Astron. Astrophys. 339, 858-871 (1998)

3. Photometric calibration of stellar parameters

3.1. Photometric calibration of surface brightness scales

The surface brightness of a star of intrinsic magnitude and angular diameter is defined by:

where the zero-point is chosen such that mag when mas. The surface brightness is currently calibrated by measuring the stellar angular diameters of non variable stars. Only recently, late-type stars with apparent angular sizes larger than few milliarcsec have become suitable targets for the modern developed Michelson interferometry techniques. Therefore, many reliable measurements of stellar diameters have become available with uncertainties better than 5% (Davis & Tango 1986; Di Benedetto & Rabbia 1987; Mozurkewich et al. 1991). Since photometric stellar diameters have to be predicted at the target accuracy better than 2% for the ISO standards, the currently available surface brightness-colour (SC) correlations should be carefully revisited in order to investigate their sensitivity to different sources of errors. Table 3 lists the sample of angular diameters which will be adopted throughout this paper for the calibration of the SC correlations. Most of these measurements related to giants cooler than the Sun have been already discussed and self-consistent comparisons showed remarkable agreement of independent visual and IR data to less than 2% of accuracy (Di Benedetto 1993).

Table 3. Nearby stars with accurate measurements of angular diameters

Now, the selected sample call for several improvements:
1. The angular diameter of the K-giant Arcturus ( Boo) is available with its well determined limb darkening (Quirrenbach et al. 1996) which was found to be in excellent agreement with the predictions from limb-darkening coefficients for a grid of model atmospheres tabulated by Manduca (1979). This grid has been currently adopted to convert measured uniform disk diameters at visual and infrared wavelengths into true photospheric stellar sizes.
2. The angular diameters of the F dwarf Procyon ( CMi) and the G0 III component of Capella ( Aur) are available from Michelson interferometry with accuracy good enough for including these spectral types into the actual revisited SC correlations.
3. The sample of A-type stars basically relies on the less accurate intensity interferometry data. But it notably includes the most relevant angular size of the star Sirius ( CMa) which has been also measured by Michelson interferometry and the two independent measurements show excellent agreement between each other.
4. Most of the calibrating stars are included in the overall sample of the ISO standards, notably the stars Vega ( Lyr) and Sirius. Sirius has been recently adopted as primary calibrator of the absolute stellar fluxes in the infrared (Cohen et al. 1992). A comparison between the broadband K magnitude of Vega in the TCS system and that of Sirius in both TCS and ESO systems shows a somewhat discrepancy with the broadband K magnitudes reported by Cohen et al.. Then, to be consistent with this absolute flux calibration, a Johnson near-infrared colour mag is adopted below for Vega rather than the value of - 0.025 mag determined from the TCS observations.

The visual surface-brightness data are plotted in Fig. 2 as a function of several standard broadband colours in the Johnson magnitude system. Reddening corrections are not applied to the data, since they would induce overall effects on the residuals much smaller than 0.1 mag, all the stars being within 100 pc. As it can be seen, the correlation is found to be affected by the smallest scatter. All the other plots show a larger dispersion which is a clear signature that the corresponding standard colours are poorer brightness indicators for the actual high-precision set of angular diameters. Most of this scatter is certainly due to photometric errors and could be likely reduced by using suitable high-precision photometry. However, there is also evidence from theoretical model atmospheres (Bell & Gustafsson 1989, hereinafter BG; Kurucz 1991) that some magnitude-colour combinations notably using optical colours may be biased by gravity and metallicity which would prevent the corresponding photometry to be adopted as a bias-free indicator of stellar sizes.

Fig. 2. Visual surface brightness for an angular diameter of 1 mas against several standard broadband colour indices on the Johnson scale. For each diagram, the solid line in the upper panel represents the least-squares quadratic fit to all data with residuals around the ridge-line plotted in the lower panel

Fig. 3 shows that a tight correlation is found for either the dwarfs which include calibrators of A-F-G spectral-type or the giants which include only stars of G-K spectral-type or both. Then, in applying the near-infrared correlation below, I shall refer to stellar luminosities of Class V and Class III with suitable calibrations given by:

where is the standard error determined from the overall scatter around the mean relation. The intrinsic colour-index is referred to the Johnson broadband magnitude system. Notice that few data with deviations greater than about 3 are omitted from the final calibration (see Table 3). There would be evidence for a small luminosity effect in the actual calibration of the stellar surface brightness. In fact, around the colour mag, the Class V and Class III correlations yield the angular diameter of the Sun in error by about + 0.7% and + 2.0%, respectively. The result from the Class III correlation would deviate by 1.5 sigma from the average ridge-line. However, in view of the relatively small number of stars involved in the calibration, this luminosity effect should not be overemphasized.

