# Charge order from the local Coulomb repulsion in undoped infinite-layer
nickelates

Hanghui Chen<EMAIL_ADDRESS>NYU-ECNU Institute of Physics, NYU
Shanghai, Shanghai 200122, China  Department of Physics, New York University,
New York, New York 10012, USA  Yi-feng Yang<EMAIL_ADDRESS>Beijing
National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China  University of Chinese
Academy of Sciences, Beijing 100190, China  Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China  Guang-Ming Zhang
<EMAIL_ADDRESS>State Key Laboratory of Low-Dimensional Quantum
Physics and Department of Physics, Tsinghua University, Beijing 100084, China
Frontier Science Center for Quantum Information, Beijing 100084, China

###### Abstract

Competition between superconductivity and other various ordered states plays a
central role in correlated electron materials. The charge order with
translational symmetry breaking has been observed in the parent compounds of
nickelate superconductors with a wave vector
$\mathbf{q}\simeq\left(\frac{1}{3},0,0\right)$. Here we combine density-
functional-theory and dynamical-mean-field-theory to study undoped NdNiO2, and
find that the charge ordered state with a periodic Ni1+-Ni2+-Ni1+ pattern can
have a lower energy than the paramagnetic state and the checkerboard
antiferromagnetic state for the strong local Coulomb repulsion on the
Ni-$d_{x^{2}-y^{2}}$ orbital. Its spectral function evidently shows that the
Ni2+-$d_{x^{2}-y^{2}}$ orbital forms a hole-singlet due to the electron
transfer to the interstitial-$s$/Nd-$d$ conduction bands, while the
Ni1+-$d_{x^{2}-y^{2}}$ orbital exhibits a sharp quasiparticle resonance near
the Fermi energy inside the Mott Hubbard gap. The charge order can be
controlled by the charge transfer energy between Ni-$d$ and
interstitial-$s$/Nd-$d$ orbitals, and there is a delicate competition between
the kinetic energy cost and the potential energy gain, compared to the
checkboard antiferromagnetic state. Our results provide a pure electronic
explanation of the charge order observed experimentally and reveal some
distinctive properties of the infinite-layer nickelates.

Motivated by cuprates, superconductivity has been proposed Anisimov et al.
(1999); Lee and Pickett (2004) and recently discovered in hole-doped infinite-
layer nickelates containing Ni1+ ions of the similar $d^{9}$ electronic
configuration Li et al. (2019, 2020); Zeng et al. (2020); Osada et al.
(2020a); Gu et al. (2020a); Osada et al. (2021); Zeng et al. (2022). While
this seems to confirm the initial expectation and support the Mott-cuprate
scenario for high-temperature superconductivity Lee et al. (2006), the
nickelates show very different properties even in their undoped parent
compounds. Instead of an antiferromagnetic (AFM) Mott insulator, the
resistivity of the parent compounds exhibits metallic behavior at high
temperatures and an upturn below around 70 K Li et al. (2019); Osada et al.
(2020b); Ikeda et al. (2016); Osada et al. (2021). So far no static long-range
magnetic order has been observed down to the lowest measured temperature
Hayward and Rosseinsky (2003); Zhao et al. (2021), although spin fluctuations
or quasi-static magnetism have been reported in experiment Lu et al. (2021).
These are all different from cuprates and have stimulated heated debate
concerning the nature of the nickelate parent state, from which
superconductivity is born upon hole doping. Although several theoretical
scenarios Sakakibara et al. (2020); Nomura et al. (2019); Botana and Norman
(2020); Kitatani et al. (2020); Karp et al. (2022) have been proposed to
emphasize the key role of Mott physics, a growing consensus is that multi-band
physics is indispensable in nickelates Hepting et al. (2020); Zhang et al.
(2020); Yang and Zhang (2022); Wang et al. (2020); Gu et al. (2020b);
Lechermann (2020); Kang and Kotliar (2021); Hu and Wu (2019); Werner and
Hoshino (2020); Zhang and Vishwanath (2020); Wan et al. (2021); Adhikary et
al. (2020). What has not been sufficiently explored is whether this multi-band
feature may lead to some unique experimental consequences in the infinite-
layer nickelates.

Lately, a charge order (CO) with broken translational symmetry has been
reported independently by three experimental groups in the parent compounds of
nickelate superconductors Rossi et al. (2021); Krieger et al. (2021); Tam et
al. (2021). Upon hole doping, the CO onset temperature is gradually reduced
and extrapolated into the superconducting dome. This raises a question
concerning the CO origin and its competition with other phases. CO in cuprates
is well-known and has been intensively studied Comin and Damascelli (2016);
Tranquada (2020). Despite the seeming similarity, the CO in nickelates has
three major differences Rossi et al. (2021); Krieger et al. (2021); Tam et al.
(2021). First, it already exists and has the highest onset temperature in
undoped compounds. Second, unlike cuprates where doped holes reside on the
oxygen atoms, they are mainly introduced into Ni $d_{x^{2}-y^{2}}$ orbitals,
as supported by the experiments Goodge et al. (2021); Rossi et al. (2021).
Third, it has an ordering wave vector
$\mathbf{q}_{\text{CO}}\simeq\left(\frac{1}{3},0,0\right)$, while in cuprates
a short-range checkerboard pattern with a wavelength around $4a_{0}$ ($a_{0}$
is the lattice constant) is most pronounced near $\frac{1}{8}$ hole doping
Hücker et al. (2011) and can be explained by the doped one-band Hubbard model
or $t$-$J$ model White and Scalapino (1998). Hence, these differences may
imply a different mechanism of the CO in nickelates, which is most probably
associated with the multi-band nature of the parent compounds Zhang et al.
(2020); Yang and Zhang (2022). Its explanation provides a key examination of
all theoretical proposals.

In this work, by using the density-functional-theory plus dynamical-mean-field
theory (DFT+DMFT) calculations, we show that strong Hubbard interaction on the
Ni ions $U_{\text{Ni}}$ can not only stabilize the
$\mathbf{q}_{\text{CO}}\simeq\left(\frac{1}{3},0,0\right)$ CO in a
prototypical parent compound of infinite-layer nickelate superconductors
NdNiO2, but also make its total energy lower than that of the usual
checkerboard AFM state when $U_{\text{Ni}}$ is reasonably large. The CO
exhibits a periodic Ni1+-Ni2+-Ni1+ pattern, in which the Ni2+ results from a
charge transfer of nearly one electron to the conduction bands. This is
different from cuprates where holes are located on the O ions Tranquada (2020)
nor the charge density wave in transition metal dichalcogenides Johannes and
Mazin (2008). We show that the stabilization of this CO state is closely
related to the existence of interstitial-$s$/Nd-$d$ conduction bands near the
Fermi energy, to which the charge transfer from the Ni atoms may yield a net
energy gain, depending on $U_{\mathrm{Ni}}$ and a charge transfer energy
between Ni-$d$ and interstitial-$s$/Nd-$d$ orbitals. Our calculations confirm
that the CO indeed emerges as a pure electronic interaction driven phenomenon
in the undoped parent compounds of infinite-layer nickelates, and reveal the
uniqueness of nickelate superconductors.

To compare different states in the infinite-layer nickelates, we take NdNiO2
as an example and use DFT and the maximally localized Wannier functions to
construct a four-orbital tight-binding model including the
Ni-$d_{x^{2}-y^{2}}$, Nd-$d_{xy}$, Nd-$d_{3z^{2}-r^{2}}$ and interstitial-$s$
orbitals per primitive cell Gu et al. (2020b). DMFT calculations are performed
after adding a local Coulomb interaction on the Ni-$d_{x^{2}-y^{2}}$ orbital.
The interacting Hamiltonian takes the form:

$\hat{H}=\sum_{\mathbf{k},mm^{\prime},\sigma}H_{0}(\mathbf{k})_{mm^{\prime}}\hat{c}^{\dagger}_{\mathbf{k}m\sigma}\hat{c}_{\mathbf{k}m^{\prime}\sigma}+U_{\text{Ni}}\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}-\hat{V}_{dc}$
(1)

where $mm^{\prime}$ label different orbitals, $i$ labels the Ni site and
$\sigma$ labels the spin. $\hat{n}_{i\sigma}$ is the occupancy operator of
Ni-$d_{x^{2}-y^{2}}$ orbital at site $i$ with spin $\sigma$.
$H_{0}(\mathbf{k})$ is the Fourier transform of the Wannier-based Hamiltonian
$H_{0}(\mathbf{R})$ and $\hat{V}_{dc}$ is the double counting potential SI .
In the CO state with three Ni ions in the unit cell, Ni1 and Ni3 are assumed
to be equivalent and belong to one type and Ni2 belongs to the other type.

