# The GALAH survey: Elemental abundances in open clusters using joint
effective temperature and surface gravity photometric priors

Kevin L. Beeson1, Janez Kos1, Richard de Grijs 2,3,4,5, Sarah L. Martell5,6,
Sven Bunder 5,7, Gregor Traven1, Geraint F. Lewis8, Tayyaba Zafar2,3, Joss
Bland-Hawthorn5,8, Ken C. Freeman5,7, Michael Hayden5,8, Sanjib Sharma9,
Gayandhi M. De Silva2,3,5
1Faculty of Mathematics and Physics,Jadranska ulica 19, Ljubljana 1000,
Slovenia
2School of Mathematical and Physical Sciences, Macquarie University, Balaclava
Road, Sydney, NSW 2109, Australia
3Astrophysics and Space Technologies Research Centre, Macquarie University,
Balaclava Road, Sydney, NSW 2109, Australia
4International Space Science Institute–Beijing, 1 Nanertiao, Zhongguancun, Hai
Dian District, Beijing 100190, China
5ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia
6School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
7Research School of Astronomy and Astrophysics, The Australian National
University, Canberra, ACT 2611, Australia
8Sydney Institute for Astronomy, School of Physics, A28, The University of
Sydney, NSW 2006, Australia
9Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The ability to measure precise and accurate stellar effective temperatures
($T_{\rm{eff}}$) and surface gravities ($\log(g)$) is essential in determining
accurate and precise abundances of chemical elements in stars. Measuring
$\log(g)$ from isochrones fitted to colour-magnitude diagrams of open clusters
is significantly more accurate and precise compared to spectroscopic
$\log(g)$. By determining the ranges of ages, metallicity, and extinction of
isochrones that fit the colour-magnitude diagram, we constructed a joint
probability distribution of $T_{\rm{eff}}$ and $\log(g)$. The joint
photometric probability shows the complex correlations between $T_{\rm{eff}}$
and $\log(g)$, which depend on the evolutionary stage of the star. We show
that by using this photometric prior while fitting spectra, we can acquire
more precise spectroscopic stellar parameters and abundances of chemical
elements. This reveals higher-order abundance trends in open clusters like
traces of atomic diffusion. We used photometry and astrometry provided by the
Gaia DR3 catalogue, Padova isochrones, and Galactic Archaeology with HERMES
(GALAH) DR4 spectra. We analysed the spectra of 1979 stars in nine open
clusters, using MCMC to fit the spectroscopic abundances of 26 elements,
$T_{\rm{eff}}$, $\log(g)$, $v_{\rm{mic}}$, and $v_{\rm{broad}}$. We found that
using photometric priors improves the accuracy of abundances and $\log(g)$,
which enables us to view higher-order trends of abundances caused by atomic
diffusion in M67 and Ruprecht 147.

###### keywords:

open clusters and associations: general – techniques: photometric– parallaxes
– Hertzsprung –Russell and colour–magnitude diagrams – stars: abundances

††pubyear: 2023††pagerange: The GALAH survey: Elemental abundances in open
clusters using joint effective temperature and surface gravity photometric
priors–G

##  1 Introduction

Open clusters provide a unique environment to study the chemical abundances of
stars. With open clusters, we are given a set of tens to a few thousand stars
that were born in the same star forming region, thus, they are chemically
homogeneous (De Silva et al., 2006, 2007; Feng & Krumholz, 2014). Open
clusters are quickly disrupted but they remain kinematically homogeneous for a
few hundred million years, with the oldest open clusters observed being 2-3
Gyrs (Bubar & King, 2010; Rangwal et al., 2019; Sariya et al., 2021). This
allows for stars in the open cluster core to be easily distinguished from
field stars using their kinematics. The kinematic and chemical homogeneity
means open clusters have been used as a place to benchmark distances (Bailer-
Jones et al., 2021), measure their ages with high precision (Jeffery et al.,
2016), study the role of physical processes such as atomic diffusion that will
affect the chemical abundances of stars (Liu et al., 2019), examine the
possibility of strong chemical tagging (Casamiquela et al., 2021a), study
stellar rotation rates of entire stellar populations (Sun et al., 2019; He et
al., 2023), and study the chemical trends of the galactic disk (Neyer et al.,
2023).

Chemical tagging is the idea that by measuring chemical abundances, one can
group stars with similar chemical signatures into physically meaningful
populations (Freeman & Bland-Hawthorn, 2002). There are two types of chemical
tagging: weak and strong. Weak chemical tagging groups together stars that
originated from the same galactic environment. An example is tagging to find
the remnants of disrupted satellite galaxies (Freeman & Bland-Hawthorn, 2002;
Buder et al., 2022). The second type is strong chemical tagging, the grouping
of stars that originated from the same birth cluster. The possibilities of
strong chemical tagging have been explored in Casamiquela et al. (2020);
Dotter et al. (2017); Ting et al. (2019); Casamiquela et al. (2021b). If
strong chemical tagging can be done, it will give major insights into the
dynamical history of stars in our Galactic disk.

The extent of chemical homogeneity and the uniqueness of each open cluster’s
abundance pattern are essential in determining how viable chemical tagging is.
Open clusters are believed to be chemically homogeneous when initially
created, as any chemical signatures in the star forming region have time to
homogenise and star formation has finished before any supernovas can occur,
which can change the chemical composition of the star forming region (Bland-
Hawthorn et al., 2010; Feng & Krumholz, 2014). However, during the star’s
life, some effects can cause chemical inhomogeneities to occur. One of the
causes of chemical inhomogeneities is the existence of planets around the
star. As rocky planets are formed, they will be made out of elements with a
high condensation temperature, causing the abundance of these elements in the
star to be lower. As some of the planets have a highly eccentric orbit, they
will be reabsorbed during the star’s life cycle, which causes the surface
abundances of the elements with a high condensation temperature to increase
(Meléndez et al., 2009; Liu et al., 2016; Spina et al., 2018).

Atomic diffusion is another factor contributing to the chemical
inhomogeneities of stars within open clusters. Atomic diffusion is a physical
process where gravitational settling driven by temperature and pressure
gradients concentrates heavier elements towards the centre of the star while
radiative acceleration pushes elements away from the centre (Michaud et al.,
1984, 2015). The effects of atomic diffusion are most significant as a star
reaches the turn-off-point on the colour-magnitude diagram. As a star reaches
its turn-off point, the effects of gravitational settling will be greater
compared to the radiative acceleration, and consequently the concentration of
heavy elements will decrease as they sink from the surface layer to the
convection region. After the star has passed the turn-off point, the
convection zone will deepen. As a result, the elements that have sunk to the
convection region will now be found in the surface layer. This change will
increase the concentration of these elements once again on the surface layer.
Atomic diffusion is one of the drivers that reduces the chemical homogeneity
of stars in open clusters. In M67, atomic diffusion will create inhomogeneity
to a 0.2 dex level for certain elements (Liu et al., 2019).

Surface gravity is used as a marker of how far along the star is in its
evolutionary stage. The uncertainty of spectroscopic measurements of $\log(g)$
is typically on the order of 0.1-0.25 dex (Jofré et al., 2019). This is
because $\log(g)$ only has a weak effect on the observed spectra. An
uncertainty of 0.25 dex in $\log(g)$ will significantly affect where we view
the star in its evolutionary stage. If the measured $\log(g)$ is imprecise,
the effect of atomic diffusion will be observed as scatter in abundances
rather than a trend in abundances with respect to $\log(g)$.

Photometry has been used to measure much more accurate and precise $\log(g)$
than spectroscopy for stars in open clusters. In Kos et al. (2019, 2021), they
fitted Padova (Bressan et al., 2012; Chen et al., 2014, 2015; Marigo et al.,
2017; Pastorelli et al., 2019; Tang et al., 2014) isochrones to HR diagrams of
open clusters. The isochrones were fitted very precisely, as the open clusters
had stars at various evolutionary stages. The nearest point in Gaia’s three
photometric bands ($G$, $G_{BP}$, and $G_{RP}$) space from the star to the
isochrone is used to measure $T_{\rm{eff}}$ and $\log(g)$. Using the
photometric $T_{\rm{eff}}$ and $\log(g)$ as priors when fitting abundances and
stellar parameters to spectra increased the accuracy of $\log(g)$ and the
precision of abundances.

Gaia (Gaia Collaboration, 2016; Gaia Collaboration et al., 2023) has
transformed the study of open clusters. The precise astrometry has
significantly improved cluster membership identification, leading to the
rejection of 50% of the pre-Gaia era open cluster listings (Cantat-Gaudin et
al., 2018). Furthermore, Gaia’s precise kinematic data has provided unique
insights into the three-dimensional structure of open clusters (Armstrong et
al., 2022). By combining precise photometry and astrometry, the mission has
enabled accurate age determinations for open clusters (Bossini et al., 2019).
In the third data release, Gaia has released the full astrometric solution
(positions on the sky, parallax, and proper motion) for 1.46 billion sources
and photometry in its three bands for 1.54 billion sources.

Many large spectroscopic surveys with a high resolution (R$>20\,000$) have
targeted open clusters as part of their observing time. A notable example is
the Gaia-ESO survey (Randich et al., 2013; Gilmore et al., 2012) which has
dedicated a significant fraction of time to target open clusters. Gaia-ESO has
provided a homogeneous set of 65 open clusters (Bragaglia et al., 2022). GALAH
(De Silva et al., 2015) and APOGEE (Majewski et al., 2017) have targeted open
clusters as part of their calibration process (Kos et al., 2018; Donor et al.,
2018) and have often observed open cluster stars serendipitously. These large
spectroscopic surveys produce cutting-edge abundance measurements and spectral
parameters for these stars. Consequently, they enable the comprehensive
exploration of abundance trends and the assessment of homogeneity within a
diverse range of open clusters.

This paper aims to explore the effect of using a joint $T_{\rm{eff}}$ and
$\log(g)$ photometric prior on spectroscopic $T_{\rm{eff}}$, $\log(g)$, micro-
turbulence ($v_{\rm{mic}}$), broadening velocity ($v_{\rm{broad}}$), and 26
elemental abundances. We first explore how to calculate better photometric
$T_{\rm{eff}}$ and $\log(g)$ by improving the calculated absolute magnitude.
The absolute magnitude is improved by optimising the precision and accuracy of
how we measure individual distances to stars in open clusters. Then, we
examine how the joint photometric prior changes between stars at different
evolutionary stages. To determine the effect of using a joint $T_{\rm{eff}}$
and $\log(g)$ for spectroscopic fitting, we fit the spectra of 1979 stars in
nine open clusters with and without a photometric prior. We compare the
scatter in abundances in these two methods and determine whether higher-order
trends such as atomic diffusion are visible in both methods. We examine the
validity of the precision of abundances and stellar parameters calculated
using Markov chain Monte Carlo (MCMC) and we examine how well GALAH DR4
measures its precision.

The structure of the paper is as follows: Sec. 2 describes how we selected the
members of the open clusters using astrometry and photometry from the Gaia DR3
catalogue, our method to improve Gaia DR3 distances to stars in open clusters
using a cluster centre distance prior, our technique to calculate the
$T_{\rm{eff}}$ and $\log(g)$ joint probability from isochrones fitted to HR
diagrams, and our approach to generate synthetic spectra and fit stellar
parameters and elemental abundances to GALAH spectra using the joint
$T_{\rm{eff}}$ and $\log(g)$ photometric prior. Sec. 3 describes the distance
improvements after using the cluster centre prior, the best-fitting
isochrones, and the resulting photometric $T_{\rm{eff}}$ and $\log(g)$ joint
probability distributions, and the effect of the photometric $T_{\rm{eff}}$
and $\log(g)$ joint probability prior on spectroscopic abundances. The section
also includes a comparison between our abundances and GALAH DR4 (Buder et al.
2023, in preparation). Sec. 4 describes our spectroscopic precisions and the
ways we verified them, we also examine how well GALAH DR4 estimates their
abundance precision. Sec. 5 is our discussion of the implications of our
results, and Sec. 6 collects our conclusions.

##  2 Methods and data

To create a joint $T_{\rm{eff}}$ and $\log(g)$ photometric prior to get better
abundances for stars in open clusters, we first have to tackle the problem of
how to acquire the best photometry, specifically the best absolute magnitudes
given the measured observed magnitudes and parallaxes. This process will be
elaborated upon in the next section, 2.1. We describe selecting the open
cluster stars through Gaia and which priors we used to make the distance
measurements more accurate and precise. After acquiring these improved
absolute magnitudes, we need to be able to get representative stellar
parameters and their uncertainties. In 2.2, we describe how we calculated
photometric stellar parameters and their precision from individual isochrones
and how to construct a joint $T_{\rm{eff}}$ and $\log(g)$ probability
distribution from the ranges of ages, metallicities, and extinction obtained
from the fitting isochrones to HR diagrams of open clusters. Lastly in Sec.
2.3, we describe how we synthesised spectra and how we fitted spectra using
photometric stellar parameters as priors.

###  2.1 Calculating better distances to open clusters

Using the Gaia DR3 catalogue, we get access to photometry in the $G$,
$G_{BP}$, and $G_{RP}$ bands, and the astrometry (ranging from proper motions
and parallax to positions) of over 1.46 billion objects using 34 months of
observation. The catalogue gives us an excellent way to probe open clusters
with high completeness of objects brighter than $G=19$ (Fabricius et al.,
2021). The Gaia DR3 catalogue provides excellent photometry with an average
uncertainty of 1, 12, and 6 mmag for the $G$, $G_{BP}$, and $G_{RP}$ bands
respectively at $G=17$ mag (Riello et al., 2021). The parallax provided has a
median uncertainty of 0.07 mas at $G=17$ (Lindegren et al., 2021).

