11institutetext: Institut für Physik und Astronomie, Universität Potsdam, Haus
28, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany
<EMAIL_ADDRESS>22institutetext: Department of Physics, University of
Warwick, Coventry, CV4 7AL, UK

# Clean catalogues of blue horizontal-branch stars using Gaia EDR3††thanks:
The catalogues are only available in electronic form at the CDS via anonymous
ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/

R. Culpan  I. Pelisoli 111122 S. Geier 11

(Received December 22, 2020; accepted July 27, 2021)

###### Abstract

Context. Blue horizontal-branch stars evolve from low-mass stars that have
completed their main-sequence lifetimes and undergone a helium flash at the
end of their red-giant phase. As such, blue horizontal-branch stars are very
old objects that can be used as markers in studies of the Galactic structure
and formation history. To create a clean sky catalogue of blue horizontal-
branch stars, we cross-matched the Gaia data release 2 (DR2) dataset with
existing reference catalogues to define selection criteria based on Gaia DR2
parameters. Following the publication of Gaia early data release 3 (EDR3),
these methods were verified and subsequently applied to this latest release.

Aims. Previous catalogues of blue horizontal-branch stars were developed using
spectral analyses or were restricted to individual globular clusters. The
purpose of this catalogue is to identify a set of blue horizontal-branch star
candidates that have been selected using photometric and astrometric
observations and exhibits a low contamination rate. This has been deemed
important as the success of the Gaia mission has changed the way that targets
are selected for large-scale spectroscopic surveys, meaning that far fewer
spectra will be acquired for blue horizontal-branch stars in the future unless
they are specifically targeted.

Methods. We cross-matched reference blue horizontal-branch datasets with the
Gaia DR2 database and defined two sets of selection criteria. Firstly, in Gaia
DR2 - colour and absolute G magnitude space, and secondly, in Gaia DR2 -
colour and reduced proper motion space. The main-sequence contamination in
both subsets of the catalogue was reduced, at the expense of completeness, by
concentrating on the Milky Way’s Galactic halo, where relatively young main-
sequence stars were not expected. The entire catalogue is limited to those
stars with no apparent neighbours within 5 arcsec. These methods were verified
and subsequently applied to the Gaia Early Data Release 3 (EDR3).

Results. We present a catalogue, based on Gaia EDR3, of 57,377 blue
horizontal-branch stars. The Gaia EDR3 parallax was used in selecting 16,794
candidates and the proper motions were used to identify a further 40,583
candidates.

###### Key Words.:

blue horizontal branch – catalogs – horizontal-branch – Hertzsprung-Russell
and C-M diagrams – halo

##  1 Introduction

Blue horizontal-branch stars are old objects found on the blue side of RR Lyra
at the bottom of the instability strip. They represent a late stage in the
evolution of low-mass stars with initial masses in the range of
${\sim}0.8M_{\odot}$ to ${\sim}2.3\,M_{\odot}$. A star will leave the main-
sequence once core hydrogen burning ceases and the inert helium core
contracts. This contraction heats the hydrogen shell around the core, thereby
allowing fusion to begin, and the star evolves onto the red giant branch
(RGB), where a large amount of the outer envelope is shed by the strong solar
wind combined with the tenuous gravitational binding forces at large radii.
The inert helium core will continue to contract and becomes electron-
degenerate. Finally the temperature in the helium core rises enough to allow
core helium burning to begin and the helium flash occurs. This increases the
temperature further and the core ceases to be degenerate. At this point, stars
move to the blue horizontal-branch with masses in the range of $\sim$0.5–1.0
$M_{\odot}$ (Montenegro, 2019) and display helium burning in their core as
well as hydrogen shell burning (Moehler, 2001; Ruhland et al., 2011; Paunzen
et al., 2019). The mass of the hydrogen shell may vary from 0.02 $M_{\odot}$
to over 0.2 $M_{\odot}$. The mass of the helium-burning core is ${\sim}$0.5
$M_{\odot}$. Blue horizontal-branch stars with a higher mass of hydrogen shell
will appear cooler and, hence, redder (Moehler, 2001; Paunzen et al., 2019).

The existence of horizontal-branch stars was first proven in photometric
studies of the M92 and M3 globular clusters in the 1950s (Arp et al., 1952;
Sandage, 1953). These observations explained the differences between the
colour-magnitude diagrams from open and globular clusters first observed by
Harlow Shapley in his study of the structure of the Galaxy between 1914 and
1919. Catalogues of blue horizontal-branch stars were first published in the
1980s (Pier, 1982; Beers et al., 1988) and have continued to develop as larger
and higher-quality datasets have become available (Beers et al., 1996, 2007;
Montenegro, 2019).

The small scatter in their absolute magnitude and their relative abundance
makes blue horizontal-branch stars a particularly useful standard candle in
determining distances (Clewey & Jarvis, 2006). This, and the fact that they
are metal-poor population II stars makes blue horizontal-branch stars ideal
for studying Galactic structure in older parts of the Galaxy (Starkenburg et
al., 2019; Newberg, 2003; Monaco, 2003; Clewey & Jarvis, 2006; Yanny, 2009;
Niederste-Ostholt, 2010) or for studying the enclosed masses of the Milky Way
within different radii (Sommer Larsen et al., 1986, 1989; Norris & Hawkins,
1991; Xue et al., 2008).

Observationally, most blue horizontal-branch stars are expected to have an
A-type (Xue et al., 2008) or very late B-type spectra. The A-type spectra seen
in blue horizontal-branch stars can be distinguished from population I main-
sequence A-type stars by their stronger Balmer jump, stronger and deeper
Balmer lines, and the lack of metal spectral lines (Smith et al., 2010). Those
with an effective temperature of T${}_{\textrm{eff}}\geq 10\,000$ K have
spectra that are similar to the much younger population I B-type main-sequence
stars but with weaker Helium lines, which can be distinguished
spectroscopically (see Figure 1). The main challenge in producing a catalogue
of low contamination is differentiating between blue horizontal-branch stars
and younger, population I main-sequence A-type and B-type stars as well as
blue stragglers, which are older, population II main-sequence stars that have
been rejuvenated through acquisition of additional mass (Rain et al., 2021)
and that display A-type spectra. The additional mass may be gained by Roche
Lobe overflow in a close binary system (McCrea, 1964), through stellar merger
(Hills & Day, 1976; Davies et al., 1994), or both. Blue stragglers, which have
similar effective temperatures but higher gravities than blue horizontal-
branch stars, are also fainter (Starkenburg et al., 2019) and are expected to
be found in older parts of the Galaxy (Santucci et al., 2015), where younger
main-sequence stars of similar effective temperatures no longer exist.

Figure 1: Typical B-type spectra displaying the difference between main-
sequence and blue horizontal-branch stars. Upper panel: Example of a typical
SDSS B-type spectrum from a blue- horizontal-branch star (SDSS
J131916.15-011404.9 from Geier et al. (2015)). Lower panel: Example of a
typical LAMOST B-type spectrum from a main-sequence star. Some reference lines
for hydrogen and helium are indicated by vertical lines. SDSS and LAMOST
spectra are comparable due to their similar resolutions and spectral coverage.

The goal of this work is to use the second data release (DR2) and early third
data release (EDR3) of Gaia to produce a catalogue of blue horizontal-branch
candidate stars with a low and quantifiable blue straggler and population I
main-sequence contamination. This method identifies likely blue horizontal-
branch stars without relying on spectra, thus providing important input
information for upcoming large spectroscopic surveys. The blue horizontal-
branch star catalogues of Xue et al. (2008) and Behr (2003) were used as
reference catalogues to calibrate the colour and absolute magnitude selection
criteria in the Gaia DR2 colour magnitude space.

