Spectrophotometric Libraries, Revised Photonic Passbands, and Zero Points for UBVRI, Hipparcos, and Tycho Photometry

The Hipparcos Catalogue (Perryman et al. 1997) is a high-precision photometric (plus parallax and proper motion) catalog of more than 100,000 stars measured with the H_p band; the Tycho2 Catalogue (Hog et al. 2000) contains 2.5 million stars measured (mostly) with lower precision in the B_T and V_T bands. The remarkable collection of data was obtained during the 4 yr (1989–1993) mission of the Hipparcos satellite. The Hipparcos and Tycho photometric systems and their measured median precisions were discussed by van Leeuwen et al. (1997b), and the passbands were given in van Leeuwen et al. (1997a). However, the detectors suffered degradation throughout the mission as a result of being launched into an incorrect orbit, and this degradation invalidated the measured prelaunch passbands. Bessell (2000) devised self-consistent Hipparcos and Tycho passbands by comparing regressions of V - H_p, V - B_T, and V - V_T versus V - I for a sample of precise E-region UBVRI standard stars with synthetic photometry computed from the R ∼ 100 Vilnius-averaged spectra (Straizys & Sviderskiene 1972). This indicated the necessity of a significant redward shift of the blue edge of the published H_p band, but only small changes for the Tycho bands. However, the passbands may not have been definitive because of the small number of averaged Vilnius spectra used and their low resolution.

In the last few years, two libraries of accurate higher-resolution (R ∼ 1000–2000) spectrophotometric data have become available—the Next Generation Spectral Library¹ (NGSL; Heap & Lindler 2007) and the Medium Resolution INT Library of Empirical Spectra² (MILES; Sanchez-Blazquez et al. 2006. Many of the stars in these spectral libraries also have Hipparcos and Tycho magnitudes—providing the opportunity to reexamine the passbands of the Hipparcos and Tycho systems. Furthermore, the high precision of the Hipparcos magnitudes (Perryman et al. 1997) enables them to be used to adjust the flux levels of the data in the NGSL libraries and to make the stars extremely valuable for whole-sky spectrophotometric calibration of imaging surveys such as SkyMapper (Keller et al. 2007).

Being space-based, a unique property of the Hipparcos photometric systems is the absence of any seasonal or hemisphere-related effects seen in some ground-based photometric systems due to variations in temperature, atmospheric extinction, and instrumental orientation. The Hipparcos photometry database can therefore be compared with databases of ground-based photometric systems to examine their magnitude zero points and to look for any systematic offsets in the photometry, as discussed by van Leeuwen et al. (1997a) and Pel & Lub (2007).

In this article, we will outline the derivation of improved UBVRI, Hipparcos, and Tycho passbands by using synthetic photometry from spectrophotometric atlases and comparing it with broadband photometry. We will also adjust the absolute levels of the spectrophotometric fluxes by comparing the synthesized H_p magnitudes with the Hipparcos Catalogue magnitudes. In addition, we will use the mean differences between the synthetic and the observed photometry to determine zero-point corrections for the UBVRI B_T and V_T bands. We will also intercompare the zero points of the SAAO UBVRI, the homogenized UBV, and the Walraven VB systems. Finally, in the Appendix we discuss confusion and inexactness concerning the derivation of mean fluxes, response functions, and the plethora of expressions for mean wavelengths and frequencies associated with broadband photometry.

The synthetic photometry in this article was computed using two photometry packages.³ For each passband x, we evaluated the quantity

where

f_ν(ν) is the observed absolute flux in erg cm^-2 s^-1 Hz^-1, f_λ(λ) is the observed absolute flux in erg cm^-2 s^-1 Å^-1, S_x(λ) are the photonic passbands (response functions), λ is the wavelength in angstroms, and ZP_x are the zero-point magnitudes for each band (see §§ 5.4 and 7 and the Appendix). For SI units the constants in the preceding equations would be different, because erg cm^-2 s^-1 is equivalent to 10^-3 W m^-2.

For accurate synthetic photometry it is important that the passbands provided to the integration routines are well sampled and smooth. Because passbands are usually published at coarse wavelength intervals (25–100 Å), it is necessary to interpolate these passband tables to a finer spacing of a few angstroms using a univariate spline or a parabolic interpolation routine. The physical passbands themselves are smooth, and the recommended interpolation recovers this. Our two packages produced identical results after this step.

There is a fundamental concern associated with the theoretical realization of the older evolved standard photometric systems in order to produce synthetic photometry from theoretical and observational fluxes. The technique used is to reverse-engineer the standard system's passband sensitivity functions by comparing synthetic photometry with observations (e.g., Straižys 1996; UBVRIJHKL: Bessell 1990a; Bessell & Brett 1988). That is, commencing with a passband based on an author's prescription of detector and filter bandpass, synthetic magnitudes are computed from absolute or relative-absolute spectrophotometric fluxes for stars with known standard colors. By slightly modifying the initial 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.

