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
routine 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.