##  2 Selection of known blue horizontal-branch stars

The blue horizontal-branch dataset of Xue et al. (2008) was considered the
reference for determining the Gaia DR2 colour and absolute magnitude selection
criteria. Xue et al. (2008) used the seventh data release of the Sloan Digital
Sky Survey (SDSS DR7, Azerbaijan et al., 2009) data to obtain 2558 blue
horizontal-branch stars claiming contamination of less than 10$\%$. The
contaminants were considered to be main-sequence stars, particularly blue
stragglers. This reference dataset was created using several steps (detailed
in Xue et al., 2008, and references therein). Firstly, by applying colour cuts
in the $(u-g)\times(g-r)$ parameter space, then by considering the Balmer line
widths at 20$\%$ below the local continuum plotted against the depth of the
Balmer line compared to the local continuum, and finally by using the ’scale
width versus shape’ method (Clewley et al., 2002) where the Balmer lines are
fit to a Sérsic profile (Sérsic, 1968).

The Behr (2003) dataset was used together with the Xue et al. (2008) dataset
for the purposes of comparing reference datasets with the Gaia DR2 selections
outlined in subsequent sections. This was done to confirm that the selection
criteria developed were valid. The Behr (2003) dataset comprises 74 blue
horizontal-branch stars selected from 6 metal-poor globular clusters in the
Globular Cluster Catalogue (Harris, 1996).

##  3 Constructing the blue horizontal-branch star catalogue from Gaia DR2

The European Space Agency’s Gaia mission was initiated with the aim of
charting a three-dimensional (3D) map of the Milky Way by surveying over one
billion stars in our Galaxy. This is about 1$\%$ of the stars in our Galaxy.
The procedure involved creating a database of the position, parallax,
brightness, colour, and proper motions of all objects visible to the satellite
(i.e. down to an apparent Gaia $G$ magnitude of $\sim$20.7 mag, Gaia
Collaboration, 2016). Each Gaia data release provides a dataset that is more
complete and with reduced uncertainties than its predecessor.

Assessments of the accuracy of the astrometric and photometric measurements in
the second data release (Gaia Collaboration, 2018a) were published by
Lindegren et al. (2018) and Evans et al. (2018), respectively. Assessments of
the accuracy of the astrometry and photometry in the early third data release
(Gaia Collaboration, 2020) were published by Lindegren et al. (2020) and
Riello et al. (2020), respectively. We used the TOPCAT application (Taylor,
2005) to cross-match the Xue et al. (2008) blue horizontal-branch stars with
the Gaia DR2 dataset to provide an initial calibration point for finding such
stars in the Gaia DR2 parameter space.

The methodology used was based on the procedure used by Gentile Fusillo et al.
(2018) to create a white dwarf catalogue, as well as by Pelisoli & Vos (2019)
to create a catalogue of extremely low-mass white dwarf candidates, and by
Geier et al. (2019) to create their catalogue of hot subluminous stars. Some
deviation from these methodologies was required to address the challenge of
main-sequence contamination as the blue horizontal-branch crosses the main-
sequence leading to similar parameter values in the Gaia DR2 and EDR3 colour
magnitude parameter space.

###  3.1 Gaia DR2 colour and absolute magnitude criteria selection

Figure 2: The result of cross-matching the Xue et al. (2008) reference dataset
with Gaia DR2. Upper panel: Xue et al. (2008) objects plotted as turquoise
dots on the Gaia DR2 colour-magnitude diagram exhibiting a large scatter in
absolute magnitude due to incorrect parallax measurements. Those with parallax
error ¡20% are plotted as red large dots. Lower panel: Lindegren et al. (2018)
’clean subset selection C’ (black) compared with Xue et al. (2008) objects
with parallax error ¡20% (red) on the same scales as the upper plot in the
Gaia DR2 colour-magnitude diagram.

The Xue et al. (2008) reference dataset was cross-matched with Gaia DR2. The
absolute G magnitude was then calculated as:

$G_{\textrm{abs}}=\verb!phot_g_mean_mag!+5\log_{10}(\verb!parallax!/1000)+5$.

As the absolute G magnitude was calculated using the Gaia DR2 parallax
measurement for distance, it was necessary to restrict the reference dataset
to only those stars with a reliable parallax measurement. Using the parallax
quality criterion of parallax error $<$ 20$\%$
($\verb!parallax_over_error!>5$), as in Geier et al. (2019) and Pelisoli & Vos
(2019), left only 39 of the original 2560 objects from the Xue et al. (2008)
reference dataset, as shown in Figure 2. These 39 stars were found to lie in
the ranges of:

$-0.1\leq(G_{BP}-G_{RP})\leq 0.83$

$-1\leq G_{\textrm{abs}}\leq 5$

in the Gaia DR2 colour absolute G magnitude parameter space.

These colour and absolute magnitude cut-off criteria (henceforth, referred to
as the initial CMD criteria) were applied to the Gaia DR2 dataset. We also
applied quality criteria that were based on the astrometry and photometry
quality criteria of the ’clean subset Selection C’ by Lindegren et al. (2018).
We modified these criteria to be less stringent by allowing parallax errors up
to 20$\%$ (as opposed to 10$\%$) and by relaxing the parallax lower limit to
the requirement of parallax being non-negative (as opposed to being greater
than 10 mas). Using the unmodified Lindegren et al. (2018) criteria would have
limited the selection to distances below $\sim 100$ pc, where predominantly
main-sequence Galactic disc objects are found. Furthermore, we did not apply
the $\verb! astrometric_excess_noise!<1$ mas criterion as we wanted to keep
the blue horizontal-branch candidates that may exist in binary systems. As can
be seen in Belokurov et al. (2020), in particular, Figure 7, there is an over-
density of sources with a re-normalised unit weight error (RUWE) ¿ 1 in the
blue horizontal-branch region of the Gaia DR2 colour-magnitude diagram,
indicating that many blue horizontal-branch candidates may exist in unresolved
binary systems. Lindegren et al. (2018) show that the astrometric excess noise
to be a proxy for RUWE.

We retained the Lindegren et al. (2018) quality criteria for the $G_{BP}$ and
$G_{RP}$ photometry, which require that the errors on the $G_{BP}$ and
$G_{RP}$ flux measurements are ¡10$\%$ and that the flux excess factor
(`phot_bp_rp_excess_factor`), namely:

$E=(I_{BP}+I_{RP})/I_{G}$

lies in the range defined by the following cut-offs:

$\verb!phot_bp_rp_excess_factor!>1.0+0.015(G_{BP}-G_{RP})^{2},$

$\verb!phot_bp_rp_excess_factor!<1.3+0.06(G_{BP}-G_{RP})^{2}.$

Applying the initial CMD criteria and the modified Lindegren et al. (2018)
criteria for astrometric and photometric quality to Gaia DR2 resulted in a set
of over 9 million objects. These stars are expected, based on their position
on the colour-magnitude diagram (CMD), to be blue horizontal-branch stars,
blue-stragglers, and population I main-sequence A-type and B-type stars. The
measured parallaxes of these objects lead us to believe that they exist in
either the Galactic disc or the Galactic halo.