Although the original system may have been based on a real set of filters and detector, 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 also have been replaced during the establishment of the system, and the later data may have been linearly transformed onto mean relations shown by 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 nonlinear transformations (e.g., Menzies 1993). To incorporate bluer or redder stars than those in the original standard lists (e.g., Kilkenny et al. 1998), 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 system and is therefore impossible to realize with a unique passband that can reproduce the standard magnitudes and colors through a linear transformation with a slope of 1.0.

In fact, perhaps 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 rather be trying to match the passbands and the linear (but nonunity slope) or nonlinear transformations used by the contemporary standard system authors to transform their natural photometry onto the standard system. The revised realization of the Geneva photometric system by Nicolet (1996) uses this philosophy, as does Bessell (2011) in the realization of the uvby system.

However, in this article we have set out in the traditional way, as outlined previously, to adjust the passbands to achieve agreement between the synthetic photometry and the standard system photometry within the errors of the standard system. It may seem desirable to do these passband adjustments in a less ad hoc way, but given the uncertainties underlying existing standard system photometry, a more accurate method is unnecessary, at present.

Because of how spectroscopic observations are normally made, spectrophotometric fluxes are calibrated mainly to determine the accurate relative-absolute fluxes (the flux variation with wavelength), but not the absolute flux (the apparent magnitude). Depending on the slit width and the seeing or other instrumental effects, the resultant absolute flux levels may be measured only to a precision of 0.1–0.2 mag. To assign an accurate absolute flux level, one is normally required to compute a synthetic magnitude from the spectrophotometric fluxes and to equate this to a standard magnitude for the object, often an existing magnitude, such as B or V. Currently, the most precise magnitudes available for the largest number of stars are the Hipparcos H_p magnitudes. There are 72,300 stars in the Hipparcos Catalogue, with median H_p magnitudes given to better than 0.002 mag. We have therefore synthesized the H_p magnitudes for all the stars in the spectrophotometric libraries and adjusted the absolute flux scale of each star (that is in the Hipparcos Catalogue) to match the catalog H_p value, thus producing spectrophotometric data with an uncertainty in the overall absolute flux level of a few millimagnitudes. In Figure 1 we show histograms of the differences for the NGSL stars between the observed and synthetic V magnitudes before and after renormalization to the H_p magnitudes. We could not show this comparison for the MILES spectra, as they published only relative-absolute fluxes (normalized to 1 at 4600 Å), but after renormalization to the H_p magnitudes, the ΔV distributions for both NGSL and MILES spectral libraries are approximately Gaussian, with a similar rms of 0.017 mag.

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The wavelength range of the NGSL spectra encompasses the wavelength range of the Hipparcos and Tycho passbands, but because the MILES spectra do not cover the complete extent of the red tails of the H_p and the R band, or any of the I band, we have extrapolated the MILES spectra from 7000 Å to 9900 Å using model atmosphere fits to the 3500 Å–7000 Å region by Kerzendorf (2011, private communication). The grids used were from ATLAS (Munari et al. 2005) for T_eff > 8000 K and MARCS (Gustafsson et al. 2008) for 8000 K > T_eff > 2500 K. The MILES spectra were also extrapolated from 3540 Å to 3000 Å to cover the U band. These extrapolations may result in a slight uncertainty in the synthetic photometry in some passbands from the MILES spectra; however, we think that it is small, as shown by the insignificant differences between the relations using the NGSL and MILES spectra, probably except for the M stars.

The 373 adjusted NGSL spectrophotometric fluxes, covering the wavelength range 1800 Å to 10,100 Å and with a precise absolute flux level, are ideally suited to calibrate whole-sky surveys, such as SkyMapper (Keller et al. 2007) and Pan-STARRS (Kaiser et al. 2010). These revised absolute fluxes are available from the authors, together with the absolute fluxes for the 836 MILES spectra that have Hipparcos photometry.

The Johnson-Cousins UBVRI system passbands have been well discussed (e.g., Azusienis & Straizys 2009; Buser & Kurucz 1978; Bessell 1990a), most recently, by Maiz Appellaniz (2006), who reconsidered the UBV passbands. Although accepting the Bessell (1990a) BV passbands, Maiz Appelaniz (2006) suggested an unusual and unphysical U passband as providing a better fit to standard U photometry. However, these previous analyses did not have available the large number of revised spectra in the NGSL and the MILES catalogs. It is very worthwhile, therefore, to reexamine the UBVRI passbands using synthetic UBVRI photometry derived from these extensive data sets. Standard system UBV data for most of the MILES and NGSL stars are available in the homogenized UBV catalog.⁴ VI data are available for many stars in the Hipparcos Catalogue, while various data sets of Cousins, Menzies, Landolt, Bessell, Kilkenny, and Koen also provided much supplementary VRI data (Cousins 1974, 1976, 1984; Cousins & Menzies 1993; Landolt 1983, 2009; Bessell 1990b; Kilkenny et al. 1998; Koen et al. 2002, 2010).