Figure 1 shows the schematics of three states in our study: (a) the uniform
paramagnetic (PM) state with dynamically fluctuating spins, (b) the AFM with
an ordering wave vector
$\mathbf{q}_{\text{AFM}}=\left(\frac{1}{2},\frac{1}{2},0\right)$, and (c) the
CO state with an ordering wave vector
$\mathbf{q}_{\text{CO}}=\left(\frac{1}{3},0,0\right)$. Starting from the
identical non-interacting Hamiltonian, we use DFT+DMFT method to calculate
their total energies for NdNiO2 SI . The results are compared in Fig. 1(d). We
find that the AFM and CO states can only be stabilized when $U_{\text{Ni}}\geq
3$ eV. When $U_{\text{Ni}}$ increases, first the AFM state shows a lower
energy than the PM, and then the CO becomes more stable than both AFM and PM
states. The inset of Fig. 1(d) plots the derived phase diagram of NdNiO2 as a
function of $U_{\text{Ni}}/t_{dd}$, where $t_{dd}=0.37$ eV is the nearest-
neighbor hopping between Ni-$d_{x^{2}-y^{2}}$ orbitals in the model. Since CO
is already observed in experiment, our calculations indicate that NdNiO2 is a
strongly correlated material with $U_{\text{Ni}}/t_{dd}$ larger than 13, which
is comparable to hole-doped cuprates Lee et al. (2006); C. et al. (2010);
Comanac et al. (2008). For such a value of $U_{\text{Ni}}/t_{dd}$, the energy
difference between the CO state and PM state is about 50 meV/f.u., a magnitude
consistent with the CO onset around the room temperature Rossi et al. (2021);
Krieger et al. (2021); Tam et al. (2021). Our results explain the lacking of
long-range AFM ordering despite the substantial AFM superexchange observed in
infinite-layer nickelates Lu et al. (2021); Zhao et al. (2021).

Figure 1: Schematics of NdNiO2 in three different states: (a) the uniform PM
state, (b) the checkerboard AFM state, and (c) the
$\mathbf{q}=\left(\frac{1}{3},0,0\right)$ CO state. (d) The total energy $E$
of NdNiO2 in the PM (green), the AFM (blue) and the CO (red). The inset shows
the phase diagram as a function of $U_{\text{Ni}}/t_{dd}$, where $t_{dd}=0.37$
eV is the nearest-neighbor hopping between Ni $d_{x^{2}-y^{2}}$ orbitals. The
DFT+DMFT calculations are carried out with $U_{\text{Ni}}=5$ eV at $T=116$ K,
and the error bar is smaller than the symbol size SI .

Figure 2 compares the many-body electronic spectra of NdNiO2 in all three
states for $U_{\text{Ni}}=5$ eV and $T=116$ K. Figure 2(a) displays the
orbital-resolved spectral function of the PM state. The red (blue) curve is
the Ni-$d$ (Nd-$d$) projected spectral function. The projection of the
interstitial-$s$ orbital is omitted here for clarity SI . The Ni-$d$ orbital
exhibits a characteristic three-peak feature with Ni lower Hubbard band (LHB)
and upper Hubbard band (UHB) separated by about 5 eV. The Nd-$d$ states barely
touches the Fermi level, leading to a small self-doping. Figure 2(b) shows the
Ni-$d$ projected spectral function close to the Fermi level, exhibiting a
clear quasi-particle peak near the Fermi energy inside a large energy gap.
Figures 2(c) and 2(d) show the spectral function of the AFM state for
completeness. Compared to the PM state, the quasi-particle peak seems more
pronounced but shifts away from the Fermi level, and the Nd-$d$ states move to
higher energy. The orbital-resolved spectral function of the CO state is
compared in Fig. 2(e), where the red and green curves are the Ni-$d$ projected
spectra (two types of Ni atoms), while the blue curve is the Nd-$d$ projected
one. The Ni1-$d$ orbital (red curve) exhibits a Mott gap, indicating that
electrons on Ni1-$d$ orbitals form a fluctuating local moment of spin
$S=\frac{1}{2}$. By contrast, the Ni2-$d$ orbital is almost empty, indicating
that a doped hole on the Ni2 site forms a hole-singlet. Thus the CO state in
NdNiO2 results in the charge transfer of almost one electron from Ni2 to the
Nd spacer layer and the Nd-$d$ states become partially occupied. Furthermore,
the peak of Ni LHB moves towards lower energy, compared to that in the PM
state. This peak shift plays an important role in the energy gain, as will be
discussed later. An enlarged plot of the Ni-$d$ projected spectral functions
near the Fermi level is shown in Fig. 2(f). We see a very narrow resonance
peak with a width of about 15 meV emerging around the Fermi level in the
Ni1-$d$ orbital, but its peak value is one order of magnitude smaller than
that in the PM state. This comparison establishes the coexistence and
competition of many-body quasi-particle resonance and the CO state. The many-
body resonance is to some extent suppressed in the presence of CO, so that it
only starts to develop when the temperature is lowered to 116 K in our
calculations. Therefore, the Ni1 and Ni3 ions are at the verge of a Mott
transition.

Figure 2: Orbital-resolved spectral function of NdNiO2 (a) in the PM, (c) in
the AFM, and (e) in the CO states. Panels (b), (d), and (f) are enlarged plots
showing the corresponding near-Fermi-level features. The DFT+DMFT calculations
are done with $U_{\text{Ni}}=5$ eV and at $T$ = 116 K. For clarity, the
spectral function projected onto the interstitial-$s$ orbital is not shown SI
.

Based on above observations, the emergence of the CO ground state can be
qualitatively understood by analyzing its energy gain compared to the PM and
AFM states. Figure 3(a) shows a schematic electronic structure of the PM state
at $U_{\mathrm{Ni}}<U_{c}$, where $U_{c}\sim 3.5\,$eV is the value of the PM-
to-CO transition. In this case, the Ni LHB and UHB are not well separated and
electrons may occupy both Ni-$d$ bands and conduction bands. As a result, the
CO is not favored due to the delocalized character of the electrons. It is
only for $U_{\mathrm{Ni}}>U_{c}$ (Fig. 3(b)), the Ni-$d$ electrons become more
localized and the Ni UHB moves towards higher energy than the Nd-$d$
conduction bands. Then the charge transfer may occur from Ni-$d$ to Nd-$d$
orbitals when there is an energy gain. Due to the localized character of
Ni-$d$ orbitals, the CO can emerge and break the translational symmetry.
However, for both PM and CO, the Ni LHB is always lower than Nd-$d$ orbitals.
The electron transfer from Ni2 site to Nd sites always results in an energy
cost (the brown arrow in Fig. 3(b)). To stabilize the CO state, an energy gain
is required to compensate the cost.

To see how this can happen, we analyze in more detail possible changes in each
term of the total energy
$E=E^{\mathrm{kin}}+E^{\mathrm{pot}}-E^{\mathrm{dc}}$, where
$E^{\mathrm{kin}}$ is the kinetic energy, $E^{\mathrm{pot}}$ is the potential
energy and $E^{\mathrm{dc}}$ is the double-counting energy SI .
$E^{\mathrm{kin}}=\text{Tr}(\hat{H_{0}}\hat{G})$ where $\hat{H}_{0}$ is the
non-interacting Hamiltonian and $\hat{G}$ is the dressed lattice Green
function. We find that the change in $E^{\text{kin}}$ is most delicate because
it involves two competing forces. As shown in Fig. 3(b), since Nd-$d$ states
are “pinned” to the Fermi level after the charge transfer, the electron that
transfers from the Ni2-$d$ to Nd-$d$ increases $E^{\mathrm{kin}}$, which is
roughly characterized by $\Delta_{1}$; while for the electrons on Ni1 and Ni3,
their Ni LHB decreases $E^{\mathrm{kin}}$ which is characterized by about
$\Delta_{2}-\Delta_{1}$. Calculations find that the energy gain from
$\Delta_{2}-\Delta_{1}$ can only overcome the energy cost that arises from
$\Delta_{1}$ when $U_{\mathrm{Ni}}$ is above 5 eV (Fig. 3(c)). On the other
hand, the change in $E^{\mathrm{pot}}$ is always negative for
$U_{\mathrm{Ni}}>U_{c}$. In the PM state, every Ni has one electron. Due to
the repulsive Coulomb interaction,
$E^{\mathrm{pot}}=\frac{1}{2}\text{Tr}(\hat{\Sigma}_{\text{cor}}\hat{G}_{\text{cor}})>0$
for each Ni. In the CO state, Ni2-$d_{x^{2}-y^{2}}$ is almost empty so that
its self-energy $\Sigma_{\mathrm{cor}}$ essentially becomes zero. This
decreases the potential energy and always leads to an energy gain $\Delta
E^{\mathrm{pot}}<0$ for $U_{\mathrm{Ni}}>U_{c}$ (Fig. 3(c)). For the double-
counting energy $E^{\text{dc}}$, we choose the fully-localized-limit (FLL)
scheme $E^{\text{dc}}=UN_{d}(N_{d}-1)$ Czyżyk and Sawatzky (1994) because it
does not favor the PM nor the CO when $U_{\mathrm{Ni}}$ is large. As
$U_{\mathrm{Ni}}$ increases, the PM approaches a half-filled configuration for
every Ni, while the CO is close to a configuration in which Ni2 has zero
occupancy and Ni1(Ni3) is half-filled. Therefore, the occupancy on Ni is
either 1 or 0, leading to a negligible double-counting correction. Figure 3(c)
shows that $\Delta E^{\mathrm{dc}}$ vanishes as $U_{\mathrm{Ni}}$ increases.
Altogether, the energy gain from the PM-to-CO transition grows with
$U_{\mathrm{Ni}}$ so that the CO has a lower energy than the PM at large $U$.