It is known that the relatively young population I main-sequence A-type and
B-type stars are not expected in the much older Galactic halo. We expected
that focusing on the Galactic halo will reduce the population I main-sequence
contamination of the dataset. To do this we applied the stringent halo
selection criteria ($V_{T}>200$ km s-1, $|b|>60^{\circ}$) to the 9 million
objects found so far. Using a tangential velocity cut-off to differentiate
between disk and halo objects is an accepted method (Gaia Collaboration, 2018;
Koppelman & Helmi, 2020) that uses only the two tangential components of
velocity. It should be noted that any halo objects with low tangential
velocities but with high radial velocities along the line of sight will not be
selected. This is more likely to be the case in the direction of the motion of
the heliocentric system within the Galactic reference frame. We calculated the
tangential velocity as:

$V_{T}=(4.74/\verb!parallax!)\times(\verb!pmra!^{2}+\verb!pmdec!^{2})^{1/2}$.

This resulted in a set of 43,040 objects that, when plotted on the Gaia DR2
CMD that showed a clear clustering in the region where blue horizontal-branch
stars would be expected. This was confirmed by overlaying the reference
datasets from Xue et al. (2008) and Behr (2003) (see Figure 3).

Figure 3: Gaia DR2 colour absolute magnitude diagram showing the blue
horizontal-branch star region. Top panel: Gaia DR2 objects that satisfy the
modified (Lindegren et al., 2018) criteria (gray points) with those with the
stringent halo criteria applied (small blue dots). Xue et al. (2008) objects
(red circles) and Behr (2003) objects (yellow triangles) are superimposed. The
green trapezium denotes the Gaia DR2 colour-magnitude coordinate space (final
CMD criteria) considered as the blue horizontal-branch region. The main-
sequence turn-off can be seen at around $(G_{BP}-G_{RP})$ = 0.6 and $G_{abs}$
= 3. Blue stragglers are found extending the main-sequence beyond that. Our
selection was designed to minimise the blue-straggler contamination and focus
on the blue horizontal-branch stars. Bottom panel: For comparison, the
Lindegren et al. (2018) ’clean subset selection C’ comprised of disc stars
within 100pc which are primarily main-sequence stars (black) with red
trapezium denoting the Gaia DR2 colour-magnitude coordinate space considered
as the blue horizontal-branch region.

We were then able to define modified CMD selection criteria that would allow
us to focus more closely on the blue horizontal-branch region in Gaia DR2
colour-magnitude space. We defined these modified criteria as:

$-0.1<(G_{BP}-G_{RP})<0.53$, $-1<G_{abs}<(2.5-2.77(G_{BP}-G_{RP})),$

henceforth the final Gaia CMD criteria. Applying the final CMD selection
criteria to the 9 million initial CMD objects reduced the selection to 305,622
objects comprising of blue horizontal-branch stars, blue-stragglers, and
population I main-sequence A-type and B-type stars. It is known that regions
of high main-sequence stellar population density and hence, high potential
population I main-sequence contamination exist in the Galactic plane and the
Magellanic Clouds. These high apparent stellar density regions are also prone
to having astrometric solutions that are not accurate, but still precise, and
therefore not easily removed by a quality assessment alone (e.g. Gentile
Fusillo et al. (2018)). Objects in these regions were removed by excluding
regions in the Galactic plane and the Magellanic Clouds with a local apparent
stellar population density $>$50,000 per deg2:

$-25^{\circ}<b<25^{\circ}$,

the Large Magellanic Cloud:

$274.5^{\circ}<l<286.5^{\circ}$ and $-37.9^{\circ}<b<-27.9^{\circ}$,

and the Small Magellanic Cloud:

$299.8^{\circ}<l<305.8^{\circ}$ and $-46.3^{\circ}<b<-42.3^{\circ}$.

This further reduced the number of objects in our selection to 104,763.
Fainter objects in crowded regions are known to be susceptible to photometric
errors in the $G_{BP}$ and $G_{RP}$ bands (Lindegren et al., 2018). Thus,
additional filtering was added whereby stars with apparent neighbours within 5
arcsec were rejected (Pelisoli & Vos, 2019; Gentile Fusillo et al., 2018;
Boubert et al., 2018). This close apparent neighbour filtering criterion had,
together with the Galactic plane and Magellanic Cloud filtering, the
additional benefit of removing the majority of objects where reddening from
extinction in dust clouds is not negligible. We did not apply extinction
corrections to the Gaia DR2 colours for these objects due to the low
resolution of existing dust maps (Pelisoli & Vos, 2019). Removing any
remaining objects with close apparent neighbours further reduced the number of
selected objects to 91,719 which are expected to comprise of blue horizontal-
branch stars, blue stragglers, and a reduced proportion of population I main-
sequence A-type and B-type stars (see Figure 4).

Figure 4: Sky distribution of parallax selection candidates. Regions of high
stellar number density (¿50,000 stars per square degree) have been excluded.
Initial Gaia CMD criteria:
---
$-0.1\leq(G_{BP}-G_{RP})\leq 0.83$
$-1\leq G_{abs}\leq 5$
Final Gaia CMD criteria:
$-0.1<(G_{BP}-G_{RP})<0.53$
$-1<G_{abs}<(2.5-2.77(G_{BP}-G_{RP}))$
Modified Lindegren et al. (2018) criteria for astrometric quality:
$\verb!parallax!>0$
$\verb!parallax_over_error!>5$
Modified Lindegren et al. (2018) criteria for photometric quality:
$\verb!phot_bp_mean_flux_over_error!>10$
$\verb!phot_rp_mean_flux_over_error!>10$
$\verb!phot_bp_rp_excess_factor!>1.0+0.015(G_{BP}-G_{RP})^{2}$
$\verb!phot_bp_rp_excess_factor!<1.3+0.06(G_{BP}-G_{RP})^{2}$
Stringent halo selection criteria:
$V_{T}>200$ km s-1
$|b|>60^{\circ}$
Crowded region criteria of
$>$50,000 stars within 1 deg2 for:
Galactic plane:
($-25^{\circ}<b<25^{\circ}$)
Large Magellanic Cloud:
$274.5^{\circ}<l<286.5^{\circ}$ and $-37.9^{\circ}<b<-27.9^{\circ}$
Small Magellanic Cloud:
$299.8^{\circ}<l<305.8^{\circ}$ and $-46.3^{\circ}<b<-42.3^{\circ}$
Close apparent neighbour criterion:
no apparent neighbour within 5 arcsec
Table 1: Table of selection criteria applied to Gaia DR2 to define blue
horizontal-branch star candidate selection from colour and absolute G
magnitude (parallax selection)

####  3.1.1 Galactic halo selection

The stringent halo selection criteria were relaxed in small iterative
increments in order to find the least restrictive cut-offs before a
disproportionate increase in the number of candidate objects was observed
indicating population I main-sequence and non-halo object contamination. A
cut-off for Galactic latitude was not applied and a value of tangential
velocity of $V_{T}>145$ km s-1 was found for the parallax candidate objects.
This was considered the Galactic halo selection criterion for the parallax
error $<$ 20$\%$ dataset (see Table 2) and was applied to the parallax
candidates to produce a cleaner set of halo, blue horizontal-branch stars and,
thus, a cleaner selection of parallax candidates.

####  3.1.2 Verification of the Galactic halo selection using LAMOST spectra

The 91,719 objects found prior to applying any halo selection criteria were
cross-matched with the SDSS data release 16 (DR16, Ahumada et al., 2020) and
LAMOST fifth data release (DR5) (Yao et al., 2018) surveys to find spectra for
the blue horizontal-branch candidate objects. The relatively close proximity
of the parallax candidates leads to their being brighter than more distant
candidates. The low apparent magnitudes (lower than the SDSS DR16 photometric
saturation limit) of the objects being cross-matched with the SDSS DR16
spectra meant that less than 100 matches were found which we considered to be
not statistically significant and thus, may not be representative of the
dataset. LAMOST DR5 spectra were, however, found for $\sim$24,000 stars. This
represented roughly one third of the objects found had a similar magnitude
distribution and were thus considered to be a statistically significant and
representative population.

