Part C - inaoe

Fundamental stellar parameters
flux received at Earth f º =
direct determination of Teff
R = radius of the spherical star, D = distance to the star.
Luminosity : L = 4π R 2 ∫ F º dº
T eff
4
= 4π R 2 F = 4π R 2 σ
4π R 2 ∫ F º dº = 4π D 2 ∫ f º dº → 4π R 2 F = 4π D 2 f
F = (D/R) 2 f → F = 4/Θ 2 f Θ = angular diameter = (2R/D)
►if Θ measured, (interferometry)
►If f computed from observations over the all spectral range
→ F determined → T eff computed f = (Θ 2 / 4 ) σ T eff
4
If the parallaxe is mesured → D → R computed, L computed
meaning of Θ ? R ?

seminal paper : Code et al. 1976, ApJ, 203, 417
32 stars O5 – F8 : SED measured
angular diameter from interferometry ( at ~440 nm)
Hanbury-Brown et al 1974, MNRAS, 167, 121
accuraccy rely upon :
♦ measurements of flux, angular diameters
♦ model of interstellar redenning to correct the observations
♦ atmosphere model :
to correct the missing flux at short wavelength (less than 110 nm)
to correct for line blocking (energy mesured in « continuum » windows)
∆ Teff / Teff = ∆ f / (4 f ) + ∆Θ / (2 Θ )
(an accuracy of 2% on Θ is needed for an error of 1% on Teff)
difficult to apply this method to a large set of stars
measurements of angular diameters still limited ( interferometry )
→ alternative method to interferometry to determine the angular diameter
IRFM (infra-red flux method)

Davis, 1997, IAU Symposium 189, 31.

Infrared Flux Method : IRFM
Blackwell & Shallis (1977, MNRAS, 180,177)
based on Gray D. 1967, 1968 ApJ 149,317; AJ 73, 769
(1) σ T eff4
=(4 / Θ 2 ) f
(2) F λ
= (4 / Θ 2 ) f λ
spectral domain in the infrared
to derive (Θ and T eff
) to satisfy (1) and (2) iterative method.
based on assumption, valid : the flux in the infra-red has a weak dependance on Teff.
SED, f λ
observed, F λ
given by a model, but low dependance on the model.
♦ not applicable to “exotic” star
♦ rely on accurate flux calibration in the IR
♦ uncertainty due to the de-reddening to compute f
an error on E(B-V) of 0.02mag gives 1 to 5% on Teff
♦ unsuitable for hot stars (above 10000K)
lack of UV flux measured.
for Teff ~25000K 20% of the flux is emitted at < 110 nm
discussion of the accuraccy :
Blackwell et al, 1979 MNRAS, 188, 847
Brooth, 1997, Symposium 189, 147; Megessier , 1997, Symposium 189, 153
a « few » % on all the required parameters
too few « fundamental » values, specially θ , for real testing (the Code’s stars)

2 = equation (1)
3 = equation (2)
Blackwell et al 1979, MNRAS, 188,847
Blackwell & Shallis 1977, MNRAS, 180, 177
(angular diameter in arcsec)

Alternative method (Blackwell et al 1980, A&A, 82, 249)
(1) σ T eff4 =(4 / Θ 2 ) f
(2) F λ = (4 / Θ 2 ) f λ → σ T eff4 = (F λ / f λ ) f
σ T
4
eff = F λ R
with R = f / f λ
F λ given by a model, λ is in the infrared domain.
R computed from a grid of models : R = σ T
4
eff / F λ
→ a grid R ( Teff, log g, λ, Fe/H)
the measure quantity : f / f λ is compared to R computed → Teff
by product : θ
accuraccy : roughly of 2.5% in Teff for star later than A5, and ~6% for Θ.
Needs good calibration of the IR and good energy distribution over λ; results influenced
by the atmosphere model and log g and Fe/H of the star (has to be known!)
Publications in the 90’s : (...Blackwell… 1986, A&A, 159,217; 1994, A&A, 282, 899; 1998, A&AS,
129, 505)
Discussion of the accuracy of the method, impact of the atmosphere model : Megessier 1994,
A&A, 289, 202; 1997, Symposium 189, 153.
IRFM : Teff accurate if log g anf Fe/H also known; rely upon IR calibration
best domain : stars cooler than F0 (most of the flux emitted in the visible)
Note:
several Teff can be determined if several λ in the IR are used, increased the accuracy.
IRFM is closed to a color-index determination of Teff, one « filter » cover the all range

Extension of the IRFM
Application of the IRFM to a large sample of F0-K5 dwarfs and giants.
Alonso et al, 1996, A&AS, 117, 227 (F0V – K5V);
1999, A&AS, 139, 335 (F0-K5 giant) (and ref. in it)
→ Teff determination for ~500 dwarfs and ~500 giants
3500K ≤ Teff ≤ 8000K; -3.0 ≤ [Fe/H] ≤ +0.5; 0.5 ≤ log g ≤ 5
(Alonso et al. 1994, A&A, 282,684: calibrate the IR flux using stars with direct measures of their
angular diameter)
♦Bolometric fluxes (SED) from broad-band photometry (its calibration taking into
account Fe/H effects) and bolometric correction
♦ Kurucz (1993) model atmosphere.
♦ Detailed analysis of the uncertainties coming from each step of computation
♦ IR photometry (broad-band), its calibration, IR monochromatic fluxes
♦ Reddening correction
♦ Determination of log g (low sensitivity to g for dwarf stars log g = 5 for the cooler, and 4 for hottest; for the
giant log g =2 or from litterature) and Fe/H from other calibrations
Internal error on Teff ~1-2 %
Comparison with other determinations : not many « direct » determination of Teff
Comparison of the mean Teff of solar analogue star to that of the Sun 5743 ± 100K
compared to 5780K.

