11institutetext: Department of Astronomy, Stockholm University, AlbaNova
University Centre, Roslagstullsbacken, SE-106 91 Stockholm, Sweden
11email<EMAIL_ADDRESS>22institutetext: Theoretical Astrophysics,
Department of Physics and Astronomy, Uppsala University, Box 516, 751 20,
Uppsala, Sweden

#  Titanium abundances in late-type stars

II. Grid of departure coefficients and application to a sample of $70\,000$
stars

J. W. E. Mallinson Titanium abundances in late-type starsTitanium abundances
in late-type stars K. Lind Titanium abundances in late-type starsTitanium
abundances in late-type stars A. M. Amarsi Titanium abundances in late-type
starsTitanium abundances in late-type stars K. Youakim Titanium abundances in
late-type starsTitanium abundances in late-type stars

###### Abstract

Context. Rapidly growing datasets from stellar spectroscopic surveys are
providing unprecedented opportunities to analyse the chemical evolution
history of our Galaxy. However, spectral analysis requires accurate modelling
of synthetic stellar spectra for late-type stars, for which the assumption of
local thermodynamic equilibrium (LTE) has been shown to be insufficient in
many cases. Errors associated with LTE can be particularly large for Ti I,
which is susceptible to over-ionisation, particularly in metal-poor stars.

Aims. The aims of this work are to study and quantify the 1D non-LTE effects
on titanium abundances across the Hertzsprung-Russell diagram for a large
sample of stars.

Methods. A large grid of departure coefficients, $\beta_{\nu}$, were computed
on standard MARCS model atmospheres. The grid extends from $3000\,\mathrm{K}$
to $8000\,\mathrm{K}$ in $T_{\mathrm{eff}}$, $-0.5\,\mathrm{dex}$ to
$+5.5\,\mathrm{dex}$ in $\log{g}$, and $-5.0$ to $+1.0$ in $\mathrm{[Fe/H]}$,
with non-LTE effects in this grid reaching up to 0.4 dex. This was used to
compute abundance corrections that were subsequently applied to the LTE
abundances of over 70 000 stars selected from the GALAH survey in addition to
a smaller sample of literature Keck data for metal-poor dwarfs.

Results. The non-LTE effects grow towards lower $\mathrm{[Fe/H]}$, lower
$\log{g}$, and higher $T_{\mathrm{eff}}$, with a minimum and maximum
$\Delta\,\mathrm{A(Ti)_{Ti\,I}}$ of $0.02$ and $0.19$ in the GALAH sample. For
metal-poor giants, the non-LTE modelling reduces the average ionisation
imbalance ($\Delta_{\textsc{I}-\textsc{II}}\,$) from $-0.11\,\mathrm{dex}$ to
$-0.01\,\mathrm{dex}$ at [Fe/H] = $-1.7$, and the enhancement in titanium
abundances from Ti I lines results in a $\mathrm{[Ti/Fe]}$ versus
$\mathrm{[Fe/H]}$ trend that more closely resembles the behaviour of Ti II at
low metallicities. At higher metallicities, the results are limited by the
precision of the GALAH DR3 LTE abundances and the effects are within the
errors. For the most metal-poor dwarfs from the Keck sample, the average
ionisation imbalance increases from $-0.1$ dex to $+0.2$ dex, a shortcoming
that is consistent with previous 1D non-LTE studies and which we speculate
could be related to 3D effects.

Conclusions. Non-LTE effects on titanium abundances are significant.
Neglecting them may alter our understanding of Galactic chemical evolution. We
have made our grid of departure coefficients publicly available, with the
caveat that the Ti abundances of metal-poor dwarfs need further study in 3D
non-LTE.

###### Key Words.:

Atomic processes - Line: formation - Radiative transfer - Stars: abundances -
Stars: atmospheres - Galaxy: abundances

##  1 Introduction

Figure 1: Kiel diagrams showing the values of $T_{\mathrm{eff}}$ and $\log{g}$
at which the non-LTE calculations were performed with PySME, as well as the
selected GALAH stars for different metallicity ranges. Table 1: Line list of
GALAH lines.