The LAMOST stellar parameter pipeline (Xiang et al., 2015) found that
$\sim$21,500 stars had A-type spectra, as would be expected for most blue
horizontal-branch stars, for blue stragglers, and for population I main-
sequence A-type stars (Xue et al., 2008). Furthermore, about 880 were found to
have B-type spectra, as would be expected for hotter blue horizontal-branch
stars (see the upper panel of Figure 1 for a typical example) and population I
main-sequence B-type stars (see the lower panel of Figure 1 for a typical
example). The remaining $\sim$1600 spectra were considered to be too noisy for
a spectral classification. Visual inspection of these spectra confirmed the
LAMOST classification for the A-type and B-type spectra and less than 1% of
the LAMOST objects were manually re-categorised.

The spectra of objects categorised by the LAMOST stellar parameter pipeline as
being B-type stars were found to have distinct helium lines as would be
expected for population I main-sequence B stars. Blue horizontal-branch B-type
stars display much subtler helium lines. Thus, we used the presence of large
numbers of LAMOST B-type spectra as a proxy for the presence of population I
main-sequence stars and, hence, as a proxy for non-halo objects.

The application of the halo selection criterion of $V_{T}>145$ km s-1 reduced
the number candidate objects by around 90% while reducing the number of LAMOST
cross-matched objects with B-type spectra from $\sim$880 to 6, a reduction of
over 99%. This represents an effective filter with which we can discern
between disc and halo objects, thereby facilitating the removal of a large
potential source of population I main-sequence contamination.

###  3.2 Gaia reduced proper motion criteria selection

Selecting only the Gaia DR2 stars with parallax errors of $<$ 20$\%$ results
in stars that are more than a few hundred pc away being rejected. New
selection criteria that do not use the Gaia parallax measurement needed to be
found in order to find candidate stars in this large population of more
distant objects. The parallax was used in calculating the absolute magnitude
and the tangential velocity in the parallax_error ¡ 20$\%$ dataset.

Reduced proper motion (H) is calculated from total proper motion ($\mu$) and
can be used as a proxy for absolute magnitude (Gentile Fusillo et al., 2015;
Geier et al., 2019):

$\mu=(\verb!pmra!^{2}+\verb!pmdec!^{2})^{1/2}$,

$H={\verb!phot_g_mean_mag!}+5+(5(log_{10}(\mu/1000)),$

$\sigma_{\mu}=(\verb!pmra_error!^{2}+\verb!pmdec_error!^{2})^{1/2}$,

${\verb!total_proper_motion_over_error!}=\mu/\sigma_{\mu}$.

Figure 5: The Gaia DR2 reduced proper motion versus colour diagram and reduced
proper motion frequency plot used to define the selection criteria for the
objects with a parallax error $\geq$ 20$\%$. Left panel: Reduced proper motion
(H) versus colour ($G_{BP}-G_{RP})$ diagram for the blue horizontal-branch
candidates with parallax error $<$ 20$\%$. Right panel: Distribution of
reduced proper motion (H) seen in the blue horizontal-branch candidates with a
parallax error ¡ $20\%$.

Figure 5 shows the distribution of reduced proper motions seen in the objects
from the parallax selection. This was used to define the selection criteria
for the blue horizontal-branch candidates from the parallax error $\geq$
20$\%$ objects (henceforth, called the proper motion candidates).

The `total_proper_motion_over_error` was also calculated in order to provide a
quality criterion to prevent stars with large proper motion uncertainties
being used (Gentile Fusillo et al., 2015). This is analogous to using
`parallax_over_error` as a quality criterion in the parallax error $<$ 20$\%$
dataset. A quality cut-off of total proper motion error ¡ 20% was used in line
with the parallax error $<$ 20$\%$ cut-off used previously.

The plot of Gaia DR2 colour versus reduced proper motion for the parallax
candidates dataset (Figure 5) gave a distribution of reduced proper motions
with $\overline{H}=9.36,\sigma_{H}=0.81$. A range of $\pm 3\sigma$ about the
mean reduced proper motion gave a 99$\%$ certainty of capturing all blue
horizontal-branch candidates using reduced proper motion as a proxy for
absolute magnitude.

The same photometric quality criteria as used in selecting the parallax
candidate dataset were also applied for the proper motion candidate dataset.
The astrometric quality criteria used for the parallax candidate dataset were
removed but the criterion $\verb!parallax_over_error!\leq 5$ was applied to
ensure that no stars were to be found in both datasets.

Application of the total proper motion over error cut-off, the reduced proper
motion and colour criteria, the photometric quality criteria, crowded region
criterion, the Galactic latitude criterion, and the parallax error cut-off to
the Gaia DR2 dataset resulted in the proper motion selection dataset.

Figure 6 shows the Gaia DR2 objects that conform to the total proper motion
over error cut-off, the photometric quality and the crowded region criteria,
the subset that conforms to the blue horizontal-branch and halo criteria with
the reference objects (Xue et al., 2008) overlain.

Figure 6: Gaia DR2 colour and reduced proper motion plot: Objects conforming
to the crowded region criterion as well as the total proper motion and
photometric quality criteria (grey dots), objects also conforming to the blue
horizontal-branch and halo criteria (blue crosses), and Xue et al. (2008)
objects (red squares).

####  3.2.1 Galactic halo selection

As the parallax is used in calculating the tangential velocity this could no
longer be reliably used as the halo selection criterion. We used the Galactic
latitude for this purpose. As done previously, we applied the stringent halo
selection criterion (absolute Galactic latitude $|b|>60^{\circ}$) and we
relaxed this criterion in small iterative increments in order to find the
least stringent cut-offs before a disproportionate increase in the number of
candidate objects was observed indicating population I main-sequence and non-
halo object contamination. The halo selection criterion found was
$|b|>50^{\circ}$.

####  3.2.2 Verification of the Galactic halo selection using LAMOST spectra

As previously done for the parallax selection, the halo selection criterion
was verified by comparing the number of LAMOST B-spectra removed to the total
number of objects removed. Application of the halo selection criterion of
$|b|>50^{\circ}$ to the $\sim$24,000 LAMOST cross-matched stars, presented in
Section 3.1.2, reduced the number candidate objects by around 92%, while
reducing the number of LAMOST cross-matched objects with B-type spectra from
$\sim$880 to 2, a reduction of over 99.8%. This represents an effective filter
for discerning between disc and halo objects, thereby facilitating the removal
of a large potential source of population I main-sequence contamination.

##  4 Application of the Gaia GR2 selection criteria to Gaia EDR3

Upon the release of Gaia EDR3, the processing steps outlined in the sections
above were performed using Gaia EDR3 and the results were compared to those
achieved with Gaia DR2. Firstly, we did this to verify that the methodology
and criteria developed for the Gaia DR2 dataset was still applicable in the
latest release. Secondly, this was done to create a catalogue of blue
horizontal-branch stars using Gaia EDR3.

The 2558 blue horizontal-branch reference objects from Xue et al. (2008) were
found in Gaia EDR3 absolute magnitude-colour space. The improved accuracy of
the parallax measurements compared to Gaia DR2 meant that 121 objects were
found to have a parallax error of less than 20$\%$ in Gaia EDR3 (compared to
39 in Gaia DR2). See Figure 7. The position of these reference objects in the
Gaia EDR3 colour and absolute magnitude space verified that using the same
colour and absolute magnitude criteria that were found using Gaia DR2 was
valid.