Figure 3: Schematics of the interacting electronic structure of NdNiO2 for (a)
$U_{\mathrm{Ni}}<U_{c}$ and (b) $U_{\mathrm{Ni}}>U_{c}$, where $U_{c}\sim
3.5\,$eV is the value of the PM-to-CO transition. (c) The total energy
difference between the CO and the PM decomposed into kinetic energy (kin),
potential energy (pot) and double-counting energy (dc) as a function of
$U_{\mathrm{Ni}}$. (d) The total energy difference between the CO and the PM
as a function of $\delta E_{CT}$ for a fixed value of $U_{\mathrm{Ni}}$, where
$E_{CT}$ is the energy separation between the bare energy of Ni-$d$ and Nd-$d$
states (see the inset). (e) The decomposition of the energy difference between
the CO and the PM as a function of $\delta E_{CT}$ for $U_{\mathrm{Ni}}=5$ eV.

We note that a prerequisite of the CO is the charge transfer from Ni-$d$ to
other non-Ni conduction orbitals (interstitial-$s$/rare-earth-$d$ orbitals).
The inset of Fig. 3(d) shows the schematic non-interacting electronic
structure of NdNiO2 (for clarity, interstitial-$s$ orbital is omitted). We may
introduce a charge transfer energy $E_{CT}$ (different from that of cuprates)
as the energy separation between the bare energy of Ni-$d$ and Nd-$d$
orbitals. Calculations find that in addition to $U_{\mathrm{Ni}}$, $E_{CT}$
also controls the energy gain from the PM-to-CO transition. To see this more
clearly, we examine the energy difference between the PM and the CO by tuning
$E_{CT}$. This can be done by simultaneously increasing the onsite energies of
interstitial-$s$, Nd-$d_{xy}$ and Nd-$d_{3z^{2}-r^{2}}$ orbitals by the same
amount $\delta E_{CT}$. The PM and CO states are then recalculated and their
energy differences are plotted in Fig. 3(e). We study a few representative
$U_{\mathrm{Ni}}$ and find a similar trend. Namely, as $E_{CT}$ increases, the
CO becomes less energetically favorable than the PM state and $\Delta E$
changes sign when $\delta E_{CT}$ is above a critical value (typically
$0.1-0.2$ eV). Specifically, as shown in Fig. 3(c), tuning $E_{CT}$ has
marginal effects on $\Delta E^{\mathrm{pot}}$ and $\Delta E^{\mathrm{dc}}$,
but has a substantial effect on $\Delta E^{\mathrm{kin}}$. This is because
tuning $E_{CT}$ mainly affects the charge transfer energy but does not
significantly change the charge distribution when $U_{\mathrm{Ni}}$ is large.
The energy cost in $\Delta E^{\mathrm{kin}}$ is therefore increased, which
eventually makes the CO energetically more unfavorable than the PM. In the CO
state, its sensitivity to $E_{CT}$ implies that the infinite-layer nickelates
have a different energy scale as the control parameter and thus belong to a
new class of correlated materials, expanding the famous Zaanen-Sawatzky-Allen
(ZSA) classification Zaanen et al. (1985).

Having established why the CO has a lower energy than the PM at large
$U_{\text{Ni}}$, we now discuss its competition with the AFM. Figure 4 shows
the total energy difference $\Delta E$ between the CO and the AFM as a
function of $U_{\text{Ni}}$, as well as its decomposition into $\Delta
E^{\text{kin}}$, $\Delta E^{\text{pot}}$ and $\Delta E^{\text{dc}}$. When
$U_{\text{Ni}}$ is reasonably large, the Ni-$d$ occupancy is close to 1 in the
AFM. Therefore, the FLL double counting does not favor the AFM nor the CO.
However, we find a delicate competition between the kinetic energy cost and
the potential energy gain. Different from the CO-PM case in which both $\Delta
E^{\text{kin}}$ and $\Delta E^{\text{pot}}$ show energy gain at large
$U_{\text{Ni}}$, $\Delta E$ between the CO and the AFM exhibits kinetic energy
cost $\Delta E^{\mathrm{kin}}>0$ and potential energy gain $\Delta
E^{\mathrm{pot}}<0$ (Fig. 4(a)). Both of them are substantial, resulting in a
cancellation and a subtle total energy difference between the two states (Fig.
4(b)). Compared to the PM in Fig. 3, the kinetic energy part of the AFM is
reduced due to spin correlations. Hence, the subtle energy balance reveals a
strong competition between the AFM and the CO. While the AFM tends to save
more spin correlation energy as included in the kinetic part, the CO tends to
save more Coulomb energy in the potential part. The true ground state is
therefore determined by the competition of these two tendencies. As a result,
the CO only becomes the ground state when the spin correlation energy is
reduced at sufficiently large $U_{\text{Ni}}\approx 5\,$eV where the kinetic
energy gain of the AFM becomes smaller than the potential energy gain of the
CO.

Figure 4: The energy difference $\Delta E$ between the CO and the AFM as a
function of $U_{\text{Ni}}$. (a) $\Delta E$ decomposed into the kinetic part
(red symbols), the potential part (blue symbols) and the double-counting part
(green symbols). (b) The total energy difference $\Delta E$ (purple symbols).

We therefore argue that the CO in infinite-layer nickelates may have an
electronic origin driven by the Hubbard interaction on the
Ni-3$d_{x^{2}-y^{2}}$ orbitals in the presence of conduction bands near the
Fermi level. It is different from the usual charge density wave ordering due
to Fermi surface nesting in weakly correlated electron systems. Its presence
in the undoped parent compounds reflects a unique property of nickelate
systems. Compared to the PM, the electron transfer at Ni2 orbitals in the CO
state not only enhances the self-doping effect by increasing the number of
conduction electrons, but also drives other Ni-3$d_{x^{2}-y^{2}}$ orbitals to
a more localized regime and form fluctuating local moments of spin
$S=\frac{1}{2}$. In this sense, the CO state may practically produce an
electronic background supporting incoherent Kondo scattering proposed earlier
to explain the resistivity upturn at low temperatures in undoped nickelates
Zhang et al. (2020); Yang and Zhang (2022); Wang et al. (2020). To what extent
these phenomena are related and whether or not the CO state may have other
important consequences require further experimental scrutiny.

To summarize, we provide numerical evidences supporting the possibility of
$\mathbf{q}_{\text{CO}}=\left(\frac{1}{3},0,0\right)$ CO as the ground state
in undoped NdNiO2. We find a delicate competition between the CO and AFM
states, and the CO can be stabilized in a Ni1+-Ni2+-Ni1+ pattern for a
sufficiently large but reasonable value of $U_{\text{Ni}}$. We have only
considered onsite Coulomb interaction and an ideal crystal structure.
Electron-phonon interaction and lattice distortion may also play a role to
promote the CO state Xia et al. (2022); Sui et al. (2022) but these need to be
elaborated in future studies. In any case, we conclude that the CO in
infinite-layer nickelates results from the special multi-band electronic
structure around the Fermi energy. It supports the nickelates as another class
of unconventional superconductors with its own unique correlated physics
distinct from the cuprates.

###### Acknowledgements.

We thank Wei-Sheng Lee, Andrew Millis, Pu Yu, and Fu-Chun Zhang for
stimulating discussions. This work was supported by the National Natural
Science Foundation of China (Grant No. 11774236, No. 11974397), the Ministry
of Science and Technology of China (Grant No. SQ2020YFE010418, No.
2017YFA0302902, No. 2017YFA0303103), Open Grant of State Key Laboratory of Low
Dimensional Quantum Physics at Tsinghua University, and the Strategic Priority
Research Program of the Chinese Academy of Sciences (Grant No. XDB33010100).

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# RGB photometric calibration of 15 million _Gaia_ stars

Nicolás Cardiel,1,2 Jaime Zamorano,1,2 Josep Manel Carrasco,6 Eduard Masana,6
Salvador Bará,3 Rafael González,1 Jaime Izquierdo,1 Sergio Pascual,1,2 and
Alejandro Sánchez de Miguel,1,4,5
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
6Departament Física Quàntica i Astrofisica. Institut de Ciències del Cosmos
(ICC-UB-IEEC), C Martí Franquès 1, Barcelona 08028, Spain E-mail:
<EMAIL_ADDRESS>(NC)

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

###### Abstract

Although a catalogue of synthetic RGB magnitudes, providing photometric data
for a sample of 1346 bright stars, has been recently published, its usefulness
is still limited due to the small number of reference stars available,
considering that they are distributed throughout the whole celestial sphere,
and the fact that they are restricted to Johnson $V<6.6$ mag. This work
presents synthetic RGB magnitudes for $\sim 15$ million stars brighter than
_Gaia_ $G\\!=\\!18$ mag, making use of a calibration between the RGB
magnitudes of the reference bright star sample and the corresponding high
quality photometric $G$, $G_{\rm BP}$ and $G_{\rm RP}$ magnitudes provided by
the _Gaia_ EDR3. The calibration has been restricted to stars exhibiting
$-0.5<G_{\rm BP}\\!-\\!G_{\rm RP}<2.0$ mag, and aims to predict RGB magnitudes
within an error interval of $\pm 0.1$ mag. Since the reference bright star
sample is dominated by nearby stars with slightly undersolar metallicity,
systematic variations in the predictions are expected, as modelled with the
help of stellar atmosphere models. These deviations are constrained to the
$\pm 0.1$ mag interval when applying the calibration only to stars scarcely
affected by interstellar extinction and with metallicity compatible with the
median value for the bright star sample. The large number of _Gaia_ sources
available in each region of the sky should guarantee high-quality RGB
photometric calibrations.