Fig. 3. Calibration of the visual surface brightness for an angular diameter of 1 mas against the intrinsic near-infrared colour. Top: solid and dotted lines represent least squares quadratic fits to data of giants and dwarfs, respectively. Open symbols: data falling more than 3 sigmas away from the lines. Bottom: residuals around each best fitting regression line

The residual scatter mag around the best fitting regression lines is due to random errors in the observed colours and angular diameter measurements. By assuming that there are no errors in the colours, it would lead to an overall uncertainty = 1.4% for angular sizes predicted by observational photometry. If the colours are fitted to surface-brightness data, assuming all of the error in the colours, it would correspond to an error mag, quite consistent with the expected uncertainty in the near-infrared photometry of the ISO standard stars. By combining the relations (2) and (3), the true photospheric stellar angular diameter is given by:

where is the visual absorption derived from the Hipparcos parallaxes. According to a first-order approximation of as a function of the colour, the relation (4) can also be rewritten as:

where the slope increases from about 1.2 for K-type stars to 1.5 for A-type stars. The relation (5) emphasizes that the error on the K magnitude will critically contribute to the uncertainty . In fact, the same photometric error in the magnitude K or in the absorption would yield as a negligible contribution as 0.5 %. Notice that an absorption mag together with a reddening coefficient of 0.8 mag/kpc corresponds to a distance of pc.

There are several potential sources of systematic error which can affect the photometric diameters predictable by the near-infrared correlation. First, the correlation strictly refer to either G-K giants or A-F-G dwarfs. To cover the whole A-F-G-K spectral range, the overall correlation must be adopted which fully ignores the small luminosity effects. However, systematic deviations would likely be no more than about 1%, as shown for the Sun angular diameter. Second, the correlation may suffer from stellar variability, multiplicity, etc. which could induce somewhat significant variations of the visual magnitude V . However, these effects are likely to be reduced to a negligible level within the subset of 537 ISO standard stars according to the adopted selection criteria and to the coefficients of the relation (5). Third, the magnitude-colour combination in the relation (5) is expected to be slightly sensitive to stellar metallicity through line blanketing. This effect can be evaluated by using model-atmosphere results of BG. For a metallicity as poor as dex, the photometric diameter would change by no more than 1% with respect to that of metal-normal content over a range of gravities and effective temperatures K.

It can be concluded that the correlation would certainly benefit from a much larger sample of calibrating angular diameters as accurate as 5% or better for deeper investigating systematic and random errors in the predicted stellar sizes by high-precision near-infrared photometry. However, there seems to be observational and theoretical evidence that the actual carefully calibrated correlation can likely provide reliable angular diameters with errors smaller than 2%, implying a target accuracy of less than 1% for temperature determinations of the ISO standard stars.

3.2. Photometric calibration of bolometric flux parameters

The absolute integrated fluxes of 420 ISO standard stars have been determined by BL for applications of the IRFM. Also, BL widely investigated the several sources of potential errors which can affect flux measurements, and then no further discussion will be done below. I shall adopt throughout this paper these bolometric fluxes for a more straightforward comparison with the IRFM which notably relies on the very recent absolute flux calibration (Cohen et al. 1992). Now, the major concern is related to the bolometric flux representations by the broadband near-infrared colour which play a relevant role in the calibration of temperature scales by the actual method. For this purpose, only a subset of 327 stars is available with broadband K magnitudes in either the TCS or ESO photometric systems and 22 stars in common. In addition, the bolometric fluxes of 35 giants (Blackwell et al. 1990) have been included in the main list to properly sample the Class III stars of K spectral type. According to BL, these fluxes should be slightly increased by 1.1% to take into account the change of the absolute calibration. To be consistent with the calibration of the surface brightness scales, the bolometric fluxes are related to the broadband colour on the Johnson magnitude system through the convenient flux parameter defined by:

where is the measured bolometric flux in and RFLUX = is the corresponding "reduced flux" used by BL. The star-by-star values of the parameter are plotted in Fig. 4 as a function of the intrinsic colour . Regression lines according to standard least-squares quadratic fits have been obtained by assuming accurate intrinsic colour determinations. The regression lines are given by:

where is the standard error determined from the overall scatter around the mean relation. Twenty-one stars have been excluded from the fits (see Tables 1 and 2), since they showed deviations significantly larger than 3 sigmas from the mean fits. The regression line for Class V stars has been obtained over two separate spectral ranges. The advantage is that the residual scatter around the best fitting ridge-line of F-G-K stars is found to be significantly smaller than that of A-type stars. In addition, one can best compare results for Class V and III stars over the same F-G-K spectral range.

Fig. 4. Calibration of the infrared flux parameter for Class V and III stars against the intrinsic near-infrared colour. Top: plots of individual data with solid and dotted lines representing least-squares quadratic fits to dwarfs and giants, respectively. Middle and bottom: residuals around each best fitting regression line. Crosses indicate A-type stars

The residual scatter about each best-fitting regression line displayied in the lower panels of Fig. 4 is due to random errors in the observed colours and bolometric flux measurements. By assuming that there are no errors in the colours, the scatter yields , i.e. 1.1% and 2.6% for the integrated fluxes of dwarfs and giants, respectively, over the F-G-K spectral-range. If the colours are fitted to the flux data, assuming all of the error in the colours, the scatter becomes = 0.018 mag for dwarfs, fully consistent with the errors on the near-infrared photometry of the ISO standard stars. But, it would be as large as = 0.049 mag for giants. As the near-infrared broadband colours in either the TCS or ESO magnitude systems have been measured with the same accuracy of mag regardless of the stellar luminosity class, it follows that the increased component of scatter in the diagram of Class III stars must be likely ascribed to the bolometric flux measurements.