Figure 7: Xue et al. (2008) blue horizontal-branch reference objects (red
dots) displayed in DR2 (upper panel) and EDR3 (lower panel). Those objects
with a parallax error of less than 20$\%$ are shown as blue squares.

Thus, the same selection criteria were applied to Gaia EDR3 and the same
stringent halo selection criteria were applied (see Figure 8). We inspected
the results and found that the final DR2 CMD criteria as defined in Section
3.1

$-0.1<(G_{BP}-G_{RP})<0.53$;

$-1<G_{\textrm{abs}}<(2.5-2.77(G_{BP}-G_{RP}))$

were also applicable to EDR3.

Figure 8: Top panel: Objects that satisfy the modified (Lindegren et al.,
2018) criteria (gray points) with those with the stringent halo criteria and
the crowded region criterion applied (small light blue dots). Xue et al.
(2008) objects (red circles) and Behr (2003) objects (yellow triangles) are
superimposed.The green trapezium denotes the Gaia DR2 colour-magnitude
coordinate space (final CMD criteria) considered as the blue horizontal-branch
region.) displayed in DR2 (upper panel) and EDR3 (lower panel). Figure 9:
Distribution of reduced proper motions for the parallax selection in Gaia DR2
(red) and Gaia EDR3 (blue).

The observed reduced proper motion distribution from Gaia EDR3 parallax
selection was then used in defining the proper motion candidate selection
criteria. We compared the reduced proper motion distribution of these objects
from the Gaia DR2 and Gaia EDR3 datasets and found a close agreement despite
the larger number of objects found in Gaia EDR3 (see Figure 9).

###  4.1 Comparison of Gaia DR2 and EDR3 results

Selection Criteria | DR2 | EDR3
---|---|---
1\. Parallax error ¡ 20$\%$, initial Gaia CMD Criteria | 9,019,816 | 10,921,642
2\. Position 1 + final Gaia CMD Criteria | 305,622 | 336,911
3\. Position 2 + crowded region criterion | 104,763 | 121,669
4\. Position 3 + no apparent neighbour within 5 arcsec criterion | 91,719 | 101,913
5\. Position 4 + $V_{T}>145$ km s-1 | 8,556 | 16,794
6\. Reduced proper motion selection | $6.94<H<11.78$ | $6.94<H<11.81$
7\. Position 6 + parallax error ¿= 20$\%$ \+ $|b|>50^{\circ}$ | 31,957 | 42,091
8\. Position 7 + no apparent neighbour within 5 arcsec criterion | 30,919 | 40,583
Table 2: Table showing a comparison of the Gaia DR2 and EDR3 selection
criteria and results

We found that 7,307 of the 8,556 objects in the Gaia DR2 parallax selection
were also present in the Gaia EDR3 parallax selection. The 1,249 parallax
candidate objects that are in the DR2 dataset but no longer in the EDR3
dataset are found on the Gaia CMD with increasing concentration towards the
$-1<G_{abs}<(2.5-2.77(G_{BP}-G_{RP}))$ cut-off criterion. This criterion
removes millions of objects, some of which will have measured values that
deviate from their true value by multiple standard deviations causing an
increase in contamination towards this highly populated region in the Gaia
colour-magnitude space.

By comparing the distance (calculated as the reciprocal of parallax)
distributions of the parallax selections from Gaia DR2 and Gaia EDR3, we can
demonstrate that we have been able to identify more distant candidate objects
associated with higher apparent magnitudes (see Figure 10).

Figure 10: Distance distribution of parallax selection objects in Gaia DR2
(red) and Gaia EDR3 (blue).

Making such a comparison for objects with high uncertainties on their
parallaxes may be misleading. As blue horizontal-branch stars are considered
to have nearly constant absolute magnitudes we can consider apparent magnitude
as a proxy for distance. The comparison of Gaia DR2 and Gaia EDR3 catalogues
can be meaningfully made in both the parallax selection and the proper motions
selection in Figure 11.

Figure 11: Distribution of apparent magnitudes for the parallax selection
(upper panel) for Gaia DR2 (red) and Gaia EDR3 (blue). Distribution of
apparent magnitudes for proper motion selection (lower panel) in Gaia DR2
(red) and Gaia EDR3 (blue).

We compared the proper motions candidate objects found in Gaia DR2 and the
parallax candidate objects found in Gaia EDR3. We observed that 1,737 objects
that were found in the Gaia DR2 proper motions catalogue were now to be found
in the Gaia EDR3 parallax dataset. Our examination of the apparent magnitudes
of these 1,737 objects showed them to be at the fainter (average apparent G
magnitude 14.38 mag) and, hence, at the more distant end of the dataset. Thus,
the increased accuracy of the parallax measurements made in Gaia EDR3 enables
the identification of halo blue horizontal-branch objects over the full sky
out to a greater distance than was previously possible using these methods in
Gaia DR2.

##  5 Estimation of catalogue contamination and completeness

We made an estimation of the population I main-sequence and blue-straggler
contamination levels using the SDSS SEGUE Stellar Parameter Pipeline (SSPP)
DR10 (Lee et al., 2008; Smolinski et al., 2011), based on its effective
temperature ($T_{eff}$) and surface gravity (log g) parameters . The method
used and the cut-offs applied were taken from Santucci et al. (2015 and
references therein), as noted in Section 3.1.1.

Cross-matching the parallax candidate objects with the SDSS SEGUE database
found 556 stars within the final Gaia CMD criteria (Table 1) and 276 stars
within the Galactic halo selection criterion (Table 2). Applying the cut-offs
used in Santucci et al. (2015), we found that 70% plotted in the blue
horizontal-branch region, 10% plotted in the population I main-sequence
region, and 20% plotted in the blue straggler region (Figure 12).

Figure 12: Differentiating between blue horizontal-branch stars and blue
stragglers. Upper panel: Gaia EDR3 objects that satisfy the modified
(Lindegren et al. 2018) and the final CMD criteria (grey dots) with those with
the stringent halo criteria and the crowded region criterion applied (blue
circles); SDSS SEGUE objects plotting in the blue horizontal-branch region
(yellow squares); and SDSS SEGUE objects plotting in the blue straggler region
(magenta triangles) in TEFF-LOGG space. Lower panel: SDSS SEGUE TEFF x LOGG
plot with blue horizontal-branch (yellow square) and blue straggler stars
(magenta triangles).

We found that modifying the final Gaia CMD cut-off from:

$-0.1<(G_{BP}-G_{RP})<0.53$;

$-1<G_{\textrm{abs}}<(2.5-2.77(G_{BP}-G_{RP}))$

to the more restrictive

$-0.1<(G_{BP}-G_{RP})<0.53$;

$-1<G_{\textrm{abs}}<(1.5-2.0(G_{BP}-G_{RP}))$

gave a smaller catalogue but with a lower contamination level, with 86% of
objects in the blue horizontal-branch region, 8% of objects in the population
I main-sequence region, and 6% in the blue-straggler region. This more
restrictive selection was not used in the final catalogue, but it demonstrates
that a higher level of purity can be achieved at the expense of completeness.

Cross-matching the proper motion candidate objects with the SDSS SEGUE
database found 488 stars within the Gaia colour and reduced proper motion
criteria (see Section 3.2) and 107 stars within the Galactic halo selection
criterion ($b<-50^{\circ}$ or $b>50^{\circ}$). Applying the same cut-offs, we
found that 68% plotted in the blue horizontal-branch region, 8% plotted in the
population I main-sequence region, and 24% plotted in the blue-straggler
region.