###### keywords:

instrumentation: photometers – catalogues – techniques: photometric – stars:
general

††pubyear: 2021††pagerange: RGB photometric calibration of 15 million _Gaia_
stars–A.2

##  1 Introduction

Recently, Cardiel et al. (2021, hereafter C21)111The main results from that
work are available online at http://guaix.ucm.es/rgbphot, and through VizieR
at http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/MNRAS/504/3730 have
established a standard RGB photometric system by setting its three basic
characteristics: i) a well-defined set of RGB spectral sensitivity curves,
determined from the median of a library of sensitivity curves corresponding to
28 cameras analyzed by Jiang et al. (2013); ii) the use of photon-based
photometric magnitudes; and iii) the adoption of zero points defined in the
absolute (AB) scale. In addition, C21 have computed a catalogue of synthetic
RGB star magnitudes for 1346 bright stars belonging to the Bright Star
Catalogue (Hoffleit, 1964), using for that purpose historical but very
reliable 13-colour medium-narrow-band photometric data gathered by Johnson &
Mitchell (1975), Schuster (1976) and Bravo Alfaro et al. (1997), covering the
3370–11090 Å interval. The RGB magnitudes in that work were determined from
stellar atmosphere models fitted to the 13-colour photometric data. The
reliability of the resulting spectral energy distributions was asserted
through both the comparison of synthetic Johnson $B$ and $V$ magnitudes with
the corresponding magnitudes in the Simbad
database222http://simbad.u-strasbg.fr/simbad/ (with a $3\sigma$ dispersion of
0.11 and 0.08 mag in the $B$ and $V$ band, respectively), and by direct
comparison with flux calibrated spectra from Kiehling (1987) (showing
discrepancies below $\pm 0.05$ mag in the historical 13-colour photometric
bandpasses). Even though there are non-negligible variations of the RGB
spectral sensitivity curves between different cameras, C21 have also shown
that simple polynomial transformations can be employed to transform RGB
measurements performed with a typical camera to the mentioned standard system
(see their Fig. 18), facilitating the use of the proposed system.

Figure 1: Distribution in the celestial sphere (Galactic coordinates) of the
C21 bright star sample, constituted by 1346 stars. Simple constellation 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 green RGB magnitude, as derived by
C21.

Although the C21 catalogue of RGB magnitudes is suitable for calibration
purposes, it only contains a small number of stars (on average 1 star for each
30 square degrees, although they tend to concentrate towards the Galactic
plane; see Fig. 1). Not only that, this catalogue is constituted by stars
brighter than Johnson $V=6.6$ mag, with a magnitude distribution whose 16th,
50th and 84th percentiles are 3.3, 4.4 and 5.0 mag, respectively. This is
specially problematic when considering astronomical projects that could
seriously benefit from the exponential growth of the number of professional
and amateur astronomers equipped with commercial-grade RGB cameras, who can
potentially generate a huge amount of useful data in many astronomical fields
(see the Introduction section in C21 and references therein).

The aim of this paper is to exploit the superb photometric data provided by
the _Gaia_ mission (Gaia Collaboration et al., 2016), through the third
intermediate _Gaia_ data release (Gaia Collaboration et al., 2021, _Gaia_
EDR3), to estimate RGB magnitudes from _Gaia_ $G$, $G_{\rm BP}$ and $G_{\rm
RP}$ photometric data. For that purpose, we have derived simple
transformations between those magnitudes and the synthetic RGB photometry in
the sample of bright stars published by C21. In order to avoid confusion
between _Gaia_ $G$ magnitudes and the ones corresponding to the green RGB
filter, from this point we are using $G^{\rm Gaia}$, $G_{\rm BP}^{\rm Gaia}$
and $G_{\rm RP}^{\rm Gaia}$ to indicate the use of the _Gaia_ magnitudes, and
$B^{\rm rgb}$, $G^{\rm rgb}$ and $R^{\rm rgb}$ when referring to the RGB
photometric measurements.

Since the C21 star sample is constituted by bright and nearby stars of
metallicity slightly undersolar (as shown in Sect. 2.1), we have also employed
synthetic _Gaia_ and RGB magnitudes measured in stellar atmosphere models to
constrain the systematic uncertainties introduced by the use of the mentioned
transformations with stars of different metallicity and also affected by
interstellar extinction. In addition, we have applied the fitted
transformations to estimate RGB magnitudes for a sample of $\sim 15$ million
stars extracted from the sample of Anders et al. (2019). These authors
published improved photo-astrometric distances, extinctions and astrophysical
parameters for _Gaia_ DR2 stars brighter than $G=18$ mag, using for that
purpose the Bayesian tool StarHorse (Queiroz et al., 2018), complementing the
_Gaia_ measurements with Pan-STARRS1, 2MASS and AllWISE photometric data. Here
we have restricted their stellar sample to minimize the impact of systematic
uncertainties, introduced by metallicity variations and interstellar
extinction, in the predictions of RGB magnitudes.

Synthetic magnitudes in this work have been determined using the Python
package synphot (STScI development Team,
2018)333https://synphot.readthedocs.io/en/latest/, which facilitates the
computation of photometric properties from user-defined bandpasses and spectra
(see Section 2 of C21 for additional computational details).

The description of the _Gaia_ –RGB calibration is presented in Section 2. Its
application to the StarHorse subsample of 15 million stars is described in
Section 3, while the final discussion and conclusions are summarized in
Section 4. Appendix A describes how to extend the estimation of RGB magnitudes
beyond the 15 million star sample, by using ADQL queries to the _Gaia_
catalogue and an auxiliary Python package specially written for that purpose.

##  2 RGB calibration from Gaia EDR3 photometry

Figure 2: Comparison of the transmissivity of the _Gaia_ EDR3 passbands (thin
blue, green and red lines), derived by Riello et al. (2021), with the standard
RGB passbands defined in C21 (thick blue, green and red lines encompassing
shaded regions). The flux density (in photon s-1 cm-2 Å-1) of the reference
spectra used to define magnitudes in the common AB, ST and Vega systems are
shown as dashed, dotted and full black lines, respectively.

The RGB sensitivity curves of the standard photometric system defined by C21
are encompassed by the transmissivity of the _Gaia_ EDR3 passbands derived by
Riello et al. (2021)444Available at
https://www.cosmos.esa.int/web/gaia/edr3-passbands (see Fig. 2). In
particular, the three RGB transmissivity curves cover a similar wavelength
range as $G_{\rm BP}^{\rm Gaia}$, and approximately half of the range spanned
by $G^{\rm Gaia}$. The fact that $G_{\rm RP}^{\rm Gaia}$ covers an additional
range towards longer wavelengths, makes the colour $G_{\rm BP}^{\rm
Gaia}-G_{\rm RP}^{\rm Gaia}$ a good proxy to estimate variations in the
spectral energy distribution covered in the visible range. For that reason, it
is expected that a good calibration of RGB magnitudes can be derived from the
accurate _Gaia_ data. In this section we describe the procedure followed to
achieve this task, starting by collecting the _Gaia_ magnitudes available for
the C21 bright star sample, creating colour-colour diagrams involving _Gaia_
and RGB magnitudes, and finding a simple mathematical relationship between
both magnitude sets.

###  2.1 _Gaia_ EDR3 data for the C21 bright star sample

Figure 3: _Panels (a)–(c):_ histograms displaying the distributions in
distance [panel (a)], metallicity [panel (b)] and $A_{V}$ extinction [panel
(c)] for the C21 bright star subsample in common with Anders et al. (2019).
These three parameters are the ones derived by these authors using StarHorse.
Although initially 785 stars had parameter estimates, 172 stars were discarded
by imposing the quality flag SH_OUTFLAG=’00000’, which guarantees the
statistical reliability of the StarHorse determinations. The histograms have
been computed separately for the non-variable (in blue) and variable (orange)
stars, with the red line delineating the coadded histogram. The three vertical
dashed lines indicate the 16th, 50th and 84th percentiles (from left to right,
respectively) of the whole displayed sample, whose values are provided under
the plot legend. _Panel (d):_ histogram displaying the 5th, 50th and 95th
percentile of the posterior probability distribution for the StarHorse $A_{V}$
estimates. Note that the 50th percentile histogram is the same displayed in
panel (c) for the whole sample with reliable parameters, whereas the 95th
percentile is greater than zero for all the stars with SH_OUTFLAG=’00000’.
This indicates that the negative $A_{V}$ values correspond to stars whose
Bayesian posterior probability distribution is not statistically incompatible
with a null extinction.

