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Astron. Astrophys. 333, 231-250 (1998)

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Appendix A: zero-points of the broad-band synthetic photometric system

The 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 ([FORMULA], log g, Z, [FORMULA]) = 9550K, 3.95, [-0.5], 2kms-1 (Castelli & Kurucz 1994) and 9850K, 4.25, [+0.5], 2kms-1 (Kurucz, 1997).

A.1. The zeropoint for V

Using 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:

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA] are the V response function (eg. normalised passbands from Bessell 1990) and f([FORMULA]) and f([FORMULA]) are the computed flux at the earth in erg cm-2 s-1 Å-1 or in erg cm-2 s-1 hz-1 respectively. The above zeropoints realize a V magnitude of 0.03 mag for Vega.

We note that the absolute flux at 5556 [FORMULA] 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: [FORMULA] = 3.24 [FORMULA] 0.07 mas, the model implies 3.26 mas. For Sirius: [FORMULA] = 5.89 [FORMULA] 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 zeropoints

To 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.


[TABLE]

Table A1. Observed and model colors for Sirius and Vega


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 1-5µ magnitudes for 25 dwarfs of spectral-type B8-A3 relative to Vega. They found that (the asterisk refers to the narrow band index) the (V-J*, J*-K*, J*-L*, J*-M*) = (0.03, 0.006, 0.019, 0.023) colors of Vega were normal for an A0 star and if anything, 1-2 percent fainter between K* and M* than the mean A0 star; that is, Vega does not have an IR excess compared to other A stars.

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 µm relative to model spectra by Dreiling and Bell (1980) and by Kurucz (1979), which led Leggett et al (1986) to suggest that the theoretical infrared flux from A-type stars is in error. This would be difficult to understand as the dominating H opacity in A stars is thought to be very well known. It is more likely that there remains a problem with horizontal and vertical extinction corrections in the absolute flux measurements. Eitherways, until this is understood it is recommended that the theoretical fluxes of model atmospheres, such as those for Sirius and Vega be used for absolute flux calibration through the atmosphere (see also Cohen et al. 1992).

Appendix B: zeropoint fluxes for the UBVRIJHKL system

In 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.


[TABLE]

Table A2. Effective wavelengths (for an A0 star), absolute fluxes (corresponding to zero magnitude) and zeropoint magnitudes for the UBVRIJHKL Cousins-Glass-Johnson system


Appendix C: colors of the sun and solar analogs

The 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 [FORMULA] 0.015 for the sun. Hayes (1985) claims a further 0.02 mag correction for horizontal extinction yielding V = -26.76 [FORMULA] 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 0.12µm and 2.4µm. We have computed magnitudes and colors for that spectrum and these are given in Table A3.


[TABLE]

Table A3. Observed and model magnitudes and colors for the Sun and a mean solar analog


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 [FORMULA] = 4.81 for a distance modulus of -31.57.

Appendix D: bolometric corrections and the zeropoint of the bolometric magnitude scale

The definition of apparent bolometric magnitude is

[FORMULA] = -2.5 log([FORMULA]) + constant

or

[FORMULA] = -2.5 log([FORMULA] [FORMULA] d [FORMULA]) + constant

where [FORMULA] dais the total flux received from the object, outside the atmosphere. The usual definition of bolometric correction

[FORMULA]

is the number of mags to be added to the V magnitude to yield the bolometric magnitude. The value of [FORMULA] 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 [FORMULA] scale on the computed bolometric correction of a ([FORMULA] =7000, log g=1.0) model, which had the smallest BC in his grid, resulting in [FORMULA] = -0.194 for his solar model. This zero-point based on model atmospheres was adopted by Schmidt-Kaler (1982) who assigned [FORMULA] = -0.19 to the Sun.

Problems in the literature have occurred when [FORMULA] 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 [FORMULA] = 1.371 x 106 erg cm-2 s-1 or 1371W m-2 s-1 (Duncan et al. 1982), therefore the total radiation from the sun [FORMULA] = 3.855(6) x 1033 erg s-1 and the radiation emittance at the sun's surface [FORMULA] = 6.334 x 1010 erg cm-2 s-1. The effective temperature is [FORMULA] = [FORMULA] = 5781K.

Let us define the absolute bolometric magnitude of the sun [FORMULA] = 4.74.

By adopting the measured apparent V magnitude of the Sun [FORMULA], the absolute [FORMULA] magnitude of the sun is thus 4.81 and the V bolometric correction for the sun is then [FORMULA].

The absolute bolometric magnitude for any star with luminosity L, effective temperature [FORMULA], and radius R is then

[FORMULA]

[FORMULA]

The computed V magnitude, whose expression was given in A1, transforms to [FORMULA] through:

[FORMULA],

where [FORMULA] is 10 parsecs. The bolometric correction follows:

[FORMULA].