The catalogue of blue horizontal-branch candidate objects is thus estimated to
have a purity level of ${\sim}70\%$. The main contaminants are population I
main-sequence stars with a contamination level of ${\sim}10\%$, and a blue-
straggler contamination of ${\sim}20\%$. Using more restrictive CMD selection
criteria can reduce these levels of contamination for the candidate objects
selected using parallax measurements, but at the cost of reduced completeness
of the catalogue.

It should be noted that cross-matching the catalogues of blue horizontal-
branch candidates with the SDSS SEGUE database only finds a relatively small
number of matches, which are not evenly distributed over the sky. Thus, the
calculated levels of purity and contamination can only be considered to be
estimates indicating a general level achieved.

The catalogue does not include those blue horizontal-branch stars that do not
conform to the various astrometric and photometric quality criteria applied,
or those with close apparent neighbours, or those that exist in the direction
of crowded regions. The actual number of such objects has not been estimated.

The distance distribution of parallax selection objects peaks at around
$\sim$5.5 kpc and goes up to $\sim$10\. kpc. The proportion of parallax
candidates that are found at $|b|>50^{\circ}$ is 22%. This apparent
overpopulation at higher Galactic latitudes (only $\sim$8% of a sphere’s
volume is at $|b|>50^{\circ}$) indicates that the crowded region and apparent
neighbour filters, which dominate at lower Galactic latitudes, have a
beneficial effect on the catalogue’s purity at the expense of the catalogue’s
completeness. It also implies that neither the catalogue’s completeness nor
its purity is uniform in all directions.

There are 30,343 of the 336,911 final Gaia EDR3 CMD criteria objects that
conform to the $V_{T}>145$ km s-1 criterion. Applying the crowded region
criterion and the near apparent neighbour criterion to these 30,343 objects
reduces the number to 16,794. This shows that these criteria have removed
roughly half of the potential candidates and will thus have a significant
effect on the completeness of the catalogue.

A further adverse effect on the completeness is due to the high line-of-sight
radial velocity of some halo objects, as noted in Section 3.2. Using the halo
stellar velocity distribution functions given in Section 3.1.3 in Anguiano et
al. (2020), we estimate that $\sim$5% of halo objects will not pass the
$V_{T}>145$ km s-1 criterion and will, thus, be missing in the catalogue. The
number of thin-disk stars with an apparent tangential velocity greater than
$145$ km s-1 is considered to be negligible. When we make the assumption all
blue horizontal-branch stars have an absolute magnitude of 1, then we can
calculate an approximate distance to the reduced proper motion selection
objects using:

$d\approx 10^{(0.2.(\verb!phot_g_mean_mag!-1+5))}$.

Using this equation, we find that the distance distribution of these objects
peaks at $\sim$12 kpc with 50% of the objects closer than $\sim$22 kpc but
with some candidates found out to $\sim$60 kpc.

##  6 Catalogue of blue horizontal-branch objects

Table Entry | Units | Description
---|---|---
$\verb!source_id!$ | - | Gaia EDR3 unique source identifier
$\verb!RA2000!$ | degrees | FK5 J2000.0 right ascension
$\verb!DEC2000!$ | degrees | FK5 J2000.0 declination
$\verb!parallax!$ | mas | Parallax
$\verb!parallax_error!$ | mas | Parallax error
$\verb!phot_g_mean_mag!$ | mag | G-band mean apparent magnitude
$\verb!bp_rp!$ | mag | BP-RP colour
$\verb!abs_g_mag!$ | mag | G-band mean absolute magnitude
$\verb!tangential_velocity!$ | km s-1 | Tangential velocity
$\verb!pmra!$ | mas yr-1 | Proper motion in the right ascension direction
$\verb!pmra_error!$ | mas yr-1 | Standard error of proper motion in the right ascension direction
$\verb!pmdec!$ | mas yr-1 | Proper motion in the declination direction
$\verb!pmdec_error!$ | mas yr-1 | Standard error of proper motion in the declination direction
$\verb!total_proper_motion!$ | mas yr-1 | Total proper motion ($\mu$)
$\verb!total_proper_motion_over_error!$ | mas yr-1 | Standard error of total proper motion ($\sigma_{\mu}$)
$\verb!reduced_proper_motion!$ | mas yr-1 | Reduced proper motion $(H)$
$\verb!abs_mag_flag!$ | - | BHB star identified using parallax
$\verb!reduced_pm_flag!$ | - | BHB star identified using proper motions
Table 3: Table showing the structure of the online version of the blue
horizontal-branch candidate star catalogue

The ADQL query for the parallax candidate dataset given in Appendix A returns
30,343 Gaia EDR3 sources, of which 21,751 have no apparent neighbours within 5
arcsec in the Gaia database(Figure 15). The dataset of parallax candidates did
not have a cut-off applied for Galactic latitude, but the criterion that there
should be no apparent neighbour within 5 arcsec does, however, remove the many
candidate objects in the Galactic plane. This means that our selection places
a greater emphasis on having an uncontaminated catalogue rather than on a
complete full-sky catalogue. The criterion that the parallax error should be
$<$ 20$\%$ results in a catalogue of objects that reside within a limited
distance from Earth. Using the distance calculated from the parallax, we found
that the parallax candidates are found at a minimum distance of 200pc, a mean
distance of 4,657pc, and a maximum of 11,888pc (Figure 10 upper). The Gaia
EDR3 apparent G magnitude of the parallax selection objects ranged from 6.0
mag to 16.1 mag with a mean value of 14.2 (Figure 11, upper panel).

Figure 13: Distribution of Gaia EDR3 apparent G magnitudes of blue horizontal-
branch candidate objects selected using parallax (blue) and using proper
motions (turquoise).

We compared the distance calculated from parallax to the theoretical distance
calculated from the apparent magnitude while assuming that blue horizontal-
branch stars have a nearly constant absolute magnitude (Sirko et al., 2003;
Deason et al., 2011). For this purpose, we assumed a constant absolute Gaia
EDR3 G magnitude of 1 mag (taken from Figure 3, upper panel). This
approximation is not a state-of-the-art approach, but it demonstrates the
distance ranges in question and shows that the increasing scatter with
increasing parallax error is as expected (Figure 14)

Figure 14: Parallax candidates plotted comparing the distance calculated from
parallax to the theoretical distance assuming that all blue horizontal-branch
objects have the same absolute magnitude (1 mag). The parallax error is
plotted on the colour axis.

The ADQL query for the proper motion data set given in Appendix A returns
42,091 Gaia EDR3 sources, of which 40,538 have no apparent neighbours within 5
arcsec in the Gaia database (Figure 16). We applied a Galactic latitude cut-
off to the proper motion candidates. This was done to ensure that
contamination was minimised while consciously accepting that this will not be
a complete catalogue. The distribution of Gaia EDR3 apparent G magnitudes
ranges from a minimum of 11.3 mag to a maximum of 20.1 mag and a mean of 17.6
mag (Figure 13).

Figure 15: The catalogue of parallax selection objects. Upper panel: Gaia DR2
colour-magnitude diagram showing the blue horizontal-branch parallax
candidates (blue) overlain on a Lindegren et al. (2018) ’clean subset
selection C’ (black). Lower panel: Sky distribution of blue horizontal-branch
parallax candidates.

Figure 16: The catalogue of proper motion selection objects. Upper panel: Gaia
DR2 colour-reduced proper motion diagram showing the blue horizontal-branch
proper motion candidates (blue) overlain on all Gaia DR2 objects with total
proper motion errors of ¡1% (gray). Lower panel: Sky distribution of blue
horizontal-branch proper motion candidates.