$\lambda_{\mathrm{air}}$ (Å) | $\mathrm{A\left(Ti\right)}_{\text{\sun,\, LTE}}$ | $\mathrm{A\left(Ti\right)}_{\text{\sun,\, NLTE}}$ | log(gf) | $E_{\mathrm{low}}/\mathrm{eV}$ | VdW | Clean
---|---|---|---|---|---|---
Ti I |  |  |  |  |  |
$4758.12$ | $4.70\pm 0.05$ | $4.76\pm 0.05$ | $0.51$ | $2.249$ | 326.246 | Y
$4759.27$ | $4.72\pm 0.01$ | $4.78\pm 0.01$ | $0.59$ | $2.256$ | 327.246 | Y
$5689.46$ | $4.83\pm 0.08$ | $4.88\pm 0.08$ | $-0.36$ | $2.297$ | 794.242 | Y
$5716.44$ | $4.90\pm 0.10$ | $4.95\pm 0.10$ | $-0.72$ | 2.297 | $790.243$ | Y
$5739.47$ | $4.82\pm 0.09$ | $4.87\pm 0.09$ | $-0.61$ | $2.249$ | 327.246 | Y
Ti II |  |  |  |  |  |
$4719.51$ | $5.12\pm 0.22$ | $5.12\pm 0.22$ | $-3.32$ | $2.297$ | $-7.850$ | —
$4798.53$ | $4.85\pm 0.05$ | $4.85\pm 0.05$ | $-2.66$ | $1.080$ | $-7.860$ | U
$4874.01$ | $4.95\pm 0.06$ | $4.95\pm 0.06$ | $-0.86$ | $3.095$ | $-7.830$ | U
111Shown are the absorption wavelength in air, the zero-point solar abundance
of the line in LTE and non-LTE, the log(gf), the excitation energy, parameters
for broadening due to elastic hydrogen collisions, and the quality
classification from Heiter et al. (2020), where ‘Y’ is clean, ‘U’ is
uncertain, ‘N’ is not clean (blended), and ‘-’ was not tested. Oscillator
strengths are from Lawler et al. (2013) and Wood et al. (2013) for Ti I and Ti
II, respectively. Positive broadening parameters are from ABO theory (Barklem
et al. 2000); the value before the decimal point represents the collisional
cross-section $\sigma$ at $10\,\mathrm{km\,s^{-1}}$, and the value after the
decimal point the velocity parameter $\alpha$. Negative values are a logarithm
of full width half maximum per unit perturber number density at $10\,000$K.
Solar abundances for each line are from Buder et al. (2021) in LTE, and with
the non-LTE effect added for the Sun from Fig. 6. Isotopic splitting was
accounted for when data were available in VALD (Ryabchikova et al. 2015),
(line 5739.47).

Table 2: Line list for the Keck I/HIRES sample of Sneden et al. (2023), with
the same layout as Table 1.

$\lambda_{\mathrm{air}}$ (Å) | $\mathrm{A\left(Ti\right)}_{\text{\sun,\, LTE}}$ | $\mathrm{A\left(Ti\right)}_{\text{\sun,\, NLTE}}$ | log(gf) | $E_{\text{low}}/\mathrm{eV}$ | VdW | Clean
---|---|---|---|---|---|---
Ti I |  |  |  |  |  |
$3741.06$ | $4.87$ | $4.90$ | $-0.15$ | $0.021$ | 249.247 | —
$4533.24$ | $4.87$ | $4.90$ | $0.54$ | $0.848$ | 347.242 | —
$5192.97$ | $4.87$ | $4.90$ | $-0.95$ | $0.021$ | 255.243 | —
$5210.38$ | $4.87$ | $4.90$ | $-0.82$ | $0.048$ | 257.243 | —
Ti II |  |  |  |  |  |
$4443.80$ | $4.92$ | $4.92$ | $-0.71$ | $1.080$ | $-7.850$ | —
$4468.49$ | $4.92$ | $4.92$ | $-0.63$ | $1.131$ | $-7.850$ | —
$4501.27$ | $4.92$ | $4.92$ | $-0.77$ | $1.116$ | $-7.850$ | —
$4571.97$ | $4.92$ | $4.92$ | $-0.31$ | $1.572$ | $-7.820$ | —
222Lines limited to those with $\lambda>3700$Å. Solar values adopted from
Mallinson et al. (2022).

Data from increasingly expansive stellar spectroscopic surveys have
facilitated the advancement of our understanding of galactic chemical
evolution (GCE) on a large scale (Jofré et al. 2019). The imminent influx of
new data from surveys such as WEAVE (Dalton et al. 2018; Jin et al. 2023) and
4MOST (de Jong et al. 2019), which will gather of the order of $\sim 10^{7}$
high-resolution spectra, will set the stage for unprecedented investigations
of the stellar content in our Galaxy.

However, the impact and reliability of such studies are contingent on the
accuracy of the applied spectral analysis methods and the underlying modelling
of the spectral absorption lines. Local thermodynamic equilibrium (LTE) is a
common assumption in the modelling of spectral lines in late-type stars,
including our Sun. In the regime of LTE, collisional transitions dominate over
radiative ones, and as such the populations of excited and ionised states can
be described by the Saha and Boltzmann distributions (Mihalas & Athay 1973;
Hubeny & Mihalas 2015). This simplification offers the advantage of fast
computation times, which is especially important when analysing large datasets
with many spectra. However, this convenience comes at the cost of ignoring
potentially important physics.

In the case of titanium, previous 1D LTE work has demonstrated that abundances
depend heavily on the diagnostic species: neutral or singly ionised. These so-
called ionisation imbalances can be particularly severe in metal-poor stars
(Gratton & Sneden 1991). For example, 1D LTE analyses of the metal-poor giant
HD122563 have an ionisation imbalance of
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.4$ to $-0.45\,\mathrm{dex}$ (Sitnova
et al. 2020; Mallinson et al. 2022).