And finally:

[FORMULA].

This expression was used to compute the bolometric correction of our models. By adopting the synthetic absolute [FORMULA] = 4.802, the V bolometric correction for the sun is then [FORMULA]. This is very close to the above defined -0.07 and within the error bars of both observations and synthetic spectra in the V band. As models evolve with time and slight future changes in the calculated V magnitude cannot be excluded we chose not to rely on the solar model -0.062 bolometric correction but rather to keep the -0.07 value, with the adopted [FORMULA] and the measured [FORMULA] V magnitude. This returns to the pre-Kurucz 79 value for the sun. Note that the V magnitude we adopted lies midway between the synthetic V magnitudes computed from the empirical solar spectrum and the model solar spectra (see Table A3).

Table A4 summarizes the solar parameters presented here and those from several well-known reference books for comparison.


[TABLE]

Table A4. Comparisonof tabulated solar V magnitude, bolometric correction and flux


Appendix E: concerning the theoretical realisations of standard system magnitudes and colors

The 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 within the errors by only a constant that is independent of the color of the star. It is usually taken for granted that such a unique passband exists and that given a large enough set of precise spectrophotometric data and sufficient passband adjustment trials it can be recovered. However, there are several reasons why this may not be the case, at least not across the complete temperature range.

E.1. Standard systems may no longer represent a real system

Whilst 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, although well defined observationally by lists of stars with precise colors and magnitudes, may not represent any real linear system and is therefore impossible to realize with a unique passband. In fact, we should not be trying to find a unique passband with a central wavelength and shape that can reproduce the colors of a standard system but we should be trying to match the passbands and the linear or non-linear transformations used by the contemporary standard system authors such as Landolt, Cousins and Menzies et al. to transform their natural photometry onto the "standard system". That is why it is important for photometrists to publish the full details of their transformations and other details of their data reductions, such as extinction corrections.

E.2. Corrections between contemporary natural systems and thestandard UBVRI system

Menzies (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.

As the standard systems have been established from natural system colors using linear and non-linear corrections of at least a few percent, we should not be reluctant to consider similar corrections to synthetic photometry to achieve good agreement with the standard system across the whole temperature range of the models.

E.3. Corrections applied to the synthetic photometry

There 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 color

That 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 [FORMULA] and for log g=4.5 and 5.0 for [FORMULA].

[FIGURE] Fig. A1a and b. The observed U-B versus B-V diagram for the SAAO version of the UBV system; see text for references. The line represents a mean unreddened locus for dwarf stars.

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.

[FIGURE] Fig. A2a and b. The theoretical U-B versus B-V digram is plotted for log g=4.0 and 4.5 for [FORMULA] and for log g=4.5 and 5.0 for [FORMULA] (crosses). It is compared with the observed unreddened dwarf locus.

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 colors

Slight 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 differences

There 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 [FORMULA] spectrum of a star in energy units by the bandpass sensitivity function. That is, the energy measured across a bandpass X is

[EQUATION]

where [FORMULA] is the response function of the system.

If instead the number of detected photons across the passband X are counted, the number is:

[EQUATION]

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 ratios

In 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 [FORMULA] and for the sun it is 5502 [FORMULA].

The effective wavelength of the V band (response function RV ([FORMULA]) for an object with flux f([FORMULA]) is

[EQUATION]

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.


[TABLE]

Table A5. Color excess ratios to E(B-V) for E(B-V) = 0.29 - 0.024(B-V)



[TABLE]

Table A6. Absorption ratios to A(V) for A(V) = 3.26 + 0.22(B-V)


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)0 of about -0.15 or an effective temperature around 15000K. The relevant theoretical ratios are shown in line 1 of Table A5. The values given in line 3 are color corrections to be added to the value in line 2 to account for changing color excess ratios with color. These derived relations show that when correcting for interstellar extinction a mean E(B-V) excess should not be applied to all stars but that the color excess should be scaled according to the intrinsic color of the star. The ratios to A(V) (Table A6) are less sensitive to the color. The good agreement between the calculated and observed ratios indicates that if the Mathis table is a good representation of the reddening then the passbands adopted for the UBVRIJHKL system are also a reasonable representation of the real passbands. The extinction law of Ardeberg & Virdefors (1982) although in agreement with the Mathis curve for BVRI produces a higher extinction in the U band that results in a calculated E(U-B)/E(B-V) ratio of 0.82 rather than 0.71 .

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) - [FORMULA] (U-B) - 0.71(B-V) [FORMULA] /3.

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Online publication: April 15, 1998
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