As the first step for this work, we retrieved the $G^{\rm Gaia}$, $G_{\rm
BP}^{\rm Gaia}$ and $G_{\rm RP}^{\rm Gaia}$ magnitudes for the bright star
sample of C21, provided by the _Gaia_ EDR3 (Gaia Collaboration et al., 2021)
Archive at the European Space Agency555https://gea.esac.esa.int/archive/. This
process was carried out in the following way: the _Gaia_ DR2 identifier of
each star was initially obtained through the Simbad database, starting from
the initial HR number of the star in the Bright Star Catalogue (Hoffleit,
1964). Since it is not guaranteed that the same astronomical source will
always have the same source identifier in the different _Gaia_ Data Releases,
for each source_id in DR2 we searched for proximal source(s) in the auxiliary
table gaiaedr3.dr2_neighbourhood, keeping the most likely crossmatch taking
into account the parameters angular_distance and magnitude_difference when
more that one possible counterpart appeared. It is important to note that not
all the bright stars in the C21 sample appear in the _Gaia_ database because
of its level of incompleteness at the bright end (Gaia Collaboration et al.,
2021). In addition, and having in mind that our main goal is to derive a
photometric transformation, we imposed the signal-to-noise ratio for the flux
measured by _Gaia_ in each passband to be high enough to guarantee a maximum
uncertainty of 0.01 mag for both $G^{\rm Gaia}$ magnitudes and the $G_{\rm
BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$ colours. At the end of this
process, the initial sample of 1346 bright stars was finally reduced to a
subsample of 888 objects, 320 of them flagged as variable stars in Simbad, and
568 with no indication of variability. The histograms displayed in Fig. 3
confirm that the bulk of this bright star C21 subsample is dominated by nearby
stars, with metallicity slightly undersolar and scarcely affected by
interstellar extinction. Although the large negative extinction estimates
displayed in Fig. 3(c) may seem alarming, it is important to realize that they
simply correspond to the median values of the $A_{V}$ Bayesian posterior
probability distribution derived by Anders et al. (2019) for each star, and
thus, a too simplistic reduction of whole probability distributions into
single numbers. The 5th and 95th percentile $A_{V}$ values displayed in Fig.
3(d) illustrate that the credible intervals for those stars are compatible
with $A_{V}=0$.

###  2.2 Colour–colour diagram relating RGB and _Gaia_

Figure 4: _Panels (a), (b), (c) and (d):_ colour-colour diagrams representing
the differences between each RGB magnitude (and $G^{\rm rgb}_{\rm BR}$) with
$G^{\rm Gaia}$ as a function of the _Gaia_ colour $G_{\rm BP}^{\rm
Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$. The bright star sample from C21 is plotted
with filled blue (non-variable stars) and orange (stars flagged as variable in
Simbad) circles. Clear outliers, overplotted with a small red cross,
correspond to double stars in Johnson & Mitchell (1975) (JM75, see text). The
predictions of CK04 models are also shown with small black (solar metallicity)
and green (non-solar metallicity) circles. The calibrations, robust fits to
5-degree polynomials, are shown as magenta lines, with the corresponding
residuals shown (twice) in the underneath panels. _Panels (e), (f), (g) and
(h)_ : the r.m.s. of the residuals is provided separately for the non-variable
(blue) and variable (orange) stars. The horizontal dashed lines encompass the
$\pm 0.1$ mag interval. _Panels (i), (j), (k) and (l)_ : the coloured crosses
indicate the impact of interstellar reddening, as parametrized by the
indicated $A_{V}$ values, for CK04 models of solar metallicity (non-solar
predictions have not been used in this case to avoid cluttering).

We represent in the top panels of Fig. 4 the colour–colour diagrams built
using the difference between each RGB magnitude (from the C21 catalogue) and
$G^{\rm Gaia}$, as a function of the _Gaia_ colour $G_{\rm BP}^{\rm
Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$. We have also included in this comparison
an estimate of the magnitude in the green RGB filter computed simply as the
arithmetic mean of the two magnitudes in the neighbouring filters, i.e.,

$\displaystyle G^{\rm rgb}_{\rm BR}=\frac{B^{\rm rgb}+R^{\rm rgb}}{2}.$ (1)

The definition of this additional magnitude can be particularly useful for
observations performed with cameras equipped with Bayer-like colour filter
systems666See U.S. Patent No. 3,971,065, available at
https://patents.google.com/patent/US3971065, in which an array of luminance-
and chrominance-sensitivity elements (the green pixels on the one hand, and
the blue and red pixels, on the other) are superposed in registration with an
imaging array. Since the spatial sampling of each pixel is different, the
computation of an averaged (blue+red) magnitude can facilitate the estimation
of an independent green magnitude which, in addition, can help to perform
image interpolations through the comparison of the demosaiced RGB channels.

Non-variable stars in the colour-colour diagrams shown in Fig. 4 are
represented with big blue filled circles, whereas variable objects are plotted
with smaller orange filled circles. It is clear that the subsample of C21
bright stars with _Gaia_ data exhibits well-defined sequences in these
diagrams. A few objects, outliers in the four panels and marked with an
additional red cross, correspond to stars flagged as double in the initial
photometric measurements by Johnson & Mitchell (1975) (see also Table 3 in
C21) and were not used in the subsequent work. We have also overplotted the
expected location of synthetic colour estimates computed from the stellar
atmosphere models by Castelli & Kurucz (2003, hereafter CK04), which are
precomputed for abundances ${\rm[M/H]}\\!=\\!-2.5$, $-2.0$, $-1.5$, $-1.0$,
$-0.5$, $0.0$, $+0.2$ and $+0.05$, 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. Note that these are the same models employed by C21 to derive
their RGB photometric database. In particular, we have used two different
symbol colours to represent the predictions for solar metallicity stars (small
black filled circles) and for non-solar metallicity objects (small green
filled circles). It is important to highlight here that the RGB photometric
system is defined in the absolute (AB) scale, whereas the _Gaia_ magnitudes
are provided in the Vega system. The latter have been computed for the
synthetic magnitudes of the stellar models using the Vega spectrum
alpha_lyr_stys_010.fits, available at the CALSPEC
database777https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec (Bohlin et al., 2014).

###  2.3 The _Gaia_ -RGB calibration

Table 1: Coefficients of the polynomials $f_{B}$, $f_{G}$, $f_{R}$ and $f_{G_{\rm BR}}$ defined in Eqs. (2)–(5), and corresponding to the fits displayed with magenta lines in panels (a), (b), (c) and (d) of Fig. 4. Each polynomial must be evaluated as $f_{X}=\sum_{i=0}^{5}a_{i}\,(G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia})^{i}$, with $X=B,\,G,\,R$ and $G_{\rm BR}$. Coef. | $f_{B}$ | $f_{G}$ | $f_{R}$ | $f_{G_{\rm BR}}$
---|---|---|---|---
$a_{0}$ | $-0.13748689$ | $-0.02330159$ | $+0.10979647$ | $-0.01252185$
$a_{1}$ | $+0.44265552$ | $+0.12884074$ | $-0.14579334$ | $+0.13983574$
$a_{2}$ | $+0.37878846$ | $+0.22149167$ | $+0.10747392$ | $+0.23688188$
$a_{3}$ | $-0.14923841$ | $-0.14550480$ | $-0.10635920$ | $-0.10175532$
$a_{4}$ | $+0.09172474$ | $+0.10635149$ | $+0.08494556$ | $+0.07401939$
$a_{5}$ | $-0.02594726$ | $-0.02363990$ | $-0.01368962$ | $-0.01821150$

The sequences displayed in the top panels of Fig. 4 have been iteratively
fitted (rejecting outliers using a sigma-clipping algorithm) to the previously
described subsample of 888 bright stars, using fifth-order polynomials888The
final polynomial degree was determined by using an orthogonal polynomial
regression with the help of the software package R (R Core Team, 2021). This
facilitated the determination of the maximum polynomial degree that led to
polynomial coefficients that were statistically significant.. Since the stars
included in this subsample flagged as variable objects already passed the
filtering process described in C21 when comparing synthetic Johnson $B$ and
$V$ magnitudes with tabulated measurements in the Simbad database, and
considering that they do not show a different behaviour to that exhibited by
non-variable stars, we kept them in the initial set to be fitted, knowing that
the sigma-clipping algorithm would get rid off the deviant cases. With the aim
of constraining the fits at the extremes of the $G_{\rm BP}^{\rm
Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$ colour interval, the fitted data were
complemented by including the predictions of the CK04 models of solar
metallicity for $-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<-0.4$
mag and for $2.0<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.5$ mag
(no additional model data were employed except for those two small colour
ranges at the borders). The fitted relationships are displayed in the top
panels of Fig. 4 with a continuous magenta line, and the resulting residuals
are plotted (twice) in the central and bottom panels. These polynomial fits
provide the sought transformation that allows estimating RGB magnitudes from
the _Gaia_ magnitudes using

$\displaystyle B^{\rm rgb}=$ $\displaystyle\mbox{}\;\;G^{\rm
Gaia}+f_{B}(G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}),$ (2)
$\displaystyle G^{\rm rgb}=$ $\displaystyle\mbox{}\;\;G^{\rm
Gaia}+f_{G}(G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}),$ (3)
$\displaystyle R^{\rm rgb}=$ $\displaystyle\mbox{}\;\;G^{\rm
Gaia}+f_{R}(G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}),$ (4)
$\displaystyle G^{\rm rgb}_{\rm BR}=$ $\displaystyle\mbox{}\;\;G^{\rm
Gaia}+f_{G_{\rm BR}}(G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}),$ (5)

where $f_{B}$, $f_{G}$, $f_{R}$ and $f_{G_{\rm BR}}$ are the fifth-order
polynomials with independent variable $G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm
RP}^{\rm Gaia}$, whose coefficients are given in Table 1.