Non-LTE analyses have been shown to affect and often reduce this ionisation
imbalance in many cases (Bergemann 2011; Sitnova et al. 2016, 2020; Mallinson
et al. 2022), with the exception of metal-poor dwarfs. Departures from LTE can
be taken into account by solving for the statistical equilibrium (e.g. Rutten
2003). This allows external radiation to impact the local medium, which is
vital for the accurate modelling of Ti I; it is the minority species in late-
type stars, and with an ionisation energy of $6.8\,\mathrm{eV,}$ it has a
number of continua in the blue, making it sensitive to over-ionisation. These
effects are enhanced in metal-poor stars due to the lack of free electrons
from metals such that there are fewer electron collisions to bring Ti I to
LTE, and because there is less UV opacity and consequently a larger over-
ionising flux. These effects on Ti I do not strongly impact the Ti II lines,
however, given that that Ti II is typically the majority species in late-type
stars; Ti II also has a higher ionisation energy of $13.6\,\mathrm{eV,}$ which
prohibits it from experiencing analogous over-ionisation effects.

Inelastic hydrogen collisions are typically a source of large uncertainty in
non-LTE calculations, especially for metal-poor stars (Barklem 2016). Past
studies, such as Bergemann (2011) and Sitnova et al. (2016), have carried out
non-LTE calculations for titanium using the Drawin recipe for hydrogen
collisions (Steenbock & Holweger 1984; Lambert 1993), while more recent
studies (e.g. Sitnova et al. 2020; Mallinson et al. 2022) have improved upon
this by employing the asymptotic models of Belyaev et al. (2017) and Belyaev &
Voronov (2018). Mallinson et al. (2022) also implemented a more extensive
model atom and more recent data from Grumer & Barklem (2020) based on the
asymptotic model of Barklem (2016), combined with the recipe from Kaulakys
(1991) as suggested in Amarsi et al. (2018, 2019).

The calculations in Sitnova et al. (2020) and Mallinson et al. (2022) show
improvements in the ionisation balance in metal-poor giants, in the latter
case reducing the ionisation imbalance of HD122563 from
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.45\,\mathrm{dex}$ in LTE to
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.16\,\mathrm{dex}$ in non-LTE. However,
for the standard dwarf and subgiant stars considered in Mallinson et al.
(2022), departures from LTE appeared to make the ionisation imbalance more
severe. Nevertheless, that small-scale study of a few standard stars was not
sufficient to conclusively assess the impact of non-LTE modelling across a
broad range of parameter space, especially given that 1D hydrostatic model
atmospheres were employed in combination with non-spectroscopic stellar
parameters.

In this work we aim to more thoroughly study and quantify the 1D non-LTE
effects on titanium abundances via the analysis of a large dataset. Following
Amarsi et al. (2020), we constructed a grid of departure coefficients333Our
grid of departure coefficients is publicly available at
https://zenodo.org/record/10753497. (Sect. 2) and used it to correct line-by-
line LTE titanium abundances for $70\,347$ stars (Sect. 3.1) selected from the
GALAH survey (Martell et al. 2017; Buder et al. 2021) and 13 metal-poor stars
observed with Keck (Sneden et al. 2023). We used the results to assess the
impact of our non-LTE modelling on titanium ionisation imbalances as well as
on our understanding of its Galactic evolution (Sect. 3). We present our
overall conclusions in Sect. 4.

##  2 Method

###  2.1 Model atom

The model atom used in this work is based on that constructed in Mallinson et
al. (2022), and the full details are presented therein. The model contains
$459$ Ti I levels, $127$ Ti II levels, and the ground state of Ti III ($587$
levels in total). All fine structure was collapsed. All Ti I and Ti II lines
were radiatively coupled to the target ground states of the next ionisation
stage. In this work the model contains $5\,026$ bound-bound radiative
transitions, and the wavelength resolution of these lines was reduced such
that the atom contained $63$ thousand frequency points overall, compared to
the$157$ thousand in Mallinson et al. (2022); this had no noticeable impact on
final abundances. Inelastic collisions with free electrons were based on van
Regemorter (1962) and Cox (2000), and inelastic collisions with neutral
hydrogen were based on Grumer & Barklem (2020) and Kaulakys (1991) for Ti I,
and on Sitnova et al. (2020) for Ti II.

###  2.2 Grid of departure coefficients with Balder

The departure coefficient, $\beta_{i}$, for a level $i$ is the ratio of
populations between LTE and non-LTE:

$\beta_{i}=\frac{n_{i}}{n_{i}^{*}}.$ (1)

These departure coefficients for titanium were calculated in the same way as
discussed in Amarsi et al. (2022) for iron, using the MPI-parallelised non-LTE
radiative transfer code Balder, which is a modified version of Multi3D
(Leenaarts & Carlsson 2009). The equation of state and background bound-bound,
bound-free, and free-free opacities were calculated with the Blue package
(Amarsi et al. 2016). Details of the included species are provided in Zhou et
al. (2023). In particular, Thomson scattering from electrons, and Rayleigh
scattering on the red wing of Ly-$\alpha$, were included, while the opacity
from background lines was treated in pure absorption.