5.1. The B and V Passbands

After comparing the observed and synthetic photometry, very small slopes were evident in the ΔB and ΔV regressions against B - V using the Bessell (1990a) passbands with the NGSL and MILES samples. These slopes were removed by making a small redward shift to the red side of the V90 band and a very small overall redward shift in the B90 band. The regressions for the MILES sample were similar but not identical. In Figures 2 and 3 for the NGSL spectra we show the differences between the observed and synthetic V and B - V, respectively, for our adopted passbands. There are no significant color terms evident, but the synthetic Vand B - V magnitude scales have small apparent offsets associated with the initial adopted zero points (hereinafter ZPs) of the synthetic photometry. These will be addressed further in § 7.

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5.2. The U Bandpass Revisited

Standard U photometry has a bad reputation due to the much larger systematic differences in U - B between observers than is evident for V and B - V. These systematic differences arise because in stars, U measures the flux across the region of the Balmer jump, and its response is therefore much more sensitive to the exact placement of the band compared with the placement of B and V. Many observers take insufficient care to match the position and width of the standard Cousins or Johnson U passband, and attempts to standardize the resulting U - B color using a single B - V or U - B color-correction term have introduced systematic errors, especially for reddened stars. Cousins (1984) (reprised in Bessell 1990a) outlined such systematic differences evident in different versions of the U - B system.

Bessell (1986, 1990a) discussed in detail the likely response function of the U band from first principles and proposed the UX90 band as representing the original band. Bessell et al. (1998) note that the U - B based on this band should be scaled by 0.96. Although scaling of this order is common in transforming observational systems (e.g., Menzies 1993; Landolt 1983), there is a notable reluctance to use such terms in computing synthetic photometry. In spite of the evidence that most standard systems have evolved with nonlinear or bilinear correction terms (Bessell et al. 1998, Appendix E1), most astronomers believe that a passband can be found that reproduces the standard system without the need for linear and/or nonlinear correction terms. In the spirit of that quixotic endeavor, Buser & Kurucz (1978) and Maiz Appellaniz (2006) proposed U passbands that have almost identical red cutoffs to the UX90 band, but different UV cutoffs, thus shifting the effective wavelength of U slightly redward. We have also produced a slightly different U band by moving the UV cutoff of the UX90 band slightly redward. This produces an acceptable compromise for the U band that fits the observations reasonably well, although a nonlinear fit would be better.

In Figure 4, we show regressions against U - B of the differences between the U photometry synthesized with the passband from this article and those of Bessell (1990a): UX90, Buser & Kurucz (1978): U3, and Maiz Appellaniz (2006). The main difference between Bessell (1990a) and Buser & Kurucz (1978) is a small difference in slope, whereas the Maiz Appellaniz (2006) passband mainly produces an offset of about 0.05 mag for G, K, and M stars compared with the A and B stars.

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In Figure 5, we show the differences between the observed values of U - B and the synthetic U - B computed for the NGSL and MILES sample of stars. Although the scatter is quite high, the Maiz Appellaniz (2006) U passband clearly does less well and results in a systematic deviation from the standard system for the cooler stars (as anticipated in Fig. 4).

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The SAAO UBVRI photometry (Cousins 1974, 1976; Kilkenny et al. 1998; Koen et al. 2002, 2010) represents some of the best standard UBVRI photometry, and we use the U - B versus B - V relation from these data (Fig. 6, left panel) as the benchmark for comparison with the synthetic photometry. The Kilkenny et al. (1998) photometry (blue dots) extended the standard system to much bluer and redder dwarf stars than are represented in the E-region stars (black dots). The red line is a fitted mean line through the O-B-A-F-G dwarf main sequence and the K and M giants. In the right panel of Figure 6, the same line is drawn for comparison on the synthetic U - B versus B - V diagram computed for the NGSL sample of stars using our adopted UBV passbands. Considering that there are many metal-deficient F, G, and K stars in the NGSL sample that are not in the empirical sample, the synthetic diagram is in good agreement with the empirical diagram. Note also that most of the reddest stars in the NGSL sample are K and M giants, and there are only a few K and M dwarfs.

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5.3. The R and I Passbands

It was not as straightforward to check the R and I bands because of the lack of precise RI photometry for many of the NGSL and MILES stars. The V - I colors given in the Hipparcos Catalogue are of uncertain heritage, as are similar data from SIMBAD. An homogenized VRI catalog would have been very useful. Our observational data comprised the Hipparcos V - I color supplemented with VRI data mostly from Bessell (1990b) and Koen et al. (2010) for the K and M dwarfs. Although the scatter was high for the Hipparcos V - I comparison, it did indicate that a small shift in the Bessell (1990a) I90 band was needed. We eventually shifted the red edge of the R90 band a little redward and the whole I90 band a little blueward. In addition to the synthetic photometry from the NGSL and MILES samples, we also had available a small sample of unpublished single observations of mostly late-type M dwarfs taken with the double-beam spectrograph (DBS) at Siding Spring Observatory. As shown in Figure 7, the resultant synthetic V - R versus V - I relations were in excellent agreement with the empirical loci defined by the precise values in Menzies et al. (1989), Menzies (1990, private communication), Landolt (2009), Bessell (1990b), and Koen et al. (2010). The four empirical data sets are essentially coincident. The sparse redder loci in the V - R versus V - I diagram beyond V - I ∼ 1.8 are Landolt K and M giants. Note that this and later figures are best viewed magnified in the electronic version.

