## Appendix A: zero-points of the broad-band synthetic photometric systemThe zeropoints of the UBVRIJHKL broad band system have been
traditionally set by using the magnitude and color of Vega or a set of
A0 stars. However, because Vega is not in the precise photometric
catalogs of Cousins (UBVRI) and Glass-Carter (JHKL) being unobservable
from the southern hemisphere and because some doubts have been
expressed about its variability and possible IR excess, it was decided
(following Cohen et al. 1992) to adopt Sirius as an additional
fundamental color standard. We have therefore chosen to use the
observed V magnitude of Vega as defining the V zeropoint, but the
observed colors of Sirius together with the colors of Vega and the
synthetic magnitudes of the ATLAS models of Vega and Sirius to define
the zero-points for the other bands. The adopted models for Vega and
Sirius have the parameters (, log g, Z,
) = 9550K, 3.95, [-0.5], 2kms ## A.1. The zeropoint for VUsing the Vega model from Castelli & Kurucz (1994) we derived
the following relation between standard V magnitudes and the computed
absolute fluxes at the earth in the V band: where , are the V
response function (eg. normalised passbands from Bessell 1990) and
f() and f() are the
computed flux at the earth in erg cm We note that the absolute flux at 5556 of the Vega model and the observed flux for Vega (Hayes 1985) are in good agreement and consistent within the 1 sigma error bars of the angular diameter measurements (Code et al. 1976). For Vega: = 3.24 0.07 mas, the model implies 3.26 mas. For Sirius: = 5.89 0.16 mas, the model used by Cohen et al. (1992) implies 6.04 mas. It is anticipated that new Michelson measurements and parallaxes will improve the precision of the angular diameters and effective temperatures for Vega and Sirius and permit better model fittings and comparisons. ## A.2. The color indices zeropointsTo derive the zero points for U-B, B-V, V-R, and V-I color indices we averaged the differences between the observed and computed colors for Vega and Sirius. The zero points for the V-J, V-H, V-K, and V-L colors were derived by fitting the observed indices of Sirius. In Table A1 are given the observed colors and magnitudes together with the synthetic colors and the zeropoints for the synthetic color indices. In general there is excellent agreement between the observed and computed colors for Vega and Sirius. There is somewhat larger disagreement between the observed and computed V-K color for Vega, but it is well within the standard errors of the 1966 Johnson IR catalog.
The IR magnitudes for Sirius (Glass 1997; private communication) were measured in the current standard SAAO system by Carter (1990). The zero-point of the current SAAO system was determined from observations of 25 main-sequence stars between B1 and A7 spectral type. The zero-point of K was set by plotting V-K against B-V for the 25 stars and ensuring that the locus passed through V-K= 0 for B-V= 0. Zero-points and color terms for the ESO, CTIO, AAO and the MSSSO systems relative to the SAAO system are also given in Carter (1990). The possible IR excess for Vega has been investigated by Leggett et
al. (1986) who measured the narrow-band 1-5 However, the absolute infrared flux calibrations of Vega by
Mountain et al. (1985), Blackwell et al. (1983) and Selby et al.
(1983) have shown Vega to have an excess between 2 and 5
## Appendix B: zeropoint fluxes for the UBVRIJHKL systemIn Table A2 are given the mean fluxes and the zeropoints corresponding to a fictitious A0 star with a magnitude of zero in all bands. These are based on V = 0.03 mag for Vega discussed above.