As described in Amarsi et al. (2020) and Amarsi et al. (2022), the
calculations were carried out for $3\,756$ MARCS model atmospheres (Gustafsson
et al. 2008) spanning $3\,000\,\mathrm{K}$ to $8\ 000\,\mathrm{K}$ in
$T_{\mathrm{eff}}$, $-0.5\,\mathrm{dex}$ to $5.5\,\mathrm{dex}$ in $\log{g}$,
and $-5.0\,\mathrm{dex}$ to $+1.0\,\mathrm{dex}$ in $\mathrm{[Fe/H]}$. Plane-
parallel models with microturbulence
$\xi_{\mathrm{t}}=1.0\,\mathrm{km\,s^{-1}}$ were used for $\log{g}\geq 4.0$,
and spherical models with solar mass and
$\xi_{\mathrm{t}}=2.0\,\mathrm{km\,s^{-1}}$ were used otherwise for
consistency with Amarsi et al. (2021). For each of these grid nodes, the
calculations with Balder were carried out for different choices of titanium
abundance, with $\mathrm{[Ti/Fe]}\in[-2.0,-1.0,-0.5,0.0,+0.5,+1.0,+2.0]$. The
microturbulence used in Balder was fixed to that used for the construction of
the model atmosphere, namely $\xi_{\mathrm{t}}=1.0\,\mathrm{km\,s^{-1}}$ for
$\log{g}\geq 4.0$, and $\xi_{\mathrm{t}}=2.0\,\mathrm{km\,s^{-1}}$ otherwise.

It should be noted that the model atom adopts level configurations and
spectroscopic terms based on those from Kurucz (2016). Following Amarsi et al.
(2020), the departure coefficients were mapped from these levels onto those
given in NIST (Ralchenko & Kramida 2020).

###  2.3 Abundance corrections with PySME

Figure 2: Non-LTE abundance correction for
$\mathrm{A\left(Ti\right)}_{\mathrm{{Ti\textsc{I}}}}$ and
$\mathrm{A\left(Ti\right)}_{\mathrm{{Ti\textsc{II}}}}$, averaged over lines,
for the GALAH sample across the Kiel diagram. Figure 3: Non-LTE abundance
correction for $\mathrm{A\left(Ti\right)}_{\mathrm{{Ti\textsc{I}}}}$ and
$\mathrm{A\left(Ti\right)}_{\mathrm{{Ti\textsc{II}}}}$, averaged over lines,
as a function of $T_{\mathrm{eff}}$ for the GALAH sample. The trends for
dwarfs and giants are shown as solid and dashed lines, respectively, using
bins of $\mathrm{\Delta T_{\mathrm{eff}}=\pm 75K}$ and limited to bins
containing at least 30 stars. The colour gradient represents the number of
stars per pixel. Figure 4: As in Fig. 3, but with a bin size of
$\Delta\log{g}=\pm 0.1$. Figure 5: As in Fig. 3, but with a bin size of
$\Delta\mathrm{[Fe/H]}=\pm 0.1$.

Figure 6: Trend lines showing the non-LTE effects for unsaturated lines
($W_{\mathrm{red},\lambda}=\mathrm{log(W/\lambda)}<-4.9$) of Ti I across five
benchmark stars: the Sun (5773, 4.44, 0.00, 1.0), Arcturus (4286, 1.64,
$-0.53$, 1.3), HD84937 (6356, 4.06, $-2.06$, 1.2), HD140283 (5792, 3.65,
$-2.36$, 1.3), and HD122563 (4636, 1.40, $-2.50$, 1.8) with parameters of
$T_{\mathrm{eff}},\ \log{g},\ \mathrm{[Fe/H]},$ and $\xi_{\mathrm{t}}$,
respectively. Figure 7: Ionisation imbalance,
$\Delta_{\textsc{I}-\textsc{II}}\,$, of titanium at various metallicities in
LTE (left) and non-LTE (right) for giants (upper) and dwarfs (lower) from the
GALAH sample. The colour gradient and trend lines are the same as in Fig. 5.
Figure 8: Ionisation imbalance, $\Delta_{\textsc{I}-\textsc{II}}\,$, of
titanium in metal-poor dwarfs and subgiants from the Keck sample (Sneden et
al. 2023) in both LTE and non-LTE for the lines shown in Table 2. Trend lines
are the same as in Fig. 5. Figure 9: Run of $\mathrm{[Ti/Fe]}$ versus
$\mathrm{[Fe/H]}$ in LTE (left) and non-LTE (right) from Ti I (upper) and Ti
II (lower) for the GALAH sample. Trend lines are the same as in Fig. 5. Figure
10: Trend of $\mathrm{[Ti/Fe]}$ versus $\mathrm{[Fe/H]}$ in LTE and non-LTE
from Ti I and Ti II. Based on the binning from Fig. 9 for the GALAH sample,
averaged over dwarfs and giants. Figure 11: Star-by-star scatter for dwarfs
(left) and giants (right) for different lines of Ti I (upper) and Ti II
(lower) from the GALAH sample. The colour gradient and the trend lines are the
same as in Fig. 5 Figure 12: Trend lines showing the non-LTE effects for
unsaturated lines ($W_{\mathrm{red},\lambda}=\mathrm{log(W/\lambda)}<-4.9$) of
Ti I across a range of stellar parameters. All models shown have
$\xi_{\mathrm{t}}$ = 2 $\mathrm{km\,s^{-1}}$ and at least two lines from Table
1.

