Tabel C2 'Calibrate-hyper-vsn-2' A temperature calibration is given for hypergiants in subsection 1) below using the s-variable defined by Table C2.1 below (cf. also the Note below). To allow transforms between MK, s, (B-V), Teff and (to) BC, we give symbolic code and used reference data based on our calibration, and on Schmidt-Kaler (1982, in Landolt-Boernstein, Neue Serie, Group 6, Volume 2, Astronomy and Astrophysics, External Supplement to Vol. 1, Subvolume b, K. Schaifers and H.H. Voigt, editors, Springer Verlag, Berlin, also in Lang, K.R. 1992, Astrophysical Data, Planets and Stars, section 9.7 Stellar Temperature, Bolometric Correction and Absolute Luminosity, pp. 137-142, Springer Verlag New York, for spectral type > Iab) in Table C2.2 below. Other calibrations (Humphreys or Massey) do not modify the general observed trends of the main paper. This section is based on spectral classification (Morgan & Keenan 1973ARA&A..11...29M), infrared measurements (Johnson 1966A&A.....4..193J), brightest known stars (Humphreys 1978ApJS...38..309H), two-dimensional interpolation over HR-diagram using spectral class parameters s, b (De Jager & Nieuwenhuijzen 1987A&A...177..217D), general calibrations MK, (B-V), BC from Flower (1977A&A....54...31F), Boehm-Vitense (1981ArA&A..19..295B), Schmidt-Kaler(1982, Verlag New York)and Gray (1992, University Press Cambridge), together with specific data for hypergiants from De Jager (1998A&AR....8..145D). In subsection 2) below we mention calibrations for (B-V) and bolometric correction BC, and in subsection 3) below we present associated transforms between MK, Teff and (B-V) and a transform from MK to BC. The errors due to the transform and the area of validity of each transform are also given. 1) The calibration of Teff from MK classification is based on the overview on hypergiants by De Jager (1998A&AR....8..145D, Table 2). We have: s logT_Iap MK star other_name association ---------------------------------------------------------- 4.80 3.722 F8Iap IRC +10420 --- 5.20 3.670 G4Iap HD 269723 LMC 4.80 3.710 F8Iap HD 224014 rho Cas --- 6.90 3.544 M2Ia HD 206936 mu Cep --- 5.40 3.667 G8Iap HD 119796 HR5171A --- 3.30 3.920 A3Iap HD 33579 R76 LMC 2.70 4.050 B8Iap Cyg OB2 #12 --- 4.50 3.765 F5Iap CRL 2688 --- Table C2.2 below gives the constants of the chebychev approximation we used for the calibration in the main paper. Two figures are shown as Figs.C2.1 and C2.2 in http://www.aai.ee/HR8752, and in the online appendix as Figs.A2 and A3. Fig. C2.1 shows Teff as a function of s-parameter of (De Jager & Nieuwenhuijzen 1987A&A...177..217D), Fig. C2.2 gives a comparison of the calibration as given in Fig.C2.1 with earlier Teff calibration as a function of s-parameter, and positions the observations between two spectral class relations: those for spectral class Ia and Ia+ . In the figures, Cyg OB#12 follows the line for spectral class Ia, HD33579 through HD 11976 follow the line for Ia+, and {mu} Cep follows the line for Ia. In De Jager & Nieuwenhuijzen (1987A&A...177..217D), the spectral class Ia+ is represented by b-parameter b=0.0, class Ia by b-parameter b=0.6, and class Iab (or I) by b=1.0 . ------------------------------------------------------------------------------ Table C2.1: Definitions of the s-variable for range of spectral type and the step range per 0.1 step spectral range ------------------------------------------------------------------------------- MK range O1-O9 O9-B2 B2-A0 A0-F0 F0- G0 G0-K0 K0-M0 M0-M10 ------------------------------------------------------------------------------- s 0.1-0.9 0.9-1.8 1.8-3.0 3.0-4.0 4.0-5.0 5.0-5.5 5.5-6.5 6.5-8.5 s-step 0.1 0.3 0.15 0.1 0.1 0.05 0.1 0.2 ------------------------------------------------------------------------------- Note: The parameters s and b are mathematical representations of spectral resp. luminosity class, introduced by De Jager & Nieuwenhuijzen (1987A&A...177..217D) with the purpose to get rid of the usual discontinuities that occur in the gradients of those curves in which stellar parameters are plotted directly against spectral type . Except where indicated the estimated accuracy in s is 0.5-0.6 of one-tenth spectral step in MK, when transforming to and from MK spectral classification. ------------------------------------------------------------------------------- The calibration line coincides with the Ia and