The residuals are displayed in the central panels of Fig. 4, using the same
symbols employed in the top panels. The robust standard deviation of these
residuals999Computed as $\sigma_{\rm robust}=0.7413(q_{75}-q_{25})$, with
$q_{25}$ and $q_{75}$ the 25th and 75th percentiles, respectively (see e.g.,
Ivezić et al., 2020) is also provided within each panel, computed separately
for the 568 non-variable stars (blue colour) and the 320 variable stars
(orange colour), being the latter slightly larger in the four panels. In all
cases, the $\pm 3\sigma_{\rm robust}$ dispersion is constrained within the
$\pm 0.1$ mag interval (displayed with the horizontal dashed lines). The same
panels also display the residuals corresponding to the synthetic magnitudes
derived from the CK04 models: the predictions for solar metallicity stars
(small black filled circles) are well reproduced by the previous fitted
relationships, except for the reddest stars, where the dispersion increases.
The models with metallicity different from solar (small green filled circles)
exhibit a wider scatter. It is interesting to note that the model predictions
interpolated for ${\rm[M/H]}\\!=\\!-0.16$, the median value in Fig. 3(b), are
not very different from those for solar metallicity. In particular, the
comparison between all the models predictions leads to

$\displaystyle\Delta(G_{\rm BP}^{\rm Gaia}-G_{\rm RP}^{\rm Gaia})$
$\displaystyle=0.004\pm 0.014\;{\rm mag},$ (6) $\displaystyle\Delta(B^{\rm
rgb}-G^{\rm Gaia})$ $\displaystyle=0.004\pm 0.008\;{\rm mag},$ (7)
$\displaystyle\Delta(G^{\rm rgb}-G^{\rm Gaia})$ $\displaystyle=0.002\pm
0.009\;{\rm mag},$ (8) $\displaystyle\Delta(R^{\rm rgb}-G^{\rm Gaia})$
$\displaystyle=0.002\pm 0.009\;{\rm mag,\;and}$ (9)
$\displaystyle\Delta(G_{\rm BR}^{\rm rgb}-G^{\rm Gaia})$
$\displaystyle=0.002\pm 0.010\;{\rm mag},$ (10)

where each $\Delta$ value corresponds to the mean (and associated standard
deviation) colour difference between the ${\rm[M/H]}\\!=\\!0.0$ and the
${\rm[M/H]}\\!=\\!-0.16$ predictions. Interestingly, all the models fit within
the $\pm 0.1$ mag interval for $G^{\rm rgb}$ and $G^{\rm rgb}_{\rm BR}$,
whereas the same is true for $B^{\rm rgb}$ and $R^{\rm rgb}$ when restricting
to the $-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.0$ mag range.

The residuals of the fitted data are displayed again in the bottom panels of
Fig. 4, but in this case the overplotted model predictions represent the
residuals of the CK04 models after reddening them employing the extinction law
of Cardelli et al. (1989) (updated by O’Donnell, 1994), with a relative
extinction parameter $R_{V}=3.1$ and variable extinctions $A_{V}$ ranging from
0.5 to 3.0 mag (crosses of different colours). Interestingly, the extinction
reddens the data basically along the sequences already displayed in the top
panels, and therefore the residuals in the bottom panels remain constrained
within the $\pm 0.1$ mag interval. The effect is larger in $B^{\rm rgb}$ and
$R^{\rm rgb}$ than in $G^{\rm rgb}$ and $G^{\rm rgb}_{\rm BR}$, with a
systematic variation as a function of the $G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm
RP}^{\rm Gaia}$ colour.

##  3 Predicting RGB magnitudes for 15 million stars

Table 2: First ten rows of the table with the 14 854 959 stars with RGB predictions. The full table is electronically available at http://guaix.fis.ucm.es/~ncl/rgbphot/gaia. Column description: (1) source identifier in the _Gaia_ EDR3; (2)–(3) star coordinates (J2000), as provided by the EDR3; (4)–(7) RGB magnitudes computed using the transformations given by Eqs. (2)–(5); (8)–(10)  _Gaia_ magnitudes in the EDR3; (11)–(13) 50th percentile (median) of the interstellar extinction $A_{V}$ (mag), [M/H] and distance (kpc), computed by Anders et al. (2019) using StarHorse. (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13)
---|---|---|---|---|---|---|---|---|---|---|---|---
EDR3 source_id | RA (°) | DEC (°) | $B^{\rm rgb}$ | $G^{\rm rgb}$ | $R^{\rm rgb}$ | $G^{\rm rgb}_{\rm BR}$ | $G^{\rm Gaia}$ | $G^{\rm Gaia}_{\rm BP}$ | $G^{\rm Gaia}_{\rm RP}$ | av50 | met50 | dist50
2875513285079465984 | 0.000095821 | $+34.987630914$ | 17.03 | 16.46 | 16.06 | 16.55 | 15.9741 | 16.6310 | 15.2054 | $+0.120$ | $-0.188$ | $+0.612$
393915403758148352 | 0.000114124 | $+50.277031947$ | 18.58 | 18.14 | 17.84 | 18.21 | 17.7949 | 18.3144 | 17.1386 | $+0.158$ | $-0.005$ | $+1.818$
4702040864637495296 | 0.000141937 | $-71.574692515$ | 18.15 | 17.76 | 17.49 | 17.82 | 17.4528 | 17.9223 | 16.8279 | $-0.239$ | $-0.111$ | $+1.720$
384492417301984256 | 0.000168528 | $+43.261207984$ | 16.71 | 16.43 | 16.25 | 16.48 | 16.2111 | 16.5778 | 15.6839 | $+0.122$ | $+0.027$ | $+1.952$
2746773530168227328 | 0.000178628 | $+08.371651685$ | 12.85 | 12.40 | 12.09 | 12.47 | 12.0394 | 12.5638 | 11.3628 | $+0.197$ | $-0.048$ | $+0.143$
2855280484422222336 | 0.000262145 | $+29.896074086$ | 18.08 | 17.45 | 16.99 | 17.54 | 16.8718 | 17.6054 | 16.0451 | $-0.129$ | $-0.019$ | $+0.771$
2875090969535427072 | 0.000301072 | $+33.520870876$ | 16.81 | 16.53 | 16.34 | 16.57 | 16.3041 | 16.6743 | 15.7705 | $+0.209$ | $-0.224$ | $+1.637$
2443095153084654080 | 0.000434825 | $-05.494409354$ | 09.63 | 09.21 | 08.92 | 09.27 | 08.8720 | 09.3678 | 08.2151 | $+0.139$ | $-0.316$ | $+0.518$
4923847544332011520 | 0.000435338 | $-54.929637980$ | 14.29 | 14.10 | 13.99 | 14.14 | 13.9466 | 14.2257 | 13.5046 | $+0.065$ | $-0.160$ | $+0.920$
2745049530295263232 | 0.000589823 | $+05.380517265$ | 13.98 | 13.43 | 13.03 | 13.51 | 12.9555 | 13.5889 | 12.1934 | $+0.038$ | $-0.235$ | $+0.149$

Figure 5: Histograms displaying the distributions in distance (top panel),
metallicity (middle panel) and $A_{V}$ extinction (bottom panel) for the 136
606 075 stars composing the StarHorse sample restricted to SH_GAIAFLAG=’000’
and SH_OUTFLAG=’00000’ (blue colour), and for the 14 854 959 stars
constituting the filtered 15M star sample (orange colour). Figure 6: Variation
of the number of stars with RGB magnitude predictions as a function of $G^{\rm
Gaia}$. The green curve shows the histogram in bins of 0.1 mag, whereas the
black line displays the cumulative sum. The inset tabulates the cumulative sum
down to some particular $G^{\rm Gaia}$ values.

We have applied the RGB calibration derived in the previous section to the
StarHorse star sample of Anders et al. (2019). The data was retrieved from the
_Gaia_ archive hosted by the Leibniz-Institute for Astrophysics
Potsdam101010https://gaia.aip.de/. With the aim of using the star subsample
with the most reliable distance, extinction and astrophysical parameter
determinations, we restricted the initial list to 136 606 075 stars flagged
with SH_GAIAFLAG=’000’ (indicating good astrometric and photometric quality of
the _Gaia_ DR2 data) and SH_OUTFLAG=’00000’ (associated to stars with reliable
StarHorse output parameters). It is worth noting that by adopting these values
of SH_GAIAFLAG and SH_OUTFLAG we excluded from the beginning stars identified
as variable in _Gaia_ DR2 (entry phot_variable_flag in the database), as well
as objects with significantly negative extinctions or with large $A_{V}$
uncertainties. From this initial collection, 13 756 448 stars were removed due
to multiple potential candidates in the crossmatch between DR2 and EDR3
(parameter dup_max_number greater than one). Then, we selected stars with
extinction estimates compatible with zero within the 16th and 84th $A_{V}$
percentiles computed by Anders et al. (2019) when deriving the astrophysical
parameters of the StarHorse sample, which restricted the sample to 20 477 474
stars. At this point, the cut $-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm
RP}^{\rm Gaia}<2.0$ mag was imposed to match the valid colour interval where
Eqs. (2)–(5) can be employed, leading to a collection of 19 567 621 objects.
Finally, we also constrained the estimated metallicity for each star to be
compatible with the median value ${\rm[M/H]}=-0.16$ exhibited by the C21
calibrating sample (middle panel in Fig. 3) within the 16th and 84th [M/H]
percentile interval derived by Anders et al. (2019). This last step led to the
final set of 14 854 959 stars (hereafter the 15M star sample) for which we
have estimated the RGB magnitudes. The predicted values are given in Table 2.

