# Synthetic RGB photometry of bright stars: definition of the standard
photometric system and UCM library of spectrophotometric spectra

Nicolás Cardiel,1,2 Jaime Zamorano,1,2 Salvador Bará,3 Alejandro Sánchez de
Miguel1,4,5 Cristina Cabello,1,2 Jesús Gallego,1,2 Lucía García,1 Rafael
González,1 Jaime Izquierdo,1 Sergio Pascual,1,2 José Robles,1 Ainhoa Sánchez,1
and Carlos Tapia1
1Departamento de Física de la Tierra y Astrofísica, Fac. CC. Físicas,
Universidad Complutense de Madrid, Plaza de las Ciencias 1, E-28040, Spain
2Instituto de Física de Partículas y del Cosmos, IPARCOS, Fac. CC. Físicas,
Universidad Complutense de Madrid, Plaza de las Ciencias 1, E-28040 Madrid,
Spain
3Departamento de Física Aplicada, Universidade de Santiago de Compostela,
E-15782 Santiago de Compostela, Galicia, Spain
4Environment and Sustainability Institute, University of Exeter, Penryn,
Cornwall TR10 9FE, UK
5Instituto de Astrofísica de Andalucía, Glorieta de la Astronomía,
s/n,C.P.18008 Granada, Spain E-mail<EMAIL_ADDRESS>(NC)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Although the use of RGB photometry has exploded in the last decades due to the
advent of high-quality and inexpensive digital cameras equipped with Bayer-
like color filter systems, there is surprisingly no catalogue of bright stars
that can be used for calibration purposes. Since due to their excessive
brightness, accurate enough spectrophotometric measurements of bright stars
typically cannot be performed with modern large telescopes, we have employed
historical 13-color medium-narrow-band photometric data, gathered with quite
reliable photomultipliers, to fit the spectrum of 1346 bright stars using
stellar atmosphere models. This not only constitutes a useful compilation of
bright spectrophotometric standards well spread in the celestial sphere, the
UCM library of spectrophotometric spectra, but allows the generation of a
catalogue of reference RGB magnitudes, with typical random uncertainties $\sim
0.01$ mag. For that purpose, we have defined a new set of spectral sensitivity
curves, computed as the median of 28 sets of empirical sensitivity curves from
the literature, that can be used to establish a standard RGB photometric
system. Conversions between RGB magnitudes computed with any of these sets of
empirical RGB curves and those determined with the new standard photometric
system are provided. Even though particular RGB measurements from single
cameras are not expected to provide extremely accurate photometric data, the
repeatability and multiplicity of observations will allow access to a large
amount of exploitable data in many astronomical fields, such as the detailed
monitoring of light pollution and its impact on the night sky brightness, or
the study of meteors, solar system bodies, variable stars, and transient
objects. In addition, the RGB magnitudes presented here make the sky an
accessible and free laboratory for the calibration of the cameras themselves.

###### keywords:

instrumentation: photometers – catalogues – techniques: photometric

††pubyear: 2021††pagerange: Synthetic RGB photometry of bright stars:
definition of the standard photometric system and UCM library of
spectrophotometric spectra–Synthetic RGB photometry of bright stars:
definition of the standard photometric system and UCM library of
spectrophotometric spectra

##  1 Introduction

Scientific and commercial-grade RGB cameras provide affordable means for the
acquisition of quantitative radiance data with large fields-of-view, using a
large number of pixels, and moderate multispectral content. Manufactured with
different technologies, and widely different in terms of their absolute
sensitivities, noise levels, pixel sizes and processing capabilities, all of
them share the key common feature of detecting light in three spectral
channels broadly comparable across devices, centered in the red, green, and
blue regions of the visible spectrum, providing that way a relatively
homogenous basis for data sharing and processing.

The sustained improvement in the performance of the CCD and CMOS RGB sensor
arrays enabled the development of an increasing number of scientific
applications in astronomy, by both the professional and amateur communities,
which is expected to grow in the coming years. Among other examples,
commercial-grade RGB cameras have proven to be valuable science tools for
night sky brightness measurement (Hänel et al., 2018; Jechow et al., 2018;
Bertolo et al., 2019; Jechow et al., 2019a, b; Jechow, 2019; Jechow & Hölker,
2019; Kolláth et al., 2020), radiometry of artificial light polluting sources
from Earth-orbit platforms (Kyba et al., 2014; Stefanov et al., 2017; Zheng et
al., 2018; Sánchez de Miguel et al., 2019; Sánchez de Miguel et al., 2020), as
well as from airborne (Kuechly et al., 2012; Bouroussis & Topalis, 2020), and
ground based stations (Dobler et al., 2015; Meier, 2018; Bará et al., 2019),
meteor and fireball detection (Gural & Šegon, 2009), planetary astronomy
(Mousis et al., 2014), and variable stars (Blackford, 2016).

Besides, the exponential growth of the consumer optoelectronics segment opens
unprecedented opportunities for large-scale citizen science projects (see e.g.
Kyba, 2019; Zamorano, 2020): recent market studies (Nisselson et al., 2017)
estimate that by 2022 as much as 45 billion cameras (defined as the
combination of an objective lens plus a focal plane spatially resolved sensor)
will be operative worldwide in different supports, from hand-held smartphones
and classical photographic units to home appliances and artificial vision
systems.

Despite these facts, a consistent astronomical magnitude system has not yet
been defined for RGB photometry. The possibility of reporting fluxes
(irradiances) in astronomical magnitudes, and surface brightnesses (radiances)
in magnitudes per square arcsecond in the native photometric bands of these
widely used devices is appealing. The relative similarity of the $R$, $G$, and
$B$ channels across camera models, and their expected stability in the
foreseeable future (as far as cameras continue to be developed for human
vision-driven applications) are two key features enabling large-scale broadly-
homogeneous data gathering and monitoring across extended periods of time.

We develop in this work a complete RGB photometric system, characterized by a
set of basic filters, the use of photon-based quantities, and with zero points
defined in the absolute (AB) scale. We also provide a catalogue of RGB star
magnitudes corresponding to a good subset of the brightest stars in the
celestial sphere, that can be used as a reference for calibration purposes.
The focus on the bright stars comes from the need to provide observers using
wide-field digital cameras with the possibility of properly calibrating their
images, even with short exposure times. The goal is to reach the same level of
calibration accuracy attained in radiometric measurements performed in
laboratory within the optical range (see Wolfe, 1998, Table 10-4): 1–2.5% with
tungsten lamps, or even 0.1% with self-calibrating detectors.

The absolute spectrophotometric flux calibration of bright star spectra is not
an easy task with modern large telescopes and spectrographs, mainly due to the
difficulty to avoid light losses while preserving the desired spectral
resolutions using narrow slits, but also because of other observational
problems like avoiding detector saturation or lack of linearity of the
detectors themselves when used at very high count rates, the proper correction
of atmospheric extinction or differential refraction (when observing at non-
negligible airmasses, with the slit position angle different from the
parallactic angle, while covering relatively wide wavelength ranges), or the
inhomogeneous illumination of the spectrograph focal plane in very short
exposure times due to the limited speed of camera shutters, to mention a few.
Although the use of neutral density filters can help to alleviate some of
these problems, this approach not only requires a good spectrophotometric
calibration of the density filters themselves, but also translates into using
modern telescopes to observe bright stars with large exposure times, something
that typically is not easy to get approved by telescope allocation committees.

For all those reasons we decided to base our work on historical, but quite
reliable and homogeneous, medium-narrow-band photometric data, that can be
used to fit star model spectra able reproduce the observed spectral energy
distribution of bright stars. The fitted models have then been used to compute
synthetic RGB magnitudes, fulfilling our initial goal.

A brief description of the practical computation of synthetic magnitudes
followed in this work is summarised in Section 2. Section 3 describes the
photometric data and initial sample selection, while the model fitting
procedure and final sample definition, based on comparisons between synthetic
magnitudes and additional photometric measurements, are presented in Section
4. To avoid the problem of choosing the RGB sensitivity curves of a particular
camera, we decided to define a set of reference RGB spectral sensitivity
curves, using median values from existing sensitivity curves of well-known
cameras, that can be used to establish an RGB standard photometric system. The
definition of these curves is presented in Section 5, together with the RGB
magnitudes for the 1346 stars constituting the UCM library of
spectrophotometric standards, and a discussion concerning the conversion
between magnitudes measured under the standard RGB system defined here and
those derived employing the RGB sensitivity curves of individual cameras. The
conclusions of this work are summarized in Section 6, while Appendices A and B
include the graphical comparison of the results of applying the adopted
fitting technique in 39 stars with available spectrophotometric data from the
literature, and a table with polynomial coefficients that allow the
computation of the expected differences between the standard RGB system and 28
particular digital cameras, respectively.

##  2 Computation of synthetic magnitudes

Synthetic magnitudes in this work have been determined using the Python
package synphot (STScI development Team,
2018)111https://synphot.readthedocs.io/en/latest/, which facilitates the
computation of photometric properties from user-defined bandpasses and
spectra. This package follows the photon-counting formalism, expected for
modern CCD detectors (see e.g. Casagrande & VandenBerg, 2014), where the
number of photons, instead of the arriving energy, is the relevant property to
be considered. In this way, for a particular bandpass defined in the
wavelength interval ranging from $\lambda_{i}$ to $\lambda_{f}$, magnitudes
are computed following

$\displaystyle
m=-2.5\,\log_{10}\frac{\int_{\lambda_{i}}^{\lambda_{f}}n_{\gamma}(\lambda)\;T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}n_{\gamma,r}(\lambda)\;T(\lambda)\;{\rm
d}\lambda}=-2.5\,\log_{10}\frac{N_{\gamma}}{N_{\gamma,r}},$ (1)

where $n_{\gamma}(\lambda)$ is the number of photons per unit time and per
unit spectral bandwidth at the wavelength $\lambda$ through a unit area
(photons s-1 cm-2 Å-1; flux units known as PHOTLAM in synphot) for the desired
target. Similarly, $n_{\gamma,r}(\lambda)$ has the same physical meaning for
the spectrum to be used to set the $m\\!=\\!0$ reference point. In addition,
$T(\lambda)$ is the system spectral sensitivity response. Sometimes, it is
also useful to compute $N_{\gamma}$, the integrated number of photons (photons
s-1 cm-2) within the bandpass, modulated by the spectral sensitivity response,
although logically the absolute number depends on the particular normalization
of $T(\lambda)$. In this sense, it is important to note that if the averaged
number of photons within the bandpass is the sought parameter, a proper
normalization must be performed using $T(\lambda)$ as the weighting factor

$\displaystyle\langle
n_{\gamma}\rangle=\frac{\int_{\lambda_{i}}^{\lambda_{f}}n_{\gamma}(\lambda)\;T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}T(\lambda)\;{\rm
d}\lambda}=\frac{N_{\gamma}}{\int_{\lambda_{i}}^{\lambda_{f}}T(\lambda)\;{\rm
d}\lambda},$ (2)

with a similar expression for the reference spectrum $\langle
n_{\gamma,r}\rangle$, interchanging $n_{\gamma}(\lambda)$ by
$n_{\gamma,r}(\lambda)$, and $N_{\gamma}$ by $N_{\gamma,r}$, in the above
equation. The denominator of the last equation, that works as normalization
factor and is computed in synphot as the bandpass equivalent width, is
actually also present in both the numerator and denominator of Eq. 1, but
cancels out and does not appear explicitly.

Since the number of photons is directly related to the incoming flux
densities, magnitudes can also be computed as

$\displaystyle
m=-2.5\,\log_{10}\frac{\int_{\lambda_{i}}^{\lambda_{f}}f(\lambda)\;\frac{\lambda}{h\,c}\;T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}f_{r}(\lambda)\;\frac{\lambda}{h\,c}\;T(\lambda)\;{\rm
d}\lambda},$ (3)

being $f(\lambda)$ the flux density of the target (erg s-1 cm-2 Å-1; flux
units known as FLAM in synphot), and $f_{r}(\lambda)$ the flux density of the
reference spectrum. In this case, we have not simplified the $h\,c$ factor in
the last equation in order to keep the traceability of the units involved,
being both the numerator and denominator in that fraction given in photons s-1
cm-2.