##  7 Summary and Conclusions

In this work, we compiled a catalogue comprising of two subsets:

  1. 1.

16,794 blue horizontal-branch parallax candidate stars (with parallax error
$<$ 20$\%$);

  2. 2.

40,583 blue horizontal-branch proper motion candidate stars (with parallax
error $\geq$ 20$\%$).

The purity of the catalogue is estimated to be ${\sim}70\%$ with a population
I main-sequence contamination level of less than ${\sim}10\%$ seen in both
subsets of the final catalogue and a blue straggler contamination levels of
${\sim}20\%$ in both subsets of the catalogue. These purity and contamination
estimates have been made using the SDSS SEGUE results.

We used reference datasets (Xue et al., 2008; Behr, 2003) to define the colour
and absolute magnitude region of the Gaia DR2 colour-magnitude diagram where
blue horizontal-branch stars would be expected. Additional criteria were used
based on Lindegren et al. (2018) to ensure that the objects being considered
had data that were fit for the purpose. Modifications and additions that were
made to the Lindegren et al. (2018) astrometry and photometry quality criteria
were based upon similar publications, providing Gaia DR2 based catalogues of
extremely low-mass white dwarfs (Pelisoli & Vos, 2019), hot sub-dwarf stars
(Geier et al., 2019), and white dwarfs (Gentile Fusillo et al., 2018).

The population I main-sequence contamination was reduced by focusing on
objects in the Galactic halo. The great age of the Milky Way’s halo is such
that comparatively young main-sequence stars are not expected to be present in
large numbers.

These steps were subsequently repeated using Gaia EDR3, having first confirmed
that the criteria selected using Gaia DR2 were applicable to Gaia EDR3. A
comparison of the Gaia DR2 and Gaia EDR3 based catalogues showed that a richer
catalogue has been made possible with the improved accuracy of the astrometric
measurements.

Our catalogue is a first step in creating a more complete, full-sky catalogue
of blue horizontal-branch stars once more reliable distance measurements as
well as more reliable and unambiguous photometry data become available. It is
known that the Herzsprung-Russel diagram on its own is not sufficient to
disentangle the blue horizontal-branch from the main-sequence but this
catalogue may provide the selection criteria upon which further surveys can be
based. In particular, upcoming large spectroscopic surveys (e.g. 4MOST, WEAVE,
DESI) can construct their target lists using catalogues such as ours, which
identify objects without relying on existing spectra.

###### Acknowledgements.

IP was partially funded by the Deutsche Forschungsgemeinschaft under grant
GE2506/12-1 and by the UK’s Science and Technology Facilities Council (STFC),
grant ST/T000406/1. 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 research has also made
extensive use of TOPCAT (http://www.starlink.ac.uk/topcat/, Taylor 2005), and
of NASA’s Astrophysics Data System (http://adsabs.harvard.edu/).

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##  Appendix A ADQL Queries

For the parallax candidates (parallax error $<20\%$ dataset):
SELECT *
FROM gaiaedr3.gaia_source
WHERE parallax_over_error ¿ 5
AND parallax ¿ 0
AND phot_bp_mean_flux_over_error ¿ 10
AND phot_rp_mean_flux_over_error ¿ 10
AND phot_bp_rp_excess_factor ¡ 1.3+0.06*power(bp_rp,2)
AND phot_bp_rp_excess_factor ¿ 1.0+0.015*power(bp_rp,2)
AND phot_g_mean_mag+5+5*(log10(parallax/1000))¿-1
AND phot_g_mean_mag+5+5*(log10(parallax/1000))¡(2.5-(2.77*bp_rp))
AND bp_rp¿-0.1
AND bp_rp¡0.53
AND (sqrt(power(pmra,2)+power(pmdec,2)))*(4.74/parallax)¿145

For the proper motion candidates (parallax error $\geq 20\%$ dataset):
SELECT *
FROM gaiaedr3.gaia_source
WHERE parallax_over_error ¡= 5
AND phot_bp_mean_flux_over_error ¿ 10
AND phot_rp_mean_flux_over_error ¿ 10
AND phot_bp_rp_excess_factor ¡ 1.3+0.06*power(bp_rp,2)
AND phot_bp_rp_excess_factor ¿ 1.0+0.015*power(bp_rp,2)
AND phot_g_mean_mag+5+5*log10((sqrt((pmra*pmra)+(pmdec*pmdec)))/1000)¡11.88
AND phot_g_mean_mag+5+5*log10((sqrt((pmra*pmra)+(pmdec*pmdec)))/1000)¿6.80
AND
(sqrt((pmra*pmra)+(pmdec*pmdec)))/(sqrt((pmdec_error*pmdec_error)+(pmra_error*pmra_error)))¿5
AND bp_rp¿-0.1
AND bp_rp¡0.53
AND abs(b)¿=50

The following ADQL call was used to find the number of objects within Gaia DR2
that are within 5 arcsec (0.001388889 deg) of a coordinate:
SELECT mine.source_id, count(*) as within_5_arcsec
from user_username.filename as mine
join gaiadr2.gaia_source as gaia
on 1 = contains(
point(’ICRS’, mine.ra, mine.dec),
circle(’ICRS’, gaia.ra, gaia.dec, 0.00138889)
)
group by mine.source_id

This returns a file containing the Gaia DR2 source_id and the number of
objects to be found within 5 arcsec. If there are no apparent neighbours
within 5 arcsec then a value of 1 (the star itself) will be returned.


The QCD Equation of State in Small Systems

W. A. Horowitz1$\star$ and Alexander Rothkopf2

1 Department of Physics, University of Cape Town, Rondebosch 7701, South
Africa

2 Faculty of Science and Technology, University of Stavanger, 4021 Stavanger,
Norway

*<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 doi:10.21468/SciPostPhysProc.?

## Abstract

We present first results on finite system size corrections to the equation of
state, trace anomaly, and speed of sound for two model systems: 1) free,
massless scalar theory and 2) quenched QCD with periodic boundary conditions
(PBC). We further present work-in-progress results for quenched QCD with
Dirichlet boundary conditions.

##  1 Introduction

Multiparticle correlations measurements in even the smallest collision systems
are consistent with predictions from viscous relativistic hydrodynamics
calculations [1]. However, these hydrodynamics calculations use a continuum
extrapolated—i.e. infinite volume—equation of state. For the modest
temperature probed in these small collisions, the controlling dimensionless
product of the temperature and system size $T\times L\sim 400$ MeV $\times$ 2
fm / 197 MeV fm $\sim 4$ is not particularly large. One should therefore
investigate the small system size corrections to the equilibrium QCD equation
of state used in modern viscous hydrodynamics simulations.