Using absolute magnitudes of stars in open clusters, we can make HR diagrams
of said clusters and fit isochrones to them. Using the isochrones, we can
calculate the $T_{\rm eff}$ and $\log(g)$ of the stars. The two sources of
uncertainties in the absolute magnitude come from the measurement uncertainty
of apparent magnitudes and the uncertainty in the distance of the star (the
parallax). The latter is the main contributor to the absolute magnitude
uncertainty. Even in a nearby cluster such as Melotte 22 ($\sim$ 130 pc), the
contribution from the distance uncertainty is around 13 times larger than the
contribution from the apparent magnitude measurement.

In the next subsection, we explain our selection function to acquire Gaia DR3
open cluster stars, and in 2.1.2 we explain our updated cluster distance prior
to increase the accuracy and precision of the parallax.

####  2.1.1 Selecting open cluster members

We decided to only use the stars that are easily differentiable from field
stars in proper motion space to minimise contamination. The population
obtained will mainly comprise stars from the open cluster core. We selected
the stars that belong to each open cluster from the Gaia DR3 archive. To do
this, we use several criteria, we represent the searched parameters in
brackets:

  * •

Using a conical search around the centre of the cluster to limit ($\alpha$
(ra), $\delta$ (dec)) of each star. The size of the conical search was
determined by eye using the Gaia data.

  * •

Constraining the parallax by introducing a minimum and a maximum parallax. The
parallaxes and their maximum and minimum values were selected by eye using the
Gaia data.

  * •

Limiting $\mu_{\alpha}\cos(\delta)$ (pmra) and $\mu_{\delta}$ (pmdec) using
maximum $\mu_{\delta}$ and $\mu_{\alpha}\cos(\delta)$. This is done as open
cluster members still share the same kinematic values. The
$\mu_{\alpha}\cos(\delta)$ and $\mu_{\delta}$ and their maximum and minimum
values were selected by eye using the Gaia data.

Table 1: Our parameters used for finding cluster members. The centre of our positional conical limits is indicated by the $\alpha$ and $\delta$ values, and the radius of the conical search by $d$. Our centre values and limits for pmra, pmdec, and parallax are indicated in the $\mu_{\alpha}\cos(\delta)$, $\mu_{\delta}$, and $\varpi$ columns. The values of the centres of the clusters are taken from Wenger et al. (2000) except for ASCC 16 which was taken from Paunzen (2008). $N_{\it{Gaia}}$ and $N_{\rm{GALAH}}$ are the number of observations from Gaia and GALAH respectively. The last column indicates the number of unique stars observed by GALAH. Note that the parallax centres are not the actual centres of the cluster; they are just used as a boundary to exclude stars that are not in the cluster. Name | $\alpha$ | $\delta$ | $d$ | $\mu_{\alpha}\cos(\delta)$ | $\mu_{\delta}$ | $\varpi$ | $N_{\it{Gaia}}$ | $N_{\mathrm{GALAH}}$ | unique GALAH stars
---|---|---|---|---|---|---|---|---|---
| deg. | deg. | deg. | $\mathrm{mas}\,\mathrm{yr}^{-1}$ | $\mathrm{mas}\,\mathrm{yr}^{-1}$ | mas |  |  |
ASCC 16 | 81.2 | 1.8 | 0.5 | 1.4 $\pm$ 0.9 |  $-0.2$$\pm$1.5 | 2.9$\pm$0.5 | 247 | 173 | 63
Blanco 1 | 0.85 | $-30.0$ | 1.0 | 18.6$\pm$1.1 | 2.7$\pm$1.0 | 4.19$\pm$0.3 | 328 | 109 | 63
Melotte 22 | 56.6 | 24.1 | 1.5 | 19.8$\pm$ 3.5 |  $-45.3$$\pm$3.2 | 7.29$\pm$0.3 | 552 | 91 | 91
NGC 2516 | 119.5 | $-60.8$ | 0.8 |  $-4.6$$\pm$1.9 | 11.3$\pm$1.7 | 2.41$\pm$0.4 | 1686 | 204 | 146
NGC 2632 | 130.1 | 19.6 | 1.1 |  $-35.8$$\pm$5.5 |  $-12.8$$\pm$4.1 | 5.35$\pm$0.4 | 913 | 147 | 147
M67 | 132.8 | 11.8 | 0.33 |  $-11.0$$\pm$0.7 |  $-3.0$$\pm$0.55 | 1.1$\pm$0.4 | 1389 | 922 | 291
Ruprecht 147 | 289.1 | $-16.3$ | 0.3 |  $-0.9$$\pm$1.4 |  $-26.6$$\pm$1.6 | 3.0$\pm$0.3 | 118 | 130 | 59
Trumpler 10 | 131.9 | $-42.6$ | 2.25 |  $-12.4$$\pm$2.0 | 6.5$\pm$1.2 | 2.5$\pm$1.0 | 1190 | 119 | 119
Upper Scorpius | 243.0 | $-23.4$ | 2.0 |  $-12$$\pm$12.0 |  $-25.0$$\pm$5.0 | 1.5$\pm$2.0 | 401 | 84 | 61

We also implement quality cuts to make sure the chosen observations of cluster
members have the required quality:

  * •

Visibility periods (visibility_periods_used) $\geq$ 7 as recommend in
Lindegren et al. (2018).

  * •

Mean flux over error (phot_g_mean_flux_over_error,
phot_bp_mean_flux_over_error, and phot_rp_mean_flux_over_error) $\geq$
25,10,10 for the $G$, $G_{RP}$, and $G_{BP}$ bands respectively.

  * •

Limits on precision on the blue and red magnitudes (GBP and GRP) excess
factors (phot_bp_rp_excess_factor) as recommend in Gaia Collaboration et al.
(2018).

  * •

Limits on the goodness of fit of the parallaxes using the chi squared fit on
the astrometry (astrometric_chi2_al) as recommend in Gaia Collaboration et al.
(2018).

The ADQL code for selection is provided in Appendix A. The values used in the
selection criteria and quality cuts are shown in table 1. To create the list
of open clusters, we selected the nine clusters with the highest number of
observed stars in GALAH DR4.

####  2.1.2 Calculating distances to open clusters

The recommended method to calculate distances for individual stars is to use
the likelihood and priors in Bailer-Jones et al. (2021). The likelihood is
shown in equation 1. The paper introduces a geometric prior and a photometric
prior. The geometric prior uses our knowledge of the empirical distribution of
stars based on their celestial position, and the photometric prior uses the
fact that the colour space of the stars is not uniformly distributed with
respect to distance. We have decided to only use the geometric prior. The
photometric prior, as described in Bailer-Jones et al. (2021) applies to
individual stars rather than a cluster.

The following equations are used for estimating distances to individual stars
using the method outlined in Bailer-Jones et al. (2021).

$\displaystyle
P(r|\varpi,\sigma_{\varpi})=\begin{cases}K\mathcal{N}(\frac{1}{r}-\varpi+\varpi_{0},\sigma_{\varpi}^{2})&\text{if
}r>0\\\ 0&\text{if }r<0\end{cases}$ (1)
$P(r|\varpi,\sigma_{\varpi}\alpha,\beta,L)=P(r|\varpi,\sigma_{\varpi})P(r|\alpha,\beta,L)$
(2) $\displaystyle
P(r|\alpha,\beta,L)=\begin{cases}Kr^{\beta}\text{exp}(-r/L)^{\alpha}&\text{if
}r>0\\\ 0&\text{if }r<0\end{cases}$ (3)

Equation 2 describes the probability of a star being at a distance $r$, given
its parallax($\varpi$), uncertainty in parallax ($\sigma_{\varpi}$), $\alpha$,
$\beta$, and $L$. $\alpha$, $\beta$, and $L$ are the parameters of the
empirical probability distribution of stars in the galaxy modelled using a
three-parameter Generalised Gamma Distribution, their values depend on the
galactic longitude and latitude. Thus, for every star with a different line of
sight their $\alpha$, $\beta$, and $L$ values will be different. A fuller
description of $\alpha$, $\beta$, and $L$ can be found in Bailer-Jones et al.
(2021). The first component of equation 2 is likelihood shown in equation 1
for an individual star at $r$, with $\varpi$ and $\varpi_{0}$ (zero point
correction), $\mathcal{N}$ is a normal distribution, and $K$ is the
normalising constant. The zero point correction is used as the zero point of
the parallax in Gaia is not at 0. The zero point correction is dependent on
the celestial position and the number of astrometry points. It varies between
$-0.150$ and $+0.130$ mas (Lindegren et al., 2021), and it varies as a
function of stellar type (Ren et al., 2021). Geometric prior is shown in
equation 3.

To decrease the distance uncertainty we will use the fact that the stars are
in the open cluster’s core. If we assume the distribution of stars in the
cluster is a normal distribution around the cluster centre and if we know the
distance to the centre of the cluster ($c$) and the radial size of the cluster
($s$), we will be able to create a cluster distance prior for the cluster. We
decided to use a normal distribution to model the open cluster core rather
than a more complicated distribution such as the Elson, Fall, and Freeman
profile (Elson et al., 1987), as the difference between these profiles in the
resulting cluster centre distance and size will not be large enough to
significantly affect the resulting absolute magnitude.

Therefore we shall first calculate the probability distribution of $s$ and
$c$. We introduce another prior for the radial size of the cluster. This is
done so the radial cluster size can not be negative or too large; the prior
used is a gamma distribution as proposed in Luri et al. (2018). The cluster
distance prior and the cluster size prior are shown below.

$P(r|s,c)=\mathcal{N}(r-c,s^{2})$ (4)

$\text{Cluster size
prior}=\chi(s|k,\theta)=\frac{x^{k-1}\exp^{-\frac{x}{\theta}}}{\Gamma(k)\theta^{k}}$
(5)

Where shape ($k$) and scale ($\theta$) are the parameters of the cumulative
distribution function of the Gamma distribution. We have chosen a $k$ and
$\theta$ that make the mean of the Gamma distribution 10 pc and a variance of
0.05 pc, which are the recommended values in Luri et al. (2018).

By combining equation 2, 4, and 5, marginalising over $r$ we can get the
probability distribution of $s$ and $c$ for a single star:

$\begin{split}\mathbf{P}&(s,c|\varpi,\sigma_{\varpi},L,k,\theta)=\\\
&\int_{0}^{\infty}\mathbf{P}(r_{i}|\varpi,\sigma_{\varpi_{i}},\alpha,\beta,L)\cdot\mathcal{N}(r_{i}-c,s^{2})dr_{i}\cdot\chi(s|k,\theta)\end{split}$
(6)

To calculate the probability distribution of $s$ and $c$ for a cluster, we
convolve the probability distribution of each star to make:

$\begin{split}\mathbf{L}&(s,c|\boldsymbol{\varpi},\boldsymbol{\sigma_{\varpi}},\boldsymbol{\alpha},\boldsymbol{\beta},\mathbf{L},k,\theta)=\\\
&\prod_{i}\int_{0}^{\infty}\mathbf{P}(r|\varpi_{i},\sigma_{\varpi_{i}},\alpha_{i},\beta_{i},L_{i})\cdot\mathcal{N}(r_{i}-c,s^{2})dr\cdot\chi(s|k,\theta)\end{split}$
(7)

With said probability distribution, we can now find more accurate individual
distances by combining equations 2 and 4 with $L(s,c)$ and marginalising a
star’s radial position through $s$ and $c$.

$\displaystyle\mathbf{L}(r|\varpi,\sigma_{\varpi})$
$\displaystyle=\begin{cases}\qquad\qquad\text{if }r>0:\\\
\iint\mathbf{L}(s,c)P(r|s,c)\mathbf{L}(r|\varpi,\sigma_{\varpi})\,dc\,ds&\\\
\qquad\qquad\text{if }r<0:\\\ 0&\end{cases}$ (8)

To implement these equations, we used the MCMC package emcee (Foreman-Mackey
et al., 2013). We first sampled the probability space of $s$ and $c$ using
equation 7. We used four walkers in total and multi-processed the integration
process for each star. To know if the space was sampled correctly, we required
the sampling iteration over auto-correlation time to be at least 17
iterations. With a 64-core computing cluster, even with the most numerous open
clusters, it takes around 2 hours to sample. For equation 8 we used the same
convergence limits as for the previous equations. Like the distance
probability distributions found in Bailer-Jones et al. (2021), we found that
they do not follow a normal distribution; however, as the spread of the
distances is so small, we still treated them as normal distributions and found
our uncertainty in distance and our peak using a normal distribution fit.

###  2.2 Calculating photometric stellar parameters

In this section, we describe the theory and implementation behind calculating
photometric $T_{\rm{eff}}$ and $\log(g)$ and their errors for individual stars
from an isochrone. In section 2.2.2, we explain how by determining the ranges
of metallicity, ages, and extinctions of isochrones that will fit the HR
diagrams of an open cluster; we can generate the joint $T_{\rm{eff}}$ and
$\log(g)$ probability distribution of an individual star.

####  2.2.1 Calculating $T_{\rm eff}$ and $\log(g)$ from individual isochrones

Using the more precise and accurate absolute magnitudes, we now better fit
isochrones to the open cluster HR diagrams. We have decided to fit the Padova
CMD 3.7 isochrones by eye; this method has been shown to be highly effective
for fitting in Kos et al. (2021, 2019). We chose to do this rather than using
Bayesian fitting programs like BASE-9 (von Hippel et al. (2006) and see also
Jeffery et al. (2016)) as we could not provide accurate complementary data,
such as a binary star indication or individual reddening values. Another
problem is that the Padova isochrones still do not fit the HR-diagram in
certain sections, such as the lower part of the main sequence, even when using
Bayesian fitting programs (Brandner et al., 2023). Irrespective we are
confident in the fitting and the probability distribution that comes with it.
When fitting by eye, we use isochrones with a spacing of 0.01 dex for both log
ages and metallicity.