We have checked that the computed synthetic magnitudes with synphot agree with
the measurements performed with
pyphot222https://mfouesneau.github.io/docs/pyphot/, another Python package
providing tools to compute synthetic photometry. In no case the differences
found were larger than 0.0002 mag. In addition, we also performed our own
integrations by directly programming the equations shown in this section,
using trapezoidal integrations with the Numpy function
trapz333https://numpy.org/doc/stable/reference/generated/numpy.trapz.html,
being the largest differences also smaller than 0.0002 mag. However, it is
worth noticing that when applying a different integration strategy, namely the
Simpson’s rule with the scipy function
simpson444https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html,
the differences are slightly larger, reaching in some cases 0.01 mag. Since
the Simpson’s rule approximates the original function using piecewise
quadratic functions, it is clear that this fact has a non-negligible effect on
the computations. For that reason, we advocate the use of synphot or pyphot,
when delegating in third-party software packages when computing synthetic
magnitudes, or the implementation of the simple trapezoidal integration, when
employing user-defined code.

##  3 Data sample

###  3.1 Historical 13-color photometric data

Figure 1: Spectral sensitivity curves for the 13 filters, as given by Johnson & Mitchell (1975, see their Table 1). The name of each filter indicates its approximate effective wavelength. Just for illustration, we are also displaying the atmospheric telluric absorption (upper gray line) computed with the help of the ESO SKYCALC tool (Moehler et al., 2014). Table 1: Basic properties of the filters composing the 13C photometric system: (1) filter identification, (2) bandpass average wavelength, (3) equivalent width, and (4) full width at half maximum. Filter |  $\lambda_{0}$ (Å) | Equivalent width (Å) | FWHM (Å)
---|---|---|---
33 | 3374 | 116 | 143
35 | 3540 | 132 | 224
37 | 3749 | 133 | 213
40 | 4034 | 240 | 349
45 | 4603 | 285 | 383
52 | 5187 | 259 | 250
58 | 5822 | 230 | 247
63 | 6360 | 326 | 392
72 | 7239 | 589 | 463
80 | 8002 | 435 | 348
86 | 8585 | 483 | 380
99 | 9828 | 583 | 575
110 | 11088 | 830 | 872

The classical 13-color (hereafter 13C) medium-narrow-band photometric system
(Johnson et al., 1967; Mitchell & Johnson, 1969; Johnson & Mitchell, 1975;
Schuster, 1982b) was created with the aim to obtain homogeneous and calibrated
photometric measurements of bright stars, with a high level of accuracy. Each
filter is identified by a number that indicates its approximate effective
wavelength. The spectral sensitivity curves for all the filters are displayed
in Fig. 1, whereas some basic filter properties are provided in Table 1. The
seven bluest filters, with effective wavelengths ranging from 3370 Å to 5820
Å, provided quantitative information concerning the behaviour of the continuum
of early-type stars, including the Balmer jump. The six reddest filters,
covering the 6360 Å to 11090 Å region, were selected to avoid conspicuous
telluric atmospheric features.

Apart from the initial references, the usefulness of the 13C photometric
system was clearly demonstrated through numerous stellar studies derived from
its use (see e.g. Schuster, 1976; Alvarez & Schuster, 1978; Schuster, 1979a,
b, c; Marx & Lehmann, 1979; Chavarria-K. & de Lara, 1981; Schuster, 1982a, b;
Alvarez & Schuster, 1982; Schuster & Alvarez, 1983; Schuster, 1984; Schuster &
Guichard, 1984; Conconi & Mantegazza, 1985; Schuster & Guichard, 1985; Petford
& Blackwell, 1989; Bravo Alfaro et al., 1993, 1997, and references therein).

###  3.2 The star sample

We have assembled 13C photometric data of bright stars, belonging to the
Bright Star Catalog (Hoffleit, 1964), coming from three different sources:
1380 stars from Johnson & Mitchell (1975, hereafter JM75), 81 stars from
Schuster (1976, hereafter S76), and 71 stars from Bravo Alfaro et al. (1997,
hereafter BAS97)555By comparing repeated stars between JM75 and BAS97, we
noted that there is an erratum in the header description of Table 1 in BAS97:
the column labelled as 58$-$63 is actually 52$-$63.. The largest contribution
comes from the JM75 sample, that basically covered all the stars brighter than
the fifth visual magnitude north of declination $-20^{\degr}$, and most of the
stars brighter than the fourth visual magnitude below that declination. It is
important to note that 163 stars from the JM75 sample ($\sim 12$% of the
objects) do not have photometric data for the 5 reddest filters (72, 80, 86,
99 and 110). On the other hand, the S76 and BAS97 samples correspond to solar-
type and A0-K0 supergiant stars, respectively. Taking into account that there
are one and nine stars in common between JM75 and S76, and between JM75 and
BAS97, respectively, the total initial number of different stars is 1522. As
explained later (Sect. 4.3), the initial list was cleaned by removing stars
with poor results in the fitting process or by discrepancies with the
available Johnson $B$ and $V$ photometric data retrieved from the Simbad
database. The final sample, listed in Table 2, comprises a total of 1346
stars, with a magnitude distribution as shown in Fig. 2, being the stars well
spread over the whole celestial sphere, as displayed in Fig. 3.

Figure 2: Distribution of the final star selection according to the stellar
magnitude in the 52 filter. The number of stars belonging to each source is
displayed between parenthesis in the figure key. The red line delineates the
coadded histogram. It is clear that most of the stars come from JM75, with a
small contribution of S76 and BAS97 in the faint regime of the sample. Figure
3: Lambert cylindrical equal-area projection in J2000 equatorial coordinates
showing the distribution of the stars belonging to the Bright Star Catalog
(Hoffleit, 1964). The 1346 stars composing the UCM library of
spectrophotometric standards are highlighted as red filled cicles. Simple
constellations shapes (van der Sluys, 2005), displayed with light orange
lines, are also shown to facilitate the visual identification of the stars.
The size of each star symbol is inversely proportional to its Johnson $V$
magnitude. Figure 4: Differences in the photometric measurements of 1 star
from S76 (HR3750, open square) and 9 stars from BAS97 (the remaining objects)
in common with the JM75 sample. Open circles indicate variable stars. HR3750
is a binary system. The r.m.s. for HR3045, the only non-variable and non-
binary star, is $0.018$ mag.

The comparison of the common stars in the three original sources is shown in
Fig. 4. Even though most of these repeated stars are variable, the scatter of
the photometric measurements (except for the bluest filters) is compatible
with a $~{}\sim 0.1$ mag dispersion. Focusing on HR3045, the only star that is
not variable and that does not belong to a binary system, the r.m.s. for the
13 photometric measurements is 0.018 mag, which is perfectly compatible with
the probable errors of single observations in JM75 and S76, reported by
Schuster (1982b, see his Table 2) to be $\sim 0.02$ mag. The photometric
measurements for these 10 repeated stars have been averaged for the subsequent
work.

##  4 Spectrum fitting

Table 2: First 10 rows of the table with the final sample, constituted by 1346
stars, comprising the UCM library of spectrophotometric standards. The full
table is electronically available at
%****␣paper_RGB.tex␣Line␣525␣****http://guaix.ucm.es/rgbphot. Column
description: (1) HR number; (2)–(3) star coordinates (J2000), as provided by
Simbad; (4) identificacion of the companion star for double star systems (see
Table 3); (5) additional name when the star is a known variable according to
Simbad; (6) variability type from Samus’ et al. (2017), as provided by Simbad;
(7)–(19) absolute flux densities in each of the C13 photometric bandpasses (in
units of erg s-1 cm-2 Å-1); (20)–(22) effective temperature, surface gravity
and metallicity derived from the best CK04 fit to the available photometric
data, with the corresponding uncertainties derived from the bootstrapped
spectra (note that the quoted uncertainties are simply lower limits to the
expected random errors in the stellar atmospheric parameters; see discussion
in Sect. 4.2); (23) scaling factor to convert the fitted CK04 model fluxes
into absolute flux densities using Eq. (9); (24)–(25) Johnson $B$ and $V$
magnitudes (VEGA system), extracted from Simbad; (26)–(27) synthetic Johnson
$B$ and $V$ magnitudes (VEGA system), measured in the best CK04 fit, with
their corresponding uncertainties estimated from the bootstrapped spectra;
(28)–(30) synthetic RGB magnitudes (AB system), measured in the best CK04 fit,
using the bandpass definitions given in Table 5, with uncertainties estimated
from bootstrapping.

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
---|---|---|---|---|---|---|---|---|---
HR | RA (°) | DEC (°) | Double | Variable | Var.Type |  $f_{33}^{\star}$ ($\times 10^{-9}$) |  $f_{35}^{\star}$ ($\times 10^{-9}$) |  $f_{37}^{\star}$ ($\times 10^{-9}$) |  $f_{40}^{\star}$ ($\times 10^{-9}$)
0003 | 1.3339247 |  $-$5.7076189 | — | BC Psc | RS | 0.008065 | 0.009427 | 0.011144 | 0.024608
0005 | 1.5658922 | 58.4367280 | — | V640 Cas | CST: | 0.006548 | 0.006900 | 0.007224 | 0.012313
0015 | 2.0969161 | 29.0904311 | — | alf And | ACV: | 1.034568 | 0.962187 | 0.997723 | 1.354483
0021 | 2.2945217 | 59.1497811 | — | bet Cas | DSCTC | 0.272195 | 0.284138 | 0.377964 | 0.622496
0025 | 2.3526731 |  $-$45.7474253 | — | — | — | 0.015074 | 0.019376 | 0.021859 | 0.051003
0027 | 2.5801942 | 46.0722722 | — | — | — | 0.014914 | 0.016286 | 0.027913 | 0.046962
0033 | 2.8160733 |  $-$15.4679794 | — | — | — | 0.027421 | 0.028911 | 0.031828 | 0.046545
0039 | 3.3089633 | 15.1835936 | — | gam Peg | BCEP | 0.956366 | 0.839646 | 0.742393 | 0.785749
0045 | 3.6506856 | 20.2067003 | — | NSV 99 | undef | 0.001080 | 0.001705 | 0.002336 | 0.007366
0048 | 3.6600689 |  $-$18.9328653 | — | AE Cet | LB: | 0.001156 | 0.001833 | 0.002564 | 0.008143

(1) | (11) | (12) | (13) | (14) | (15) | (16) | (17) | (18) | (19)
---|---|---|---|---|---|---|---|---|---
HR |  $f_{45}^{\star}$ ($\times 10^{-9}$) |  $f_{52}^{\star}$ ($\times 10^{-9}$) |  $f_{58}^{\star}$ ($\times 10^{-9}$) |  $f_{63}^{\star}$ ($\times 10^{-9}$) |  $f_{72}^{\star}$ ($\times 10^{-9}$) |  $f_{80}^{\star}$ ($\times 10^{-9}$) |  $f_{86}^{\star}$ ($\times 10^{-9}$) |  $f_{99}^{\star}$ ($\times 10^{-9}$) |  $f_{110}^{\star}$ ($\times 10^{-9}$)
0003 | 0.045110 | 0.048056 | 0.055965 | 0.055715 | 0.047546 | 0.042668 | 0.039957 | 0.034183 | 0.026974
0005 | 0.015360 | 0.014287 | 0.014343 | 0.013207 | 0.010510 | 0.008935 | 0.007667 | 0.006164 | 0.004537
0015 | 0.949220 | 0.642224 | 0.456478 | 0.335422 | 0.225124 | 0.161805 | 0.127347 | 0.090751 | 0.061170
0021 | 0.589519 | 0.485483 | 0.416094 | 0.351972 | 0.266133 | 0.212027 | 0.175507 | 0.135903 | 0.098852
0025 | 0.086732 | 0.095887 | 0.110133 | 0.104018 | 0.087565 | 0.078870 | 0.073035 | 0.061796 | 0.052358
0027 | 0.044648 | 0.038470 | 0.034585 | 0.029620 | 0.023448 | 0.019148 | 0.016498 | 0.013264 | 0.010151
0033 | 0.047392 | 0.042263 | 0.038594 | 0.033799 | 0.027282 | 0.022794 | 0.019054 | 0.014683 | 0.011011
0039 | 0.494316 | 0.326709 | 0.220809 | 0.159055 | 0.101730 | 0.069933 | 0.053288 | 0.035159 | 0.023059
0045 | 0.026858 | 0.035779 | 0.051488 | 0.062204 | 0.079683 | 0.091648 | 0.090080 | 0.092416 | 0.082166
0048 | 0.032928 | 0.047326 | 0.070281 | 0.085556 | 0.108490 | 0.123857 | 0.123218 | 0.124770 | 0.105232