## Appendix C: colors of the sun and solar analogsThe sun has also often been used for flux and magnitude calibrations. There have been several direct measurements of the solar V and B magnitudes. One of the most respected is that of Stebbins & Kron (1957). From their paper, using recent standard V magnitudes for their comparison G dwarfs one deduces V = -26.744 0.015 for the sun. Hayes (1985) claims a further 0.02 mag correction for horizontal extinction yielding V = -26.76 0.02. Other direct solar measurements have been discussed by Hayes (1985). Direct spectrophotometric observations of the sun have also been
made, Neckel & Labs (1984) work on the visual spectrum being the
best known. These have been well discussed by Colina et al. (1996) who
have derived a combined flux spectrum (solar_reference) for the sun
between 0.12
The colors of the solar analogs have also been used for assessing solar colors. Hardorp (1980) was amongst the first to propose a list of stars whose spectra resembled that of the sun; however, the latest word is that of Cayrel de Strobel (1996) who has carefully compared hydrogen line profiles and derived temperatures, gravities and abundances for a list of possible candidate analogs. In Table A3 we give the mean U-B and B-V of the stars in her Table 6. The other colors were derived from the "solar" R-I = 0.337 (Taylor 1992) using mean color-color relations for G dwarfs from data in Allen & Tinney (1991), Carter (1990) and Bessell & Brett (1988). Finally, we have also synthesized colors and magnitudes for the ATLAS9 model atmospheres of the sun computed with the overshooting option switched both on (SUN-OVER) and off (SUN-NOVER) (Castelli et al. 1997). There is generally good agreement between the observed, solar analog and synthetic colors with the exceptions of U-B and V-I. The agreement in the absolute flux level is excellent although other attempts at direct solar flux measurements show larger scatter (eg. see Hayes 1985). We will adopt V = -26.76 for the sun which corresponds to = 4.81 for a distance modulus of -31.57. ## Appendix D: bolometric corrections and the zeropoint of the bolometric magnitude scaleThe definition of apparent bolometric magnitude is = -2.5 log() +
constant or = -2.5 log( d ) + constant where dais the total flux received from the object, outside the atmosphere. The usual definition of bolometric correction
is the number of mags to be added to the V magnitude to yield the bolometric magnitude. The value of does not change when magnitudes at the stellar surface or absolute magnitudes are considered. In fact they differ from the apparent magnitudes only for the distance, which is eliminated when the difference between the bolometric and V magnitude is taken. Although originally defined for the V magnitude only, the definition has now been generalised to all passbands (hence the V subscript above). Although the definition of bolometric magnitude is a straightforward one, there is some confusion in the literature resulting from the choice of zeropoint. Traditionally it had been generally accepted that the bolometric correction in V should be negative for all stars (but with generalisation of the correction to all passbands this rationale vanishes) and this had resulted in F dwarfs having a BC near zero and consequently the BC for solar-type stars was between -0.07 (Morton & Adams, 1968) and -0.11 mag (Aller, 1963). However, with the publication of his grid of model atmospheres, Kurucz (1979) formalised this tradition and based the zeropoint of his scale on the computed bolometric correction of a ( =7000, log g=1.0) model, which had the smallest BC in his grid, resulting in = -0.194 for his solar model. This zero-point based on model atmospheres was adopted by Schmidt-Kaler (1982) who assigned = -0.19 to the Sun. Problems in the literature have occurred when tables have been used from various empirical and theoretical sources without addressing the different zeropoints involved. As emphasized by Cayrel (1997), the traditional basis of the zeropoint is no longer useful and we should adopt a fixed zeropoint, disconnected formally from other magnitudes, but related to fundamental solar measurements for historical reasons. The solar constant is = 1.371 x
10 Let us By adopting the The absolute bolometric magnitude for any star with luminosity L,
effective temperature , and radius R is then The computed V magnitude, whose expression was given in A1,
transforms to through: , where is 10 parsecs. The bolometric
correction follows: . And finally: . This expression was used to compute the bolometric correction of
our models. By adopting the Table A4 summarizes the solar parameters presented here and those from several well-known reference books for comparison.
## Appendix E: concerning the theoretical realisations of standard system magnitudes and colorsThe theoretical colors and magnitudes presented in this paper were
computed using passband sensitivity functions claimed to represent
those of the standard UBVRIJHKL system (Bessell 1990; Bessell &
Brett 1988). These passbands were essentially reversed engineered,
that is, commencing with a passband based on an author's prescription
of detector and filter bandpass, synthetic colors were computed from
absolute or relative absolute spectrophotometric fluxes for stars with
known standard colors. By slightly modifying the starting passband
(shifting the central wavelength or altering the blue or red cutoff)
and recomputing the synthetic colors it is usually possible to devise
a bandpass that generates magnitudes that differ from the standard
magnitudes ## E.1. Standard systems may no longer represent a real systemWhilst the original system may have been based on a real set of
filters and detectors, the original set of standard stars would almost
certainly have been obtained with lower precision than is now possible
and for stars of a restricted temperature and luminosity range. The
filters may have also been replaced during the establishment of the
system and the later data linearly transformed onto mean relations
shown in the previous data. In addition, the contemporary lists of
very high precision secondary standards that essentially define the
"standard systems" have all been measured using more sensitive
equipment with different wavelength responses. Again, rather than
preserve the natural scale of the contemporary equipment the
measurements have been "transformed" to some mean representation of
the original system by applying one or more linear transformations or
even non-linear transformations. To incorporate bluer or redder stars
than those in the original standard lists, extrapolations have also
been made and these may have been unavoidably skewed by the
imprecision of the original data and the small number of stars with
extreme colors in the original lists. As a result, the contemporary
standard system, ## E.2. Corrections between contemporary natural systems and thestandard UBVRI systemMenzies (1993) details the linear and non-linear transformations that have been used over the past 10 years to correct the SAAO natural system to the standard Cousins UBVRI system. Standard non-linear corrections are first made to the raw magnitude data in V, B-V, U-B, V-R and V-I. The resultant values are then linearly transformed to the standard system. In a further refinement, Kilkenny et al. (1997) detail systematic non-linear corrections that have been necessary to correct the V, B-V and V-I colors of the bluest and reddest stars. Summarising the SAAO results, their current instrumental magnitudes require three sets of corrections, one for the O and B stars, another for the A to early K stars and another for the late K to late M stars. The maximum non-linear corrections are 0.06 to 0.10 mag for the reddest stars. The corrections for the bluest stars amount to less than 0.03 mag. After the non-linear corrections were made, the resultant linear relations were V= v + 0.012(b-v), B = b + 0.027(b-v), U = u -0.022(u-b) for blue stars and U = u - 0.005(u-b) for the red stars. Landolt (1983, 1992) has not detailed as clearly the corrections made from his instrumental system to the standard system but he does give U-B = 0.925 (u-b) for the bluest stars and 1.026(u-b) for the remaining stars. As discussed by Bessell (1990), Landolt's B passband is bluer than the Johnson/Cousins natural B and Graham (1982) needed corrections of up to 0.10 (B-V) to correct his natural colors using similar filters. Small systematic differences between Landolt and Cousins VRI colors for the reddest stars resulted from Cousins use of two linear relations for his V-R and V-I transformations while Landolt used a single relation.