Using an isochrone we can find the $T_{\rm eff}$ and $\log(g)$ for an
individual star by using:

$\textbf{{T}}_{\rm eff}=\frac{1}{\textbf{K}}\int T_{\rm
eff}(m)\prod_{\mathclap{i=G,BP-
RP}}\mathcal{N}(M_{i}-M_{i}(m),\sigma_{M_{i}})n(m)\thinspace dm$ (9)

Where $m$ is the initial mass of a star in the open cluster it is used as a
variable to parameterise the shape of the isochrone. $n(m)$ is the density of
stars per unit $m$ that we should observe at that point in the isochrone and
$K$ is the normalisation constant. $M$ is the absolute $G$ magnitude or the
$BP-RP$ colour, depending on the subscript. $\sigma_{M}$ is the uncertainty in
absolute magnitude. $n(m)$ is included as the theoretical density of stars
along the isochrone is not constant. We integrate along the isochrone. An
analogous equation is used for $\log(g)$. Note that in the equation, we are
combining $G_{BP}$ and $G_{RP}$ bands into $G_{BP-RP}$ unlike in Kos et al.
(2019, 2021), where they used the nearest point in all three photometric
bands. We did this as our testing shows at the turn-off point, some stars’
absolute magnitude’s can be close to one of the $G_{BP}$ and $G_{RP}$ bands
but not close to the $G_{BP}-G_{RP}$ axis, therefore creating anomalous
results when viewed on a colour-magnitude diagram. This line integral takes
into account the photometric uncertainty and the number density of stars one
should observe at that point in the isochrone.

After acquiring the $T_{\rm{eff}}$ and $\log(g)$, we can calculate the
precision of the derived parameters from the magnitude uncertainties by
modifying the equation above:

$\displaystyle\sigma_{T}^{2}=\frac{1}{\textbf{K}}\int(\textbf{{T}}_{\rm
eff}-T_{\rm eff}(m))^{2}\prod_{\mathclap{i=G,BP-
RP}}\mathcal{N}(M_{i}-M_{i}(m),\sigma_{M_{i}})n(m)\thinspace dm$ (10)

The Padova isochrone is sampled at discrete intervals. To make sure that the
sampling along the isochrone does not affect our measurement, we interpolated
the provided isochrone. We measured the change in $T_{\rm{eff}}$ or $\log(g)$
between the interpolated isochrone and non-interpolated one. If the change was
below 1% of the precision, we consider the isochrone to be well sampled. If
not, the interpolation process is repeated using the integrated isochrone as a
starting isochrone. When interpolating we do it with respect to the initial
starting mass of the star. For a cluster of 1000 stars, it takes, on average,
10 to 20 minutes to calculate the parameters and their associated precision on
a 144-core computer cluster.

####  2.2.2 Using the uncertainty in the isochrone fitting to create the joint
$T_{\rm{eff}}$ and $\log(g)$ probability distribution

When fitting the isochrone, we vary three main parameters: age, metallicity,
and extinction. We start by finding the optimum values of parameters, and then
we vary parameters one by one until around 85% of the stars that are in the
area of high sensitivity to the change are encompassed (an example is shown in
Fig. 1). These areas are normally around the turn-off point or the red giant
clump. The best-fitted isochrone and the two extreme isochrones define an
asymmetric normal distribution of parameters. We use an asymmetric normal
distribution as the range of metallicities, ages, and extinctions can differ
depending on whether you are increasing or decreasing the parameter. These
ranges are wide enough to encompass the extended main sequence turn-off stars.

Figure 1: Isochrones when changing the metallicity, age, and extinction for
Ruprecht 147. Blue line shows the best fit and the green lines show the
maximum and minimum values parameters we have chosen. The chosen ranges are
Age$=9.40^{+0.04}_{-0.05}$ log(age), metallicity $=0.08^{+0.12}_{-0.08}$ dex,
and extinction $\it{A_{\rm{V}}}$ $=0.40^{+0.10}_{-0.05}$ mag.

By using the range of isochrones, we can sample the probability density
($\rho(T_{\rm{eff}},\rm{log}(g))$), as each isochrone will generate a
particular $T_{\rm{eff}}$ and $\rm{log}(g)$. Each of these samples will have
its own precision associated with it, which will differ depending on the shape
of the isochrone which depends on the metallicity, age, and extinction. Thus
when calculating the probability of the star having a $T_{\rm{eff}}=T_{1}$ and
$\rm{log}(g)=g_{1}$ from the probability density
($\rho(T_{\rm{eff}},\rm{log}(g))$) each sample will contribute a probability
of

$P(T_{1},g_{1})=\mathcal{N}(T_{1}-T_{\rm{eff}},\sigma_{T_{\rm{eff}}})\mathcal{N}(g_{1}-\rm{log}(g),\sigma_{\rm{log}(g)}).$
(11)

To find the overall probability we marginalise over $T_{\rm{eff}}$ and
$\rm{log}(g)$:

$\begin{split}P(T_{1},g_{1})=&\iint\rho(T_{\rm{eff}},\rm{log}(g))\mathcal{N}(T_{1}-T_{\rm{eff}},\sigma_{T_{\rm{eff}}})\\\
&\mathcal{N}(g_{1}-\rm{log}(g),\sigma_{\rm{log}(g)})dT_{\rm{eff}}d\rm{log}(g)\end{split}$
(12)

We found that the uncertainty we got from each isochrone is much smaller than
the spread of $T_{\rm{eff}}$ and $\rm{log}(g)$ from varying ages,
metallicities, and extinctions. As a result the normal distributions act like
delta functions around $T_{1}$ and $g_{1}$ so we can approximate the equation
as:

$P(T_{1},g_{1})=\rho(T_{1},g_{1})$ (13)

where $\rho$ is obtained using scipy’s Gaussian KDE fitting function (Virtanen
et al., 2020). To take into account the individual precisions obtained during
fitting, we used them as weights during the Gaussian KDE fitting. Using
equation 13, we have a fast way to sample the probability space of photometric
stellar parameters. We decided to use 1000 isochrones to sample the space
generated from the asymmetric normal distribution of ages, metallicity, and
extinction. It took 2 hours for a 1000-star open cluster to be fully sampled
on a 72 core computing cluster.

###  2.3 Spectroscopic fitting

We have chosen to use the GALAH survey as our source of spectroscopic data.
GALAH is a spectroscopic survey employing the HERMES fibre-fed multi-object
spectrograph, which is installed on the 3.9-meter Anglo-Australian Telescope
(AAT). With the release of GALAH DR4, the survey has observed more than
$900\,000$ objects and provided stellar parameters $T_{\rm{eff}}$, $\log(g)$,
[Fe/H], $v_{\rm{mic}}$, $v_{\rm{broad}}$ and radial velocity ($v_{\rm{rad}}$),
and abundances of 30 elements. The survey observes in 4 bands: 4718-4903 Å
(blue band), 5649-5873 Å (green band), 6481-6739 Å (red band), and 7590-7890 Å
(IR band). Each band has a nominal resolving power of $R=28\,000$.

####  2.3.1 Creating synthetic spectra

This subsection gives an overview of creating the spectra, from preparing the
training data to training the neural network and evaluating the neural
network’s performance and quality.

To maximise the speed of synthesising spectra, we used the Payne algorithm
(Ting et al., 2019). This algorithm relies on an artificial neural network to
generate synthetic spectra. After it is trained, it can generate a high-
resolution spectrum with an average error of $\lesssim 10^{-3}$ in normalised
flux (Ting et al., 2019). With PySME, we need around 4 minutes to synthesise
all four HERMES bands, but using Payne, it only takes $0.01\ \mathrm{s}$ while
preserving the accuracy needed for spectroscopic fitting.

We chose to use Payne and did not use the pre-made grids method used in Kos et
al. (2021) as it would be computationally too intensive to compute a grid that
spans the 36 labels (stellar parameters and elemental abundances) we would
like to optimise. We chose not to use a generative model system as used in Rix
et al. (2016); Ness et al. (2015) as it has been shown in Ting et al. (2019)
that it cannot generate precise spectra (relative error of $<10^{-3}$) of
normalised flux when generating spectra over 25 labels (stellar parameters and
elemental abundances).

By using a neural network, we can swiftly and precisely generate spectra to
effectively address the issue of blended lines. While other methods involve
restricting the analysis to specific segments of spectra with no line blending
issues (e.g., Buder et al., 2021; Kos et al., 2021). The approach runs into
some problems. First, we are not using all the available information in the
observed spectra. Second, if that a specific point on the observed spectra is
contaminated (for example, by cosmic rays). In that case the ability to fit
that element will be lost, and lastly, a problem will be encountered if there
is a strong line just out of our specific region. This line will affect the
line inside the region, but it is not taken into account. Hence, for these
reasons, we will be fitting the majority of the wavelength range and all the
parameters at once.

We produced the training spectra using PySME (Wehrhahn et al., 2022) which is
a Python package based on the SME (Spectroscopy Made Easy; Valenti & Piskunov
(1996); Piskunov & Valenti (2017)). PySME is open source, free to use, and as
powerful as the old SME package.

To make sure we account for as many non-local thermodynamic equilibrium (non-
LTE) effects as practically possible, we used all the available non-LTE
elemental model grids in the PySME large data server111https://pysme-
astro.readthedocs.io/en/latest/usage/lfs.html. The list of the models is shown
in table 2. An explanation of the grids can be found in Amarsi et al. (2020).
In short, these non-NLTE models contain grids of pre-calculated line-by-line
intensity corrections that are used to correct the intensity of synthesised
LTE lines.

Our line list is acquired from VALD (Ryabchikova et al.,
2015)222http://vald.astro.uu.se/ and a full list of references of the line
list used is available in Appendix B. To accurately generate synthetic spectra
while cutting down on computing time, we wanted a complete but not superfluous
line list. To reduce the number of superfluous lines, we requested a line list
with an intensity of at least 0.005 in normalised flux (this value was chosen
so the errors of the GALAH spectra are always significantly larger than that),
and to make sure our line list is complete, we submitted this request using
multiple stellar parameters that will cover the parameter space of the stars
that we wish to analyse. We requested a line list using stellar parameters
($T_{\rm{eff}}$, $\log(g)$): (4500 $K$, 4.5), (4750 $K$, 2.4), (6000 $K$,
4.1), and (6000 $K$, 4.1); for each stellar $T_{\rm{eff}}$, $\log(g)$ pair, we
then requested them with three different metallicities ($-1$, 0, 1). Thus, we
had 12 line lists in total that we joined to make our final line list.

We decided to train the Payne on four stellar parameters: $T_{\rm{eff}}$,
$\log(g)$, $v_{\rm{mic}}$, $v_{\rm{broad}}$ and 32 abundances ([Fe/H],
[Li/Fe], [C/Fe], [N/Fe], [O/Fe], [Na/Fe], [Mg/Fe], [Al/Fe], [Si/Fe], [K/Fe],
[Ca/Fe], [Sc/Fe], [Ti/Fe], [V/Fe], [Cr/Fe], [Mn/Fe], [Co/Fe], [Ni/Fe],
[Cu/Fe], [Zn/Fe], [Rb/Fe], [Sr/Fe], [Y/Fe], [Zr/Fe], [Mo/Fe], [Ru/Fe],
[Ba/Fe], [La/Fe], [Ce/Fe], [Nd/Fe], [Sm/Fe], [Eu/Fe]). As in Buder et al.
(2021) and DR4, we have set macro-turbulence ($v_{\rm{mac}}$) to 0 as the
resolution of GALAH $v_{\rm{broad}}$ and $v_{\rm{mac}}$ are degenerate
broadening influences. We created a training set that is representative of the
parameter space of the stars that we will analyse. As we are fitting GALAH
spectra, we chose to use GALAH DR4 to create the parameter space of our
training set. By using the values and uncertainties measured in GALAH DR4 of
the stars in our open clusters, we were able to generate the probability space
of the training set. Our training set is more densely sampled in areas in
which there are more stars, as we would like our model to be better trained at
these points. When creating the training set, we selected maximum and minimum
values for abundances and stellar parameters, as we are not expecting to fit
any stars that have extreme abundances or stellar parameters. To make sure we
have created a training set that will span the parameter space of stars that
we want to observe, we overlay the Kiel diagrams of the fitted isochrone for
the open cluster we want to analyse. The parameter space of our training set
is shown in Fig. 2, with contours representing the density of the points made
with a Gaussian kernel density estimator.

The number of training spectra used was $10\,000$ as recommended in Ting et
al. (2019). We synthesised spectra with $0.004\ $Å steps, so we have a sample
rate of at least 10 times finer than the GALAH spectra. We chose wavelength
limits for each band to allow for at least $\pm 200\ \mathrm{km\,s^{-1}}$ of
stellar radial velocity.

The complexity of the neural networks depends on parameters called
hyperparameters. One of the hyperparameters that we can change is the number
of neurons per layer. If we have a number that is too large, we will overfit
the model, or if we do not give enough neurons, the model cannot simulate the
complexity of the training set. So, we have done hyperparameter tuning (i.e.,
trying to find the best value for a hyperparameter that will create a model
that fits the validation data the best) on the number of neurons per layer and
found that the default 300 neurons per layer is the optimum number.
Hyperparameter tuning of the number of layers was already done in Ting et al.
(2019), and it was found that extra layers only led to overfitting of the
spectra.