(1) | (20) | (21) | (22) | (23)
---|---|---|---|---
HR |  $T_{\rm eff}$ (K) | $\log g$ | [M/H] |  $c_{\rm sc}$ ($\times 10^{-17}$)
0003 | 4732$\;\pm\;$ | 21 | 3.088$\;\pm\;$ | 0.320 | 0.000$\;\pm\;$ | 0.072 | 1.8003
0005 | 5762$\;\pm\;$ | 50 | 4.809$\;\pm\;$ | 0.291 | 0.200$\;\pm\;$ | 0.095 | 0.1705
0015 | 12827$\;\pm\;$ | 101 | 4.484$\;\pm\;$ | 0.116 | 0.500$\;\pm\;$ | 0.000 | 0.5109
0021 | 6917$\;\pm\;$ | 81 | 3.229$\;\pm\;$ | 0.099 |  $-$0.085$\;\pm\;$ | 0.153 | 2.3213
0025 | 4824$\;\pm\;$ | 29 | 2.522$\;\pm\;$ | 0.312 | 0.000$\;\pm\;$ | 0.043 | 3.0634
0027 | 6181$\;\pm\;$ | 31 | 1.299$\;\pm\;$ | 0.107 |  $-$0.801$\;\pm\;$ | 0.158 | 0.3045
0033 | 6217$\;\pm\;$ | 76 | 3.912$\;\pm\;$ | 0.211 |  $-$0.386$\;\pm\;$ | 0.126 | 0.3484
0039 | 21414$\;\pm\;$ | 393 | 4.192$\;\pm\;$ | 0.330 |  $-$2.000$\;\pm\;$ | 0.000 | 0.1248
0045 | 3748$\;\pm\;$ | 16 | 1.202$\;\pm\;$ | 0.070 | 0.429$\;\pm\;$ | 0.080 | 11.7681
0048 | 3750$\;\pm\;$ | 3 | 0.770$\;\pm\;$ | 0.117 | 0.500$\;\pm\;$ | 0.010 | 15.6778

(1) | (24) | (25) | (26) | (27) | (28) | (29) | (30)
---|---|---|---|---|---|---|---
HR | Johnson $B_{\rm Simbad}$ | Johnson $V_{\rm Simbad}$ | Johnson $B_{\rm CK04}$ | Johnson $V_{\rm CK04}$ | standard $B$ | standard $G$ | standard $R$
0003 | 5.65 | 4.61 | 5.649 $\;\pm$ | 0.011 | 4.583 $\;\pm$ | 0.010 | 5.128 $\;\pm\;$ | 0.010 | 4.680 $\;\pm\;$ | 0.010 | 4.357 $\;\pm\;$ | 0.008
0005 | — | — | 6.691 $\;\pm$ | 0.013 | 5.985 $\;\pm$ | 0.009 | 6.307 $\;\pm\;$ | 0.012 | 6.044 $\;\pm\;$ | 0.009 | 5.862 $\;\pm\;$ | 0.009
0015 | 1.95 | 2.06 | 1.914 $\;\pm$ | 0.010 | 2.021 $\;\pm$ | 0.011 | 1.834 $\;\pm\;$ | 0.010 | 1.983 $\;\pm\;$ | 0.011 | 2.144 $\;\pm\;$ | 0.012
0021 | 2.61 | 2.27 | 2.580 $\;\pm$ | 0.015 | 2.248 $\;\pm$ | 0.010 | 2.347 $\;\pm\;$ | 0.014 | 2.268 $\;\pm\;$ | 0.011 | 2.234 $\;\pm\;$ | 0.008
0025 | 4.89 | 3.87 | 4.916 $\;\pm$ | 0.011 | 3.863 $\;\pm$ | 0.008 | 4.399 $\;\pm\;$ | 0.010 | 3.958 $\;\pm\;$ | 0.009 | 3.653 $\;\pm\;$ | 0.008
0027 | 5.44 | 5.04 | 5.386 $\;\pm$ | 0.012 | 4.978 $\;\pm$ | 0.009 | 5.133 $\;\pm\;$ | 0.013 | 5.009 $\;\pm\;$ | 0.010 | 4.932 $\;\pm\;$ | 0.008
0033 | 5.38 | 4.89 | 5.358 $\;\pm$ | 0.012 | 4.865 $\;\pm$ | 0.008 | 5.075 $\;\pm\;$ | 0.012 | 4.906 $\;\pm\;$ | 0.009 | 4.797 $\;\pm\;$ | 0.007
0039 | 2.61 | 2.84 | 2.568 $\;\pm$ | 0.010 | 2.770 $\;\pm$ | 0.013 | 2.529 $\;\pm\;$ | 0.011 | 2.719 $\;\pm\;$ | 0.013 | 2.917 $\;\pm\;$ | 0.013
0045 | 6.38 | 4.80 | 6.342 $\;\pm$ | 0.015 | 4.714 $\;\pm$ | 0.010 | 5.668 $\;\pm\;$ | 0.014 | 4.879 $\;\pm\;$ | 0.010 | 4.326 $\;\pm\;$ | 0.014
0048 | 6.12 | 4.46 | 6.101 $\;\pm$ | 0.015 | 4.383 $\;\pm$ | 0.009 | 5.389 $\;\pm\;$ | 0.009 | 4.554 $\;\pm\;$ | 0.009 | 3.981 $\;\pm\;$ | 0.012

###  4.1 The fitting procedure

Figure 5: Example of computation of initial atmospheric stellar parameters in
the fit of the C13 photometric data of the star HR7254 to the CK04 models.
Each plot shows the residuals obtained after determining the best scaling
factor when fitting the modeled spectra with different $T_{\rm eff}$ and $\log
g$ values at a fixed metallicity ([M/H], given in the upper right corner). The
residuals were computed as $\sqrt{\chi^{2}/N_{\rm bands}}$, with $\chi^{2}$
given by Eq. (6). The white areas indicate the stellar parameter space not
available in the models. The red cross in each panel marks a local minimum,
i.e., the $T_{\rm eff}$ and $\log g$ values that provide the best fit at the
considered [M/H]. A large red filled circle has been overplotted in the panel
where the global minimum is reached.

Figure 6: Residuals of the same fit displayed in Fig. 5, corresponding to the
star HR7254. These two panels show two perpendicular views in the 3D parameter
space around the global minimum found at $T_{\rm eff}=8750$ K,
${\rm[M/H]}=0.0$ and $\log g=3.5$ in the first step of the fitting procedure
discussed in Sect. 4.1.

The first step before attempting to estimate the synthetic RGB photometry of
the bright star sample was the fit of the spectral energy distribution of each
star to the stellar model spectrum that best matched the available 13C
photometric data. For that purpose, we have chosen the Stellar Atmosphere
Models by Castelli & Kurucz (2003, hereafter CK04), as provided by the STScI
web page666https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/castelli-and-kurucz-atlas. An
important advantage of these models is that they can be easily interpolated
using the Python package stsynphot (Lim, 2018, available
online777https://stsynphot.readthedocs.io/en/latest/index.html) for any
arbitrary $T_{\rm eff}$, [M/H] and $\log g$ selection (within the parameter
space covered by the models). Since the bright star sample is constituted by
nearby objects, it is safe to assume that most of them will not be strongly
affected by interstellar reddening. For that reason, we have not included any
correction for this effect in the subsequent fits. In any case, here we are
aiming at providing accurate photometry of the observed (i.e., uncorrected)
RGB fluxes, and the fact of excluding this type of correction will only
translate in a systematic deviation of the derived stellar parameters.

It is important to highlight that it is not the aim of this work to determine
accurate stellar atmospheric parameters for each star, but to obtain a good
fit to the available 13C photometric data in order to derive reliable spectral
energy distributions that facilitate the proper computation of synthetic RGB
magnitudes. The comparison with observed spectra, shown in section 4.4,
indicates that this is actually the case.

The initial 13C stellar magnitudes were converted into absolute flux densities
(erg s-1 cm-2 Å-1) using the conversion given by JM75 (the Fortran IV code
provided in their Table 5 was transformed to Python to facilitate this task).
The selection of the best CK04 model for each star was accomplished in two
steps:

  1. Step 1:

Initial determination of the atmospheric stellar parameters using all the
available CK04 models at their pre-computed sampling grid in the parameter
space. In particular, the CK04 atlas provides models for abundances
[M/H]=$-2.5$, $-2.0$, $-1.5$, $-1.0$, $-0.5$, $0.0$, $+0.2$ and $+0.5$, with
effective temperatures ranging from 3500 to 50000 K, and $\log g$ (surface
stellar gravity, with $g$ in cm s-2) from 0.0 to 5.0 dex. A simple chi-square
minimization process was performed, by adjusting the arbitrary constant
$c_{\rm sc}$ necessary to scale each model prediction to the absolute C13 flux
densities. For this purpose, the stellar fluxes for the star $f^{\star}_{i}$
and the model $f^{\rm CK04}_{i}$ in each of the $N_{\rm bands}$ photometric
bandpasses were normalized by their mean values

$\displaystyle f^{\star}_{{\rm nor},i}$
$\displaystyle=\frac{f^{\star}_{i}}{\langle f^{\star}\rangle},$
$\displaystyle\;\;{\rm with}$ $\displaystyle\langle f^{\star}\rangle$
$\displaystyle=\frac{1}{N_{\rm bands}}\sum_{i=1}^{N_{\rm
bands}}f^{\star}_{i},$ (4) $\displaystyle f^{\rm CK04}_{{\rm nor},i}$
$\displaystyle=\frac{f^{\rm CK04}_{i}}{\langle f^{\rm CK04}\rangle},$
$\displaystyle\;\;{\rm with}$ $\displaystyle\langle f^{\rm CK04}\rangle$
$\displaystyle=\frac{1}{N_{\rm bands}}\sum_{i=1}^{N_{\rm bands}}f^{\rm
CK04}_{i}.$ (5)

The initial stellar fluxes $f_{i}^{\star}$ (with $i=33,\,35,\ldots,110$
indicating the 13C photometric band), computed from the published photometric
data, are listed in Table 2, columns (5)–(17).