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To better appreciate the comparison, we fitted a ninth-order polynomial to the empirical V - R versus V - I locus and plotted the V - R residuals of the fit against V - I. Applying the same polynomial, we also computed V - R residuals for the synthetic photometry. In Figure 8 we overlay the synthetic residuals, which are seen to agree very well with the trends in the empirical residuals. The few hundredths-of-a-magnitude systematic differences between the MILES VRI colors for the M dwarfs compared with the empirical stars is undoubtedly due to the extrapolation of the MILES spectra from 7000 Å to 9900 Å using model spectra. However, for non-M stars, the relation defined by the extrapolated MILES spectra is indistinguishable from the others, indicating an impressive fidelity of the ATLAS (Munari et al. 2005) and MARCS (Gustafsson et al. 2008) spectra.

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5.4. Photometric Passbands: Photon-counting and Energy-integrating Response Functions

There continues to be some confusion in the definition of photometric response functions and their use in computing synthetic photometry. As discussed in Koornneef et al. (1986), Bessell et al. (Appendix 4, 1998), and Maiz Appellaniz (Appendix A, 2006), in the era before CCDs, photometry was largely done with energy-measuring detectors. The normalized response functions, , that were generally published described the relative fraction of energy detected at different wavelengths across a particular passband. Nowadays, detectors are almost all photon-integrating devices, such as CCDs, and the response functions used, S_x(λ), relate to the relative number of photons detected (or the probability of a photon being detected) at different wavelengths across the passband. These issues are outlined and explored in the Appendix, where it is also shown why the magnitudes derived from photon-counting or energy-integration observations are identical (as expected). In Table 1 we list our adopted normalized photon-counting passbands S_x(λ) for U, B, V, R, and I. In Figure 9 we show the normalized photon-counting passbands for U, B, V, R, and I compared with the Bessell (1990a) passbands converted to photon-counting. The Maiz Appellaniz (2006) U band and the converted Buser & Kurucz (1978) photon-counting U3 band are also shown.

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There are two ground-based photometric systems notable for their precision and stability. These are the Walraven VBLUW photometry of Pel & Lub (2007) and the UBVRI SAAO and Landolt (2009) photometry discussed previously. Pel (1990, private communication) also provided precise transformations between the Johnson-Cousins V and B - V and Walraven V and V - B.

We regressed V - H_p, B - B_T, and V - V_T versus B - V for these two data sets and compared them with the synthetic photometry from the NGSL and MILES. As done for UBV, the H_p, B_T, and V_T passbands were adjusted until the slopes and shapes of the regressions with the synthetic photometry matched as closely as possible those of the observed regressions. In order to remove the small color term evident in the initial regressions, the red side of the Bessell (2000) V_T passband was shifted slightly redward, while a smaller blueward shift was made to the blue side of the Bessell (2000) B_T band. Figure 10 shows the final regressions for B - B_T and V - V_T from the NGSL spectra. Figure 11 shows the adopted B_T and V_T passbands in comparison with the original passbands (van Leeuwen et al. 1997a) and the Bessell (2000) passbands. There is little obvious difference between the three passbands. The adopted photon-counting response functions for B_T and V_T are listed in Table 2.

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There have been suggestions (e.g., Grenon 2001, private communication), that the change in the H_p sensitivity function caused by the in-orbit radiation damage was unlikely to be a complete loss of the bluest sensitivity as suggested by Bessell (2000), but rather a more complicated drop in sensitivity across a wider wavelength range. We have attempted to use the two spectrophotometric samples to examine this proposition and, although the results are not unequivocal, a slightly better fit is achieved by making small modifications to the Bessell (2000) passband.

The synthetic V - H_p versus U - B, V - I, and B - V regressions are shown in Figures 12, 13, and 14, respectively, in comparison with empirical relations for these stars. These plots show the range of stars represented in the NGSL spectrophotometric catalog (few K and M dwarfs, but many FG subdwarfs) and the different distribution of stars in the comparison standard photometric SAAO catalogs. We have fitted a cubic polynomial to the V - H_p versus B - V regression for the E-region stars (Menzies et al. 1989; Menzies 1990, private communication) bluer than B - V = 1.1. The same polynomial in B - V was applied to the catalog stars of Pel & Lub (2007) and Pel (1990, private communication) and to the synthetic photometry of the NGSL and MILES stars.

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In Figure 15 we plot the residuals of the fit. It is clear that the synthetic photometry using the adopted H_p band is a very good match to the standard photometry, with the caveat that the ZPs of the synthetic V - H_p magnitudes were adjusted to achieve this. This will be discussed in the next section. Table 2 also lists the new Hipparcos passband, and Figure 16 shows the new and old photon-counting passbands.

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All standard photometric systems adopt some ZP for their magnitude scale. Historically, the ZP of V in the UBV system is generally used for other systems.