From our overall training data, 20% was used for our validation set and the
rest for our training set. The actual training took around 20 thousand
iterations to converge, which took around 2 hours per band to finish. The
training was done on NVIDIA Tesla P100 GPU provided by the free online
computing cluster Kaggle 333https://www.kaggle.com/. We found that the average
difference in flux between our neural network model and our validation set is
$0.0056$ in the blue band, $0.0060$ in the green band, $0.0069$ in the red
band, and $0.0046$ in the IR band. This is in line with the expected values in
Ting et al. (2019) of $0.007$.

These high-resolution synthetic spectra then need to be degraded to the
resolution of the observed spectra. The observed spectra have a nominal
resolution of 28000. However, this varies with wavelength, fibres, and time of
observation. To precisely degrade the spectra, we used the resolution profile
for each observed spectrum in each band. This contains information on the
resolution at each wavelength. The resolution profile is of a generalised
Gaussian profile. By using photonic combs and the Thorium-Xenon arc lamp, Kos
et al. (2017) was able to measure the resolution profile. A photonic comb can
generate spectra that are predictably spaced out with a known Lorentzian
profile. The time needed to degrade the resolution of the synthetic spectra to
that of the observed spectra is similar to the time needed to synthesise the
high resolution spectra with Payne. Some observed spectra do not have the
resolution profile available or have an invalid resolution profile (e.g., they
have values that are too large or have a negative value); thus, we used the
resolution profile of the closest observation in time done on the same fibre
as the spectrum in question. This incorrect resolution profile happens in
around 8% of spectra. For more information on GALAH resolution profiles, see
Buder et al. (2023, in preparation).

Table 2: Non-LTE and atmospheric models used in the creation of the training set. Type | Model
---|---
Atmospheric | Gustafsson et al. (2008)
Ba | Mashonkina et al. (1999)
Ca | Mashonkina et al. (2007)
Fe | Amarsi et al. (2022)
O | Amarsi et al. (2016)
Li | Lind et al. (2009)
Na | Lind et al. (2009)
Mg | Osorio & Barklem (2016)
Si | Amarsi & Asplund (2017)
Ti | Sitnova et al. (2016)

Figure 2: The parameter space of the $10\,000$ spectra used for the training
set. Each black dot represents a value used to generate a training spectrum.
The bottom 26 panels are a scatter plot of [Fe/H] against [x/Fe], with the
element indicated in the top right of the panels.

####  2.3.2 Fitting procedure

To combine both spectroscopic and photometric stellar parameters, we decided
to use a Bayesian fitting scheme. Pinsonneault et al. (2004) have shown how
photometric $T_{\rm{eff}}$ and spectroscopic $T_{\rm{eff}}$ derived without
using a photometric prior might differ. However, in our case, we found that
there were no overall strong trends in the difference between spectroscopic
temperatures derived without using a photometric prior and photometric
temperatures, so we decided not to shift the temperatures. The spread of the
temperature differences is lower than in Kos et al. (2021) where the shifting
of the temperature scale was necessary.

The likelihood used to fit stellar parameters and abundances to spectra is
expressed as:

$\rm{log}(\mathcal{L}(\textbf{{x}}))\propto-\sum_{i}\frac{(s_{\rm{i}}-Payne(\textbf{{x}}))^{2}}{\sigma_{i}^{2}},$
(14)

where x is the vector of stellar parameters and abundances, $s_{\rm{i}}$ is
the flux of an observed spectrum at wavelength $i$, and $\sigma_{i}$ is the
observational error of the normalised flux at wavelength $i$.

We have included two priors in the fitting. The first one is the photometric
stellar parameters prior, derived from equation 12, and the other is a mask.
The mask will prevent the program from fitting wavelength ranges that we do
not want to fit as we know our models do not accurately model them, while
ensuring we always fit wavelength ranges of strong non blended absorption
lines. A full description of the mask is found in Appendix C.1.

The fitting is done in four steps: The first step is an initial check of
whether the GALAH reduction process produced an estimate for $T_{\rm{eff}}$
and $\log(g)$. Estimates for $T_{\rm{eff}}$ and $\log(g)$ are made for each
band of each spectrum during the GALAH data reduction process unless the data
quality is too poor. We do not use any band that does not have these initial
estimates. Then, if there is no valid resolution profile, we acquire one. For
the infrared band, we discard the first 25% of the wavelength range as it is
prone to having extreme values in it, as the infrared is susceptible to be
completely dominated by the telluric A band.

In the second step, we find the radial velocities of the individual bands. We
synthesise stellar spectra using GALAH DR4 values as starting parameters, and
if there are no abundances available in the GALAH DR4 catalogue, we use solar
abundances. We vary the radial velocities of each band individually to find
which radial velocity for each band has the highest likelihood value. These
radial velocity values are kept for the rest of the analysis process. In this
work, we found that the average difference in radial velocities between the
bands is 0.89 $\mathrm{km\,s^{-1}}$, this is the expected value as it is
around the expected radial velocity uncertainty of GALAH DR4.

The third step is to do an initial fit of the spectra using GALAH DR4 as the
starting value. After an initial test, we chose not to fit [Rb/Fe], [Sr/Fe],
[Mo/Fe], [Ru/Fe], [La/Fe], and [Ce/Fe]. The fitting is done by using Bayesian
sampling from the MCMC package emcee. As we have 30 parameters, we run the
sampler using the minimum recommended 60 walkers.

While the GALAH reduction process provides us with pre-normalised spectra, we
opt to normalise spectra using our synthetic spectra. We did this as our
method to generate spectra differs from the reduction process, and the values
used to generate the synthetic spectra to normalise the observed spectra are
different. We normalise the spectra at the beginning of this step. The
normalisation is done using the same methods described in the reduction
process of GALAH (Kos et al., 2017). In short, we use a univariate spline
function over the entire spectrum.

The fitting process ends when the burn-in has finished for $T_{\rm{eff}}$,
$\log(g)$, $v_{\rm{mic}}$, $v_{\rm{broad}}$, and [Fe/H] plus an additional 50
iterations. To check whether the burn-in has finished, we fitted a 1D
polynomial to each walker’s samples (chain) for these five parameters. As the
values of the parameters are vastly different, we normalised them using their
mean value before fitting the polynomial. We checked every 40 iterations to
see if the gradient was below 0.01, which corresponds to a 1% change every 100
iterations. If 90% of the chains have achieved this value, we consider the
sampler to be burned in. This first process takes around 410 iterations to
finish, which is around 6-7 minutes when using 18 cores. We then create an
additional mask, as described in Appendix C.1.

Using the three combined masks, we do the final fitting run. In this run, we
re-normalise the spectra once at the beginning using the values acquired
during the mask run. Our starting parameters for the main fit come from the
last iteration of the mask fitting run. As we have just introduced a new mask,
we let the sampler run until burn-in is achieved again. We detect when burn-in
is achieved using the same method as in the previous step.

To check whether the sampler has converged, we use auto-correlation time. When
using an MCMC sampler, each sample is not independent from one another; auto-
correlation time is a measure of how independent the samples are from one
another. To calculate the number of independent samples, we divide the total
number of samples by the auto-correlation time. When the sampler has
converged, it means it has successfully explored the underlying distribution
and thus the samples are representative of the underlying distribution. By
accurately calculating the number of independent samples, we have a
quantitative measure of how well the sampler has sampled the underlying
distribution. We use an auto-regression model to calculate our auto-
correlation time. A full discussion of why is available in Appendix C.2. We
found that 400 independent samples were sufficient for calculating the mean
value of the parameter and the uncertainties associated with it. Thus, the
program will stop sampling once a mean average of 400 independent samples
across the 30 parameters has been achieved. However, even with 100 independent
samples, it was already sufficient to sample the mean value of the parameter
(as the number of independent samples only contributes an extra 10% to the
uncertainty; Goodman & Weare, 2010a).

After the initial burn-in, we estimated the auto-correlation for the first
time after 100 iterations and recalculated the auto-correlation time for every
80 independent samples. We did not recalculate the auto-correlation time after
each iteration, as it was computationally expensive. This process continued
until an average of 400 independent samples was obtained. After completing the
sampling, we calculated the number of independent samples for each parameter,
which served as a quality metric. The sampling process averaged $15\,000$
iterations and took approximately 4 hours to compute using 18 cores. We had
set a maximum iteration limit of $40\,000$. If the maximum iteration is
reached, we assume that the fitting method will not converge, so we will stop
sampling. After the fitting process is finished a mean value and precision are
calculated for each fitted parameter.

##  3 Results

The results section is divided into three parts. The first section, 3.1,
discusses how well the distance measurements are using a cluster centre
distance prior. The next section 3.2 discusses the photometric stellar
parameters and the fits of the isochrone. The last section discusses the
effect of using photometric stellar parameters as priors on the spectroscopic
stellar parameters and abundances of stars in open clusters.

###  3.1 Distance improvements

Table 3: Comparison between our cluster centres and cluster sizes to cluster distances and sizes calculated directly from geometric and photo-geometric distances. Cluster |  This work’s distance to the cluster centre |  This work’s radial cluster size |  Geometric distance to cluster centre |  Geometric cluster size |  Photo-geometric distance to cluster core |  Photo-geometric cluster size
---|---|---|---|---|---|---
|  pc |  pc |  pc |  pc |  pc |  pc
ASCC 16 |  362.4 $\begin{subarray}{c}+3.5\\\ -3.5\end{subarray}$ |  4.2 $\begin{subarray}{c}+3.3\\\ -2.6\end{subarray}$ |  343.9 |  18.0 |  308.7 |  10.7
Blanco 1 |  234.7$\begin{subarray}{c}+0.2\\\ -0.2\end{subarray}$ |  2.7$\begin{subarray}{c}+0.2\\\ -0.2\end{subarray}$ |  235.4 |  6.4 |  234.9 |  6.3
Melotte 22 |  135.2 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  1.9$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  135.3 |  2.3 |  135.2 |  2.3
NGC 2516 |  407.3 $\begin{subarray}{c}+0.2\\\ -0.3\end{subarray}$ |  7.6$\begin{subarray}{c}+0.3\\\ -0.2\end{subarray}$ |  411.9 |  22.3 |  410.8 |  25.1
NGC 2632 |  183.9 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  3.7$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  184.4 |  5.1 |  184.1 |  5.2
M67 |  838.7$\begin{subarray}{c}+1.3\\\ -1.2\end{subarray}$ |  24.5$\begin{subarray}{c}+1.3\\\ -1.0\end{subarray}$ |  865.0 |  89.3 |  833.6 |  53.8
Ruprecht 147 |  307.1 $\begin{subarray}{c}+0.7\\\ -0.6\end{subarray}$ |  6.4$\begin{subarray}{c}+0.5\\\ -0.5\end{subarray}$ |  309.4 |  11.3 |  308.7 |  10.7
Trumpler 10 |  430.9 $\begin{subarray}{c}+0.5\\\ -0.5\end{subarray}$ |  7.9$\begin{subarray}{c}+0.6\\\ -0.7\end{subarray}$ |  457.7 |  71.6 |  455.3 |  71.9
Upper Scorpius |  146 $\begin{subarray}{c}+0.4\\\ -0.5\end{subarray}$ |  8.7$\begin{subarray}{c}+0.3\\\ -0.3\end{subarray}$ |  146.0 |  8.9 |  146.0 |  8.9
Figure 3: Colour-magnitude diagram of Trumpler 10, with the colours indicating geometric distance $-$ photo-geometric distance. Table 4: Comparison between our cluster centre, the size, and the values given in BO19, HR23. Cluster |  This work’s distance to the cluster centre |  This work’s radial cluster size |  BO19 distance to the cluster centre |  HR23 distance to the cluster centre |  HR23 core cluster size |  HR23 tidal cluster size
---|---|---|---|---|---|---
|  pc |  pc |  pc |  pc |  pc |  pc
ASCC 16 |  362.4 $\begin{subarray}{c}+3.5\\\ -3.5\end{subarray}$ |  4.2 $\begin{subarray}{c}+3.3\\\ -2.6\end{subarray}$ |  317.7 $\begin{subarray}{c}+3.5\\\ -8.7\end{subarray}$ |  341.8$\begin{subarray}{c}+0.3\\\ -0.3\end{subarray}$ |  1.9 |  5.9
Blanco 1 |  234.7$\begin{subarray}{c}+0.2\\\ -0.2\end{subarray}$ |  2.7$\begin{subarray}{c}+0.2\\\ -0.2\end{subarray}$ |  237.2 $\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  234.3$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  2.4 |  7.4
Melotte 22 |  135.2 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  1.9$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  136.0 $\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  134.8$\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  2.7 |  13.7
NGC 2516 |  407.3 $\begin{subarray}{c}+0.2\\\ -0.3\end{subarray}$ |  7.6$\begin{subarray}{c}+0.3\\\ -0.2\end{subarray}$ |  415.1 $\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  407.1$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  3.2 |  13.5
NGC 2632 |  183.9 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  3.7$\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  186.2 $\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  183.5$\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  3.2 |  10.5
M67 |  838.7$\begin{subarray}{c}+1.3\\\ -1.2\end{subarray}$ |  24.5$\begin{subarray}{c}+1.3\\\ -1.0\end{subarray}$ |  881.4 $\begin{subarray}{c}+0.0\\\ -0.0\end{subarray}$ |  836.6$\begin{subarray}{c}+0.6\\\ -0.6\end{subarray}$ |  2.5 |  8.7
Ruprecht 147 |  307.1 $\begin{subarray}{c}+0.7\\\ -0.6\end{subarray}$ |  6.4$\begin{subarray}{c}+0.5\\\ -0.5\end{subarray}$ |  - |  302.5 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$ |  3.4 |  15.2
Trumpler 10 |  430.9 $\begin{subarray}{c}+0.5\\\ -0.5\end{subarray}$ |  7.9$\begin{subarray}{c}+0.6\\\ -0.7\end{subarray}$ |  365.8 $\begin{subarray}{c}+4.7\\\ -3.6\end{subarray}$ |  428.0$\begin{subarray}{c}+0.2\\\ -0.2\end{subarray}$ |  4.5 |  10.3
Upper Scorpius |  146 $\begin{subarray}{c}+0.4\\\ -0.5\end{subarray}$ |  8.7$\begin{subarray}{c}+0.3\\\ -0.3\end{subarray}$ |  - |  - |  - |  -
Table 5: Membership numbers from our work compared to the membership numbers in HR23, showing the possible field stars. The second and fifth columns shows the number of member stars in our study obtained from Gaia and GALAH catalogues, respectively. The third column shows the number of open cluster members from HR23. The fourth and last columns show the possible field stars in our study from Gaia and GALAH catalogues, respectively. Cluster | $N_{\it{Gaia}}$ | $N_{\mathrm{HR23}}$ |  Possible $N_{\it{Gaia}}$ field stars | $N_{\mathrm{GALAH}}$ |  Possible $N_{\mathrm{GALAH}}$ field stars
---|---|---|---|---|---
ASCC 16 | 247 | 171 |  126 | 173 |  75
Blanco 1 | 328 | 841 |  0 | 109 |  0
M67 | 913 | 1314 |  4 | 147 |  0
Melotte 22 | 552 | 1721 |  0 | 91 |  0
NGC 2516 | 1686 | 3784 |  29 | 204 |  2
NGC 2682 | 1389 | 1844 |  7 | 922 |  0
Ruprecht 147 | 118 | 279 |  4 | 130 |  0
Trumpler 10 | 1190 | 1425 |  466 | 119 |  18