An objective function to be minimized was defined at this point as

$\displaystyle\chi^{2}(c_{\rm nor})=\sum_{i=1}^{N_{\rm
bands}}\left(f^{\star}_{{\rm nor},i}-c_{\rm nor}f^{\rm CK04}_{{\rm
nor},i}\right)^{2},$ (6)

where $c_{\rm nor}$ is an intermediate scaling factor between the normalized
stellar and model fluxes. By equating to zero the first derivative of the last
equation, it is immediate to obtain

$\displaystyle c_{\rm nor}=\frac{\sum_{i=1}^{N_{\rm bands}}f^{\star}_{{\rm
nor},i}\,f^{\rm CK04}_{{\rm nor},i}}{\sum_{i=1}^{N_{\rm bands}}\left(f^{\rm
CK04}_{{\rm nor},i}\right)^{2}}.$ (7)

Finally, the sought scaling factor $c_{\rm sc}$ is computed as

$\displaystyle c_{\rm sc}=c_{\rm nor}\frac{\langle f^{\star}\rangle}{\langle
f^{\rm CK04}\rangle},$ (8)

which allows to convert the CK04 model fluxes into absolute flux densities as

$\displaystyle f^{\star}=c_{\rm sc}f^{\rm CK04}.$ (9)

At this stage, the whole parameter space was sampled using all the initially
available combinations of the three stellar parameters. The global minimum of
the objective function in this comprehensive search led to an initial guess
for $T_{\rm eff}$, [M/H], and $\log g$, which were refined in the subsequent
step. A graphical example of this initial computation is shown in Figs. 5 and
6. The comparison of these plots indicates that, not surprisingly, $T_{\rm
eff}$ is the main parameter governing the variation of the $\chi^{2}$ value,
followed by $\log g$. These two parameters are correlated, as shown by the
oblique orientation of the minimum valley in Fig. 5. In addition, the role of
[M/H] is quite small, as revealed by the almost undetectable variation of the
residuals with this parameter in Fig. 6. This result gives support to the idea
that the derived stellar parameters should not be considered as extremely
accurate.

  2. Step 2:

Refinement of the atmospheric stellar parameters: the initial $T_{\rm eff}$,
[M/H], and $\log g$ values were employed as the starting point in the
parameter space to compute a more refined solution. For that purpose, we used
a numerical minimization process based on the Nelder-Mead method (Nelder &
Mead, 1965) with the help of the Python package lmfit (Newville et al., 2014,
available online888https://lmfit.github.io/lmfit-py/). During this process,
the CK04 model predictions were interpolated at arbitrary locations within the
valid stellar atmospheric parameter space, as required by the objective
function to be minimized. The use of a good starting point, obtained in the
previous step, facilitated the convergence of this process. The final $T_{\rm
eff}$, [M/H] and $\log g$ parameters for each star are given in Table 2,
columns (18)–(20).

###  4.2 Uncertainties in the fitting procedure

Figure 7: Distribution of random uncertainties in the derived stellar
atmospheric parameters in the final sample of fitted CK04 models, estimated
from the bootstrapping method described in Sect. 4.2. Panel (a): relative
errors in effective temperature. Panels (b) and (c): absolute errors in
metallicity and surface gravity. It is important to highlight that these
uncertainties are simply lower limits to the expected random errors in the
stellar atmospheric parameters, as explained in the text.

Two main sources of uncertainties have been considered: i) systematic errors
due to the inability of the adopted CK04 models to properly reproduce the
spectral energy distribution of the stars, and ii) random errors in the fitted
C13 photometric data.

Since there is not an easy way to fix model fits exhibiting C13 residuals with
a systematic variation as a function of wavelength, we decided to get rid of
those stars with unreliable fits, following the criteria described in Sect.
4.3.

On the other hand, the impact of random errors in the fitting process has been
determined by generating bootstrapped absolute flux densities, using for that
purpose the already mentioned probable error of 0.02 mag in the C13
photometric data, and repeating the whole fitting procedure in each simulated
set of C13 photometric data for every star. This allowed obtaining an initial
estimate of the uncertainties in the fitted stellar parameters, which are
given in the corresponding columns (18)–(20) of Table 2, and graphically
displayed in Fig. 7. It is important to stress that these random uncertainties
must not be understood as the actual uncertainties associated to the derived
atmospheric stellar parameters. As previously mentioned, no attempt to
introduce an interstellar reddening correction has been performed.
Furthermore, we are constrained to the capability of CK04 models to provide an
accurate modelling of actual stellar spectra, without considering, for
example, the impact of the assumed line opacities and chemical abundances, to
mention some of the additional relevant parameters that should be considered
to obtain reliable physical descriptions of the stellar atmospheres. Once the
CK04 set of stellar atmosphere models has been adopted, the quoted random
errors are just a lower limit to the actual $T_{\rm eff}$, [M/H] and $\log g$
uncertainties for each star. Nevertheless, the quoted values can be compared
with the sampling of the atmospheric parameters at which the CK04 models are
provided. In particular, the effective temperature is sampled with $\Delta
T_{\rm eff}/T_{\rm eff}$ ranging from 0.02 (hot stars) to 0.07 (cool stars),
whereas gravity and metallicity are sampled at $\Delta(\log g)=0.5$, and
$\Delta{\rm[M/H]}$ ranging from 0.2 to 0.5, respectively. Comparing these
numbers with the distributions displayed in Fig. 7, it is clear that in most
cases the derived uncertainties are within the considered sampling steps,
which indicates that, at least from the point of view of the fitting
procedure, each one of the CK04 models is different enough from its
neighbouring models within the 3D parameter space even after bootstrapping the
C13 photometric data. This reinforces the usefulness of Step 2 (see Sect. 4.1)
devoted to the refinement of the atmospheric stellar parameters. In addition,
the bootstrapping method provided a collection of bootstrapped fitted spectra
associated to every single star, that were employed later to estimate random
uncertainties in the synthetic photometry performed on the CK04 model fits.

It is important to highlight that although the numerical minimization
procedure adopted for this work is robust, the computation time spent on each
individual fit is not negligible. In particular, the adopted Nelder-Mead
method required the evaluation of the objective function in many points
(typically a few hundreds) within the 3D parameter space defined by $T_{\rm
eff}$, $\log g$ and [M/H], which in turn translated into the corresponding
number of interpolations of the CK04 models. The median computation time for
every fit amounted to a few minutes. For that reason, we decided to generate a
number of bootstrapped spectra not excessively large, in order to apply this
technique to the whole final sample and not only to a representative subsample
of stars. Finally, this total number of bootstrapped spectra for each
individual star was set to 30, which leads to uncertainties in the
uncertainties999Here we are using the approximation provided by normally
distributed data, for which the fractional uncertainty on the standard
deviation can be approximated by $(2n-2)^{-1/2}$ (see e.g. Squires, 2001,
Section 3.7), with $n$ the number of samples. of $\sim 13$%.

Figure 8: Histogram of minimum values of the objective function obtained
during the fitting procedure of CK04 models. The initial sample (1522 stars)
were segregated according to the number of available photometric bandpasses
(13 or 8). The black lines indicate the robust fit to Normal distributions.
The vertical dashed lines mark the 3$\sigma$ location above the median value,
which corresponds to $\chi^{2}_{3\sigma}\simeq 0.105$ for the 1359 stars
observed with 13 filters, and $\chi^{2}_{3\sigma}\simeq 0.047$ for the 163
stars observed with 8 filters.

###  4.3 Cleaning the sample

Figure 9: Distribution of random uncertainties in the synthetic Johnson $B$
and $V$ measurements performed on the final sample of fitted CK04 models,
estimated from the bootstrapping method described in Sec. 4.2. The median
values are $\Delta B=0.011$ mag (top panel) and $\Delta V=0.010$ mag (bottom
panel).

Figure 10: Differences between the synthetic Johnson $B$ and $V$ magnitudes computed using the CK04 model fits and the tabulated $B$ and $V$ data available in Simbad. Known variable stars (according to Simbad) are plotted with orange crosses, whereas normal stars (i.e., non-variable) are represented with blue filled circles. Panel (a) shows that, not surprisingly, the errors in $B$ and $V$ are correlated. The individual analysis of the differences within each photometric band are shown in panels (b) and (c). The dashed lines in the last two panels correspond to the robust $\pm 3\sigma$ region around the median. A few stars, exhibiting large deviations, are labelled in each panel. Table 3: Double stars from the JM75 sample for which the individual fluxes of the two stars have been coadded to derive the resulting integrated $B$ and $V$ magnitudes (last two columns). Star#1 is the initial star identification provided in JM75. Star#2 is the companion star (in some cases the companion star is brighter than Star#1). For some stars the magnitude in one of the two filters is missing. The quoted $B$ and $V$ magnitudes for the individual stars, as well as the spectral type, correspond to the data retrieved from the Simbad database. Stars marked with an asterisk in the first column were finally removed from the final sample due to large discrepancies between the Simbad data and the predicted $B$ and $V$ magnitudes, as explained in Sect. 4.3. The fourth column in Table 2 provides the name of the companion star for the double star systems incorporated in the final stellar sample. Star#1 | Star#2 | Star#1+#2
---|---|---
Name | $B_{1}$ | $V_{1}$ | SpT1 | X | Name | $B_{2}$ | $V_{2}$ | SpT2 | X | $B_{1+2}$ | $V_{1+2}$
HR0545 | 4.558 | 4.589 | A0V |  | HR0546 | 4.490 | 4.520 | A2IV |  | 3.771 | 3.801
∗HR0897 | 3.330 | 3.180 | A3IV–V |  | HR0898 | 4.200 | 4.110 | A1V |  | 2.928 | 2.796
HR1211 | 6.190 | 6.090 | A1V |  | HR1212 | 5.590 | 4.700 | G6.5III |  | 5.096 | 4.434
∗HR1897 | 6.300 | 6.390 | O9.5IV |  |  $\theta^{2}$ Ori B | 6.290 | 6.380 | B2–B5 |  | 5.542 | 5.632
∗HR1948 | 1.790 | 1.880 | O9.2Ib |  | HR1949 | 3.550 | 3.730 | O9.5II–III |  | 1.594 | 1.698
HR2298 | 4.583 | 4.398 | A8V |  | HR2299 | 6.990 | 6.600 | F5V |  | 4.471 | 4.264
HR2735 | 6.046 | 5.623 | F0/3 |  | HR2736 | 4.774 | 3.746 | K0III |  | 4.481 | 3.569
HR2890 | — | 3.000 | A0–A2 |  | HR2891 | — | 1.900 | A1.5IV |  | — | 1.564
HR3890 | 3.250 | 2.990 | A9 |  | HR3891 | 6.080 | 5.990 | B7III |  | 3.173 | 2.924
∗HR3925 | 5.680 | 5.810 | B5V |  | HD85980B | 8.260 | 8.230 | — |  | 5.584 | 5.699
∗HR4057 | 3.130 | — | K1III |  | HR4058 | 3.400 | — | G7III |  | 2.504 | —
HR4374 | 5.410 | 4.770 | G2V |  | HR4375 | 4.790 | 4.250 | F8.5V |  | 4.304 | 3.727
∗HR4730 | 1.100 | 1.280 | B0.5IV |  | HR4731 | 1.410 | 1.580 | B1V |  | 0.491 | 0.667
HR4825 | 3.800 | 3.440 | F1–F2V |  | HR4826 | 3.850 | 3.490 | F0–F2V |  | 3.072 | 2.712
HR5054 | 2.271 | 2.220 | A1.5V |  | HR5055 | 4.050 | 3.880 | A1–A7IV–V |  | 2.078 | 2.007
HR5328 | 7.080 | 6.690 | F2V |  | HR5329 | 4.740 | 4.510 | A7IV |  | 4.621 | 4.373
HR5459 | 0.720 | 0.010 | G2V |  | HR5460 | 2.210 | 1.330 | K1V |  | 0.475 |  $-$0.272
HR5475 | 4.792 | 4.893 | B9III |  | HR5476 | 5.979 | 5.761 | A6V |  | 4.478 | 4.490
HR5477 | 4.590 | 4.510 | — |  | HR5478 | 4.560 | 4.510 | — |  | 3.822 | 3.757
HR5505 | 4.853 | 4.801 | A0V |  | HR5506 | 3.610 | 2.450 | K0II–III |  | 3.310 | 2.332
HR5788 | 5.380 | 5.130 | F0IV |  | HR5789 | 4.390 | 4.140 | F0IV |  | 4.023 | 3.773
∗HR5977 | 5.350 | 4.870 | F5V |  | HR5978 | 5.640 | 5.160 | F4(V) |  | 4.733 | 4.253
∗HR5984 | 2.550 | 2.620 | B1V |  | HR5985 | 4.870 | 4.890 | B2V |  | 2.429 | 2.493
∗HR6406 | 4.670 | 3.330 | M5Ib–II |  | HR6407 | 6.030 | 5.322 | G5III+F2V |  | 4.397 | 3.169
HR6484 | 5.395 | 5.398 | A0V |  | HR6485 | 4.476 | 5.510 | B9.5III |  | 4.088 | 4.113
HR6896 | 6.320 | 4.860 | K1/2III |  | 21 Sgr B | 7.680 | 7.390 | — |  | 6.047 | 4.759
HR7293 | 7.400 | — | G3V |  | HR7294 | 7.220 | — | G2V |  | 6.554 | —
HR7921 | 6.672 | 5.638 | G8IIb |  | 49 Cyg B | 8.150 | 8.090 | B9.9 |  | 6.424 | 5.530
HR7947 | 5.530 | 4.960 | F8V |  | HR7948 | 5.260 | 4.250 | K1IV |  | 4.634 | 3.795
∗HR8148 | 7.500 | 6.680 | G7V |  | HD202940B | 10.980 | 9.960 | K5 |  | 7.457 | 6.609
HR8309 | 5.140 | 4.700 | F7V |  | HR8310 | 6.670 | 6.120 | F3V |  | 4.903 | 4.440
HR8545 | 6.830 | 6.220 | G2V |  | 53 Aqr B | 6.960 | 6.320 | G3V |  | 6.140 | 5.516
HR8558 | 4.890 | 4.490 | F2IV/V |  | HR8559 | 4.790 | 4.340 | F2IV/V |  | 4.086 | 3.660
HR9074 | 6.910 | 6.400 | F8 |  | BD+32 4747B | 7.210 | 6.660 | G1 |  | 6.297 | 5.770
Table 4: References for the $B$ and $V$ photometric data extracted from the Simbad database and employed for the comparison with the synthetic magnitudes computed in the fitted CK04 models. (1) ADS bibcode, (2) number of stars with $B$ photometry, (3) number of stars with $V$ photometry, and (4) reference. Bibcode | $B$ | $V$ | Reference
---|---|---|---
1965LowOB...6..167A | 1 | 1 | Alcaino (1965)
1966CoLPL...4...99J | 4 | 4 | Johnson et al. (1966)
1967ArA.....4..375L | 2 | 2 | Lodén (1967)
1968ArA.....4..425L | 1 | 1 | Lodén (1968)
1969ArA.....5..149L | 1 | 1 | Lodén (1969a)
1969ArA.....5..161L | 2 | 2 | Lodén (1969b)
1969ArA.....5..231L | 1 | 1 | Lodén & Nordström (1969)
1978A&AS...34....1N | 3 | 3 | Nicolet (1978)
1982A&AS...47..221R | 2 | 1 | Rakos et al. (1982)
1985A&AS...61..331O | 2 | 2 | Oja (1985)
1991A&AS...89..415O | 10 | 7 | Oja (1991)
1993A&AS..100..591O | 52 | 51 | Oja (1993)
1997JApA...18..161Y | 5 | 6 | Yoss & Griffin (1997)
2000A&A...355L..27H | 391 | 390 | Høg et al. (2000)
2001AJ....122.3466M | 0 | 1 | Mason et al. (2001)
2002A&A...384..180F | 37 | 39 | Fabricius et al. (2002)
2002yCat.2237....0D | 788 | 765 | Ducati (2002)
2006AJ....132..111J | 6 | 6 | Joner et al. (2006)
2009ApJ...694.1085V | 0 | 17 | van Belle & von Braun (2009)
2011A&A...531A..92R | 7 | 7 | Röser et al. (2011)
2012yCat.1322....0Z | 4 | 4 | Zacharias et al. (2012)
2013ApJ...764..114H | 1 | 1 | Hsu et al. (2013)
2014ApJ...794...36H | 1 | 1 | Hernández et al. (2014)