Ia+ lines at high temperature (low value of s), then after s=2 stays close to the Ia line, between s=3.5-5 stays close to Ia+, and then stays close to the Ia line around s=6.8 ({mu} Cep). For lower temperatures we might add the observation of KY Cyg (De Jager 1998A&AR....8..145D), and see that the calibration line would also follow the Ia line from {mu} Cep to KY Cyg, resulting in a (future) calibration going to somewhat lower temperatures, but this has not been done here. In our work this has little importance, as we have observations of spectral class in general earlier than about G5. The one exception is a measurement in 1973 of a cooler spectrum. We use the given calibration in the main paper for HR8752. For practical purposes we use a piecewise (one-dimensional) numerical fit to the observational s-range of s=2.6-7.0. The resulting temperature calibration is valid within the limited spectral range B8 - M2. 2) Calibrations for (B-V) and BC as function of s-parameter The calibration transforms between MK (s) and (B-V) and between MK (s) and BC for hypergiants in 'Calibrate-Hyper' are based on Schmidt-Kaler (1982, Verlag New York, (Ia) spectral type > Iab). The different interpolation parameters and their accuracies are given in subsection 3) below. 3) Description of transforms between (MK) s, Teff, (B-V) and BC for hypergiants as used in the main paper. Note: In this paper we use chebychev polynomials for the transforms used in Section 3 of the main paper. Chebychev polynomials of the first kind can be written as T0=0, T1=x, T2=2x^2, T3=4x^3, T4=8x^4-8x^2-1, ... (Carnahan et al. 1969), and require normalization of data to [-1,1], eg. as follows: For a variable z on [minimum, maximum] the variable x to use as variable in the chebychev polynomial is normalized as x=(2z -minimum -maximum)/(maximum - minimum). It is possible to use a recursive definition of the terms, but we use as alternative for the 4th degree chebychev polynomial, which is sufficiently accurate for the transforms used here: y=a-c+e+x*(b-3*d+x*(2*c-8e+x*(4*d+x*8*e))). Errors created in the transform can be summed quadratically with the errors in the measurement, assuming independent statistical errors. The transforms are based on a 5-parameter chebychev polynomial (4th order) interpolations. We give these details so that the results allow verification of the transforms of the original MK, Teff and (B-V) observations. The local variable x is normalized over the used range of interpolation. For z on [min,max], x= (2z -min -max) /( max -min) is the normalization to x on [-1,1] or alternatively: x=(z-mean)/rdev with mean=(max+min)/2; rdev=(max-min)/2. The chebychev computational formula is y=a-c+e+x*(b-3*d+x*(2*c-8*e+x*(4*d+x*8*e))), where the constants are given in Table C2.2 below. Table C2.2 contains the constants a, b, c, d, and e for each of the indicated transforms (from --> to), accuracy of interpolation in the result, minimum and maximum bounds of the independent variable x, the constants a-e to be used in the formula, and finally the mean and rdev as defined above. -------------------------------------------------------------------------------- Table C2.2 coefficients of interpolating formulas used for transformations -------------------------------------------------------------------------------- var -> res err [min,max] a b c d e mean rdev -------------------------------------------------------------------------------- s ->logTeff 0.012 [2.6,7.0] 3.75896 -0.25985 0.04368 -0.00722 0.00532 4.8 2.2 logTeff ->s 0.098 [3.5,4.1] 4.61292 -2.47427 0.54217 -0.06304 0.10708 3.8 0.3 s ->BC 0.021 [2.6,6.2] -0.48642 -0.23007 -0.49348 -0.00741 -0.01042 4.4 1.8 s ->(B-V) 0.028 [3.0,6.0] 0.63114 0.82265 0.21170 -0.03161 -0.05704 4.5 1.5 (B-V) ->s 0.038 [0.0,1.6] 4.84940 1.28415 -0.29079 0.25259 -0.10847 0.8 0.8 -------------------------------------------------------------------------------- Using Table C2.2 above, the measurement error and the transform error are combined: The error in the result due to errors in (the independent variable) x is dy/dx=b-3*d+x*(4*c-16*e+x*(12*d+x*32*e)) and dx/dz= 2/(max-min) , which leads to err(y)=dy/dx*dx/dz*err(z), which is the error in the result due to the measurement error. We quadratically add to this the error in the transformation, given in column 2 (err), and take the square root to obtain the total error. (end)