We compared our cluster distance centres and sizes obtained by using equation
7 to geometric and photo-geometric distances obtained in Bailer-Jones et al.
(2021). We calculated the cluster distance centres and sizes from the
distances in Bailer-Jones et al. (2021) by fitting a normal distribution to
the geometric and photo-geometric distances of the open cluster members. The
results are shown in Table 3. We found that faint stars’ photo-geometric and
geometric distances differ by a significant amount, with some stars having up
to $\sim 7\,000\,\rm{pc}$ in difference. An example of this difference is
shown in Fig. 3. Thus, we have excluded stars that have more than a 100 pc
difference between their photo-geometric and geometric distances when
calculating the distance and size of the open cluster.

We observed that the geometric and the photo-geometric distances to the
cluster centre were within one cluster size for all open clusters except for
ASCC 16. We find that our distances to the cluster centre are within one
cluster size compared to the cluster distances and sizes calculated directly
using the geometric and photo-geometric distances for all the open clusters
except for ASCC 16.

We found that the cluster sizes calculated directly from geometric and photo-
geometric distances are larger compared to the cluster sizes calculated using
equation 7. For Trumpler 10, M67, and NGC 2516 we find that our cluster sizes
are significantly smaller, we think this is caused by our prior on cluster
size (i.e., equation 5).

We Further compared our cluster centre distances and sizes obtained by using
equation 7 to two different catalogues Bossini et al. (2019) and Hunt &
Reffert (2023); called BO19 and HR23 hereafter. We have chosen these two
catalogues because BO19 uses an automated isochrone fitter to get distances;
hence, it is relevant to see how different our distance results and isochrone
parameters are. We chose HR23 as it uses a clustering algorithm on Gaia DR3
data to find distances; hence, we should find similar results as we have the
same data. The tabulated distance results with these two catalogues are shown
in table 4. We found that the majority of our calculated cluster centres are
within 3 pc of BO19 and HR23 values. In BO19, M67 and Trumpler 10 were
outliers; this could be caused by the difference in priors, methods, and the
raw data used (as we used Gaia DR3 rather than Gaia DR2, which BO19 used,
which has 20% smaller uncertainties than Gaia DR2 (Lindegren et al., 2021)).
M67 is also the furthest cluster we studied in this paper at 839 pc. Our
values correspond more with HR23, as we both use the same data from Gaia DR3.
The cluster with the largest difference was ASCC 16.

We decided not to study a specific group in the Upper Scorpius region but to
study it as a whole. Our Upper Scorpius distances are within the distances of
the groups inside the Upper Scorpius region. The distances of the groups
inside the Upper Scorpius regions range from 142 pc to 159 pc in Ratzenböck et
al. (2022) or ranges from 126 pc to 153 pc in Miret-Roig et al. (2022). Our
distance of 146 pc lies within both of these ranges.

Figure 4: Corner plot of the probability distribution of the cluster distance
prior calculated for NGC 2632. The vertical dashed lines in the histogram show
the mean and $\pm 1\sigma$ for each parameter Figure 5: Histogram showing the
distance distribution of stars in M67 without the cluster distance prior
(i.e., geometric distances, shown in equation 2) shown in blue and after using
the cluster prior (i.e., equation 8) shown in yellow. The red line indicates
the centre of the cluster according to the cluster distance prior and the
yellow lines are the radial size of the cluster according to the cluster
distance prior

Our radial cluster sizes are as expected, with all but one of the clusters
being around 10 pc. M67 does have a much larger cluster size; this could be
caused by the pencil effect that happens when the distance to the cluster
centre increases. The increase in distance increases the uncertainty in the
parallax. Comparing our radial cluster sizes with the HR23 size values, we
found that the sizes of our clusters are within the bounds of the core radius
and the tidal radius for all but M67.

One of the reasons that our distances and cluster sizes could differ is the
different membership of the open clusters. We have compared our membership
with the membership in HR23 in table 5. We expect HR23 to have a higher
membership count than us, as their membership includes stars in the tidal
tail, whereas ours focuses on stars in the open cluster core. The possible
field stars are the stars in our study but not found in HR23. The amount of
possible field stars does not exceed more than 4 % of the total member stars
in our study in all open clusters except for ASCC 16 and Trumpler 10. In ASCC
16, the possible field stars lie on the best-fitting isochrone; thus, we think
these possible field stars are open cluster members. However, in Trumpler 10,
these possible field stars do not lie on the best-fitting isochrone; thus, we
think our membership criteria for Trumpler 10 are not strict enough. Even
though 39 % of stars in Trumpler 10 are possible field stars, we found that
our distance to the cluster core is within 3 pc compared to the distance in
HR23. Thus, we do not think that the field stars have impacted the distance in
a significant way. We also confirmed that none of these possible field stars
in ASCC 16 and Trumpler 10 are members of any other cluster studied in HR23.

Overall, we found that we estimated our cluster core distances and sizes well
for our purpose of calculating absolute magnitudes. Even though some of our
cluster core distances and sizes are not within the ranges in BO19 and HR23,
the resulting distance distribution of the stars in the open clusters will be
similar to our values. Furthermore the difference between our cluster core
distances and the catalogues are within one cluster size.

The corner plots of the probability distribution of the cluster distance
priors (e.g., Fig. 4) are as expected, with the probability centred around one
point. If there was a lot of contamination with other field stars or other
clusters, we could have found multiple peaks in the PDF.

After the introduction of the cluster distance prior, the stars are normally
distributed within the calculated size of the cluster; this is shown in Fig.
5. The new distribution of stars is in line with our expected results when
using the cluster distance and shows how our cluster distance prior is working
as intended.

By using the distance cluster prior we saw an average increase in the
precision of the individual star distance of 31% compared to the distances
calculated without using the distance cluster prior. i.e., the geometric
distances calculated in Bailer-Jones et al. (2021). We calculated the
geometric distance precision of individual stars by using:

$\frac{\texttt{r\\_hi\\_geo}-\texttt{r\\_lo\\_geo}}{2}$ (15)

Where r_hi_geo and r_lo_geo are the 84th and 16th percentiles of the distance
posterior calculated using the geometric prior.

The stars in Upper Scorpius were the only ones that did not see an average
increase in precision. We think that this is caused by the size of the region
estimated by the cluster distance prior (8.7 pc), which simulates well the
original size of the region, thus leading to the prior having a negligible
effect on this cluster. The radial cluster size was expected, as Upper
Scorpius is a large region with multiple groups in it, and we did not choose a
specific group to study. In Miret-Roig et al. (2022) they estimates the groups
having sizes of $\sim 1$ pc to $\sim 10$ pc. Our estimation of 8.7 pc is
within the ranges of these groups.

Our method of using a cluster distance prior to fit individual distances has
increased the accuracy of the individual distances as the distances now lie
inside the expected size of an open cluster core. The method has increased the
precision of the individual distance measurement. With both improvements to
the distance measurements, it will improve our absolute magnitudes, thus
improving the photometric $T_{\rm{eff}}$ and $\log(g)$ measurements.

###  3.2 Isochrone fitting and photometric stellar parameters

Using the method explained in Sec. 2.2.1, we fitted isochrones to the HR
diagrams and selected appropriate ranges for the ages, metallicities, and
extinctions for each cluster. The ranges are in the expected bounds for an
open cluster. The ages are younger than $10^{10}$ years, and the metallicities
are around solar metallicity. The ranges selected are shown in table 6. The HR
diagrams fitted best fitting isochrone are shown in Appendix D.

We did extensive checks of the validity of the ranges of ages, metallicities,
and extinctions for each cluster, as demonstrated in Appendix E. We also
verified the method of calculating photometric $T_{\rm{eff}}$ and $\log(g)$
(i.e., equation 9) in Appendix F.

Figure 6: $T_{\mathrm{eff}}$ and $\log(g)$ joint probability distribution for
four different stars in M67. The four stars represent the typical PDF for
different types of stars in the cluster. The blue line is the best-fitting
isochrone.

We constructed a joint PDF for $T_{\rm{eff}}$ and $\log(g)$ for each star in
the cluster. An example for M67 is shown in Fig. 6. We found that for stars
that are in the lower half of the main sequence, the PDFs were uncorrelated in
$T_{\rm{eff}}$ vs. $\log(g)$. This is as expected because when we plot a Kiel
diagram, the gradient of $T_{\rm{eff}}$ and $\log(g)$ with respect to the
starting masses of the stars at that point along the isochrone is small, so
when we change the isochrone parameters, the change in $T_{\rm{eff}}$ and
$\log(g)$ will be quite small. Another factor is that the isochrone changes
shape in a symmetrical way when increasing or decreasing a parameter in the
isochrone. Further along the isochrone, the correlation of $T_{\rm{eff}}$ and
$\log(g)$ becomes stronger. Stars at the turn-off point have a very
unsymmetrical PDF shape, which is because of the complex shape of the
isochrone at the turn-off point. Therefore, when we change the parameters of
the isochrones, the newest closest point to the star can have a very different
$T_{\rm{eff}}$ and $\log(g)$. In the lower red giant branch, the trends become
very strong as shifts in the isochrone parameters cause large changes in the
shape of the isochrone, thus causing a large change in $T_{\rm{eff}}$ and
$\log(g)$.

The average photometric precision of $T_{\rm{eff}}$ and $\log(g)$ is 38.2 K
and 0.031 dex. This precision is calculated from PDF in equation 13. The
average precision measured when using an individual isochrone (i.e., equation
10) is 3.15 K and 0.0016 dex. As the final photometric precision is much worse
than the precision measured using an individual isochrone, it shows that the
ranges of metallicities, ages, and extinctions are a much larger source of
imprecision than the absolute magnitudes.

Spectroscopic uncertainty in GALAH is 70 K in $T_{\rm{eff}}$ and 0.064 dex in
$\log(g)$. The size of GALAH uncertainty is comparable to the size of the
joint $T_{\rm{eff}}$ and $\log(g)$ probability distribution. This comparable
size is an indication that the shape of the photometric prior will have an
impact on our spectroscopic fit.