The departure coefficients from Sect. 2.2 were constructed in such a way as to
be compatible with the spectrum synthesis codes SME and PySME (Wehrhahn et al.
2023). For each Ti line, PySME searches for its upper and lower level in the
grid of departure coefficients, and uses them to correct the LTE level
populations before synthesising the line with the correct non-LTE line opacity
and line source function (Piskunov & Valenti 2017).

As in Lind et al. (2022), PySME was used to generate abundance corrections
using the departure coefficients for more values of $\mathrm{[Ti/Fe]}$ and
$\xi_{\mathrm{t}}$. Namely, synthetic spectra were calculated for
$\mathrm{[Ti/Fe]}$ between $-2.0\,\mathrm{dex}$ and $+2.0\,\mathrm{dex}$ in
steps of $0.25\,\mathrm{dex}$, and $\xi_{\mathrm{t}}=1\,\mathrm{km\,s^{-1}}$
and $2\,\mathrm{km\,s^{-1}}$. These were performed on a subset of the MARCS
model atmospheres; Fig. 1 shows these grid nodes, as well as the selected
GALAH stars (described in Sect. 2.4). For a given titanium line, equivalent
widths were calculated in LTE and non-LTE and linearly interpolated to
generate a grid of abundance corrections shown in Fig. 12. Finally, these
abundance corrections were linearly interpolated onto the line-by-line LTE
GALAH abundances, as functions of $T_{\mathrm{eff}}$, $\log{g}$,
$\mathrm{[Fe/H]}$, $\xi_{\mathrm{t}}$, and abundance
A$(\mathrm{Ti})_{\text{line, \,LTE}}$. The star selection is described in
Sect. 2.4 and the line selection is described in Sect. 2.5, below.

###  2.4 GALAH DR3 star selection

This work builds upon the few benchmark stars analysed in Mallinson et al.
(2022) and examines a subsample of $70\,347$ stars, all of which have some
measurement of titanium abundance from either Ti I or Ti II (or both), out of
the $588\,571$ stars in GALAH DR3 (Buder et al. 2021). These data contain
$T_{\mathrm{eff}}$, $\log{g}$, and $\mathrm{[Fe/H]}$ (both the initial
atmospheric and final estimate of abundance, with the latter being used in
this work), $\xi_{\mathrm{t}}$, and line-by-line LTE titanium abundances,
$\mathrm{A\left(Ti\right)}_{\text{line,\,LTE}}$. The way in which these
parameters were determined are described in Buder et al. (2021). To give a
short summary, these parameters were determined by normalising the spectra
based on some initial guess, and optimising the stellar parameters. Forming an
initial $\log{g}$ by the mass-luminosity relation, $T_{\mathrm{eff}}$,
$\mathrm{[Fe/H]}$, $v_{\mathrm{broad}}$, and $v_{\mathrm{rad}}$ are optimised
with double-sided partial derivatives and $\chi^{2}$, updating $\log{g}$ and
$v_{\mathrm{mic}}$ with each change, repeating this until the change in
$\chi^{2}$ was below 0.001 for each iteration. [Fe/H] was finally redetermined
in non-LTE via analysing Fe lines directly (and this value of [Fe/H] is
adopted for the present study).

The selected stars are shown in Fig. 1. The selection was made by limiting to
those stars with the most reliable parameters and highest signal-to-noise
ratio. This was done in several steps. First, only the stars with quality
flags of 0 for $\mathrm{[Fe/H]}$ and $\mathrm{[Ti/Fe]}$ (‘flag_fe_h’,
‘flag_ti_fe’, and ‘flag_ti2_fe’) and the stellar parameter flag ‘flag_sp’ were
selected. Secondly, as recommended in the best practices for using the data
from the GALAH consortium444https://www.galah-survey.org, the average signal-
to-noise ratio per pixel for the third charged couple device was limited to
$30$ or above ($\text{snr\\_c3\\_iraf}>30$) for very metal-poor stars
($\mathrm{[Fe/H]}<-1.5$), in order to keep enough low metallicity stars in the
sample for a robust analysis, and average signal-to-noise ratio over all lines
of $\mathrm{S/N}>70$ for the stars at higher metallicities. Finally, only
stars with at least two lines with a reduced equivalent width
$\mathrm{W_{red,\lambda}}=\mathrm{log(W}_{\lambda}/\lambda)$ in LTE between
$-6.0<\mathrm{W_{red,\,\lambda}}<-4.9$ were kept.

###  2.5 Line selection

As the GALAH reanalysis of absolute abundances, $\mathrm{A\left(Ti\right)}{}$,
was done on a line-by-line basis, the line-by-line scatter can be used as
another diagnostic of the non-LTE modelling. The line selection strongly
influences the final results, and care was taken to try to select the best
available lines out of those considered by GALAH.

Ti I lines were initially limited to those that are reported as unblended in
Heiter et al. (2020) in the Sun and Arcturus, although the 6599.11Å line was
also removed due to a lack of titanium abundance data in the available stars.

In the case of Ti II, no lines were reported as unblended, so lines were
instead restricted to those that were reported as uncertain. Of the remaining
available lines, 4865.6$\AA$ can be strongly affected by H$\beta$, causing
blending and continuum normalisation, primarily in dwarfs. However, to keep
the line list identical for all stars, it was removed entirely. To increase
the line list to more than two lines, the least visibly blended line in the
Sun and Arcturus that was not examined in Heiter et al. (2020), at
4719.51$\AA$, was added.