The AB-magnitude system (Oke & Gunn 1983; also see the Appendix) was defined as a monochromatic-magnitude system for spectrophotometry, where AB_ν = -2.5 log f_ν + 48.60, and f_ν is the flux in erg cm^-2 s^-1 Hz^-1. This has now been generalized for use with broadband photometric bands. In the AB system, a flat-spectrum star (in f_ν) has the same AB magnitude in all passbands.

The AB_λ- or ST-magnitude system (see the Appendix) was defined in terms of f_λ, where ST = -2.5 log f_λ + 21.10

The so-called VEGAMAG system (like the UBV system) is one where Vega (α Lyrae) has colors (magnitude differences), such as U - B and B - V, that are identically zero. This is equivalent to dividing all the observed fluxes by the flux of Vega but adjusting the V ZP to give the adopted V magnitude for Vega. For the Vega spectrum we used the CALSPEC⁵ spectrum alpha_lyr_stis_005, which is distributed in the synphot and pysynphot software packages (see the Appendix).

7.1. Observed Zero Points

7.2. Synthetic Photometry Zero Points

We carried out synthetic photometry on the Vega STIS005 (alpha_lyr_stis_005) spectrum and assigned ZPs to force H_p = V_T = B_T = U = B = 0.03 (see the Appendix). These Vega-based f_ν and f_λ ZPs are listed in Table 3. All ZPs in this article are to be subtracted from the AB magnitudes (eq. [1]).

With these ZPs we computed synthetic photometry for all NGSL and MILES stars that had Hipparcos photometry. The ZPs from the synthetic photometry will check (1) whether there are systematic differences between the mean MILES and NGSL flux calibrations and (2) whether the STIS005 spectrum correctly represents the empirical ZPs of the UBVRI and H_p, V_T, and B_T systems. We compared the synthetic magnitudes and/or colors with the observed magnitudes and colors and derived the mean differences. The few stars with exceptionally large differences were not used in the means. There were about 700 stars in the MILES sample and 300 stars in the NGSL sample. We also computed synthetic photometry for 46 of the CALSPEC spectra,⁹ 27 of which have UBVRI photometry from Landolt & Uomoto (2007) and Landolt (2009), 16 had H_p photometry, and 10 had Tycho photometry.

In Table 4 we list the mean differences. The standard errors of the means for the NGSL and MILES samples are less than 0.001 mag. There was good agreement between the NGSL and MILES VRI, V_T, and H_p results; however, the differences for B and B_T appear to be small but real. We chose to adopt the NGSL values for B and B_T in preference to the MILES values, as the NGSL data were taken outside the atmosphere and are unaffected by atmospheric extinction. For the far fewer CALSPEC spectra, the errors in the mean were between 0.004 (H_p) and 0.04 (B_T and V_T). Given the small number of CALSPEC stars with photometry, the mean differences in the colors of the much fainter CALSPEC spectra were in reasonable agreement with those for the NGSL sample, except for an unexplained difference of a few hundredths of a magnitude between the V and H_p magnitudes.

In Table 5 we list the additional ZP-magnitude offsets that will place synthetic photometry computed with the AB-magnitude ZPs from Table 3 on the same scale as the homogeneous UBV system, the Cousins-Landolt UBVRI system, and the Hipparcos and Tycho systems. We also list two wavelengths associated with each passband, which are defined independently of the flux: the pivot wavelength λ_p and the mean photon wavelength λ₀ (see the Appendix for details) and the FWHM of the passband; the wavelengths are given in angstroms. Note that these are the wavelengths that should be associated with published UBVRI photometry, not the natural passbands used by various observers, as their photometry has been transformed onto the standard system. The uncertainties in these additional zero points should only be a few millimagnitudes, except for R and I, where it is more uncertain, as the available V - I photometry was of lower precision. Using these total ZP corrections we recomputed the magnitudes for the Vega stis005 spectrum and obtained V = 0.027, U - B = 0.018, B - V = -0.004, V - R = 0.000, and V - I = -0.001. In addition, the 1994 ATLAS Vega spectrum of Castelli¹⁰ gives U - B = -0.017, B - V = -0.017, V - R = -0.004, and V - I = -0.009. For comparison, Bessell (1983) measured for Vega, B - V = -0.01, V - R = -0.009, and V - I = -0.005.

Excellent spectrophotometric catalogs are now available from NGSL¹¹ (Heap & Lindler 2007) and MILES¹² (Sanchez-Blazquez et al. 2006). In addition to their intrinsic worth, such stars are very useful to use to calibrate all-sky surveys, such as SkyMapper (Keller et al. 2007). However, the published absolute flux levels are imprecise or nonexistent, so we have renormalized the spectra to their precise H_p magnitudes. In order to do this it was necessary to determine the best H_p passband. We also decided to reexamine the passbands representing the UBVRI and Tycho B_T and V_T standard photometric systems using the renormalized NGSL and MILES spectra. We used the CALSPEC stis005 spectrum of Vega to derive the nominal ZP corrections to the AB-magnitude fluxes and synthesized the various magnitudes and slightly adjusted the photonic passbands, achieving better agreement between the synthetic and standard magnitudes than was possible with the previous passbands. In Tables 3 and 4 we present the adopted photonic passbands for UBVRI and H_p, B_T, and V_T, respectively. Table 5 lists the ZP-magnitude corrections based on the stis005 Vega spectrum and V = 0.03.