The photometric correlations show also evidence for a real differential effect between Class III and V stars at the same intrinsic colour. The average difference derived according to the flux representation by the polynomials (7) is drawn in Fig. 5. The figure indicates as well that this effect can be reasonably well compensated for by a colour shift of:

This shift would enable the Class V and III second-order polynomials (7) to be interchanged for bolometric flux representations of all the F-G-K stars.

Fig. 5. Solid line: ratio of Class V to Class III bolometric flux representations by second-order polynomials against the intrinsic near-infrared colour. Dotted line: the same ratio with colour shift of one class with respect to other [see text Eq. (8)]

The goodness of the bolometric flux representation by the second-order polynomials (7) can be compared with recent representations by third-order polynomials of the colour on the Johnson magnitude scale (Alonso et al. 1995; BL). Fig. 6 shows the bolometric flux residuals for each representation of Class V stars less affected by random errors. The diagrams indicate that a second- order polynomial function suffices to best fit the flux data even at the level of noise as low as that of dwarfs. This improvement may be due either to the use of the flux parameter or to the adoption of two separate spectral ranges for regression lines or both.

Fig. 6. Comparison of residuals from bolometric flux representations by the broadband near-infrared colour. Top and middle: residuals according to third-order polynomials from Alonso et al. (1995) and BL, respectively. Bottom: residuals according to second-order polynomials with crosses indicate A-type stars

3.3. Photometric calibration of effective temperature scales

According to the fundamental method, the true photospheric stellar angular diameter should be combined with the integrated flux , in order to derive the effective temperature T from the Stefan-Boltzmann law. Then, the temperature determination is given by:

Since both the correlations and are well-represented by quadratic fits in the colour , the calibration of the scale T through the coefficients A, B, C immediately follows as a second-order polynomial function of the same colour. Table 4 reports these coefficients suitable over specific colour ranges together with the overall accuracies for predicted temperatures.

Table 4. Calibration of scales as a function of the Johnson broadband colour

These results call for several remarks.
1. The first three colour ranges have both the correlations and properly sampled by suitable calibrating stars and are selected to maintain as far as possible the peculiarities of the spectral ranges notably the small luminosity effects. According to these effects, a temperature shift of T(V) - T(III) = 57 K would be observed around the colour = 1.5, corresponding to a difference of about 1 % for the Sun temperature mainly due to the contribution from the correlation.
2. The next three colour ranges are not properly sampled by calibrating angular diameters and adopt the overall surface-brightness correlation (3) which fully ignores the luminosity effects on the data.
3. The colour range 0.7-3.7 for Class V and III stars adopts the overall surface-brightness correlation (3) along with the correlation derived by averaging the corresponding correlations of Class V and III stars. Then, it fully ignores luminosity effects on both the and data. All temperatures of the ISO standards lacking luminosity classification have been derived according to this scale.
4. The overall accuracy T / T quoted in Table 4 takes into account independent sources of error, added quadratically, intrinsic to the calibration. These include a conservative error 0.02 mag consistent with the lower dispersion observed in the and data and the errors and from the calibrations themselves. As it can be seen, the required target accuracy of 1% is achievable for all temperatures of the ISO standards, but not for A-type stars with mag. For these stars, the observational photometry becomes the dominant source of uncertainty owing to the steeper slope of the temperature scale .

The star-by-star representations of the scales are plotted in Fig. 7 along with the residuals around the average correlations of Table 4. The scales are as tight as the corresponding () scales drawn in Fig. 4, since a smoothed () correlation is applied in any case for deriving temperatures by the relation (9). The relation (9) makes also clear the overall strategy adopted to provide the final effective temperatures as a function of the near-infrared broadband colour for all the 537 ISO standard stars. Accordingly, all individual temperatures appear to be well represented by second-order polynomial functions with the scatter induced by observational photometry and bolometric flux measurements alone. The worse results for Class III stars due to the increased scatter in the bolometric flux measurements are also evident. All the individual temperatures are reported in the last column of Tables 1 and 2.

Fig. 7. Top: plots of individual temperatures for Class V and III stars as a function of the intrinsic near-infrared colour. Middle and bottom: residuals from temperature scales represented by the second-order polynomials of Table 4. Crosses indicate A-type stars

The broadband near-infrared temperatures can also be exploited as high-precision calibrating data for assessing random and systematic errors affecting other photometric temperature scales. For instance, the scale shown in Fig. 8 looks quite different because of the increased errors affecting the optical colour-index. These include systematic effects due to stellar gravity and metallicity biasing the colours of giants and dwarfs. Therefore, the scale must include evaluations of gravity and metallicity of stars in order to get reliable and accurate individual results. Notice that the superimposed systematic effects tend to compensate for the small temperature shift between the luminosity classes observed in the near-infrared diagram .

Fig. 8. The same as the top panel of Fig. 7 as a function of the optical colour index

© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998
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