The relative abundances $\mathrm{[Ti/H]}$ and $\mathrm{[Ti/Fe]}$ were
determined on a line-by-line basis using the GALAH abundances, with non-LTE
solar corrections applied to their values when in non-LTE. Average values of
$\mathrm{[Ti/H]}$ were calculated for different species,
$\mathrm{[Ti/H]}_{\mathrm{{Ti\textsc{I}}}}$ and
$\mathrm{[Ti/H]}_{\mathrm{{Ti\textsc{II}}}}$, via a mean of the available
lines for a given star weighted by their error. Weak lines and strong lines
were cut using the reduced equivalent width criterion of
$-6.0<\mathrm{W_{red,\,\lambda}}<-4.9$. The upper limit prevents the use of
saturated lines, which would have both given an under-abundance and increase
the sensitivity to microturbulence error. The lower limit reduces the impact
of noise fluctuation. The un-normalised weights were given by
$w_{\text{line}}=1/\sigma_{\text{line}}^{2}$, using the uncertainties provided
by GALAH. Finally, where illustrated in Sect. 3.1 below, the absolute
abundances use the non-LTE line-averaged solar correction:
$0.052\,\mathrm{dex}$ for Ti I and $-0.002\,\mathrm{dex}$ for Ti II.

##  3 Results and discussion

###  3.1 Non-LTE effects across the Hertzsprung-Russell diagram

Figures 2–5 show the line-averaged non-LTE corrections of Ti I and Ti II as a
function of stellar parameters. The non-LTE effects on both species grow in
severity towards lower metallicities, lower surface gravities, and higher
effective temperatures, qualitatively similar to that found for iron in Lind
et al. (2012).

For Ti I, the abundance corrections rise to $+0.18\,\mathrm{dex}$ on average
for giants at $\mathrm{[Fe/H]}=-2.0$ (Fig. 5). The analysis of benchmark stars
in Mallinson et al. (2022) suggests that the dwarfs should show a similar
trend with $\mathrm{[Fe/H]}$ as the giants, namely increasing non-LTE
abundance corrections with decreasing metallicity. However, the dwarfs in this
sample only extend down to $\mathrm{[Fe/H]}=-1.0$, reflecting the low number
of very metal-poor dwarfs in the dataset of Buder et al. (2021). At
$\mathrm{[Fe/H]}=-1.0$, the abundance corrections reach $+0.08\,\mathrm{dex}$,
compared to $+0.13\,\mathrm{dex}$ for giants at this same metallicity.

The abundance corrections for Ti II are generally much less severe than for Ti
I. This is because of the majority status of Ti II: due to the larger number
of ions, the effect of over-ionisation by UV radiation is smaller overall,
which implies that the assumption of LTE for both dwarfs and giants is a more
reasonable one for this species. The corrections are typically $-0.02$ dex for
giants but rising at low metallicities, and consistent with zero for the
dwarfs in the GALAH sample. However, at the lowest metallicities around
$\mathrm{[Fe/H]}=-2.0$ the abundance corrections reverse sign and tend to
become slightly positive (Fig. 5), consistent with Mallinson et al. (2022). It
reflects how over-excitation due to radiative pumping becomes increasingly
important for this species towards lower metallicities (Sitnova et al. 2020).

Additionally, the impact of titanium abundance itself on the non-LTE
corrections for Ti I was investigated in Fig. 6 for benchmark stars from
Mallinson et al. (2022). Between $-1.00$ ¡ [Ti/Fe] ¡ $0.75$ the correction
changes by only 0.05 dex for the Sun, and 0.09 dex for HD122563. Due to the
large difference in abundance, this is not a significant change and within the
error of this paper.

###  3.2 Ionisation imbalances

####  3.2.1 GALAH stars

The difference in the predictions based on Ti I and Ti II is a long-standing
problem in studies of titanium abundances, with Ti I predicting lower
abundances than Ti II. Neglected non-LTE effects are a contributing factor to
this problem (Bergemann 2011). For both dwarfs and giants, the abundance
corrections become larger towards lower metallicities for Ti I, while
remaining rather close to zero for Ti II.

Consequently, the ionisation imbalance is reduced in non-LTE in giants from
the GALAH sample at low metallicities, as shown in Fig. 7. For giants in LTE,
the ionisation imbalance reaches on average
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.11,\mathrm{dex}$ at
$\mathrm{[Fe/H]}\approx-1.7$. In non-LTE, this is reduced to
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.01$, demonstrating a non-LTE impact on
the imbalance of $0.10$ dex.

For dwarfs in the GALAH sample, departures from LTE do not significantly alter
the ionisation imbalance on average, with an imbalance of
$\Delta_{\textsc{I}-\textsc{II}}\,$$=-0.000$ for LTE and
$\Delta_{\textsc{I}-\textsc{II}}\,$$=0.001$ in non-LTE, at
$\mathrm{[Fe/H]}\approx-0.9$. It should be noted that this results in a solar
abundance correction of $0.05\,\mathrm{dex}$ for Ti I and
$-0.002\,\mathrm{dex}$ for Ti II (see Table 1 and Sect. 2.5); the square
bracket notation thus hides an ionisation imbalance of around
$0.05\,\mathrm{dex}$.