## E.3. Corrections applied to the synthetic photometryThere are too few stars with accurate spectrophotometric fluxes and standard colors to be able to compare empirical synthetic colors with observed standard colors with a high degree of certainty for all spectral types and for all colors. We decided therefore to use synthetic colors computed from the models and compare them with the mean observed color-color relations and the mean color-temperature relations. In this way, even if the model colors are not perfect, we can use them to interpolate between stars of different gravity and different abundance with a high degree of confidence. Preliminary comparisons indicated that the computed model V, B-V, V-R and V-I colors relevant to A to K stars were in reasonable agreement with observations while the U-B comparison was much poorer ## E.3.1. The U-B colorThat the U-B results were not as good as other colors should not have been too unexpected given the uncertainty in the U passband (Bessell 1986). Buser (1978) had devised a U passband by arbitrarily shifting the U3 passband of Azusienis and Straizys (1966); Bessell (1986, 1990) had attempted to reproduce the U response function by combining the transmission of the U filter glass, the response of a 1P21 phototube and included some atmospheric extinction. This passband matched the red cutoff of the Buser response but had a blue cutoff about 100 further to the UV. The Buser realisations of the UBV bandpasses had been used by Kurucz for the CDROM 13 colors. The intrisic UBV colors for dwarf stars should be well established, but because of differences between versions of the standard system and uncertainties in the interstellar reddening for early type stars it is in fact not so clear. In Figs. A1a,b we plot the collected E-region photometry of Menzies et al (1980) and the selected bluest and reddest dwarf stars from Kilkenny et al. (1997): (solid circles), together with some older bright star photometry by Cousins of stars in common with Johnson (crosses). The smooth curve drawn in these figures is an attempt to represent the mean unreddened locus for dwarf stars so that we can use it in comparison with model colors. We note that this locus could perhaps have been drawn slightly bluer in B-V to make a greater allowance for reddening. In Figs. A2a,b we show such diagrams. The theoretical data (crosses) are plotted for log g=4.0 and 4.5 for and for log g=4.5 and 5.0 for .
To achieve the agreement in the range of the U-B colors between O and K stars shown in Figs A2a,b we have multiplied the U-B colors computed using the Bessell (1990) UBV passbands by 0.96. The resulting agreement is excellent except for the mid-A stars where the observed U-B colors appear slightly redder than the computed colors; however, there are several reasons why in this restricted color regime U-B colors in particular are uncertain. It is the temperature range where the hydrogen lines and the Balmer discontinuity are near their maximum strengths and the computed U and B magnitudes are correspondingly very sensitive to the exact positioning of the edges of the U and B passbands. There are also some uncertainties in the handling of the computation of the overlapping hydrogen lines near the Balmer series limit. It is also the color range where small non-linear corrections are often made in U-B transformations and where systematic differences in photometry result from how atmospheric extinction corrections are made, that is, how the discontinuity in the U or U-B extinction coefficient is handled (Cousins 1997). So we should not let a slight disagreement in the U-B colors of the A stars detract from the good agreement at other temperatures.
