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J/A+A/405/1095   Limb-darkening coefficients from ATLAS9 models  (Barban+, 2003)
New grids of ATLAS9 atmospheres. II.
Limb-darkening coefficients for the Stroemgren photometric system for A-F stars.
    Barban C., Goupil M.J., Van't Veer-Menneret C., Garrido R., Kupka F.,
    Heiter U.
   <Astron. Astrophys. 405, 1095 (2003)>
   =2003A&A...405.1095B
ADC_Keywords: Models, atmosphere ; Photometry, uvby

Keywords: stars: atmospheres - photometry, uvby - stars: oscillations -
          convection

Description:
    Using up-to-date model atmospheres (Heiter et al. 2002A&A...392..619H)
    with the turbulent convection approach developed by Canuto, Goldman &
    Mazzitelli (1996ApJ...473..550C, CGM), quadratic, cubic and square
    root limb darkening coefficients (LDC) are calculated with a least
    square fit method for the Stroemgren photometric system. This is done
    for a sample of solar metallicity models with effective temperatures
    between 6000 and 8500K and with logg between 2.5 and 4.5. A comparison
    is made between these LDC and the ones computed from model atmospheres
    using the classical mixing length prescription with a mixing length
    parameter α=1.25 and α=0.5. For CGM model atmospheres, the
    law which reproduces better the model intensity is found to be the
    square root one for the u band and the cubic law for the v band. The
    results are more complex for the b and y bands depending on the
    temperature and gravity of the model. Similar conclusions are reached
    for Mixing Length Theory (MLT) α=0.5 models. As expected much
    larger differences are found between CGM and MLT with α=1.25. In
    a second part, the weighted limb-darkening integrals, bell, and
    their derivatives with respect to temperature and gravity, are then
    computed using the best limb-darkening law. These integrals are known
    to be very important in the context of photometric mode identification
    of non-radial pulsating stars. The effect of convection treatment on
    these quantities is discussed and as expected differences in the
    bell coefficients and derivatives computed with CGM and MLT
    α=0.5 are much smaller than differences obtained between
    computations with CGM and MLT α=1.25.

    The limb darkening coefficients are given here for the u, v, b and y
    bands and for CGM models, MLT α=0.5 models and MLT α=1.25
    models.

File Summary:

       FileName      Lrecl  Records   Explanations

ReadMe            80        .   This file
cgm.dat           78      924   Quadratic (a,b), cubic (c,d,e), square root (f,g)
                            limb darkening coefficients and σ (sd)
                            corresponding to the law for uvby bands and for
                            CGM (1996ApJ...473..550C) models. (tables 1-4)
mlt050.dat        78      924   Quadratic (a,b), cubic (c,d,e), square root (f,g)
                            limb darkening coefficients and σ (sd)
                            corresponding to the law for uvby bands and for
                            Mixing Length Theory (MLT) α=0.5 models.
                            (tables 5-8)
mlt125.dat        78      924   Quadratic (a,b), cubic (c,d,e), square root (f,g)
                            limb darkening coefficients and σ (sd)
                            corresponding to the law for uvby band and for
                            Mixing Length Theory (MLT) α=1.25 models.
                            (tables 9-12)
tables.tex       173     2933   LaTeX version of the tables


See also:
 J/A+A/401/657  : Non-linear limb-darkening law for LTE models II (Claret, 2003)
 J/A+A/363/1081 : Non-linear limb-darkening law for LTE models (Claret, 2000)
 J/A+A/335/647  : Limb-darkening coefficients for ubvyUBVRIJHK (Claret 1998)
 J/A+AS/110/329 : LTE model atmospheres coeff. (Diaz-cordoves+, 1995)
 J/A+AS/114/247 : Limb-darkening coefficients for RIJHK (Claret+, 1995)


Byte-by-byte Description of file: *.dat

   Bytes Format Units   Label     Explanations

       1  A1    ---     Band      [uvby] Band (Stroemgren photometry)
   3-  5  F3.1 [cm/s+2] logg      Surface gravity
   7- 10  I4    K       Teff      Effective temperature
  12- 16  F5.3  ---     a         Quadratic limb-darkening coefficient (1)
  18- 23  F6.3  ---     b         Quadratic limb-darkening coefficient (1)
  25- 30  F6.4  ---     sigmaq    Standard deviation for the quadratic law (2)
  32- 36  F5.3  ---     c         Cubic limb-darkening coefficient (1)
  38- 43  F6.3  ---     d         Cubic limb-darkening coefficient (1)
  45- 51  F7.4  ---     e         Cubic limb-darkening coefficient (1)
  53- 58  F6.4  ---     sigmac    Standard deviation for the cubic law (2)
  60- 64  F5.3  ---     f         Square root limb-darkening coefficient (1)
  66- 71  F6.3  ---     g         Square root limb-darkening coefficient (1)
  73- 78  F6.4  ---     sigmasr   Standard deviation for the square root law (2)

Note (1): Limb-darkening coefficients:
    a, b (quadratic):
        I(µ)/I(1) = 1 - a(1-µ) - b(1-µ)2
    c, d, e (cubic):
        I(µ)/I(1) = 1 - c(1-µ) - d(1-µ)2 -e(1-µ)3
    f, g (square root):
        I(µ)/I(1) = 1 - f(1-µ) - g(1-sqrt(µ))
    where I(1) is the specific intensity at the center of the disk, a, b,
    c, d, e, f, g are the corresponding limb-darkening coefficients and
    µ=cos(γ), γ being the angle between the line of sight
    and the emergent intensity
Note (2): Same definition of the standard deviation as in
    Diaz-Cordoves Gimenez (1992A&A...259..227D).
       sigma2=1/20 {sum(i=1to20)}{[I(mu)/I(1)th - I(mu)/I(1)ap]i}2,
    where the subindex th denotes the values derived from the models and
    ap corresponds to those derived from the corresponding approximation.
    We consider sigma as a measure of the quality of the actual fit.


Acknowledgements: Caroline Barban 


References:
    Heiter et al., Paper I                 2002A&A...392..619H
(End)                                          Patricia Bauer [CDS]  22-May-2003