare and Hound exercices)
with a validated tool available for all the CoRoT SWG.
http://www.lesia.obspm.fr/~corotswg/simulightcurve.html
Main features:
Theoretical mode excitation rates are calculated according to Samadi et al
(2003, A&A, 404, 1129)
➔ Theoretical mode damping rates are obtained from the tables calculated by
Houdek et al (1999, A&A 351, 582)
➔ The mode light-curves are simulated according to Anderson et al (1990, Apj,
364, 699)
➔ Stellar granulation simulation is based on : Harvey (1985, ESA-SP235,
p.199).
➔ Activity signal modelled from Aigrain et al (2004, A&A, 414, 1139)
➔ Instrumental photon noise is computed in the case of COROT but can been
changed.
➔
Simulated signal = modes + photon noise + granulation signal + activity signal
Instrumental noise (orbital periodicities) not yet included (next version)
Simulation inputs:
• Duration of the time series and sampling
• Characteristics (magnitude, age, etc…) of the star
• Option : characteristics of the instrument performances (photon noise
level for the given star magnitude)
Simulation outputs: time series and spectra for :
• mode signal (solar-like oscillations)
• photon noise, granulation signal, activity signal
Modeling the solar-like oscillations spectrum (1/4)
Each solar-like oscillation is a superposition of a large number of excited and damped proper
modes:
 A j exp[ 2i 0 ] exp[  (t  t j )]H [t  t j ]  c.c.
j
Aj : amplitude at which the mode « j » is
excited by turbulent convection
tj : instant at which its is excited
0 : mode frequency
 : mode (linear) damping rate
H : Heaviside function
Modeling the solar-like oscillations spectrum (2/4)
 A j exp[ 2i 0 ] exp[  (t  t j )]H [t  t j ]  c.c.
j
Fourier spectrum :
A
Line-width :
U ( )
F ( ) 

1  2i (   0 ) /  1  2i (   0 ) / 
j
j
U ( )   A j
j
Power spectrum :
P( )  F ( )
  /
2

U ( )
2
1  [2(   0 ) / ]2
 The stochastic fluctuations from the mean Lorentzian profil are simulated by
generating the imaginary and real parts of U according to a normal distribution
(Anderson et al , 1990).
Modeling the solar-like oscillations spectrum (3/4)
P( )  F ( )
2

U ( )
2
1  [2(   0 ) / ]2
We have
necesseraly:
L²
Line-width
(L) 2   P( )d    k Pk ( k )
k
where <(L)²> is the rms value of
the mode amplitude

constraints on: U ( )
2
•  and L/L predicted on the base of theoretical models
• Excitations rates according to Samadi & Goupil(2001) model
• Damping rates computed by G. Houdek on the base of Gough’s
formulation of convection
Modeling the solar-like oscillations spectrum (4/4)
Simulated spectrum of solar-like oscillations for a stellar model with M=1.20 MO
located at the end of MS.
Photon noise
Flat (white) noise
COROT specification:
For a star of magnitude m0=5.7, the photon noise in an
amplitude spectrum of a time series of 5 days is
B0 = 0.6 ppm
For a given magnitude m, B = B010(m – m0)/5
Granulation and activity signals
Non white noise, characterised by its auto-correlation
function: AC = A2 exp(-|t|/t)
A: “amplitude”
t: characteristic time scale
Fourier transform of the auto-correlation function
=> Fourier spectrum of the initial signal:
P() = 2A2t/(1 + (2t)2)
s (or « rms variation ») from s2 =
 P() d
> s  A/2 and P() = 4s2t/(1 + (2t)2)
Modelling the granulation characteristics (continue)
Granulation spectrum = function of:
• Eddies contrast (border/center of the granule) : (dL/L)granul
• Eddie size at the surface : dgranul
• Overturn time of the eddies at the surface : tgranul
• Number of eddies at the surface : Ngranul
Modelling the granulation characteristics
Eddie size : dgranul = dgranul,Sun (H*/HSun)
● Number of eddies : N
granul = 2(R*/dgranul)²
● Overturn time :
tgranul = dgranul / V
● Convective velocity : V,
from Mixing-length Theory (MLT)
●
Modelling the granulation characteristics (continue)
●
Eddies contrast : (dL/L)granul = function (T)
● T : temperature difference between the granule and the
medium ; function of the convective efficiency, .
● 
and T from MLT
● The relation is finally calibrated to match the Solar
constraints.
Granulation signal
Inputs:
● characteristic time scale (t)
● dL/L for a granule
|
● size of a granule
| (s)
|
Modelled on the base of the Mixing-length theory
All calculations based on 1D stellar models computed with
CESAM, assuming standard physics and solar metallicity
Inputs from models
Activity signal
Inputs:
● characteristic time scale of variability t
[Aigrain et al 2004, A&A]
t = intrinsic spot lifetime (solar case) or Prot
How to do better?
●
standard deviation of variability s
[Aigrain et al 2004, A&A, Noyes et al 1984, ApJ]
s = f1 (CaII H & K flux)
CaII H & K flux = f2 (Rossby number: Prot /tbcz)
Prot = f3 (age, B-V) and B-V = f4 (Teff)
tbcz , age, Teff from models
fi are empirical (as t)
The solar case
Not too bad, but has to be improved
Example: a Sun at m=8
Example: a Sun at m=8
Example: a young 1.2MO star (m=9)
Example: a young 1.2MO star (m=9)
Prospectives
Next steps:
- improvement of granulation and
activity modelling
- rotation (José Dias & José Renan)
- orbital instrumental perturbations
Simulation are not always close to reality,
but they prepare you to face reality
```
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