We intercompared the ZPs of the V-magnitude scale of the SAAO Cousins-Landolt UBVRI system, the homogenized UBV system,¹³, the Walraven Pel & Lub (2007) system, and the Hipparcos H_p and Tycho B_T and V_T systems. We found small differences of less than 0.01 mag between them. The H_p-magnitude ZP differs by 0.008 mag from the ZP of the UBVRI system.

We analyzed the mean magnitude and color differences between the synthetic photometry and the standard photometry and proposed small additional ZP corrections to place the synthetic photometry computed using the AB-magnitude ZPs in Table 5 onto the same ZPs as the standard system photometry. These additional ZP corrections are given in Table 3, together with the passband parameters that should be used to characterize the standard systems.

There was good agreement between the mean differences from the NGSL and MILES catalogs, although the mean level of the MILES blue fluxes deviated slightly, but systematically, from those of the NGSL spectra. The synthetic colors of the fainter CALSPEC spectra also support the proposed additional ZP corrections, except for an unexplained difference in the relative V and H_p magnitudes.

Finally, in the Appendix, we present an extensive discussion on the confusion in the literature concerning measured magnitudes, fluxes, and response functions when broadband photometry is involved, and we provide equations that clearly set out the derivation of photometric quantities. We also cross-reference parameters and definitions used in the Hubble Space Telescope photometric packages synphot and its successor pysynphot (§ A2.3).

We wish to thank Sally Heap for correspondence concerning the rereduction of the Next Generation Spectral Library (NGSL) spectra, Jan Willem Pel and Jan Lub for a digital version of their Walraven photometry catalog and for helpful discussions, and Wolfgang Kerzendorf for fitting and extrapolating the Medium Resolution INT Library of Empirical Spectra (MILES) spectra to cover the 3000 Å–1,200 Å wavelength region. We thank the referee for suggestions to make the article more accessible to physicists. VizieR, Simbad, TOPCAT, and Kaleidagraph were used in preparing this article.

• Table 1
• Table 2

Unfortunately, there is some confusion in the literature concerning measured magnitudes, fluxes, and response functions when broadband photometry is discussed. The definitions concerning monochromatic fluxes are clear—but see Soffer & Lynch (1999) concerning the paradoxes, errors, and confusions that arise when density distributions are involved—but the clarity is lost when these definitions are generalized to involve mean magnitudes, mean fluxes, and the choice of the "effective" wavelength or frequency most appropriately associated with them.

A1. Photometric Quantities and Definitions

In astronomy, flux (f) refers to the radiative flux density, a quantity in physics referred to as the spectral irradiance. In astronomy, flux is also referred to as the monochromatic flux f_ν or f_λ, to distinguish it from the total flux F, which is summed over all wavelengths or frequencies. In SI units, f_λ is measured in W m^-3 or, more practically, in W m^-2 Å^-1, W m^-2 nm^-1, or W m^-2 μm^-1, depending on the part of the spectrum being considered. In cgs units it is measured in erg cm^-2 s^-1 Å^-1, erg cm^-2 s^-1 nm^-1 or erg cm^-2 s^-1 μm^-1. (10³ erg cm^-2 s^-1 = 1 W m^-2). In radio astronomy, fluxes are usually expressed in terms of a non-SI unit, the flux unit or jansky (Jy), which is equivalent to 10^-26 W m^-2 Hz^-1 or 10^-23 erg cm^-2 s^-1 Hz^-1.

A good starting point for the relevant formulae and definitions used in photometry is Rufener & Nicolet (1988), Koornneef et al. (1986), and Tokunaga & Vacca (2005). The stellar flux is normally given in terms of f_ν or f_λ, and the units, respectively, are erg cm^-2 s^-1 Hz^-1 and erg cm^-2 s^-1 Å^-1 or, in the SI system of units, W m^-2 Hz^-1 and W m^-2 nm^-1; although, rather than energy, the photon flux n_p in photon m^-2 s^-1 Hz^-1 or photon m^-2 s^-1 Å^-1 is also used. The relations between these quantities are precisely defined for monochromatic light: namely,

and

The AB (absolute) magnitude scale was introduced by Oke (1965), who proposed the f_ν definition, having noted that a plot of f_ν versus 1/λ for hot stars, was approximately linear in the optical part of the spectrum. The monochromatic-magnitude AB was later defined by Oke & Gunn (1983) using the flux measurement adopted by Oke & Schild (1970) for Vega at 5480 Å and an apparent magnitude of V = +0.035. The Vega flux was considered measured to an accuracy of about 2%. Oke & Schild (1970) measured the flux of Vega at a set of discrete 50 Å bands. A mean value of f_ν = 3.46 × 10^-20 erg cm^-2 s^-1 Hz^-1 or 3.36 × 10^-9 erg cm^-2 s^-1 Å^-1 or 940 photon cm^-2 s^-1 Å^-1 was measured at 5556 Å. They then interpolated Vega's flux to the value of 3.65 × 10^-20 erg cm^-2 s^-1 Hz^-1 at 5480 Å, assumed to be the "effective" wavelength of the V band, and using this value together with V = +0.035, they derived the constant -48.60 associated with definition for the AB magnitude: namely,

It is somewhat unfortunate that Oke (1965) chose to define the AB magnitude in terms of f_ν rather than f_λ, which is more appropriate for most stars—but the conversions, at least for monochromatic light, are straightforward:

AB_λ is called STMAG in synphot and pysynphot. Note that these ZPs are based on the nominal wavelength of 5480 Å for the V band.