The U-B versus B-V comparison for colors computed using the Buser (1978) passbands are similar. They fit the B stars' relation well (except perhaps for the bluest stars) but diverge for the late-F to K star models. The differences for the reddest stars can be removed by scaling the U-B colors for those models redder than B-V=0.4 by 0.96. Given the uncertainties in the observed U-B versus B-V relation both sets of scaled colors can be said to fit the data, but overall, the scaled Bessell U-B colors produce slightly better agreement than the scaled Buser U-B colors. The Landolt (1983) version of the U-B system differs systematically from the SAAO (Cousins) U-B system (see eg. Bessell 1995). The differences range from -0.10 mag for the bluest stars (Landolt values are bluer) to +0.05 for the reddest stars. A scale factor of 1.03 removes some of this difference but leaves systematic residuals with B-V (or U-B) color. These residuals exhibit the same shaped variation with color as seen in the differences Bessell - Buser but have higher amplitude. This suggests that the Landolt U-B system has a U band whose blue wing extends much further to the UV than does the SAAO U band. The unscaled U-B colors computed using the Bessell (1990) passbands represent the Landolt U-B versus B-V relation quite well although again the observed U-B colors of the mid-A stars are redder than the models. Although for this paper we have decided to use the scaled U-B colors computed using the Bessell (1990) U passband, it would certainly be worthwhile to experiment more with other U passbands and the theoretical fluxes to try and better fit the observed U-B versus B-V relation. ## E.3.2. Other colorsSlight adjustments to the B-V colors could be considered based on the U-B versus B-V diagram comparisons that indicate that the hottest models may need their B-V colors corrected by -0.01 or -0.02 mag, but given the uncertainties and the insensitivity of the B-V color to temperature we have made no changes. The slope change made by many observers in the B-V transformation for stars redder than B-V=1.5 suggests that we should perhaps also consider increasing the B-V colors of the redder models; we have not yet done so. The theoretical V-R, R-I and V-I colors should probably also be adjusted for the M stars as most natural systems required two slopes for transformation onto the standard system. But we will await on better comparison data for the reddest stars before doing so. ## E.4. Energy integration versus photon counting: observational and computational differencesThere is another subtle reason why band-pass matching, linear transformations and synthetic photometry can be confusing in modern photometry. This is a result of a switch from flux measurements by energy integration across a wavelength band to photon integration across the same band. The resulting colors and magnitudes are not the same. Most standard system photometry was carried using with photomultiplier tubes with current integration. This is equivalent to convolving the spectrum of a star in energy units by the bandpass sensitivity function. That is, the energy measured across a bandpass X is where is the response function of the system. If instead the number of detected photons across the passband X are counted, the number is: This in essence weights the fluxes by the wavelength. The net effect is to shift the apparent effective wavelength of a passband for stars of different temperature more when photons are counted than when the energy is measured. The equi-energy effective wavelength will be different to the equi-photon effective wavelength indicating that a small linear transformation is required to the photon counting colors to match the energy derived colors. The effects are larger for broad bands and usually larger in the UV because the same width bandpass is a larger fraction of the wavelength for smaller wavelengths. CCDs and IR array detectors are all photon counting devices and this effect should be considered in comparisons between synthetic photometry and observations made with these devices. ## Appendix F: effective wavelengths and reddening ratiosIn broad-band photometry the nominal wavelength associated with a passband (the effective wavelength) shifts with the color of the star. For Vega the effective wavelength of the V band is 5448 and for the sun it is 5502 . The effective wavelength of the V band (response function
R The effective wavelengths of the V band for different spectral types are B0: 5430Å , A0: 5450Å , F0: 5475Å , G2: 5502Å , K0: 5515Å , M0: 5597Å , M5: 5580Å . Some bands, such as the R band show much greater shifts (eg. Bessell 1986). Interstellar reddening estimates are also affected by effective wavelength shifts which result in different values of reddening being derived from stars of different color. That is, a given amount of dust obscuration will produce a larger E(B-V) value for OB stars than for GK stars. We have investigated quantitatively the effects of interstellar extinction on colors in the UBVRI system by multiplying model spectral fluxes with the extinction law summarised by Mathis (1990). Comparison between the synthetic photometry of the ATLAS9 models before and after application of an amount of interstellar extinction (corresponding to a nominal E(B-V)=0.30) gave us the color excess and magnitude ratios given in Tables A5 and A6. Note that the color excess ratios increase with color mainly as a result of E(B-V) decreasing with color.
The reddening ratios quoted in the literature are usually based on
empirical reddening measurements using OB stars so would probably
represent a mean (B-V) The reddening independent parameter Q is often used as a temperature index for OB stars. If we derive Q = (U-B) - 0.71(B-V) from our theoretical colors and regress against B-V, we find Q= 3(B-V). Therefore, we predict for OB dwarfs that E(B-V) = (B-V) - (U-B) - 0.71(B-V) /3. © European Southern Observatory (ESO) 1998 Online publication: April 15, 1998 |