After applying the previously described fitting method to the initial sample
of 1522 stars, we discovered that some of the CK04 model fits were not
reliable. After sorting the stars by the minimum value of the objective
function obtained at the end of the minimization process, we visually examined
all the individual fits to establish potential biases in the fitting
procedure. At this point the following criteria were sequentially employed to
remove stars from the initial sample:

  1. 1.

Stars with large objective function values in the minimization process: we
analysed the histograms of the resulting $\chi^{2}$ (Fig. 8), segregating the
sample in two groups, depending on the number of photometric bands available
for each star (13 and 8 bands for 1359 and 163 stars, respectively). In both
cases, although the distributions are roughly normal, there is a tail of stars
with large $\chi^{2}$ values. The $3\sigma$ cut (shown as the dotted vertical
lines) is $\chi^{2}_{3\sigma}\simeq 0.105$ and $0.047$, for the subsamples
with 13 and 8 bands, respectively. We decided to remove the 48 stars with
$\chi^{2}$ above these threshold values. Among them, 43 stars are known
variables according to Simbad or exhibit emission lines (not considered in our
fitting procedure), and 5 show large red or blue fluxes that could not be
properly fitted with the CK04 models.

  2. 2.

Stars with large discrepancies in synthetic Johnson $B$ and $V$ magnitudes
when compared with the available data in the Simbad
database101010http://simbad.u-strasbg.fr/simbad/. These magnitudes were
computed in the VEGA system, using as reference the flux density (factor
$f_{r}(\lambda)$ in Eq. 3) of the Vega spectrum alpha_lyr_stys_010.fits,
available at the CALSPEC
database111111https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec (Bohlin et al., 2014). In
addition, the spectral sensitivity curves for the $B$ and $V$ filters from
Bessell & Murphy (2012, see their Table 1) were employed121212With this
election of flux density reference spectrum and spectral sensitivity curves,
the integrated number of photons $N_{\gamma,r}$ (see Eq. 1) are 1286455 and
873896 photons s-1 cm-2, for the $B$ and $V$ bands, respectivetly. In
addition, the averaged number of photons for the reference spectrum $\langle
n_{\gamma,r}\rangle$ (see Eq. 2) are 1401.67 and 996.80 photons s-1 cm-2 Å-1,
for the $B$ and $V$ bands.. We have chosen these two classical bandpasses
because their wavelength coverage overlaps with that of typical RGB Bayer-like
photometric systems. The expected random errors in these synthetic
measurements, estimated from the bootstrapping strategy described in Sect. 4.2
and displayed in Fig. 9, are small: $\Delta B\sim 0.011$ mag and $\Delta V\sim
0.010$ mag. Since some stars in the original JM75 sample were flagged as
double (i.e., more than one star were observed simultaneously), the
corresponding flux coaddition was performed to compute the expected $B$ and
$V$ magnitudes in those cases (see Table 3; the name of the companion stars
are also provided in the fourth column of Table 2). Interestingly, the spectra
of these double stars did not lead to large $\chi^{2}$ values in Fig. 8,
mainly because the light of the combined spectrum was dominated by the
brightest star in the system or because in several cases the difference in
spectral type was not large. The comparison between the synthetic and the
tabulated Simbad magnitudes is shown in Fig. 10. It is important to highlight
that the Simbad measurements come from a relatively high number of different
sources (see Table 4). Thus, it is expected that the compiled data are
heterogeneous in photometric quality and not completely free from systematic
offsets. This problem, together with additional sources of systematic errors,
such as the inability of the adopted CK04 model fitting procedure to reproduce
the actual star spectra in all cases, the presence of unaccounted stellar
variability, and the inherent photometric uncertainties within the C13
photometric data themselves, make the correlation exhibited by the magnitude
differences in Fig. 10(a) not unexpected. In any case, an independent analysis
of $B_{\rm CK04}\\!-\\!B_{\rm Simbad}$ and $V_{\rm CK04}\\!-\\!V_{\rm Simbad}$
has been performed, as displayed in Figs. 10(b) and (c). We decided to follow
a statistical approach to remove from the star sample those objects with large
deviations in either $B$ or $V$. For that purpose, we first computed the
median values ($-0.014$ mag and $-0.021$ mag for $B_{\rm CK04}\\!-\\!B_{\rm
Simbad}$ and $V_{\rm CK04}\\!-\\!V_{\rm Simbad}$, respectively) and rejected
those stars outside the robust $\pm 3\sigma$ interval around the median value
($\pm 0.107$ mag and $\pm 0.077$ mag). A total of 128 stars were removed from
the sample, being $\sim 2/3$ of this rejected subsample (82 stars) constituted
by known variables.

Figure 11: Examples of spectrum fits to stars with different effective
temperature. The filled circles correspond to the C13 photometric data (flux
densities in arbitrary units) from JM75. The continuous lines are the fitted
CK04 models. The spectra have been numbered by decreasing fitted $T_{\rm eff}$
(from Table 2), and their identifications are given in the upper key.

After this cleaning process, the final sample of fitted CK04 models is formed
by 1346 stars, that constitute the UCM (Universidad Complutense de Madrid)
library of spectrophotometric standards. Although an important fraction of
them ($\sim 44$%, 594 stars) are still classified as known variables, we have
decided to keep these objects in the final list since no significant
differences with the Simbad tabulated measurements have been found. Some of
these variables are eclipsing binaries, and most of the time they constitute
suitable reference stars. In any case, columns (5) and (6) in Table 2 provide
an additional name and the corresponding classification (according to Samus’
et al., 2017)131313See also a detailed description of the variability types in
http://cdsarc.u-strasbg.fr/viz-
bin/getCatFile_Redirect/?-plus=-%2b&B/gcvs/vartype.txt for the variable stars,
and potential users of the derived RGB magnitudes should be well aware of
this.

A selection of spectrum fits for stars exhibiting a wide range in effective
temperature is shown in Fig. 11.

###  4.4 Comparison with Kiehling 1987

Figure 12: Comparison between the best CK04 fitted models and the
spectrophotometric data from K87, for 4 stars (out of 39) in common with our
final sample (all the plots corresponding to the 39 stars in common with K87
are provided in Appendix A). CK04 models are represented with thick blue lines
for non-variable stars and thick orange lines for variable stars, while the
K87 spectra are plotted with thin black lines. For each star, the upper panel
displays the flux densities. The lower panel shows the corresponding
residuals, in magnitudes, obtained when dividing the fluxes from CK04 models
by the K87 measurements (continuous blue/orange line) or by the JM75
photometric data (red crosses); the dotted line marks the $\Delta$mag=0.0
level, whereas the dashed lines encompass the $\pm 0.1$ mag interval. For the
variable stars one additional identification name is given in parenthesis. The
C13 photometric data from JM75 are plotted with red crosses. The standard RGB
spectral sensitivity curves, as defined in Section 5.1, are shown with blue,
green, and red shaded areas, whereas the vertical grey bands correspond to the
wavelength intervals where conspicuous telluric absorptions appear in the K87
spectra. See Sec. 4.4 for additional details.

To assess the quality of the fitting procedure just described, we have
compared the compiled C13 photometric data and the fitted CK04 models with
observed flux calibrated spectra. For that purpose, we have chosen the data
from Kiehling (1987, hereafter K87), who published high-quality spectral
energy distributions for 60 bright stars in the [3200–8600] Å wavelength
interval, with typical internal flux errors of 0.02 mag (above 4000 Å) and
0.05 mag (below 4000 Å), when comparing observations from different nights.
Fortunately, there were 39 stars in common between our final star sample and
that from K87. Although 25 of them are known variable stars, according to
Simbad, there is a very good agreement between K87 with both the C13 color
photometric data and the corresponding CK04 fitted models.