The histograms displayed in Fig. 5 compare the distributions in distance (top
panel), metallicity [M/H] (middle panel) and interstellar extinction $A_{V}$
(bottom panel) of the 15M star subsample (orange colour) with those exhibited
by the initial $\sim 136$ million stars corresponding to the StarHorse sample
(blue colour). Although the strong constraint introduced when selecting stars
with extinction estimates compatible with zero does not introduce a systematic
bias in the metallicity distribution, it imposes a clear restriction in the
distance coverage: most of the 15M star sample is composed of stars closer
than 3 kpc to the Sun.

Figure 7: _Panel (a):_ star density map, in Galactic coordinates,
corresponding to the 136 606 075 stars in the StarHorse sample of Anders et
al. (2019) flagged with SH_GAIAFLAG=’000’ and SH_OUTFLAG=’00000’. _Panels (b)
(c) and (d):_ star density maps corresponding to the stars with estimated RGB
magnitudes (i.e., those in the 15M star sample) for different limiting $G^{\rm
Gaia}$ magnitudes: 10, 14 and 18 mag for panels (b), (c) and (d),
respectively. These maps have been created using HEALPix of level 6 (providing
a pixel size of 0.84 square degrees), and are colour coded depending on the
number of stars within the pixel. _Panels (e) and (f):_ extinction maps
representing the mean $A_{V}$ values within each HEALPix of level 6, exhibited
by the 15M sample, panel (e), and the 136 606 075 stars in the StarHorse
sample, panel (f). _Panels (g) and (h):_ mean distance (kpc) within each
HEALPix of level 6 for the stars in the 15M sample, panel (g), and the
mentioned StarHorse sample, panel (h). The coordinate origin and grid in all
the panels are the same employed in Fig. 1.

The number of stars in the 15M sample, as a function of the $G^{\rm Gaia}$
magnitude, is shown in Fig. 6 (green line). As expected, this number increases
when moving to fainter objects. The cumulative number of stars down to a given
magnitude is also displayed (black line), with the total number of stars down
to some particular $G^{\rm gaia}$ values listed within the figure.

The global distribution in the celestial sphere of the 136 606 075 stars in
the StarHorse sample of Anders et al. (2019) flagged with SH_GAIAFLAG=’000’
and SH_OUTFLAG=’00000’ is shown in panel (a) of Fig. 7 (using Galactic
coordinates, with HEALPix111111https://healpix.sourceforge.io/ of level 6). It
is clear that most of the stars concentrate towards the Galactic plane. For
comparison, panels (b), (c) and (d) of Fig. 7 show the same map for the 15M
star sample restricted to different limiting $G^{\rm gaia}$ magnitudes. In
addition, panels (e) and (f), on one hand, and panels (g) and (h), on the
other, compare, respectively, the mean extinction $A_{V}$ (mag) and distance
(kpc) exhibited by the 15M sample and the initial StarHorse star collection.
These maps show that the stars in the 15M sample are unevenly spread because
high-extinction regions, clearly seen in panel (f), have been purposely
avoided, being the $A_{V}$ values of the 15M star sample, panel (e), quite
small. This behaviour is consistent with the distance maps, which confirm that
the 15M sample, panel (g), is constituted by nearby stars, as already shown in
the histogram of Fig. 5(a), specially at low Galactic latitudes.

##  4 Final discussion and conclusions

The work presented in the previous sections has shown that it is possible to
derive simple mathematical transformations between _Gaia_ EDR3 photometric
data and RGB magnitudes, and to employ them to substantially extend, both in
quantity and magnitude coverage, the number of stars in the celestial sphere
with RGB magnitude estimates. The comparison with measurements performed using
stellar atmosphere models (covering [M/H] from $-2.5$ to $+0.5$, effective
temperatures from 3500 to 50000 K, and $\log g$ from 0.0 to 5.0 dex) has
revealed that Eqs. (2)–(5) are expected to provide RGB estimates within a $\pm
0.1$ mag uncertainty when imposing the colour cut $-0.5<G_{\rm BP}^{\rm
Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.0$ mag. The predicted magnitudes remain
within the $\pm 0.1$ mag interval even when considering interstellar
extinctions in the range $A_{V}\in[0.0,3.0]$ mag, although there are some
systematic deviations depending on the $G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm
RP}^{\rm Gaia}$ colour, being $G^{\rm rgb}$ and $G_{\rm BR}^{\rm rgb}$ more
robust to this effect than $B^{\rm rgb}$ and $R^{\rm rgb}$. For that reason,
we have restricted the star sample to nearby objects for which the
interstellar extinction estimate derived by Anders et al. (2019) was
statistically compatible with zero.

The new catalogue, encompassing $\sim 15$ million stars, should smooth the way
for:

  * i)

The use of the standard RGB photometric system proposed by C21: the
homogenization of RGB measurements, derived from data obtained with a
potentially very large number of different cameras, will become essential.
This is a critical aspect that can hardly be overestimated.

  * ii)

The proper calibration of commercial-grade RGB cameras: this task will be
facilitated by considering the wide magnitude range exhibited by the 15M star
sample, leaving ample room for the use of RGB imaging instruments with
different field of views, as well as their exploitation in scientific projects
requiring distinct exposure times.

  * iii)

The calibration of observations carried out in any region of the sky, even at
high Galactic latitudes. Note, however, that the distribution in the celestial
sphere is inhomogeneous and depends on the interstellar extinction in the
direction of observation, being the total number of available calibrated stars
unavoidably tied to the adopted limiting magnitude.

  * iv)

The correction of the measurements for atmospheric extinction: this data
reduction step is particularly important in wide-field exposures, where stars
at different airmasses are simultaneously observed.

It is important to highlight that the RGB magnitude predictions computed for
the 15M star sample should not be considered to be extremely accurate on a
star by star basis. In particular, although stars detected as variables in DR2
have been removed, it is likely that variable sources are still hidden in the
selected sample. In any case, the large number of calibrating stars available
should facilitate the computation of statistical averages that allow the
rejection of potential outliers, guaranteeing adequate calibrations.

Although in this work we have restricted the RGB estimates to a subsample of
the stars published by Anders et al. (2019) with good photo-astrometric
distances, extinctions and astrophysical parameters, it is possible to apply
the derived calibrations to many more _Gaia_ EDR3 stars for which these
parameters are not even available. In principle, this should only be employed
for stars with high Galactic latitude in order to minimize systematic errors
introduced by interstellar extinction (although there are regions with low
galactic latitude and low extinction), and within the colour interval
$-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.0$ mag. In this
regard, it is interesting to note that $\sim 82$ per cent of the 304 602 695
stars available in _Gaia_ EDR3 with $G^{\rm Gaia}\leq 18$ mag (251 118 359
stars) verify the last colour cut. This means that, down to a given limiting
magnitude, the total number of observable stars is expected to be much larger
than the number of stars belonging to the 15M star sample. The initial
calibration obtained by employing the predicted RGB magnitudes presented here,
using only a subsample of the observed stars, can be applied, in a second
iteration, to the remaining stars verifying the appropriate colour cut. The
inclusion of additional stars (after removing outliers) should facilitate the
computation of a more reliable calibration. In addition, the relative
robustness of $G^{\rm rgb}$ and $G_{\rm BR}^{\rm rgb}$ to moderate amounts of
interstellar extinction, should even allow the use of observations performed
in regions closer to the Galactic plane (although this should always be
double-checked through comparison with calibrations performed at higher
Galactic latitudes). With the aim of helping on the use (and extension) of the
15M star sample, in Appendix A we illustrate how to estimate RGB magnitudes
for all the _Gaia_ EDR3 stars within a particular cone search.

The synthetic RGB photometry presented here fills an important gap that can
help to provide a firm ground for accurate camera calibrations and the
systematic exploitation of RGB photometry.

## Acknowledgements

The authors are grateful for the 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. The participation of
ICCUB researchers was (partially) supported by the Spanish Ministry of
Science, Innovation and University (MICIU/FEDER, UE) through grant
RTI2018-095076-B-C21, and the Institute of Cosmos Sciences University of
Barcelona (ICCUB, Unidad de Excelencia ’María de Maeztu’) through grant
CEX2019-000918-M. 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 made use of data
from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This work has been possible
thanks to the extensive use of IPython and Jupyter notebooks (Pérez & Granger,
2007), as well as the software package R121212https://www.R-project.org/ (R
Core Team, 2021). This research made use of
astropy131313http://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), matplotlib (Hunter, 2007), pandas
(Wes McKinney, 2010), and vaex (Breddels & Veljanoski, 2018). Some of the
results in this paper have been derived using the healpy and HEALPix packages
(Górski et al., 2005; Zonca et al., 2019). 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 _Gaia_ DR2 and EDR3, provided by the
European Space Agency141414https://gea.esac.esa.int/archive/, the StarHorse
database hosted by the Leibniz-Institute for Astrophysics
Potsdam151515https://data.aip.de/projects/starhorse2019.html, the Stellar
Atmopshere Models of Castelli & Kurucz (2003), as provided by the STScI web
page161616https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/castelli-and-kurucz-atlas, the
Bright Star Catalogue (Hoffleit, 1964, available
online171717https://vizier.u-strasbg.fr/viz-
bin/VizieR-3?-source=V/50/catalog), and the catalogue of synthethic RGB
magnitudes published by Cardiel et al. (2021).