More recent measurements of Vega's flux are about 2% brighter, and retaining the preceding values of the ZPs in the definition of AB magnitude and ST magnitude will mean that these scales will necessarily have different ZPs from the V system. And if a different nominal wavelength for the V band is adopted, this will introduce an additional systematic difference between the f_ν and f_λ ZPs.

A1.1. The Flux and V Magnitude of Vega

Summaries of the direct measurements of the optical flux of Vega are given by Hayes (1985) and Megessier (1995), who proposed f_λ = 3.44 ± 0.05 × 10^-9 and 3.46 ± 0.01 × 10^-9 erg cm^-2 s^-1 Å^-1, respectively, for Vega at 5556 Å. Cohen et al. (1992) adopted the Hayes value, together with the flux spectrum of a Vega ATLAS 9 model for their spectral irradiance calibration. More recently, Bohlin & Gilliland (2004) measured the flux for Vega using STIS spectra, and Bohlin (2007) refined these observations and discussed model fits, including rapidly rotating pole-on models. Bohlin (2007) quoted an absolute flux at 5556 Å, the same as Megessier (1995), and V = 0.023 and adopted for Vega a combination of various source fluxes to produce the CALSPEC spectrum alpha_lyr_stis_005 that is now generally used by pysynphot and other routines.

Many direct V measurements of Vega have been made over the years. An obvious problem has been its extreme brightness, making it difficult to measure with sensitive photomultipliers on 1 m telescopes; however, Bessell (1983) measured V = 0.03 in comparison with Cousins bright equatorial stars using an Inconel-coated 1% neutral density filter and a GaAs photomultiplier tube at Kitt Peak. This value is in exact agreement with Johnson et al. (1966). Hayes (1985) discussed measurements of the V magnitude of Vega and discounted reports of its variability. More recently, Gray (2007) also discussed observations of Vega. Mermilliod¹⁴ gave V = 0.033 ± 0.012 for Vega.

We have computed V = 0.007 from the CALSPEC spectrum of Vega using our passband and the ZP of -48.60 in equation (A3). This implies that ZPs of -48.58 and -21.08, respectively, would put the AB_ν and AB_λ-magnitude scale on the same ZP as the V-magnitude system, but see § 7.2.

A2. Issues Arising from a Broad Passband

Photometric observations are normally made by summing the flux over discrete wavelength intervals defined by a window (filter) function. A generalized filter function is a dimensionless (unitless) quantity R representing the fraction of the flux f at each wavelength, which is incident on the detector. It is the product of the atmospheric transmission, the mirror reflectivity, the optics transmission, and the glass filter transmission. It is usually used in the form of a normalized function. The mean f_λ flux would be expressed by the following equation:

and a similar equation for 〈f_ν〉 with all λ replaced by ν. All integrals are nominally from zero to infinity, but are sensibly evaluated over the defined range of the filter passband.

One source of confusion is the fact that the flux is evaluated after the detector, not before it. This means that the filter function must be multiplied by the response function of the detector to give the system response function, as the detector converts the incident light into electrons, which are then amplified and measured. In the case of a photon-counting detector, such as a CCD, the function R is multiplied by the quantum efficiency η(λ) of the CCD to give the system photon response function S. In the case of detectors with photocathodes using DC techniques and current integration, the function R is multiplied by the photocathode radiant response σ(λ) (in units of mA/W) to give the system energy response function S^'. Incorrect equations for photon-counting and energy integration (e.g., Buser 1986, eqs. [1] and [2]) result from overlooking this difference. The system response functions are then generally renormalized.

The relations between S(λ), S^'(λ), η(λ), and σ(λ) are

where λ is in angstroms, η is in percent, and σ is in mA W^-1, or in frequency units,

Ignoring the atomic constants for the moment (and noting that the response functions are usually normalized), we can write

and

So for an energy-integrating detector we can express the measured mean energy flux as

and for 〈f_ν〉,

For photon-counting detectors, one electron is collected for every detected photon, so the mean photon flux is given by

where

Equation (A13) is very important because it shows that in broadband photometry, the mean photon flux is proportional to the mean energy flux and counting photons is equivalent to integrating the energy. Furthermore, the wavelength λ₀ is the representative wavelength of the mean photons and could be called the mean photon wavelength of the passband.