The graphical comparison is shown in Fig. 12 (extended by Fig. A1). The
wavelength regions of strong O2 and H2O telluric absorptions (wavelength
intervals [6850–6950] Å, [7150–7350] Å, [7550–7650] Å, and [8150–8350] Å) were
not corrected in the K87 data, and have been marked with a grey background in
the displayed plots. The C13 measurements are plotted with filled red circles.
The best CK04 model fits to these data are shown with thick blue (for non-
variable stars) or orange (for variable stars) lines. The K87 spectra are
overplotted with thin black lines. It is very important to highlight that the
K87 spectra displayed in this figure are not fits to the C13 data, but
calibrated spectral energy distributions, independently calibrated from the
JM75 data. The resolution of these spectra was slightly reduced using a
Gaussian kernel of 600 km/s in order to match the spectral resolution
exhibited by the CK04 models141414The kernel width was determined using the
movel utility of the REDucm E package (Cardiel, 1999);
https://reduceme.readthedocs.io/en/latest/. The 39 stars are sorted by HR
name, and for each one two panels are shown: the top plot represents the flux
density (erg s-1 cm-2 Å-1); the lower panel shows the corresponding residuals,
in magnitudes, obtained when dividing the fluxes from CK04 models by the K87
measurements (continuous blue/orange line) or by the JM75 photometric data
(filled red circles). The dotted horizontal line in the residuals panel sets
the $\Delta{\rm mag}=0$ level, while the two dashed lines encompass the $\pm
0.1$ mag interval. For each star, the residuals panel also provides two
residuals summaries: $\Delta{\rm mag}_{\rm K87}$, the median and robust
standard deviation when dividing the best CK04 model by the corresponding K87
spectrum (avoiding the marked telluric absorption regions), and $\Delta{\rm
mag}_{\rm C13}$, the median and robust standard deviation when computing the
ratio between the synthetic C13 fluxes (computed with the best CK04 model and
the transmission curves displayed in Fig. 1) and the C13 measurements compiled
in this work. In this sense, $\Delta{\rm mag}_{\rm K87}$ provides an
indication of how well the K87 spectra agree with the JM75 photometric
measurements, while $\Delta{\rm mag}_{\rm C13}$ summarizes the quality of the
CK04 model fit to the JM75 data. The scatter in both parameters is in most
cases (for non-variable stars) below $\pm 0.05$ mag.

##  5 Synthetic RGB magnitudes

###  5.1 Definition of a camera-independent RGB standard system

Figure 13: Comparison of RGB spectral sensitivity curves: the thin lines correspond to the 28 cameras measured by Jiang et al. (2013): Canon 1DMarkIII, Canon 20D, Canon 300D, Canon 40D, Canon 500D, Canon 50D, Canon 5DMarkII, Canon 600D, Canon 60D, Hasselblad H2, Nikon D3X, Nikon D200, Nikon D3, Nikon D300s, Nikon D40, Nikon D50, Nikon D5100, Nikon D700, Nikon D80, Nikon D90, Nokia N900, Olympus E-PL2, Pentax K-5, Pentax Q, Point Grey Grasshopper 50S5C, Point Grey Grasshopper2 14S5C, Phase One, and SONY NEX-5N. Then median values at each sampled wavelength are plotted as the thick lines, and the corresponding values are listed in Table 5. Table 5: Renormalized median spectral sensitivity curves computed from the empirical sensitivity curves corresponding to the 28 cameras measured by Jiang et al. (2013) and displayed in Fig. 13. Column description: (1) wavelength (in Å); and (2)–(4) spectral sensitivity at each considered wavelength, for the median B, G and R bandpasses, respectively. In order to constraint the curves at the borders of the initial 4000–7200 Å interval sampled by the Jiang et al. (2013) measurements, two additional rows have been introduced in the table, at 3990 and 7210 Å, forcing the curves to drop rapidly to zero at these wavelengths. The information in this table is electronically available at http://guaix.ucm.es/rgbphot. (1) | (2) | (3) | (4)
---|---|---|---
$\lambda$ (Å) | standard $B$ | standard $G$ | standard $R$
3990 | 0.0000000 | 0.0000000 | 0.0000000
4000 | 0.0150428 | 0.0030840 | 0.0034970
4100 | 0.1039736 | 0.0113570 | 0.0103892
4200 | 0.4892935 | 0.0388725 | 0.0238270
4300 | 0.7202255 | 0.0575395 | 0.0277668
4400 | 0.8216436 | 0.0791910 | 0.0180056
4500 | 0.9308637 | 0.1006650 | 0.0180360
4600 | 1.0000000 | 0.1360232 | 0.0218683
4700 | 0.9802917 | 0.2571178 | 0.0299132
4800 | 0.9275882 | 0.3809050 | 0.0339620
4900 | 0.7807393 | 0.4251800 | 0.0401877
5000 | 0.6143757 | 0.6113000 | 0.0430846
5100 | 0.4338580 | 0.7933000 | 0.0625152
5200 | 0.2491595 | 0.9033850 | 0.1111756
5300 | 0.1594246 | 1.0000000 | 0.1419566
5400 | 0.0947855 | 0.9064100 | 0.0849897
5500 | 0.0567221 | 0.8807233 | 0.0478130
5600 | 0.0273214 | 0.7437300 | 0.0521587
5700 | 0.0166126 | 0.6428150 | 0.1533735
5800 | 0.0114100 | 0.4597650 | 0.6503433
5900 | 0.0084778 | 0.3175050 | 1.0000000
6000 | 0.0048916 | 0.1819950 | 0.9353758
6100 | 0.0034230 | 0.0897230 | 0.8337379
6200 | 0.0029658 | 0.0485390 | 0.6858826
6300 | 0.0032853 | 0.0316045 | 0.5929939
6400 | 0.0037959 | 0.0229870 | 0.4600072
6500 | 0.0051010 | 0.0167670 | 0.3717754
6600 | 0.0050765 | 0.0112830 | 0.2205769
6700 | 0.0032660 | 0.0088190 | 0.1200198
6800 | 0.0011294 | 0.0046771 | 0.0371512
6900 | 0.0005127 | 0.0016871 | 0.0126072
7000 | 0.0002948 | 0.0007490 | 0.0037169
7100 | 0.0001017 | 0.0003077 | 0.0012105
7200 | 0.0000616 | 0.0001488 | 0.0005449
7210 | 0.0000000 | 0.0000000 | 0.0000000

Since the spectral sensitivity curves of Bayer-like color filter systems vary
between different camera models, we have decided to define a particular set of
median sensitivity curves that can be adopted as a camera-independent RGB
standard system. For that purpose, we initially compared the 28 spectral
sensitivity curves measured by Jiang et al. (2013), which are displayed in
Fig. 13 (thin lines), and then we computed the median value at each sampled
wavelength (from 4000 Å to 7200 Å, with $\Delta\lambda=100$ Å). The resulting
median curves, that are represented with thick lines in the same figure and
are tabulated in Table 5, can be adopted to define a new standard RGB system.

We have checked that the spectral sensitivity curves gathered by Jiang et al.
(2013) already included response curves that are quite similar to those found
in more recent camera models (see e.g. Fig. 1 in Sánchez de Miguel et al.,
2019), and we have decided not to include these additional curves in order to
avoid biasing the median responses to a particular camera manufacturer.

Table 6: Characteristic parameters of the standard photometric system employed to determine the synthetic RGB magnitudes. The quoted numbers correspond to the spectral sensitivity curves listed in Table 5. Column description (see more details in Sect. 5.1): (1) bandpass name; (2) wavelength at the peak of the spectral sensitivity curve; (3) average wavelength; (4) pivot wavelength (5) bandpass equivalent width; (6) bandpass r.m.s. width; (7) integrated number of photons, as defined in Eq. 1, for the reference flux density of the AB system; (8) averaged number of photons, given by Eq. 2, for the reference flux density of the AB system; (9) magnitude difference when using the reference flux density of the ST system (Eq. 15) instead of the one for the AB system (Eq. 14); and (10) magnitude difference when using the Vega spectrum alpha_lyr_stys_010.fits as the reference flux density instead of the reference flux density of the AB system (Eq. 14). (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
---|---|---|---|---|---|---|---|---|---
RGB | wpeak | avgwave | pivot | equivwidth | rmswidth | $N_{\gamma,r}$ | $\langle n_{\gamma,r}\rangle$ | $m_{\rm AB}\\!-\\!m_{\rm ST}$ | $m_{\rm AB}\\!-\\!m_{\rm Vega}$
bandpass | (Å) | (Å) | (Å) | (Å) | (Å) | (photons s-1 cm-2) | (photons s-1 cm-2 Å-1) | (mag) | (mag)
standard $B$ | 4600.00 | 4691.29 | 4680.11 | 846.89 | 331.52 | 993989 | 1173.69 |  $-$0.341 |  $-$0.124
standard $G$ | 5300.00 | 5323.53 | 5308.56 | 916.67 | 395.38 | 948922 | 1035.18 |  $-$0.067 |  $-$0.024
standard $R$ | 5900.00 | 6006.96 | 5989.64 | 684.86 | 426.98 | 628378 |  1917.52 |  $-$0.195 |  $-$0.103

Some basic bandpass properties that can be easily derived, once the spectral
sensitivity curves are defined, are provided in Table 6: in particular, the
wavelength at the peak of the spectral sensititivy curve (wpeak property in
synphot), the average and pivot wavelengths, as defined in Koornneef et al.
(1986, avgwave and pivot properties in synphot) computed as

$\displaystyle{\rm
avgwave}=\frac{\int_{\lambda_{i}}^{\lambda_{f}}\lambda\;T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}T(\lambda)\;{\rm d}\lambda}$ (10)

and

$\displaystyle{\rm
pivot}=\frac{\int_{\lambda_{i}}^{\lambda_{f}}\lambda\;T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}\frac{T(\lambda)}{\lambda}\;{\rm
d}\lambda},$ (11)

the bandpass equivalent width (equivwidth in synphot), which is the
normalization factor in Eq. 2, and the r.m.s. of the bandpass following
Koornneef et al. (1986, rmswidth in synphot)

$\displaystyle{\rm
rmswidth}=\sqrt{\frac{\int_{\lambda_{i}}^{\lambda_{f}}(\lambda-{\rm
avgwave})^{2}\,T(\lambda)\;{\rm
d}\lambda}{\int_{\lambda_{i}}^{\lambda_{f}}T(\lambda)\;{\rm d}\lambda}}.$ (12)

The synthetic RGB magnitudes computed in this work, and described in the next
section, were computed in the AB system, in which the flux density reference
spectrum is defined to exhibit a constant flux density per unit frequency (Oke
& Gunn, 1983), given by

$\displaystyle f_{r}(\nu)=10^{-48.60/2.5}\simeq 3.6308\times 10^{-20}~{}{\rm
erg}~{}{\rm s}^{-1}~{}{\rm cm}^{-2}~{}{\rm Hz}^{-1}.$ (13)

This expression can easily be converted into a flux density per unit
wavelength as

$\displaystyle
f_{r}(\lambda)=\frac{c}{\lambda^{2}}\,f_{r}(\nu)\simeq\frac{0.10885}{\lambda^{2}}\;\text{erg}~{}\text{s}^{-1}~{}\text{cm}^{-2}~{}{\text{\AA}}^{-1},$
(14)

with $\lambda$ expressed in Å. Inserting the last expression in Eqs. 1 and 2,
we can compute additional relevant parameters for each RGB spectral
sensitivity curve defined in this section, such as $N_{\gamma,r}$ and $\langle
n_{\gamma,r}\rangle$, which are also given in Table 6.

Figure 14: Comparison between the flux density (in photons s-1 cm-2 Å-1) of
the reference spectrum used to define the AB magnitudes (dashed black line;
Eq. 14), the ST magnitudes (dotted black line; Eq. 15), and Vega magnitudes
(full black line). The standard RGB spectral sensitivity curves, defined in
Table 5, are shown with blue, green and red shaded areas. The units employed
in this figure (known in synphot as PHOTLAM) allow a direct visual comparison
with the $\langle n_{\gamma,r}\rangle$ values listed in Table 6.