####  3.2.2 Keck stars

Due to the lack of low metallicity dwarfs available in the GALAH data sample,
an additional set of stars were taken from Sneden et al. (2023), consisting of
13 dwarfs of the original 37 due to line selection and saturation; the
majority of which have a metallicity between
$-3.0<\mathrm{\mathrm{[Fe/H]}}<-2.0,$ as shown in Fig. 8. The spectra of these
stars were obtained with Keck I/HIRES spectrograph (Vogt et al. 1994), and
originally used in Boesgaard et al. (2011). The lines available for these
stars do not overlap with those in the GALAH data, and the majority instead
are below $4000\AA$. The inclusion of these lines dramatically reduces the
abundance of both species of titanium.

In HD84937, Lawler et al. (2013) found a strong reduction in Ti I abundances
of around $0.15-0.2$ dex when moving to the near-UV range of
$\lambda=3100-3700\AA$. Wood et al. (2013) finds similarly for the high
excitation energy lines. The overabundance of UV lines in the Ti II line list
may result in the heavily positive LTE ionisation imbalance
$\Delta_{\textsc{I}-\textsc{II}}\,$seen in Sneden et al. (2023) that is in
disagreement with the expectations from previous work that focused on the
longer optical wavelength lines (Bergemann 2011; Sitnova et al. 2020;
Mallinson et al. 2022). The cause of this may be related to the Balmer
continuum opacity (Wood et al. 2013). In Sneden et al. (2016), the removal of
these lines raises the Ti II-based abundance and results in a smaller
imbalance, similar to that found for HD84937 in Mallinson et al. (2022).

By including only unsaturated and reasonably strong lines
($-6.0<\mathrm{W_{red,\,\lambda}}<-4.8$) above this limit (i.e. $>3700\AA,$ as
shown in Table 2), only 13 of the original 34 stars in Sneden et al. (2023)
can be analysed. However, this gives the ionisation imbalance for dwarfs at
these metallicities a closer agreement with Mallinson et al. (2022) and
remains close to 0 in LTE, worsening in non-LTE and prompting 3D analysis to
further investigate.

####  3.2.3 3D impact

The inclusion of 3D effects is a potential solution to the ionisation
imbalance of Ti. To estimate its impact in LTE, we employed four snapshots of
two 3D STAGGER radiation-hydrodynamic models (Magic et al. 2013). The models
pertain to metal-poor dwarfs with $T_{\mathrm{eff}}$ = 6000K, $\log{g}$ = 4.5,
[Fe/H] = $-3.0$ and [Fe/H] = $-2.0$, respectively. [Ti/Fe] was set to 0.4, and
we adopted $\xi=1$ km s-1 in the corresponding 1D calculations. The spectral
lines listed in Table 2 were computed both in 1D and 3D under LTE conditions
using the SCATE (Hayek et al. 2011) code. The mean 3D effects (i.e. 3D minus
1D abundance differences) were found to be $-0.41$ dex and 0.01 dex for Ti I
and Ti II, respectively, in the [Fe/H] = $-3$ model, and $-0.36$ dex and
$-0.05$ dex, respectively, in the [Fe/H] = $-2$ model.

For giants characterised by $T_{\mathrm{eff}}$ = 4000K, $\log{g}$ = 2.0, and
[Fe/H] = $-2.0$, a considerably lesser impact of $-0.04$ and +0.06 is found
for Ti I and Ti II, respectively. These findings align with prior research
(Dobrovolskas et al. 2013) that reports a large abundance change in 3D for
resonant lines only at around $-0.45$ dex at [Fe/H] = $-3$, but that steeply
decreases to -0.05 dex in abundance change as the excitation energy increases
to 2eV.

These results underscore the significant influence of 3D effects on Ti I in
the inverse direction of non-LTE, offering a potential resolution to the dwarf
metallicity discrepancy at lower metallicities. However, the interaction
between 3D and non-LTE is not necessarily additive (Lagae et al. 2023) and a
full 3D non-LTE investigation would be needed to make strong conclusions.

###  3.3 Star-by-star scatter

The abundance results in Fig. 9 also suggest a slight reduction in the star-
by-star scatter in the $\mathrm{[Ti/Fe]}$ versus $\mathrm{[Fe/H]}$ plane when
departures from LTE are taken into account. This is also illustrated in Fig.
11, where we plot the standard deviation in the $\mathrm{[Ti/Fe]}$ versus
$\mathrm{[Fe/H]}$ plane in different $\mathrm{[Fe/H]}$ bins for given titanium
lines. There is indeed a slight reduction in the star-by-star scatter,
reaching over $0.01\,\mathrm{dex}$ for the Ti I $4759.27\AA$ line in low-
metallicity giants. As a result, the offset between dwarfs and giants is
slightly reduced when comparing the upper left and upper right panels of Fig.
9.