##  2 Analytic Results

A natural first analytic model to test the relevance of system size on
thermodynamic quantities is the massless, free scalar field put into a box,
i.e. $\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}$ with Dirichlet boundary
conditions (DBCs) in 1, 2, or 3 directions of lengths $L_{1}$, $L_{2}$, and
$L_{3}$, respectively. One may straightforwardly compute the partition
function [2], then from the free energy one may compute the pressure, energy
density, etc. We show in Fig. (1) the finite system size effects on the
pressure as a function of temperature: the presence of the boundary limits the
modes of the field, reducing the pressure from the infinite volume, Stefan-
Boltzmann (SB) limit. As the dimensionless parameter $T\times L$ increases,
the finite size effects decrease, as they must. As $T\times L\rightarrow 0$,
the pressure converges to the $T=0$ Casimir pressure. One can see that the
finite size effects are $\sim 10\%$ for even $T\times L\sim 20$ and grow as
$T\times L$ decreases. It’s not straightforward to estimate $T\times L$ for
hadronic collisions: after the initial nuclear overlap, the system expands
longitudinally at the speed of light and transversely at half the speed of
light, all while the temperature drops. For the Au and Pb nuclei collided at
RHIC and LHC, the radii are $\lesssim 7$ fm. Relativistic viscous
hydrodynamics suggests the initial temperature of the plasma is $\sim 400$ MeV
at time $t_{0}\sim 1$ fm [1]. Using the initial transverse size to roughly set
$L$ we have that for central collisions, with impact parameters $b\sim 4$ fm,
yield an initial $T\times L\sim 8$ and semi-central collisions, with $b\sim 9$
fm, imply $T\times L\sim 4$. Protons have a radius $\sim 1$ fm, and so for
high multiplicity p+p collisions one has with $T\sim 400$ MeV that $T\times
L\sim 2$. Thus with the caveats that the real systems are dynamical and
nuclear, as opposed to static and scalar, the scalar field theory results
suggest that the finite size effects on the thermodynamics (and thus
hydrodynamics) in hadronic collisions might be significant.

Figure 1:  (Left) Finite volume pressure $p$ divided by the infinite volume,
Stefan-Boltzmann limit $p_{SB}$ of a free, massless scalar field with
Dirichlet boundary conditions in 1, 2, and 3 dimensions as a function of
temperature $T$ in units of $1/L$, where $L$ is the length of any of the
fixed, compactified directions [2]. (Right) The scaled trace anomaly
$\Delta/T^{4}=(\epsilon-3p)/T^{4}$ as a function of temperature $T$ from
$N_{f}=0$ pQCD to order $g_{s}^{5}$ with scale set by $\pi T,\,2\pi T$, and
$4\pi T$ in infinite volume as given in [3] and then with finite size running
coupling; see text for details.

One may then examine the trace anomaly $\Delta\equiv\epsilon-3p$. The trace
anomaly controls the speed of sound through $c_{s}^{2}=\partial
p/\partial\epsilon$: the larger the trace anomaly, the slower the speed of
sound (and the further the speed of sound is from the conformal limit of
$c_{s}=1/\surd 3$); the smaller the trace anomaly, the larger the speed of
sound. The larger the speed of sound, the more responsive is the plasma to
pressure gradients. Thus the smaller the trace anomaly, the larger the
particle flow. For fixed experimentally measured flow, a smaller trace anomaly
therefore implies a larger viscosity required to describe the data.

One may show that the trace anomaly in the case of a compactified massless
free scalar field is identically 0, independent of any choice of conformality-
breaking non-trivial finite geometry. One may further see that for a coupled
$\lambda\phi^{4}$ theory, the trace anomaly in infinite volume is identically
0 up to fixed order $\lambda^{2}$; note that the dynamically generated Debye
mass $m_{D}\sim\lambda T$ impacts the pressure and energy density at order
$\lambda^{3/2}$ [5].

The trace anomaly becomes non-zero when the coupling is allowed to run and
when the order of the expansion of the partition function yields logs of the
temperature. Inspired by the finite size correction to the running coupling in
$\lambda\phi^{4}$ theory [6], we took as an ansatz
$g_{s}(\mu,L)=g_{s}(\mu)-\frac{3}{2}g^{2}_{s}(\mu)\frac{3}{2}\big{[}(2\pi)^{3}m_{D}(\mu)L\big{]}^{-1/2}$
in the $g_{s}^{5}$ QCD partition function as given in [3]. We plotted the
results for the scaled trace anomaly $\Delta/T^{4}\equiv(\epsilon-3p)/T^{4}$
in infinite volume and in a box of sides $L=1$ fm in Fig. (1). We
conservatively estimated the uncertainty in the result by varying the scale
$\mu$ from $\pi T$ to $4\pi T$. One can see that the reduction in the coupling
from the infinite volume limit due to the finite system size significantly
reduces the trace anomaly, which will lead to a faster speed of sound.

Figure 2:  (Top left) Scaled pressure $p/T^{4}$ and (top right) energy density
$\epsilon/T^{4}$ of quenched QCD with periodic boundary conditions with one
side of fixed (small) length $L$ [7]. (Bottom) The scaled trace anomaly
$\Delta/T^{4}\equiv(\epsilon-3p)/T^{4}$ as a function of temperature $T$ in
quenched QCD with periodic boundary conditions with sides $N_{x}=$ 24 (red),16
(green), 8 (orange) and 4 (blue) at fixed coupling [8]

.

##  3 Lattice QCD Results

With the intuition from our analytic calculations in hand, we may now quantify
the finite system size effects on thermodynamic quantities using the lattice.
Quenched QCD pure SU(3) gauge lattice simulations were performed on lattices
with anisotropic spatial volumes with periodic boundary conditions (PBCs) [7].
The energy-momentum tensor defined through the gradient flow was used for the
analysis of the stress tensor on the lattice. We show in Fig. (2) clear
finite-size effects in the pressure and the energy density. Note that in
quenched QCD $T_{c}\approx 270$ MeV. As anticipated from the analytic results,
the pressure and energy density are reduced compared to their infinite volume
limit as the dimensionless parameter $T\times L_{x}$ decreases.

We performed additional lattice calculations in quenched QCD with PBCs to
examine the trace anomaly as a function of the temperature alone [8]. One can
see in Fig. (2) that as the system size decreases, the phase transition
broadens and the trace anomaly is reduced, as expected from our analytic
results.

##  4 Discussion and Conclusions

We showed that for a free, massless scalar field, placing the system in a
finite-sized box reduces the pressure and energy density as compared to the
infinite volume limit. In p+p collisions, $L\sim 1$ fm, so $T\sim 400$ MeV
implies $T\times L\sim 2$, which is far from large and we can see from Fig.
(1) that finite size corrections are $\sim 50\%$. These finite size effects
persist out to surprisingly large $T\times L\sim 20$; in semi-central A+A
collisions with $L\sim 10$ fm, $T\times L\sim 20$ corresponds to a QGP
temperature of $\sim 400$ MeV. Thus finite size effects on the thermodynamics
may have non-trivial implications for hydrodynamic modelling of heavy-ion
collisions, especially in small- to moderate-sized systems. We next
analytically examined the finite size effects on the $N_{f}=0$ QCD trace
anomaly, finding that the finite system size reduced the anomaly compared to
the infinite volume limit, suggesting a faster-than-expected speed of sound.
Detailed quenched QCD calculations with periodic boundary conditions agreed
qualitatively with the analytic results: pressure, energy density, and the
trace anomaly were all reduced compared to their infinite volume limits.

Future work includes rigorously determining the finite size effects on the
running coupling in QCD, implementing Dirichlet boundary conditions (DBCs) in
the lattice simulations [9], and extending the lattice simulations to full
QCD. We anticipate that both DBCs and the presence of fermions will increase
the magnitude of the finite size corrections seen in the lattice results
presented here.

## Acknowledgements

WAH gratefully acknowledges support from the South African National Research
Foundation and the SA-CERN Collaboration. AR gratefully acknowledges support
from the Research Council of Norway under the FRIPRO Young Research Talent
grant 286883. The numerical simulations have been partially carried out on
computing resources provided by UNINETT Sigma2 – the National Infrastructure
for High Performance Computing and Data Storage in Norway under project
NN9578K-QCDrtX “Real-time dynamics of nuclear matter under extreme
conditions.”

## References

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