For completeness, and although in this paper we strongly advocate the use of
AB magnitudes, we also indicate, in the last two columns of Table 6, the
expected offsets when measuring ST magnitudes, in which the reference flux
densitity per unit wavelength is constant (Koornneef et al., 1986) and equal
to

$\displaystyle f_{r}(\lambda)=10^{-21.10/2.5}\simeq 3.6308\times
10^{-9}~{}\text{erg}~{}\text{s}^{-1}~{}\text{cm}^{-2}~{}{\text{\AA}}^{-1},$
(15)

or when employing Vega magnitudes, using for that purpose the flux density
provided by the spectrum alpha_lyr_stys_010.fits as reference. As expected,
the differences between the three different systems are smaller in the $G$
band, where the corresponding reference flux densities intersect (See Fig.
14).

###  5.2 Synthetic RGB magnitudes for the bright star sample

Figure 15: Distribution of random uncertainties in the synthetic standard RGB
measurements performed on the final sample of fitted CK04 models, estimated
from the bootstrapped spectra generated as explained in Sect. 4.2. The median
values are $\Delta B=0.011$ mag (top panel), $\Delta G=0.010$ mag (middle
panel) and $\Delta R=0.010$ mag (bottom panel).

Figure 16: Differences between the synthetic RGB magnitudes computed using the best CK04 model fit and those determined from the spectra available in the K87 sample. Known variable stars are plotted with orange crosses, whereas normal stars (i.e., non-variable) are represented with blue filled circles. The horizontal dotted line in each plot marks the $\Delta{\rm mag}$=0.0 level, and the two horizontal dashed lines encompass the $\pm 0.1$ mag region. All the non-variable stars are clearly within the latter interval (the variable stars beyond the $\pm 0.1$ region have been labelled). The median and standard deviation, using the non-variable stars, are $0.000\pm 0.028$ mag for the $B$ bandpass (top panel), $-0.014\pm 0.022$ mag for the $G$ bandpass (middle panel), and $-0.023\pm 0.022$ mag for the $R$ bandpass (bottom panel). Figure 17: Color–color diagram, using the RGB magnitudes measured in the final sample of 1346 stars, using the standard photometric system defined in Sect. 5.2. The black line shows the 7-degree polynomial fit given in Eq. 5.2. The residuals around the fit are displayed in the lower panel, where the horizontal dashed lines encompass the $\pm 0.01$ mag interval. For stars with $G-R\leq 0.2$ mag, the standard deviation of the residuals is 0.004 mag for both normal (i.e. non-variable) and variable stars, whereas for stars with redder colors the standard deviations are 0.012 mag and 0.032 mag for the normal and variable stars, respectively. Table 7: Comparison between the synthetic RGB magnitudes measured in the best CK04 fit models and those determined from the K87 flux calibrated spectra. The 39 stars are those already shown in Fig. 12. Column description: (1) HR number; (2) additional name when the star is a known variable according to Simbad; (3)–(5) synthetic RGB magnitudes (AB system), measured in the best CK04 fit, using the bandpass definitions given in Table 5, with uncertainties estimated from bootstrapping; (6)–(8) synthetic RGB magnitudes (AB system), measured in the flux calibrated K87 spectrum. (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
---|---|---|---|---|---|---|---
HR | Var.Name | standard $B$ (CK04) | standard $G$ (CK04) | standard $R$ (CK04) | standard $B$ (K87) | standard $G$ (K87) | standard $R$ (K87)
1325 | — | 4.794$\;\;\pm\;$ | 0.010 | 4.462$\;\;\pm\;$ | 0.009 | 4.215$\;\;\pm\;$ | 0.009 | 4.787 | 4.498 | 4.265
1457 | alf Tau | 1.758$\;\;\pm\;$ | 0.018 | 1.001$\;\;\pm\;$ | 0.011 | 0.445$\;\;\pm\;$ | 0.012 | 1.715 | 1.012 | 0.497
1654 | eps Lep | 3.969$\;\;\pm\;$ | 0.012 | 3.296$\;\;\pm\;$ | 0.012 | 2.819$\;\;\pm\;$ | 0.011 | 3.972 | 3.310 | 2.818
1829 | NSV 2008 | 3.198$\;\;\pm\;$ | 0.013 | 2.866$\;\;\pm\;$ | 0.011 | 2.641$\;\;\pm\;$ | 0.010 | 3.177 | 2.867 | 2.649
1865 | NSV 2128 | 2.566$\;\;\pm\;$ | 0.012 | 2.559$\;\;\pm\;$ | 0.012 | 2.578$\;\;\pm\;$ | 0.013 | 2.581 | 2.576 | 2.594
1983 | — | 3.758$\;\;\pm\;$ | 0.010 | 3.600$\;\;\pm\;$ | 0.008 | 3.502$\;\;\pm\;$ | 0.008 | 3.761 | 3.613 | 3.516
2085 | — | 3.804$\;\;\pm\;$ | 0.010 | 3.740$\;\;\pm\;$ | 0.009 | 3.722$\;\;\pm\;$ | 0.010 | 3.826 | 3.756 | 3.724
2326 | — |  $-$0.797$\;\;\pm\;$ | 0.011 |  $-$0.774$\;\;\pm\;$ | 0.010 |  $-$0.728$\;\;\pm\;$ | 0.010 |  $-$0.781 |  $-$0.768 |  $-$0.727
2646 | sig CMa | 4.464$\;\;\pm\;$ | 0.017 | 3.619$\;\;\pm\;$ | 0.013 | 3.017$\;\;\pm\;$ | 0.011 | 4.502 | 3.687 | 3.124
2693 | NSV 3424 | 2.078$\;\;\pm\;$ | 0.011 | 1.861$\;\;\pm\;$ | 0.009 | 1.720$\;\;\pm\;$ | 0.008 | 2.019 | 1.774 | 1.610
2943 | alf CMi A | 0.481$\;\;\pm\;$ | 0.009 | 0.358$\;\;\pm\;$ | 0.007 | 0.287$\;\;\pm\;$ | 0.006 | 0.495 | 0.378 | 0.307
3153 | V460 Car | 6.256$\;\;\pm\;$ | 0.022 | 5.348$\;\;\pm\;$ | 0.013 | 4.691$\;\;\pm\;$ | 0.015 | 6.150 | 5.311 | 4.714
3185 | rho Pup | 2.884$\;\;\pm\;$ | 0.010 | 2.771$\;\;\pm\;$ | 0.008 | 2.709$\;\;\pm\;$ | 0.008 | 2.941 | 2.824 | 2.760
3188 | — | 4.837$\;\;\pm\;$ | 0.010 | 4.427$\;\;\pm\;$ | 0.009 | 4.151$\;\;\pm\;$ | 0.009 | 4.789 | 4.414 | 4.156
3249 | NSV 3973 | 4.338$\;\;\pm\;$ | 0.012 | 3.643$\;\;\pm\;$ | 0.013 | 3.148$\;\;\pm\;$ | 0.012 | 4.331 | 3.665 | 3.172
3438 | — | 4.389$\;\;\pm\;$ | 0.013 | 4.000$\;\;\pm\;$ | 0.011 | 3.737$\;\;\pm\;$ | 0.010 | 4.409 | 4.031 | 3.769
3547 | — | 3.643$\;\;\pm\;$ | 0.011 | 3.216$\;\;\pm\;$ | 0.010 | 2.920$\;\;\pm\;$ | 0.009 | 3.599 | 3.193 | 2.913
3748 | alf Hya | 2.761$\;\;\pm\;$ | 0.015 | 2.103$\;\;\pm\;$ | 0.010 | 1.637$\;\;\pm\;$ | 0.011 | 2.730 | 2.089 | 1.632
3873 | NSV 4613 | 3.368$\;\;\pm\;$ | 0.013 | 3.045$\;\;\pm\;$ | 0.011 | 2.827$\;\;\pm\;$ | 0.010 | 3.390 | 3.067 | 2.842
4050 | V337 Car | 4.226$\;\;\pm\;$ | 0.012 | 3.511$\;\;\pm\;$ | 0.012 | 3.014$\;\;\pm\;$ | 0.011 | 4.289 | 3.624 | 3.164
4114 | NSV 4869 | 3.798$\;\;\pm\;$ | 0.013 | 3.748$\;\;\pm\;$ | 0.010 | 3.733$\;\;\pm\;$ | 0.009 | 3.805 | 3.795 | 3.802
4362 | FN Leo | 5.565$\;\;\pm\;$ | 0.017 | 4.731$\;\;\pm\;$ | 0.009 | 4.147$\;\;\pm\;$ | 0.010 | 5.540 | 4.774 | 4.245
4517 | nu. Vir | 4.873$\;\;\pm\;$ | 0.032 | 4.143$\;\;\pm\;$ | 0.019 | 3.617$\;\;\pm\;$ | 0.018 | 4.869 | 4.176 | 3.675
4540 | — | 3.776$\;\;\pm\;$ | 0.011 | 3.579$\;\;\pm\;$ | 0.009 | 3.450$\;\;\pm\;$ | 0.009 | 3.818 | 3.638 | 3.519
4630 | eps Crv | 3.719$\;\;\pm\;$ | 0.008 | 3.117$\;\;\pm\;$ | 0.008 | 2.691$\;\;\pm\;$ | 0.011 | 3.695 | 3.104 | 2.686
4763 | gam Cru | 2.591$\;\;\pm\;$ | 0.013 | 1.763$\;\;\pm\;$ | 0.014 | 1.205$\;\;\pm\;$ | 0.019 | 2.520 | 1.782 | 1.281
4786 | bet Crv | 3.082$\;\;\pm\;$ | 0.009 | 2.710$\;\;\pm\;$ | 0.008 | 2.456$\;\;\pm\;$ | 0.008 | 3.045 | 2.698 | 2.469
4932 | NSV 6064 | 3.314$\;\;\pm\;$ | 0.012 | 2.921$\;\;\pm\;$ | 0.011 | 2.649$\;\;\pm\;$ | 0.011 | 3.285 | 2.928 | 2.673
5019 | — | 5.061$\;\;\pm\;$ | 0.011 | 4.774$\;\;\pm\;$ | 0.009 | 4.574$\;\;\pm\;$ | 0.008 | 5.044 | 4.787 | 4.604
5072 | — | 5.322$\;\;\pm\;$ | 0.012 | 5.037$\;\;\pm\;$ | 0.010 | 4.841$\;\;\pm\;$ | 0.009 | 5.299 | 5.027 | 4.836
5235 | NSV 19993 | 2.887$\;\;\pm\;$ | 0.012 | 2.682$\;\;\pm\;$ | 0.010 | 2.546$\;\;\pm\;$ | 0.010 | 2.919 | 2.720 | 2.585
5340 | alf Boo | 0.573$\;\;\pm\;$ | 0.010 | 0.008$\;\;\pm\;$ | 0.008 |  $-$0.397$\;\;\pm\;$ | 0.010 | 0.520 | 0.007 |  $-$0.375
5459 | — | 0.039$\;\;\pm\;$ | 0.014 |  $-$0.244$\;\;\pm\;$ | 0.012 |  $-$0.441$\;\;\pm\;$ | 0.011 | 0.014 |  $-$0.241 |  $-$0.424
5854 | NSV 20391 | 3.232$\;\;\pm\;$ | 0.009 | 2.718$\;\;\pm\;$ | 0.008 | 2.356$\;\;\pm\;$ | 0.010 | 3.219 | 2.726 | 2.371
5868 | lam Ser | 4.640$\;\;\pm\;$ | 0.014 | 4.431$\;\;\pm\;$ | 0.011 | 4.292$\;\;\pm\;$ | 0.010 | 4.673 | 4.462 | 4.316
6030 | — | 4.372$\;\;\pm\;$ | 0.008 | 3.901$\;\;\pm\;$ | 0.008 | 3.583$\;\;\pm\;$ | 0.008 | 4.397 | 3.935 | 3.635
6102 | — | 4.305$\;\;\pm\;$ | 0.012 | 3.925$\;\;\pm\;$ | 0.008 | 3.653$\;\;\pm\;$ | 0.008 | 4.303 | 3.940 | 3.684
6159 | NSV 7812 | 5.644$\;\;\pm\;$ | 0.012 | 4.933$\;\;\pm\;$ | 0.011 | 4.418$\;\;\pm\;$ | 0.010 | 5.648 | 4.972 | 4.467
6623 | — | 3.752$\;\;\pm\;$ | 0.009 | 3.467$\;\;\pm\;$ | 0.008 | 3.269$\;\;\pm\;$ | 0.008 | 3.792 | 3.502 | 3.297

Using the median sensitivity curves defined in the previous subsection, we
measured synthetic RGB magnitudes in the AB system over the final sample of
1346 CK04 models fitted to the C13 photometric data. The results are given in
columns (26)–(28) of Table 2. The uncertainties in each case were derived from
the standard deviations of the different values computed when using the
bootstrapped spectra generated during the fitting procedure (Sect. 4.2).
Histograms with these uncertainties are displayed in Fig. 15. Not
surprisingly, the median uncertainties are similar to those previously found
for the Johnson $B$ and $V$ filters, in particular $\Delta B=0.011$ mag,
$\Delta G=0.010$ mag, and $\Delta R=0.010$ mag.