Table 6: The parameters and their ranges for isochrones for each open cluster. Age is given in $\log(\rm{age})$. Cluster | Age | Metallicity | Extinction ($\it{A_{\rm{V}}}$)
---|---|---|---
| $\log\,\rm{age\,(yr)}$ | dex | mag
ASCC 16 | 7.10 $\begin{subarray}{c}+0.09\\\ -0.08\end{subarray}$ | 0.05 $\begin{subarray}{c}+0.20\\\ -0.15\end{subarray}$ | 0.25 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$
Blanco 1 | 7.98 $\begin{subarray}{c}+0.12\\\ -0.08\end{subarray}$ | 0.10 $\begin{subarray}{c}+0.08\\\ -0.10\end{subarray}$ | 0.05 $\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$
Melotte 22 | 8.00 $\begin{subarray}{c}+0.15\\\ -0.20\end{subarray}$ | 0.15 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 0.25 $\begin{subarray}{c}+0.10\\\ -0.15\end{subarray}$
NGC 2516 | 8.30 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 0.20 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 0.3 $\begin{subarray}{c}+0.1\\\ -0.1\end{subarray}$
NGC 2632 | 8.90 $\begin{subarray}{c}+0.05\\\ -0.09\end{subarray}$ | 0.19 $\begin{subarray}{c}+0.11\\\ -0.08\end{subarray}$ | 0.08 $\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$
M67 | 9.59 $\begin{subarray}{c}+0.06\\\ -0.05\end{subarray}$ | 0.11 $\begin{subarray}{c}+0.07\\\ -0.05\end{subarray}$ | 0.09 $\begin{subarray}{c}+0.06\\\ -0.04\end{subarray}$
Ruprecht 147 | 9.40 $\begin{subarray}{c}+0.04\\\ -0.05\end{subarray}$ | 0.08 $\begin{subarray}{c}+0.12\\\ -0.08\end{subarray}$ | 0.40 $\begin{subarray}{c}+0.1\\\ -0.05\end{subarray}$
Trumpler 10 | 7.65 $\begin{subarray}{c}+0.12\\\ -0.08\end{subarray}$ | 0.10 $\begin{subarray}{c}+0.20\\\ -0.20\end{subarray}$ | 0.15 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$
Upper Scorpous | 6.90 $\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 0.15 $\begin{subarray}{c}+0.30\\\ -0.25\end{subarray}$ | 0.60 $\begin{subarray}{c}+0.20\\\ -0.20\end{subarray}$

###  3.3 Abundances

Table 7: An overview of the number of stars successfully fitted by the cluster fitting process. $N_{\mathrm{GALAH}}$ is the total number of spectra we fitted from the DR4 GALAH catalogue. $N_{100\rm{np}>}$ and $N_{400\rm{np}>}$ are the number of stars with more than 100 and 400 independent samples on average for each fitted parameter without using the prior. $N_{100\rm{p}>}$ and $N_{400\rm{p}>}$ are the number of stars with more than 100 and 400 independent samples on average for each fitted parameter using the prior.  Cluster | $N_{\mathrm{GALAH}}$ | $N_{100\rm{np}>}$ | $N_{100\rm{p}>}$ | $N_{400\rm{np}>}$ | $N_{400\rm{p}>}$ | SNR
---|---|---|---|---|---|---
ASCC 16 | 173 | 18 | 27 | 2 | 4 | 40.0
Blanco 1 | 109 | 53 | 52 | 21 | 22 | 94.0
Melotte 22 | 91 | 26 | 27 | 11 | 9 | 75.4
NGC 2516 | 204 | 32 | 25 | 6 | 10 | 40.3
NGC 2632 | 147 | 58 | 49 | 18 | 19 | 60.3
NGC 2682 | 922 | 608 | 580 | 152 | 163 | 61.9
Ruprecht 147 | 130 | 85 | 86 | 16 | 17 | 91.6
Trumpler 10 | 119 | 58 | 57 | 8 | 8 | 32.2
Upper Scorpius | 84 | 9 | 4 | 1 | 0 | 31.8
Table 8: Precision values of spectroscopic abundances and stellar parameters calculated using different methods and a comparison with GALAH DR4. The MCMC columns show the average uncertainty acquired from MCMC fit when fitting using the photometric prior, and no photometric prior. The overall scatter shows the mean scatter of the parameters when fitted using the prior in each cluster weighted with the number of stars. M67 scatter is the spread of the parameters in M67 with $4.0<\log(g)<4.1$. Precision GALAH is the precision given in GALAH DR3. Accuracy GALAH is the estimated accuracy of the parameters in GALAH DR4. Uncertainty GALAH is the total uncertainty estimation in GALAH DR4 acquired by adding in quadrature the precision and the accuracy. Parameter | Units | MCMC | Scatter | Repeat observations | Precision GALAH | Accuracy GALAH | Uncertainty GALAH
---|---|---|---|---|---|---|---
Prior | No prior | Overall | M67
$T_{\rm{eff}}$ | K | 10.27 | 10.284 | - | - | 20.975 | 26.254 | 66 | 71.03
$\log(g)$ | dex | 0.026 | 0.031 | - | - | 0.037 | 0.057 | 0.042 | 0.071
[Fe/H] | dex | 0.01 | 0.01 | 0.078 | 0.059 | 0.019 | 0.038 | 0.044 | 0.058
$v_{\rm{mic}}$ | $\mathrm{km\,s^{-1}}$ | 0.04 | 0.04 | - | - | 0.071 | 0.075 | 0.28 | 0.29
$v_{\rm{broad}}$ | $\mathrm{km\,s^{-1}}$ | 0.135 | 0.133 | - | - | 0.212 | 0.555 | 1.4 | 1.506
[Li/Fe] | dex | 0.291 | 0.244 | 1.302 | 0.100 | 0.025 | 0.154 | - | 0.154
[C/Fe] | dex | 0.041 | 0.032 | 0.089 | 0.052 | 0.028 | 0.086 | - | 0.086
[N/Fe] | dex | 0.063 | 0.063 | 0.251 | 0.120 | 0.093 | 0.181 | - | 0.181
[O/Fe] | dex | 0.055 | 0.056 | 0.16 | 0.096 | 0.039 | 0.084 | - | 0.084
[Na/Fe] | dex | 0.048 | 0.051 | 0.097 | 0.076 | 0.035 | 0.037 | - | 0.037
[Mg/Fe] | dex | 0.028 | 0.028 | 0.086 | 0.042 | 0.036 | 0.035 | - | 0.035
[Al/Fe] | dex | 0.052 | 0.055 | 0.135 | 0.091 | 0.090 | 0.07 | - | 0.07
[Si/Fe] | dex | 0.02 | 0.02 | 0.054 | 0.028 | 0.037 | 0.025 | - | 0.025
[K/Fe] | dex | 0.059 | 0.06 | 0.263 | 0.293 | 0.053 | 0.062 | - | 0.062
[Ca/Fe] | dex | 0.032 | 0.034 | 0.069 | 0.041 | 0.019 | 0.041 | - | 0.041
[Sc/Fe] | dex | 0.036 | 0.037 | 0.074 | 0.061 | 0.036 | 0.04 | - | 0.04
[Ti/Fe] | dex | 0.025 | 0.026 | 0.064 | 0.032 | 0.022 | 0.033 | - | 0.033
[V/Fe] | dex | 0.043 | 0.044 | 0.112 | 0.063 | 0.046 | 0.057 | - | 0.057
[Cr/Fe] | dex | 0.022 | 0.024 | 0.049 | 0.037 | 0.025 | 0.024 | - | 0.024
[Mn/Fe] | dex | 0.052 | 0.057 | 0.11 | 0.046 | 0.038 | 0.037 | - | 0.037
[Co/Fe] | dex | 0.044 | 0.043 | 0.112 | 0.070 | 0.031 | 0.055 | - | 0.055
[Ni/Fe] | dex | 0.018 | 0.018 | 0.042 | 0.021 | 0.014 | 0.018 | - | 0.018
[Cu/Fe] | dex | 0.099 | 0.107 | 0.244 | 0.195 | 0.113 | 0.056 | - | 0.056
[Zn/Fe] | dex | 0.096 | 0.095 | 0.194 | 0.155 | 0.082 | 0.096 | - | 0.096
[Y/Fe] | dex | 0.074 | 0.075 | 0.144 | 0.121 | 0.090 | 0.074 | - | 0.074
[Zr/Fe] | dex | 0.209 | 0.208 | 0.493 | 0.402 | 0.247 | 0.564 | - | 0.564
[Ba/Fe] | dex | 0.064 | 0.065 | 0.16 | 0.083 | 0.055 | 0.073 | - | 0.073
[Nd/Fe] | dex | 0.08 | 0.078 | 0.191 | 0.199 | 0.098 | 0.228 | - | 0.228
[Sm/Fe] | dex | 0.185 | 0.191 | 0.449 | 0.391 | 0.312 | 0.294 | - | 0.294
[Eu/Fe] | dex | 0.299 | 0.252 | 0.59 | 0.477 | 0.111 | 0.223 | - | 0.223

Table 7 shows the overall number of stars that were fitted for each cluster
and how many stars were successfully fitted. To see whether the star is
successfully fitted, we used some basic cuts. The cuts we used are
$v_{\mathrm{mic}}>0\,\mathrm{km\,s^{-1}}$,
$15\,\mathrm{km\,s^{-1}}\,>v_{\mathrm{broad}}>0\,\mathrm{km\,s^{-1}}$, $5$ dex
$>\log(g)>$ $2$ dex. We chose these values for the quality cuts as the stars
outside of these ranges have unreliable abundances, thus contributing
erroneous scatter to our data. As we are looking at the trends of abundances,
we decided to only use stars that was fitted with more than 100 independent
samples on average, as we found it to be a sufficient number to compare
abundance trends. In ASCC 16 and Upper Scorpius, our fitting software
struggled to fit spectra successfully. This is because these two clusters are
our two youngest clusters at $10^{7.10}$ and $10^{6.90}$ years, respectively;
thus, the stars will be more likely to have a higher rotation (Nielsen et al.,
2013; Godoy-Rivera et al., 2021), as a result not pass our
$v_{\mathrm{broad}}$ of our quality cuts. The other reason is that the SNRs
for those clusters are among the lowest, at 47.0 and 35.5, respectively. SNR
is defined as the average ratio of the uncertainty over normalised flux in the
spectra.

In Fig. 7, we can see an example of the effect of using a joint $T_{\rm{eff}}$
and $\log(g)$ prior when fitting spectra. The probability distributions are
displayed using a Gaussian KDE, of a star that fitted with more than an
average of 400 individual samples for each parameter. We can clearly see that
the photometric prior affects the value of $T_{\rm{eff}}$ and $\log(g)$. As
the fit using the prior has a different $T_{\rm{eff}}$ and $\log(g)$ to the
fit without using the prior, we can see that the photometric prior has
affected the probably distribution of [Fe/H], [O/Fe], and [Si/Fe]. We chose
these three abundances as they show dependence on $T_{\rm{eff}}$ and $\log(g)$
most clearly. Fig. 7 also demonstrates that the $T_{\rm{eff}}$ and $\log(g)$
prior cannot be approximated by a multivariate normal distribution.

Figure 7: Corner plot of a star with a Gaia id $=604907110979352576$. The
photometric prior is shown in green with it being marginalised values on the
$T_{\rm{eff}}$ and $\log(g)$ histogram plots. The probability distribution of
the fit with and without the photometric prior is shown in blue and red
respectively.

M67 and Ruprecht 147 fitted abundances and stellar parameters can be seen in
Fig. 9 and Fig. 10, the other clusters can be seen in Appendix G. In the
figures comparing stellar parameters and abundances fitted with and without
the photometric prior, we only show stars that have passed the quality cuts of
both types of fitting. We see a large effect on the accuracy of $\log(g)$, as
seen in Fig. 9. This is especially true in areas that are not most sensitive
to isochrone changes (i.e., not at the turn-off point). This increase in
accuracy causes a new population of stars located at $4.20-4.55$ dex to
appear, as shown in Fig. 11. The $\log(g)$ change in M67 is mostly below a
$\log(g)$ of 0.1, with a small population of stars that has a change in
$\log(g)$ above 0.1 dex. This shift in $\log(g)$ will be the source of the
different abundance trends in the abundances fitted using the photometric
prior and without. Thus, we do not expect a change in abundance in stars
between $3.5<\log(g)<4.20$ as the majority of the stars have the same
$\log(g)$ when fitted with or without the photometric prior. The effect of the
photometric prior on the fitting regime will be most apparent in the
$4.20<\log(g)<4.55$ range as there lies the shifted population. From the Kiel
diagrams, we found that our fitting regime struggled to fit stars with
$T_{\rm{eff}}$<$4\,500$ K and a $\log(g)$ >4.20 dex. Our model tended to
vastly underestimate the $\log(g)$ of those stars. This underestimation is
also shown in Fig. 8. The figure shows the difference between photometric
$\log(g)$ and $T_{\rm{eff}}$ to spectroscopic $\log(g)$ and $T_{\rm{eff}}$
calculated with and without the prior, in all open clusters. We see that when
calculating spectroscopic $\log(g)$ without using the photometric prior, we
underestimate $\log(g)$ of stars with a $\log(g)$ >4.20 dex. We found that for
stars that have a photometric temperature of lower than $5\,800$ K, the
photometric prior had a more substantial effect on spectroscopic
$T_{\rm{eff}}$ compared to stars with a photometric temperature of more than
$5\,800$ K.

Figure 8: The top panel is a scatter plot of photometric $\log(g)$ vs
spectroscopic $\log(g)$ minus photometric $\log(g)$. The bottom is a scatter
plot of photometric $T_{\rm{eff}}$ vs spectroscopic $T_{\rm{eff}}$ minus
photometric $T_{\rm{eff}}$. For both panels the spectroscopic $\log(g)$ in the
blue points are calculated using the photometric prior and the red points
without. For both panels we used all the stars from all open clusters that
passed the quality cuts. The solid lines are the median of a bin containing 5%
of the stars in the cluster. The coloured area shows $\pm$ 1 $\sigma$ Figure
9: Stellar parameters and abundance spread with respect to $\log(g)$ for M67.
Blue dots are the stars that were analysed using the photometric prior and red
without the prior. The number on the top right corner of each graph indicates
the number of stars present in each graph. The Kiel diagram panel has the best
fitting isochrone used in getting the photometric parameters. The solid lines
are the median of a bin containing 10% of the stars in the cluster, except for
the Li panel where it is 5% of the stars in the cluster. The coloured area
shows $\pm$ 1 $\sigma$. Figure 9: Continuation of Fig. 9 Figure 9:
Continuation of Fig. 9 Figure 10: Stellar parameters and abundance spread with
respect to $\log(g)$ for Ruprecht 147. Blue dots are the stars that were
analysed using the photometric prior and red without the prior. The number on
the top right corner of each graph indicates the number of stars present in
each graph. The Kiel diagram panel has the best fitting isochrone used in
getting the photometric parameters. The solid lines are the median of a bin
containing 7 stars, with the coloured area showing $\pm$ 1 $\sigma$.