All the results of this paper, together with future additional material, is
available online at http://guaix.fis.ucm.es/~ncl/rgbphot/gaia, and will be
also available through VizieR.

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##  Appendix A Estimation of RGB magnitudes beyond the 15 million star sample

We illustrate in this appendix how to estimate RGB magnitudes for all the
_Gaia_ EDR3 stars within an arbitrary cone search in the celestial sphere.
Their use in any calibration procedure should always be accompanied by the
observation of reference stars belonging to the 15M star sample.

It is important to remember that _Gaia_ has a bright limit around $G^{\rm
Gaia}\sim 3$ mag, and thus, bright stars will be missing.

###  A.1 Use of an ADQL query

Figure 8: Screenshots of the results of using the ADQL query described in
Sect. A.1 in the European Agency Portal of the Gaia EDR3 database. Top panel:
interface for advanced queries. The cone search is centrered at RA=56.66°,
DEC=24.10° (line number 14 of the displayed query), with a search radius of 1°
(line 15), and a limiting $G^{\rm Gaia}$ magnitude of 12 mag (line 21). Bottom
panel: table with results. A total of 291 stars match the selection criteria
(only the first 20 are displayed), with the RGB magnitude estimates $B^{\rm
rgb}$, $G^{\rm rgb}$, $R^{\rm rgb}$ and $G_{\rm BR}^{\rm rgb}$ listed in the
last four columns, respectively.

A direct way to retrieve RGB magnitudes is to employ an ADQL (Astronomical
Data Query Language; see e.g. Osuna et al., 2008) query to access the _Gaia_
EDR3 database. This language allows to evaluate mathematical expressions on
the relevant parameters. For illustration, a simple cone search of _Gaia_ EDR3
stars brighter than $G^{\rm Gaia}\\!=\\!12$ mag, within a circular region of
radius 1°, with centre at RA=56.66°and DEC=24.10° (corresponding to the
Pleiades star cluster), imposing the colour selection $-0.5<G_{\rm BP}^{\rm
Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.0$ mag, and using the polynomial functions
given Eqs. (2)–(5) to estimate the RGB magnitudes, can be performed employing:

    SELECT source_id, ra, dec,
    phot_g_mean_mag, phot_bp_mean_mag, phot_rp_mean_mag,
    phot_bp_mean_mag-phot_rp_mean_mag as bp_rp,
    round(phot_g_mean_mag
          -0.13748689
          +0.44265552*power(bp_rp, 1)
          +0.37878846*power(bp_rp, 2)
          -0.14923841*power(bp_rp, 3)
          +0.09172474*power(bp_rp, 4)
          -0.02594726*power(bp_rp, 5), 2) as b_rgb,
    round(phot_g_mean_mag
          -0.02330159
          +0.12884074*power(bp_rp, 1)
          +0.22149167*power(bp_rp, 2)
          -0.14550480*power(bp_rp, 3)
          +0.10635149*power(bp_rp, 4)
          -0.02363990*power(bp_rp, 5), 2) as g_rgb,
    round(phot_g_mean_mag
          +0.10979647
          -0.14579334*power(bp_rp, 1)
          +0.10747392*power(bp_rp, 2)
          -0.10635920*power(bp_rp, 3)
          +0.08494556*power(bp_rp, 4)
          -0.01368962*power(bp_rp, 5), 2) as r_rgb,
    round(phot_g_mean_mag
          -0.01252185
          +0.13983574*power(bp_rp, 1)
          +0.23688188*power(bp_rp, 2)
          -0.10175532*power(bp_rp, 3)
          +0.07401939*power(bp_rp, 4)
          -0.01821150*power(bp_rp, 5), 2) as g_br_rgb

FROM gaiaedr3.gaia_source
WHERE 1=CONTAINS(
POINT(’ICRS’, 56.66, 24.10),
CIRCLE(’ICRS’,ra, dec, 1.00))
AND phot_g_mean_mag IS NOT NULL
AND phot_bp_mean_mag IS NOT NULL
AND phot_rp_mean_mag IS NOT NULL
AND phot_bp_mean_mag - phot_rp_mean_mag > -0.5
AND phot_bp_mean_mag - phot_rp_mean_mag < 2.0
AND phot_g_mean_mag < 12

ORDER BY ra

Note that in the previous example the user-defined central coordinates, search
radius and limiting $G^{\rm Gaia}$ magnitude are shown in boldface. Fig. 8
illustrates the execution of this query through the _Gaia_ EDR3 Archive at the
European Space Agency181818https://gea.esac.esa.int/archive/, which returns
RGB magnitude estimates for the sample of 291 stars matching the selection
criteria. Note, however, that at this point the resulting star list should be
cross-matched with the 15M star sample built in this work in order to
segregate the star sample into reference stars (those belonging to the 15M
sample) and secondary calibrating stars (those that do not belong). Since this
cross-matching goes beyond the idea of the simple ADQL query employed here, in
the next subsection we describe an auxiliary Python package that performs this
task automatically.

###  A.2 The Python rgblues package

Figure 9: Example of finding chart generated by the Python package rgblues
after performing a cone search centred in the Pleiades star cluster, with a
search radius of 1°. The stars in this plot are colour coded based on the
$G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$ colour. A red circle has
been overplotted on the stars belonging to the 15M star sample, a blue square
on the variable objects in DR2, and a grey diamond on EDR3 stars outside the
$-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}<2.0$ mag colour
interval. Stars brighter than a pre-defined threshold are displayed with big
star symbols. The stars are numbered with labels of different colours (red,
blue and black for stars in the 15M star sample, variable objects in DR2, and
remaining stars in EDR3, respectively), matching the order of the stars in the
three output CSV files generated during the execution of the program.

Trying to help future users of the 15M star sample to perform cone search
queries as that shown in the previous subsection, we have also created a
Python package, called rgblues191919Available at https://github.com/guaix-
ucm/rgblues, that executes this type of queries and performs the additional
extra work required to automatically discriminate between reference stars,
belonging to the 15M star sample, from secondary calibrating stars (additional
objects in EDR3), flagging objects detected to be variable in Gaia DR2 and
those outside the colour range $-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm
RP}^{\rm Gaia}<2.0$ mag.

Once installed, the software can be easily executed from the command line:
$ rgblues 56.66 24.10 1.0 12
The four positional arguments correspond to RA, DEC, search radius (these
three parameters in decimal degrees) and limiting $G^{\rm Gaia}$ magnitude.

The steps followed by rgblues to complete its tasks are the following:

  1. Step 1:

cone search in Gaia EDR3 down to a pre-defined limiting $G^{\rm gaia}$
magnitude, gathering the following parameters: source_id, ra, dec,
phot_g_mean_mag, phot_bp_mean_mag and phot_rp_mean_mag. In this case, an ADQL
query similar to that shown in Sect. A.1 is performed, without imposing any
$G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm Gaia}$ colour restriction nor
evaluating the polynomial transformations given in Eqs. (2)–(5).

  2. Step 2:

cone search in the StarHorse sample through the _Gaia_ archive hosted by the
Leibniz-Institute for Astrophysics Potsdam. This step, which is optional,
allows the compilation of stellar parameters associated with each star, such
as interstellar extinction, metallicity and distance.

  3. Step 3:

cross-matching of the previous EDR3 sample with the list of 15M star sample
from this work. This step determines the subsample of EDR3 stars for which the
RGB photometric calibration is reliable.

  4. Step 4:

cone search in Gaia DR2. This additional step is performed to retrieve the
phot_variable_flag parameter indicating whether the star was flagged as
variable in DR2. Note that this flag is not available in EDR3.

  5. Step 5:

cross-matching between DR2 and EDR3 to identify the variable stars in EDR3.
This step is required because it is not guaranteed that the same astronomical
source will always have the same source identifier in the different Gaia Data
Releases.

  6. Step 6:

computation of the RGB magnitudes using the polynomial transformations given
in Eqs. (2)–(5).

  7. Step 7:

generation of the output files. Three files (in CSV format) are generated,
segregating the star list in i) stars belonging to the 15M star sample (with
reliable RGB magnitude estimates), ii) objects flagged as variable in Gaia
DR2, and iii) remaining objects in Gaia EDR3. Note that the RGB magnitudes
estimated for the latter can be potentially biased due to systematic effects
introduced by interstellar extinction, or by exhibiting non-solar metallicity
or a colour outside the $-0.5<G_{\rm BP}^{\rm Gaia}\\!-\\!G_{\rm RP}^{\rm
Gaia}<2.0$ mag interval.

  8. Step 8:

creation of a finding chart. Fig. 9 illustrates the resulting plot after
executing the Python code with the same cone-search parameters employed in
Sect. A.1.