Another interesting exercise is the comparison of the synthetic RGB magnitudes
with those computed using the K87 sample. The synthetic RGB magnitudes for the
39 stars in common are listed in Table 7. The differences in each bandpass, as
a function of the $B-R$ color, are plotted in Fig. 16. The median and standard
deviation of the differences, using only the 14 non-variable stars in this
subsample, are $0.000\pm 0.028$ mag, $-0.014\pm 0.022$ mag, and $-0.023\pm
0.022$ mag, for the $B$, $G$, and $R$ bandpasses respectively. The dispersion
is up to 3 times larger than the typical uncertainties previously estimated
from the bootstrapping analysis, which should be at least in part attributable
to the internal flux errors in the K87 spectra (already mentioned in Sect.
4.4), and we cannot discard the presence of a small systematic offset between
the $G$ and $R$ measurements when comparing the CK04 and K87 spectra. In
addition, most variable stars in Fig. 16 follow the same trend exhibited by
the non-variable stars, with a larger dispersion towards redder colors. This
behaviour is consistent with the results shown in Figs. 12 and A1, where
several variable stars (plotted with orange lines) already exhibited non-
negligible differences when comparing the best CK04 fitted model with the K87
spectrum.

A color–color diagram is shown in Fig. 17, where a tight correlation between
the standard $B-G$ and $G-R$ colors is manifest, that can be well reproduced
by the 7-degree polynomial given by

$\displaystyle(B-G)=$ $\displaystyle\mbox{}\;\;0.04162407+1.08718859\,(G-R)+$
$\displaystyle+0.31438309\,(G-R)^{2}+6.07961811\,(G-R)^{3}+$
$\displaystyle-10.8882237\,(G-R)^{4}-65.9762145\,(G-R)^{5}+$
$\displaystyle+216.580798\,(G-R)^{6}-174.464510\,(G-R)^{7},$ (16)

which is valid for $-0.22<G\\!-\\!R<0.59$. The residuals around this fit,
computed using only the normal (i.e. non-variable) stars, exhibit a scatter
well constrained within $\pm 0.01$ mag for stars with $G-R\leq 0.2$ mag. Note,
however, that the scatter increases towards redder colors, where the $\log g$
range covered by common stars is considerably large.

###  5.3 Comparison between different RGB systems

Figure 18: Differences between the RGB magnitudes computed using the median
sensitivity curves of Table 5 (subindex ’standard’) and the ones determined
with each one of the 28 individual sets of RGB sensitivity curves displayed in
Fig. 13 (subindex ’camera’), as a function of the $B-R$ color of the
considered camera. The three RGB filters are represented separately ($B$, $G$,
and $R$ from top to bottom). The differences have been computed for the 1346
stars composing the final star sample listed in Table 2, and for the 28
cameras listed in the key that appears in the inset of the bottom panel. A
9-degree polynomial has been fitted to each filter and camera (continuous
lines; polynomial coefficients given in Appendix B), while the fit residuals
are displayed at the bottom panel of each filter plot.

In order to test the potential usability of the standard RGB system defined
with the median sensitivity curves, it is important to check how easy will be
the transformation from RGB magnitudes measured with a typical camera, to the
mentioned standard system. For that purpose, we computed the synthetic RGB
magnitudes over the whole final sample of 1346 stars, using the 28 sets of RGB
filters compiled by Jiang et al. (2013) and displayed in Fig. 13.

In Fig. 18 we represent the difference between the RGB magnitudes computed
using the median spectral sensitivity curves provided in Table 5 and the ones
measured with each of the 28 individual camera sets. In most cases, the
differences are of the order of a few tenths of a magnitude, although larger
corrections are necessary for the bluest stars in the $B$ band and for the
reddest stars in the $R$ band. Without entering into unnecessary details, this
behaviour is not surprising considering the variations of the different RGB
sensitivity curves in Fig. 13. It is important to emphasize here that, more
important than how large the differences between a particular RGB filter set
and the standard RGB system are, the relevant result is that these differences
can be reasonably well modelled by smooth polynomials. For illustration, the
displayed data for each filter and camera have been fitted using a 9-degree
polynomial, overplotted as continuous line with the same color as the one
chosen for each RGB set, with the fit residuals displayed at the bottom plot
of all the panels. The resulting polynomial coefficients for 28 considered
cameras are listed in Table B1, together with the median and standard
deviation of the residuals. In most cases, the residuals scatter is below 0.01
mag. Based on these results, we conclude that observations performed with most
cameras can be transformed into the standard RGB system by using smooth
polynomial fits.

##  6 Conclusions

The availability of a catalogue of bright standard stars with RGB magnitudes,
measured in a well-defined photometric system, is essential to scientifically
exploit the large amount of data that at present can be gathered with high-
quality and inexpensive digital cameras equipped with Bayer-like color
filters.

For the work presented in this paper we have used high-quality 13 medium-
narrow-band photometric data to fit stellar atmosphere models, in order to
obtain spectral energy distributions for bright stars. The reliability of
these fits has been checked by comparing the synthetic Johnson $B$ and $V$
magnitudes computed on the fitted models with the corresponding data available
through the Simbad database. This comparison has shown that the $\pm 3\sigma$
scatter in both bands is around or below 0.10 mag, being part of this
dispersion attributable to the heterogenous source compilation in the Simbad
data. The initial star sample, composed by 1522 objects, was cleaned by
removing bad fits and stars with discrepant Johnson $B$ and $V$ magnitudes,
reducing the final sample to 1346 objects. As an additional check, we have
compared a subset of 39 fitted models with actual absolute flux calibrated
spectra from the literature, which revealed that the typical standard
deviation in the fitted 13 photometric bandpasses is 0.05 mag, being part of
this dispersion attributable to the flux calibrated spectra. Furthermore, the
fitted 13 band fluxes for each star were randomly modified (assuming typical
uncertainties of 0.02 mag) in order to generate bootstrapped versions of the
fitted models.

The whole set of fitted star models, that constitute the UCM library of
spectrophotometric standards, together with their associated bootstrapped
versions, have been employed to compute synthetic RGB magnitudes and
uncertainties. Prior to this step, a new RGB photometric system, that can be
used as a standard reference, has been established, by defining standard RGB
sensitivity curves as the median of the corresponding curves of 28 commercial
cameras. The typical random uncertainties in the synthetic RGB magnitudes,
computed only from the bootstrapped spectra, are close to $0.01$ mag. These
uncertainties are 3 times larger when we compare the synthetic RGB magnitudes
with the corresponding values in the subsample of 39 stars with flux
calibrated spectra. However, the unaccounted uncertainties in the flux
calibrated data should have a contribution to this budget. In addition, we
cannot discard a small systematic deviation of $-0.01$ and $-0.03$ mag in the
$G$ and $R$ bandpasses, although these numbers rely on the analysis of only 14
non-variable stars.

The feasibility of using the new RGB photometric bandpasses defined in this
work as a standard RGB system, has been demonstrated by computing simple
polynomial transformations that model the differences between the RGB
magnitudes derived employing the standard system and the ones obtained using
28 individual sets of RGB sensitivity curves of real cameras, with a typical
scatter around these polynomial fits within 0.01 mag.

Since non-variable stars constitue ideal radiometric references, one immediate
application of this work is the transformation of the sky in an accessible and
free laboratory for the proper calibration of the high volume of already
existing (and future) digital cameras. Obviously, a proper calibration will
require the corresponding observational effort and subsequent data reduction
and analysis, but the repeatability of measurements should facilitate to reach
calibration accuracies comparable with those achievable in radiometric
laboratories. It is important to emphasize that the synthetic stellar library
include a non-negligible number of variable stars (594 out of 1346 objects),
but they have been kept in the final sample because they did not present
discrepant $B$ and $V$ magnitudes in the Simbad database. In any case, their
use and validity should be subject to a careful analysis.

We hope that the catalogue of 1346 flux calibrated stellar spectra presented
here, that by itself already constitutes a library of bright
spectrophotometric standards suitable for spectroscopic calibrations, and the
corresponding synthetic RGB magnitudes, can be used as a reference for future
work on several astronomical fields, where the collaboration of many observers
equipped with high-quality digital cameras may provide data that facilitate
the research advancement. In addition, this could help to make citizen science
a reality in the realm of astronomy, increasing the public’s interest and
understanding of science, highlighting the fact that scientific research
matters.

## Acknowledgements

The authors are grateful for the exceptionally careful reading by the referee,
whose constructive remarks have helped to improve the paper, making the text
more precise and readable. The authors acknowledge financial support from the
Spanish Programa Estatal de I+D+i Orientada a los Retos de la Sociedad under
grant RTI2018-096188-B-I00, which is partly funded by the European Regional
Development Fund (ERDF), S2018/NMT-4291 (TEC2SPACE-CM), and ACTION, a project
funded by the European Union H2020-SwafS-2018-1-824603. SB acknowledges Xunta
de Galicia for financial support under grant ED431B 2020/29. The participation
of ASdM was (partially) supported by the EMISSI@N project (NERC grant
NE/P01156X/1). This work has been possible thanks to the extensive use of
IPython and Jupyter notebooks (Pérez & Granger, 2007). This research made use
of Astropy,151515http://www.astropy.org a community-developed core Python
package for Astronomy (Astropy Collaboration et al., 2013, 2018), Numpy
(Harris et al., 2020), Scipy (Virtanen et al., 2020), and Matplotlib (Hunter,
2007). This research has made use of the Simbad database and the VizieR
catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093/cds/vizier). The
original description of the VizieR service was published in A&AS 143, 23.

## Data Availability

The work in this paper has made use of the photometric data published by
Johnson & Mitchell (1975, Table 7, available
online161616http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=II/84), Schuster
(1976, Table 1), Bravo Alfaro et al. (1997, Table 1), the Stellar Atmopshere
Models of Castelli & Kurucz (2003), as provided by the STScI web
page171717https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/castelli-and-kurucz-atlas, the
database of camera spectral sensitivity database of Jiang et al. (2013,
available online181818http://www.gujinwei.org/research/camspec/db.html), the
Bright Star Catalogue (Hoffleit, 1964, available
online191919https://vizier.u-strasbg.fr/viz-
bin/VizieR-3?-source=V/50/catalog), and the General Catalogue of Variable
Stars (Samus’ et al., 2017, available
online202020https://vizier.u-strasbg.fr/viz-bin/VizieR?-source=B/gcvs).

The supplementary material described as Appendices A and B is available online
only.

All the results of this paper, together with future additional material, will
be available online at http://guaix.ucm.es/rgbphot.

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