The average spectroscopic values of [Fe/H] did not change significantly when
comparing stars fitted with or without the photometric prior. The small change
that occurred did not shift the spectroscopic [Fe/H] towards the average
metallicity chosen to generate the isochrones.

Figure 11: The top and middle panel shows the spread of $\log(g)$ in M67 and
Ruprecht 147 when calculated with and without the photometric prior. The
bottom panel shows the change in $\log(g)$ between the stars fitted with and
without the photometric prior.

In clusters with stars with a small range of $\log(g)$, the effect of the
photometric prior is large enough that there is no overlap in $\log(g)$ within
the population fitted with and without the photometric prior, thus making it
hard to compare trends in those clusters. Consequently, we have concentrated
our analysis mostly on M67 and Ruprecht 147, which have a wide range of stars
in $\log(g)$ and so a large overlap.

Within each open cluster, our abundances fitted with or without the
photometric prior do not have a linear trend with respect to $T_{\rm{eff}}$ or
$\log(g)$. We attribute this to using non-LTE models to create the training
set data and fitting the majority of the wavelength range. We found that we
fit most abundances within the ranges set out in the training set of our
neural model. [Eu/Fe] and [Zr/Fe] were the elements that were most likely to
fit with a value that was outside of our training set. The [Fe/H] of M67 and
Ruprecht 147 decreases as $\log(g)$ decreases, which we attribute to
systematic errors in our model. The higher-order abundance trends in M67 and
Ruprecht 147, we attribute to atomic diffusion. This is because the higher-
order trends present themselves as dips around the turn-off point, which is at
a $\log(g)$ of $\sim 4$ for M67 and Ruprecht 147 (Liu et al., 2019; Gao et
al., 2018). These dips are present in studies that have modelled atomic
diffusion in M67 (Liu et al., 2019; Gao et al., 2018).

The higher-order trends within M67 are visible in [Fe/H], [Ni/Fe], [Cu/Fe],
[O/Fe], [V/Fe], [Mg/Fe], [Ti/Fe], [Sc/Fe], [Al/Fe], [V/Fe] , [Ba/Fe] and
[Mn/Fe]. The higher-order trends are visible in the abundances fitted with and
without the photometric prior. For these elements, there is a clear dip at
around $\log(g)$=4. The depth varies, with [Ni/Fe] having a magnitude of 0.05
dex, while [Mn/Fe] has a magnitude of 0.1 dex. These depths match the expected
dip shown in Liu et al. (2019), with shallower dips in [Ni/Fe] with a depth of
0.05. Our [Ca/Fe] abundances stay flat, as predicted by Liu et al. (2019).

In M67 there is an increased range in $\log(g)$ in the data fitted using the
photometric prior compared to the data fitted without the photometric prior.
The increased range in $\log(g)$ enables the observation of how the trends in
abundances at a $\log(g)$ larger than 4.2 dex. In most of the abundances, the
trends above a $\log(g)$ of 4.2 dex in the fit using the photometric prior are
just a continuation of the trends in the fit without using the photometric
prior. However, in [Si/Fe] and [Nd/Fe], the trend differs. Furthermore the
stars that were analysed without using the photometric prior and were shifted
to higher than 4.2 dex in the analysis using the photometric prior, will add
scatter to the abundances. As a result, it becomes more challenging to discern
higher-order trends in the analysis without the photometric prior.

In Ruprecht 147, the effect of the photometric prior on the spread of
$\log(g)$ is also observable. Stars with $4.0<\log(g)$ show that the fit
without the photometric prior underestimates $\log(g)$. The ranges of
$\log(g)$ of the stars fitted using the prior and without are similar in
Ruprecht 147. This stands in contrast to M67 which has a population of stars
with a $\log(g)$ of 4.5 in the analysis using the photometric prior which is
not present in the analysis without the photometric prior, as illustrated in
Fig. 11. An atomic diffusion dip at around $\log(g)$ of $4.1$ dex is visible
in [Fe/H], [Al/Fe], [Mg/Fe], [Ni/Fe], [Cu/Fe], and [Mn/Fe]. However, in
[Fe/H], [Al/Fe], and [Mg/Fe], the dip was only present in the abundances that
were fitted using the photometric prior.

The abundances calculated using the photometric prior in the M67 and Ruprecht
147 open clusters were both more accurate than the abundances calculated
without using the photometric prior. In M67, the increase in accuracy was
caused by the population’s $\log(g)$ of $4.2-4.55$ dex, which only occurs when
fitting using the photometric prior; however, other than the stars that get
shifted to this population, the rest of the population’s $\log(g)$ did not
change significantly. The new population causes the effects of atomic
diffusion to be more visible, as we could see the effect of atomic diffusion
on stars with a $\log(g)$$>4.2$. Unlike in M67 in Ruprecht 147, most of the
stellar population shifted its $\log(g)$ when fitted using the photometric
prior and without. This caused larger differences in the abundance trends. We
could see evidence of atomic diffusion more clearly using the photometric
prior compared to without. These two clusters show that using photometric
priors causes better $\log(g)$ compared to the analysis without using the
photometric priors, thus enabling us to see atomic diffusion.

A comparison of our stellar parameters and abundances fitted using the
photometric prior (blue) and GALAH DR4 (pink) in M67 and Ruprecht 147 can be
seen in Fig. 12 and Fig. 13, and for the other clusters they can be seen in
Appendix G. Both data set uses the same quality cuts, but with the GALAH DR4
data has an additional quality cut of $\texttt{flag\\_sp}==0$. The medians and
standard deviations of the stellar parameter and abundances in Fig. 12 and
Fig. 13 differs to Fig. 9 and Fig. 10 slightly. This is because the Fig. 9 and
Fig. 10 the data has to pass the quality cuts for both the data analysed with
and without the photometric prior, but in Fig. 12 and Fig. 13 it only has to
pass the quality cuts of the data analysed with the photometric prior.

Comparing our results to GALAH DR4, we found that GALAH DR4 $T_{\rm{eff}}$ and
$\log(g)$ followed the Kiel diagram of the best fitting isochrone used in
getting the photometric prior better than our work. This could be caused by
how in GALAH DR4, a different neural network is used to calculate stellar
parameters and abundances depending on its $T_{\rm{eff}}$, $\log(g)$, and
[Fe/H]. i.e., a different neural network is used to analyse Sun-like stars,
red clump stars, etc. Unlike in this work, where we used a single neural
network per band for all the stars analysed. The difference between our
$\log(g)$ and GALAH DR4 $\log(g)$ could be due to GALAH DR4 using astrometry
from Gaia and photometry from 2MASS to calculate $\log(g)$, which is different
from this work. A 0.1 dex decrease in [Fe/H] was found in most clusters. The
absolute average values of [Mn/Fe], [K/Fe], and [Ni/Fe] are significantly
different. For Ni, Mn, and K, this can be explained by this work not using a
non-LTE model for these specific elements when generating spectra for training
our neural network. We also found significant differences in the trends of
Carbon and Nitrogen; this is explained by the differences in the methods of
measuring Carbon. GALAH DR4 uses molecular features such as the $\rm{C_{2}}$
and CN molecules, and this work uses weak Carbon and Nitrogen lines. Thus
leading to the better abundances in GALAH DR4 for those two elements. This
work’s lithium spread is also much higher than GALAH DR4. When comparing the
abundances of stars in M67 with $\log(g)$ above 4.4 dex, we can see that the
trends of [Si/Fe], [Ni/Fe], [Cu/Fe], [Sc/Fe], [Ti/Fe] and [Cr/Fe] are
significantly different. Atomic diffusion is much more visible in this work in
[Mg/Fe], [Si/Fe], [Ni/Fe], [O/Fe], [Sc/Fe], [Ti/Fe], [Cu/Fe], [Sm/Fe],
[Ba/Fe], and [Al/Fe] in M67. In Ruprecht 147, the difference is larger, with
atomic diffusion only visible in this work in [Cu/Fe], [Co/Fe] and [Ni/Fe].
The spread of Calcium in all open clusters is lower in this work compared to
GALAH DR4.

Figure 12: Stellar parameters and abundance spread with respect to $\log(g)$
for M67. Blue dots are the stars that were analysed using the photometric
prior and green dots are from GALAH DR4. The number on the top right corner of
each graph indicates the number of stars present used in the photometric prior
graph. The Kiel diagram panel has the best fitting isochrone used in getting
the photometric parameters. The solid lines are the median of a bin containing
10% of the stars in the cluster, except for the Li panel where it is 5% of the
stars in the cluster. The coloured area shows $\pm$ 1 $\sigma$. Figure 12:
Continuation of Fig. 12 Figure 12: Continuation of Fig. 12 Figure 13: Stellar
parameters and abundance spread with respect to $\log(g)$ for Ruprecht 147.
Blue dots are the stars that were analysed using the photometric prior and
green dots are from GALAH DR4. The number on the top right corner of each
graph indicates the number of stars present used in the photometric prior
graph. The Kiel diagram panel has the best fitting isochrone used in getting
the photometric parameters. The solid lines are the median of a bin containing
7 stars, with the coloured area showing $\pm$ 1 $\sigma$.

##  4 Spectroscopic precision

MCMC provides an opportunity to more meticulously sample the probability space
while fitting, as it gives the exact shape of the PDF compared to methods such
as using the covariance matrix. Thus, we have explored how well MCMC estimates
precision. In this work we only measure the precision, defined as the
statistical variability of a measurement. We do not measure accuracy, which is
defined as how close a measurement is to the true value.

We first investigated the role that the SNR of spectra has on the precision
and the scatter of the fitted spectroscopic parameter. We investigated this by
picking 10 spectra with an SNR of higher than 100 with
$6100\,K>T_{\rm{eff}}>5000\,K$ and $3.9$ dex $>\log(g)>2.9$ dex. We added
Gaussian noise to the spectra to create spectra with an SNR ranging from 10 to
90 in steps of 10. We fitted them without using the photometric prior. To
investigate the effect of SNR on the average precision, we took the average
precision of [Ti/Fe] in bins of 10 in SNR, as shown in the top panel in Fig.
14. In these spectra the deepest blended Titanium lines are $\sim 0.25$ and
$\sim 0.15$ in normalised flux for non-blended lines. When the SNR is above
$\sim$ 30, there is a minimal decrease in uncertainty with the increase in
SNR. To examine the change in scatter, we first calculated the mean [Ti/Fe]
for each star. Subsequently, we measure the absolute difference between the
mean [Ti/Fe] and the corresponding fitted [Ti/Fe] value at different SNRs.
Lastly, we calculate the average change in [Ti/Fe] values within bins of 10
SNR units. The average change is shown in the bottom panel in Fig. 14. At all
SNRs, the average change is smaller than the average uncertainty. Under an SNR
of 30, there is a significant increase in uncertainty compared to the higher
SNR spectra. The effect of SNR on the other fitted parameters is similar to
[Ti/Fe] except for the abundances of elements with the weakest lines.

Figure 14: The top panel shows the effect of changing the SNR of a star on the
precision of [Ti/Fe]. The bottom graph shows the effect of changing the SNR of
a star on the scatter of [Ti/Fe]

Table 8 shows the effect of the prior on the precision of individual
abundances. These precisions are the average precision of the spectra that
fitted with more than 400 individual samples. We do not see worse precision of
the abundances fitted using the photometric prior compared to abundances
fitted without using the photometric prior. This is because even though the
photometric prior’s spread is much larger than the value of the precision in
the spectroscopic fit without using the photometric prior, their most probable
values differ from each other. Therefore, the overlap of the probability space
is smaller. This is shown in the precision of $\log(g)$. The difference
between the photometric $\log(g)$ and the $\log(g)$ obtained using
spectroscopy without using the photometric prior is large. Hence, we see a
decrease in the uncertainty in $\log(g)$ when fitting using spectroscopy
without the photometric prior. As the precision in $T_{\rm{eff}}$ and
$\log(g)$ do not change significantly, we do not observe a significant change
in the precision of the abundances when fitting with the photometric prior
compared to without using the photometric prior for the rest of the
parameters.

To validate our spectroscopic precision calculated using MCMC, we have
estimated precision in two additional ways. The first is to use the scatter of
abundances as a measure of precision. To take into account the trends caused
by effects such as atomic diffusion, we have selected stars within a narrow
range of $\log(g)$ where their abundances should not change significantly. We
selected the range between $4.0<\log(g)<4.1$, as the abundances should
theoretically change by less than 0.005 dex (Liu et al., 2019). Thus, by
calculating the scatter of abundances in that section, we can compare the
precision from MCMC and the scatter. We decided to only use M67 as the
abundance trends calculated in Liu et al. (2019) are specifically made for the
ages and metallicity of M67, and as we are only using a small range of
$\log(g)$, all the other open clusters do not have enough fitted members in
this range. To reduce the scatter introduced by the range of SNR, we have only
used stars with an SNR of $35-45$, decreasing the role the SNR will have on
the scatter of abundances. Using those cuts we have 88 stars to calculate the
spread. The spread of abundances is shown in Table 8. We found that the median
abundance precision estimated by MCMC is underestimated by 30% compared to
precision measured from the scatter in M67. Thus MCMC is more optimistic in
precision compared to the precision calculated using M67 scatter.