Alonso, 1996, A&AS, 117, 227

Alonso et al, 1999, A&AS139, 335

outine to determine Teff with such method not in the public domain due
to the numerous “correcting” steps needed to apply it; keep the internal
consistency of these results.
→ calibration of color indices in term of Teff, [Fe/H]
for the dwarf F0V – K5V : (B-V), (R-I), (V-R), (V-I), (V-K), (J-H), (J-K), β
for the giant F0 – K5 : (U-V), (B-V), (R-I), (V-R), (V-I), (V-K), (J-H), (J-K),
(I-K), (V-L’), (b-y), (u-b)
using the IRFM determinations of Teff
Alonso et al 1996, A&A, 313, 873; 1999, A&AS, 140, 261.
log g , [Fe/H] not explicitly taken into account but :
♦ gravity effect on calibration of (B-V), (V-R), (V-I), (J-H), (J-K)
♦ negligible dependance on gravity for (R-I), (V-K)
♦ for giant stars non-dependance on metallicity for (V-I), (I-K), (J-K)
Extensive comparison with previous work of Teff determination including theoretical
calibration.
(±0.1 dex on [Fe/H] → ±0.1 5% on Teff)

Alonso et al, 1996, A&A, 313, 873 (F0V – K5V)

Angular diameters from IRFM
by-product
σ T eff4 =(4 / Θ 2 ) f → determination of the angular diameter Θ
compared to that determined by interferometric technics
for giant stars F0-K5 domain, (with internal error below 5%)
↔ to compare the « IRFM » Teff with « direct » Teff
Alonso et al, 2000, A&A, 355,1060
no systematic differences → no « zero point »correction for the Teff scale
But:
direct angular diameter Θ determination may be affected by limb
darkening corrections and/or possible circumstellar dust shells giving
larger Θ; circumstellar dust shells may also affect IRFM via the SED
If parallaxe mesured → mean radius values computed
(for solar metallicity stars)
note : trend of the radius according to the metallicity
(R = Θ(D/2). error on parallaxes: 2-20%, expected error on Θ →error on radius 5-27%)

Effective temperature scale for FGK stars
dwarf and giant
Extension of Alonso et al Teff calibration by Ramírez & Meléndez
(2005, ApJ, 626, 446, ApJ,626,465)
3600K ≤ Teff ≤ 8000K -4.0 ≤ Fe/H] ≤ +0.5
Consistency checked with angular diameters by interferometry and lunar
occultation and transit
Updated and extension of the previous calibration of photometric indices
UBV, uvby, Vilnius, Geneva, RI (Cousins), DDO, Tycho, 2MASS
→ for the very metal-poor dwarf star, (V-K) is not completely metallicity independent
(if IRFM Teff is realiable for these stars)

….Teff of cooler stars ?
molecules and dust in the stellar atmosphere complicate greatly any
modelisation
for dwarf M, L, T type stars, no Θ measured through interferometry,
→ no direct Teff mesured (yet)
Teff and absolute magnitude determined from theoretical color indices
computed from model atmosphere
→ subject in continuous evolution
e.g. Jones et al. 1996, MNRAS, 280, 77; Leggett et al 2002, ApJ, 564, 452;
Burrows et al, 2003, ApJ, 596, 587; Dahn et al. 2002, AJ, 124,1170.

for giant K and M type stars Θ measured through interferometry, Teff
computed with IRFM
give the basis for a direct determination of Teff
(e.g. Dyck et al, 1996, AJ, 111, 1705)
IRFM method affected by the strong molecular bands in the IR
→ define « continuum » region
→ bolometric correction difficult to compute (strong absorption bands,
large variation with Teff)
measure of Θ affected by circumstellar dust shells: this dust is difficult
to separe from the photospheric radiation (e.g. van Belle et al 1999, AJ,
117, 521)
Calibration in Teff of color indices UBV, RI(Cousins), JHK
(e.g. Houdashelt et al. 2000, AJ, 119, 1424)

Theoretical colors, Bolometric Corrections
and colors versus Teff relations
from stellar atmosphere models
→ theoretical colors for any photometric system
synthetic photometry
→ produce extended grids of Teff calibration over extended ranges of
Teff, log g and [Fe/H] which represend real photometric systems
(if « zero point » carrefully determined)
Hints : the passband sensitivity functions of a photometric system
may vary … and the one used in the computations
may not represent any real system

these grids can be used only with de-reddenned observed colors;
log g, [Fe/H] must be known.
■ Bessell, Castelli & Plez, 1998, A&A 333, 231 UBVRIJHKL
O-M stars Teff versus : (BCP98)
(U-B) B0-A0 dwarfs; (V-K) A-G dwarfs; (V-I) A-M dwarfs; (I-K) M dwarfs; (V-K) G-M giants
■ Bessell, 2001, PASP, 113, 66 Washington system
6000K ≤ Teff ≤ 40000K log g : 2.0 – 5.0
3000K ≤ Teff ≤ 550K log g : -0.5 – 5.0 [Fe/H] -3 to +0.3
■ Houdashelt & Bell, 2000, AJ, 119, 1448 UBVRIJHK
F-K stars Teff versus colors
Cousins, Johnson-Glass, CIT/CTIO systems calibrated with IRFM
4000 K ≤ Teff ≤ 6500 K, 0.0 ≤ log g ≤ 4.5, and -3.0 ≤ [Fe/H] ≤ 0.0
■ Clem et al, 2004, AJ, 127, 1227 Strömgren uvby : analysis of the correction to be added
to the purely synthetic colors to satisfy various situations : globular clusters analysis etc.
combination of color indices have been produced in order to
determine from photometry log g, [Fe/H] and the reddenning.