A2.1. Definitions of Other Wavelengths and Frequencies Associated with a Passband

Now, because monochromatically f_ν = fλ(λ²/c), we can write that

where λ_p is called the pivot wavelength and from equations (A11) and (A12) can be shown to be

As noted by Koornneef et al. (1986), the pivot wavelength is convenient because it allows an exact conversion between the mean broadband fluxes 〈f_ν〉 and 〈f_λ〉.

We previously defined the mean photon wavelength λ₀; we could also define the mean energy wavelength: that is,

This mean wavelength was discussed by King (1952), who cites it as being favored as a flux-independent "effective" wavelength for broadband systems. Two other wavelengths are commonly used: the isophotal wavelength and the effective wavelength.

The isophotal wavelength λ_iso, recommended by Cohen et al. (1992), is the wavelength at which the interpolated, smoothed monochromatic flux has the same value as the mean flux integrated across the band. That is,

A similar expression can be written for the isophotal frequency:

Note that both definitions relate to the energy flux, but a different pair of equations could be defined in terms of the photon flux:

The effective wavelength is usually defined as the flux-weighted mean wavelength. In terms of photons,

which is the same in terms of energy:

and

Note that λ_iso and λ_eff depend explicitly on the underlying flux distribution through the filter.

The definitions and labels of the various wavelengths and frequencies have long stirred passions. King (1952) argued strongly against the currently accepted use of "effective wavelength" and "effective frequency" as defined in equations (A22) and (A23), noting that the meaning of "effective wavelength" is better served by the isophotal wavelength. He further proposed that the mean wavelengths, defined in equations (A14) and (A17) to a first approximation, act as effective wavelengths for all stars, being independent of the flux distribution.

Finally, for aesthetic reasons, Schneider et al. (1983) defined the effective frequency for a passband to be

and it follows that

We note that this is not what Fukugita et al. (1996) claimed Schneider et al. (1983) defined as λ_eff. Fukugita et al. (1996) defined

which is not flux-averaged as is the λ_eff of Schneider et al. (1983). To further complicate definitions, Doi et al. (2010) defined

where

Equation (A28) is the definition of the mean energy-weighted frequency, not the mean (effective [sic]) photon-weighted frequency as stated by Doi et al. (2010). Footnote 13 in that article is also in error.

The "effective" frequency defined by Doi et al. (2010) is the "mean" frequency defined by Koornneef et al. (1986) and not the usual definition of the flux-weighted "effective" frequency similar to the Schneider et al. (1983) "effective" frequency.

For comparison we evaluate the labeled wavelengths for the V band and Vega. Some, such as λ_eff, involve the product of the stellar flux and the system response function, while others such as λ_p and λ₀ concern the system response passband only. Some of the different labeled photon-counting wavelength values are (λ_eff,λ_iso,λ_p) = (5455 Å,5486 Å,5488 Å). The mean photon wavelength λ₀ = 5499 Å, compared with the mean energy wavelength . The Schneider et al. (1983) , the Fukugita et al. (1996) λ_eff = 5464 Å, and the Doi et al. (2010) λ_eff = 5453 Å. We have marked some of these wavelengths in Figure 17, showing the f_ν flux (in magnitudes) of Vega between 5400 Å and 5600 Å.

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This illustrates the unnecessary confusion of these weighted wavelengths. We recommend the retention of only two: the pivot wavelength, λ_p, which is a property of the passband only, and the isophotal wavelength, λ_iso, which takes into account the spectrum measured. To better quantify the derivation of the isophotal wavelength, we recommend that the flux be smoothed to a resolution of one-third of the FWHM of the passband. The pivot wavelength should be used as part of a description of the filter system, while the isophotal wavelength should be used to plot the fluxes as broadband magnitudes against wavelength.

A2.2. Equations Involving Observed Fluxes

Following Oke & Gunn (1983), Fukugita et al. (1996), and Doi et al. (2010), we defined the broadband AB magnitude as

Fukugita et al. (1996) imply that this is a photon-counting magnitude; however, the preceding equation is the energy-integration equation (see eq. [A12]; as discussed previously, based on the Vega STIS005 spectrum, the constant should be -48.577 to be on the same magnitude scale as the V system for V_vega = 0.03).

Equation (A13) shows that photon-counting and energy-integration magnitudes were equivalent, but with different offset constants that are subsumed in the standardization process. That is, the apparent observed magnitude is usually defined as

with the constants C_j or found from photometric standards. Note that the first integral describes photon-counting, while the second integral describes energy integration. The regular appearance of the (normalized) product S(λ)λ in integrals pertaining to photometric magnitudes is often explained simply as wavelength weighting the response function to account for photon-counting; however, it is primarily a consequence of the modern practice of using the photon response functions, rather than the energy response functions used in the past.

The most important reason for maintaining the contemporary practice of using photonic response functions is the fact that they are the default response functions in commonly used data reductions packages, such as synphot and pysynphot. All the passbands published in this article are photonic passbands S(λ).

A2.3. Synphot and Pysynphot

The HST photometry packages synphot¹⁵ and pysynphot¹⁶ are commonly used for planning HST observations and synthetic photometry. It is useful to relate the definitions, variable names, and labels in these packages to those used in this article. Table 6 is a cross-reference list of terms.