The second method to evaluate our abundance precision uses the scatter in
abundances derived from repeat observations of stars. We selected repeat
observations where their SNR only changes by 10% between the observations. We
found that the median precision estimated by MCMC is underestimated by 9%
compared to the precision estimated by the repeated measurements. However, if
we only compare abundances, the precision is only underestimated by 1%. To
compare precisions we are using the median value rather than the mean as there
is an outlier with the estimate of the precision of [Li/Fe]. We found that
these two results show that the abundance precision calculated using MCMC is
accurate, as their value is comparable to the repeat observation. The
precision calculated by MCMC for $T_{\rm{eff}}$, [Fe/H] and $v_{\rm{mic}}$ is
underestimated. Thus MCMC is more optimistic in the precision of these
parameters compared to the precision calculated in the repeat observations.

In GALAH DR4, we calculated the total uncertainty of all fitted parameters by
adding in quadrature, its precision, and its accuracy. Note that accuracy is
only available for $T_{\rm{eff}}$, $\log(g)$, $v_{\rm{mic}}$, [Fe/H], and
$v_{\rm{broad}}$. In this work, we only measure precision, so we will not
compare accuracy. GALAH measures the precision of the fitted parameters in two
main ways. The first is by using repeat observations and seeing how the fitted
values differ in this observation. The second method is by using the precision
derived from the covariance matrix calculated by SME while fitting. Whichever
of the two methods produces the biggest value becomes precision. The average
precision, accuracy, and total uncertainty for the stars used in this work and
GALAH DR4 are shown in table 8. We find that GALAH DR4 precision is 25% worse
compared to the precision calculated by MCMC, 20% better compared to our
precision calculated using repeat observations, and 17% better compared to the
precision calculated by scatter in M67.

MCMC provides a more meticulous way to measure precision as it gives the exact
shape of the probability distribution compared to using a covariance matrix
like in GALAH. We find that GALAH estimates its precision well in abundances
and other stellar parameters by combining both the precision acquired using
repeat observations and its fitting procedure. GALAH has a good estimate of
both abundance and the stellar parameters’ precision.

##  5 Discussion

Calculating accurate and precise photometric $T_{\rm{eff}}$ and $\log(g)$ was
one of the key reasons we were able to measure accurate abundances for open
cluster stars. One of the ways we increased the accuracy and precision of the
photometric $T_{\rm{eff}}$ and $\log(g)$ measurements was to use a cluster
distance prior. This method will only be valid when determining distances for
stars that are inside the open cluster core. However, with the increased
accuracy and precision of astrometry provided by Gaia DR3, Hunt & Reffert
(2023); Meingast et al. (2021); Tarricq et al. (2022) have shown the
possibility of determining members that are in the tidal tail of the open
cluster. With these members, our method of determining distance might not be
applicable as the members can be many pc away from the cluster centre.

Our method of using the cluster distance prior relies on having an accurate
membership of each open cluster. Upper Scorpius and Orion (which contains the
open cluster ASCC 16) contain multiple groups of open clusters with a very
complex spatial structure. In Upper Scorpius, we did not pick a specific group
inside; thus, the cluster distance prior did not succeed in decreasing the
uncertainty of the stars in that cluster. In these complex regions, we note
the importance of having radial velocities to be able to calculate accurate
memberships to the open clusters (Kos et al., 2019; Miret-Roig et al., 2022).

The complex relationship between $T_{\rm{eff}}$ and $\log(g)$ in the
photometric prior shown in this work is the result of how varying ages,
metallicities, and extinction affect the shape of the isochrone. Fu et al.
(2018) has shown that the $\alpha-$element abundance can have a significant
effect on the shape of the isochrone. Further studies should also be done into
how changing the $\alpha$-element abundances in stellar models will affect the
shape of the isochrone, thus affecting the shape of the joint $T_{\rm{eff}}$
and $\log(g)$ probability distribution.

We have chosen our stars to be far from the galactic plane reducing the effect
of the reddening. We have also assumed that the extinction is the same for all
stars in the open cluster. However, if we would like to more accurately fit
isochrone to the HR diagram of these stars, especially those that are closer
to the galactic plane. In that case, we should take into account the effects
of differential extinction.

The accuracy of generating synthetic spectra is greatly affected by the line
list from which the synthetic spectra are generated from. As discussed in
Jofré et al. (2019), many transition data ($\log(gf)$ values) are not known
accurately, and the line list available is incomplete with transitions
missing. It is crucial to have an accurate and complete line list since we are
fitting the majority of the wavelength range, rather than just the areas with
the strongest non-blended lines. We also do not quantify the uncertainty that
is associated with inaccurate atomic characteristics, such as the wavelength,
transition probabilities, and properties of the atomic states responsible for
these atomic transitions.

To increase the accuracy of generating synthetic spectra, we would also need
to increase the accuracy of stellar models. Effects such as magnetic fields
and stellar spots are not taken into account in this work. Depending on the
spectral type, magnetic fields, and stellar spots can affect the equivalent
width and depth of the absorption lines. The strength of magnetic fields and
stellar spots varies during the activity cycle. Thus, measurements of
$T_{\rm{eff}}$, [Fe/H], $v_{\rm{mic}}$, and chemical abundances will vary
through the activity cycle (Spina et al., 2020). When not taken into account,
the magnetic fields and stellar spots can increase the scatter of the measured
abundances.

Another effect that we did not take into account is the effect of diffuse
interstellar bands (DIBs). These bands can affect the observed spectra at
visible and near-infrared wavelengths. These effects although small ($\sim
0.5$ per cent of normalised flux (Vogrinčič et al., 2023)) are still
measurable in high precision spectroscopy. Further studies should use
catalogues such as Vogrinčič et al. (2023) to correctly model DIBs when
generating synthetic spectra.

These inaccuracies when generating synthetic spectra could be one of the
reasons why we overestimated the precision of $T_{\rm{eff}}$, $\log(g)$,
$v_{\rm{broad}}$, and $v_{\rm{mic}}$ using MCMC compared to the repeat
precision measurements. With further improvements to the accuracy of
generating synthetic spectra, we can reduce the need to mask certain
wavelengths of the spectra, allowing us to use more of the wavelength range.

With the continued use of Bayesian statistics in fitting spectra, as
demonstrated in Buder et al. (2021); Gent et al. (2022), works like this show
that MCMC provides an accurate way to measure the probability space of the
fitted spectra. However, MCMC is very computationally heavy, making it
impractical to use in large surveys compared to our work. In GALAH DR4, an
average spectrum takes 2 minutes to fit, compared to our 4 hours. In our work,
the two functions that consume the most time are scipy’s fast Fourier
transform, which is used when equalising the resolution of the synthetic
spectra with the observed spectra. The second function is Numpy’s einsum,
which is used to multiply vectors and matrices to generate the synthetic
spectra. These two functions are well optimised, so the way to increase the
sampling speed must come from somewhere else. Hamiltonian Monte Carlo (HMC) is
a more efficient sampling method compared to MCMC. This is because HMC uses
the probability gradient at each point to calculate its next step. This causes
the auto-correlation time to be much shorter, so fewer samples are needed to
acquire one independent sample. This has not been used in fitting spectra, as
the method to generate spectra has traditionally not been able to be
analytically differentiated. However, with the use of neural networks to
generate spectra, we can analytically differentiate the spectra with respect
to its training variables, thus getting the gradient of the probability space
while fitting analytically. In further studies, this fact should be used to
increase the efficiency of the fitting procedure.

Liu et al. (2019); Dotter (2016) and our work have further shown that the idea
of chemically tagging stars with a wide range of $\log(g)$ and $T_{\rm{eff}}$
will only be possible with a greater understanding of how abundances in stars
change as they age. This is because effects such as atomic diffusion are large
enough that without taking them into account, it will be impossible to
accurately group stars that come from the same star forming region. The
effects of atomic diffusion on abundances depend on many variables, such as
the age of the star, which element is being studied, the metallicity of the
star, and the current evolutionary stage of the star. This complex
relationship means that modelling the effect of atomic diffusion is a non-
trivial thing to do for a large range of stars. Our work shows the necessity
of having spectroscopic observations of stars in different evolutionary stages
in each open cluster to be able to see the effect.

The effects of atomic diffusion are clear when looking at the higher-order
trends of abundances with respect to $\log(g)$. This further highlights the
importance of getting an accurate $\log(g)$; without the photometric prior,
our $\log(g)$ would not have been accurate enough for us to see the effects of
atomic diffusion. A better understanding of atomic diffusion will also allow
us to measure the precision of abundances in open clusters, as we will be able
to measure the scatter of abundances compared to the theoretical abundance
trend lines.

##  6 Conclusions

In this paper, we examined the effects of using a joint $T_{\rm{eff}}$ and
$\log(g)$ photometric prior on spectroscopic $T_{\rm{eff}}$, $\log(g)$,
$v_{\rm{mic}}$, $v_{\rm{broad}}$, and 26 elemental abundances of stars in open
clusters. We used photometry and astrometry from the Gaia DR3 catalogue. We
selected members in the core of the open cluster, which were selected by
simple cuts in position and proper motion space. To generate more accurate and
precise photometric priors, we first reduced the uncertainty of the absolute
magnitude of the stars by reducing the uncertainty in their distance
measurement. We achieved this by assuming that the stars are normally
distributed around the cluster centre. We computed the joint probability
distribution of the location of the centre of each cluster and the size of
each cluster. Using this prior, we recalculated the distances of the stars,
achieving a reduction in uncertainties by a factor of 31% compared to
geometric distances in Gaia DR3. The reduction of the uncertainties is
important as it allows us to fit isochrones more precisely to the HR diagrams
of the open clusters and consequently it allows us to calculate more precise
$T_{\rm{eff}}$ and $\log(g)$ from each isochrone.

To create a joint $T_{\rm{eff}}$ and $\log(g)$ photometric prior, we first
fitted isochrones to HR diagrams of the open clusters to see which ranges of
ages, metallicities, and extinctions would encompass the majority of stars in
the regions sensitive to the change in the isochrone parameters in the HR
diagram. By using the absolute magnitude of individual stars, we calculated
$T_{\rm{eff}}$ and $\log(g)$ and their precision for that star from an
isochrone. By calculating the ranges of ages, metallicities, and extinctions
for each open cluster, we were able to sample the joint $T_{\rm{eff}}$ and
$\log(g)$ probability spaces. There was a complex relationship between
$T_{\rm{eff}}$ and $\log(g)$ that differed depending on where the star was
located on the HR diagram. We concluded that the shape of the $T_{\rm{eff}}$
and $\log(g)$ joint probability distribution was primarily due to the size of
the ranges of ages, metallicities, and extinctions rather than the
contribution of the precision of $T_{\rm{eff}}$ and $\log(g)$ from individual
isochrones (i.e., equation 10).

We used a single neural network generated using Payne for each spectral band
and trained the neural network on spectra that are representative of the
population of stars in the open clusters. We fitted GALAH spectra of 1979
stars across nine open clusters with and without using the joint photometric
prior. We found that using photometric prior led to a more accurate and
precise $\log(g)$. This allowed us to see higher-order trends like the
signature of atomic diffusion in the abundances analysed using the photometric
prior. The atomic diffusion was visible in [Fe/H], [Ni/Fe], [Cu/Fe], [O/Fe],
[V/Fe], [Mg/Fe], [Ti/Fe], [Sc/Fe], [Al/Fe], [V/Fe], [Ba/Fe] and [Mn/Fe] in M67
and [Fe/H], [Al/Fe], [Mg/Fe], [Ni/Fe], [Cu/Fe], and [Mn/Fe] in Ruprecht 147.
However, in [Fe/H], [Al/Fe], and [Mg/Fe], the dip was only present in the
abundances that were fitted using the photometric prior.

We validated the uncertainty generated by MCMC using a variety of methods. We
compared the precision to the spread of abundances in M67 in a small range of
$\log(g)$. We also compared our precision to the precision calculated using
repeated measurements. We found that the estimate of the precision of
abundances is well estimated by MCMC, but tends to inflate the precision
values of $T_{\rm{eff}}$, $\log(g)$, $v_{\rm{mic}}$, and $v_{\rm{broad}}$. We
investigated the precision of GALAH values calculated for DR4 by comparing it
to our measures of precision. GALAH’s method of picking the largest precision
between the precision calculated using repeat observations and the precision
calculated from its fitting procedure provides a reliable value for precision.

## Acknowledgements

This work made use of the Fourth Data Release of the GALAH survey (Buder et
al. 2023, in preparation). The GALAH survey is based on data acquired through
the Australian Astronomical Observatory, under programs: A/2013B/13 (GALAH
pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (GALAH survey, Phase 1);
A2018A/18 (Open clusters with HERMES); A2019A/1 (Hierarchical star formation
in Ori OB1); A2019A/15 (GALAH survey, Phase 2); A/2015B/19, A/2016A/22,
A/2016B/10, A/2017B/16, A/2018B/15 (HERMESTESS programme); and A/2015A/3,
A/2015B/1, A/2015B/19, A/2016A/22, A/2016B/12, A/2017A/14 (HERMES K2-follow-up
programme). We acknowledge the traditional owners of the land on which the
Anglo-Australian Telescope stands, the Gamilaraay people, and pay our respects
to elders past and present. This paper includes data that have been provided
by AAO Data Central (datacentral.org.au). This work has made use of the VALD
database, operated at Uppsala University, the Institute of Astronomy RAS in
Moscow, and the University of Vienna. KLB, JK, and GT are grateful for the
financial support of the Slovenian Research Agency (research core funding No.
P10188). SLM acknowledges funding support from the UNSW Scientia program and
from the Australian Research Council through Discovery Project grant
220102254. 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.

## Data Availability

Data from this paper can be accessed upon reasonable request through private
communication. Please contact Kevin Beeson at<EMAIL_ADDRESS>for data
inquiries.

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