It is plausible that the scatter of these results is dominated by issues with
the LTE GALAH abundances. As such, the non-LTE model presented here may lead
to a further reduction in the star-by-star and line-by-line scatter when
applied to more precise measurements, but no strong conclusion can be made
from such small effects.

###  3.4 Departures from LTE for different stellar parameters

The impact that stellar parameters have on the magnitude of the non-LTE effect
is shown in Fig. 12, computed using the MARCS atmospheres at specific grid
points with (PySME), and plotted as the average non-LTE impact on A(Ti),
averaged over the chosen GALAH Ti I lines. As is found in previous work, a
decrease in metallicity increases the magnitude of $\Delta$
A(Ti)${}_{\mathrm{Ti\,I}}$ in most cases, particularly between
$-3<\mathrm{[Fe/H]}<0$. Similarly, a decreasing $\log{g}$ results in a larger
non-LTE effect as collisional rates — scaling with the number density of both
titanium atoms and the ionising impact species — reduce faster than radiative
rates, which scales only with the number density of titanium. As seen in Fig.
12, this effect also increases with $T_{\mathrm{eff}}$ until around
$T_{\mathrm{eff}}\approx 6000$K, as the increased number of photons outweigh
the increased number and speed of colliders. Beyond this, as with most
titanium lines in Sitnova et al. (2016), the increasing temperature begins to
reduce the non-LTE effect up to our limit of 7000K. The exact temperature at
which this decline occurs varies with $\log{g}$ and $\mathrm{[Fe/H]}$. Of the
available grid points, the most extreme non-LTE effects range from -0.004 (at
$T_{\mathrm{eff}}=4000$K, $\log{g}=2.5$, [Fe/H] $=0.5$) to 0.442 dex(at
$T_{\mathrm{eff}}=5000K$, $\log{g}=1.0$, [Fe/H] $=-4.0$).

###  3.5 The Galactic evolution of titanium

Precision measurements of chemical abundances are crucial to accurately
interpreting and constraining theories of GCE. While titanium is formally
considered an iron peak element, it often behaves as an alpha element in
observations (Zhao et al. 2016; Kobayashi et al. 2020) Thus, it is useful in
disentangling Galactic components and tracing back their evolution histories.
For example, [Ti/Fe] can be used as a chemical tracer to separate the chemical
thin and thick discs, stellar halo and accreted stellar populations (e.g.
Bensby et al. 2005, 2014; Mashonkina et al. 2019). In addition, the Galactic
Anticenter Stellar Structure, or ‘Monoceros Ring’, has been shown to exhibit
abundance patterns in [Ti/Fe] and [Y/Fe] that are similar to those of
Sagittarius stars and distinct from Milky Way stars, allowing for constraints
to be placed on the origin of this structure (Chou et al. 2010).

Interestingly, some recent GCE models show that [Ti/Fe] is under-predicted by
$\sim 0.4$ dex over a wide metallicity range when compared to observational
data (Kobayashi et al. 2006; Romano et al. 2010; Kobayashi et al. 2020), even
in non-LTE (Mashonkina et al. 2019). It has been shown that [Ti/Fe] levels can
be increased by including yields from asymmetric 2D models of high-entropy,
jet-like ejecta from hypernovae (Maeda & Nomoto 2003; Tominaga 2009) in these
GCE models (see model K15, as presented in Sneden et al. 2016; Kobayashi et
al. 2020), but high precision measurements of [Ti/Fe] are needed to constrain
the parameters of these ejecting supernovae. Notably, the non-LTE corrections
presented in this work have shown that average giant [Ti/Fe] abundances
measured with Ti I become larger by $\sim 0.1$ dex as compared to 1D-LTE
analyses, thereby exacerbating the discrepancy between models and
observations, particularly at low metallicities of [Fe/H] ¡ $-0.7$.

In a study designed to investigate the impact of modelling assumptions on GCE
yields, Côté et al. (2017) emphasise the significant influence of the choice
of stellar yields on the trends in chemical evolution. With different stellar
models within their set of stellar yields, they substantially reduce the
titanium underprediction of Côté et al. (2016) and Kobayashi et al. (2020) and
successfully replicate an approximately +0.25 enhancement in [Ti/Fe] at low
metallicities, within the framework of a one-zone model, and consistent with
observations in the Sculptor dwarf galaxy. However, they point out that these
abundance trends can be realised through various combinations of input
parameters, making it challenging to distinguish among several degenerate
models, a caveat that was also described in previous work (Romano &
Starkenburg 2013).

Clearly, the abundance trends from these models are highly dependent on the
choice of input parameters and the adopted stellar yields. This implies a
fundamental limitation in one-zone GCE models, which cannot be overcome by
simply increasing the precision of observed abundance data.

Therefore, while the non-LTE corrections reduce the discrepancy between
abundances measured using Ti I and Ti II for giants at low metallicities, they
are not sufficient to offer further insight needed to constrain GCE models.
They nevertheless emphasise the need for improved precision in theoretical
yields for these models.

##  4 Conclusion

This study is the first to investigate the impact of 1D non-LTE effects on
titanium abundances across a large sample of stars, building upon previous
work that had primarily examined a small number of benchmark stars. It was
expected that larger non-LTE effects would be found for Ti I, in metal-poor
stars, and in giants: all three of these predictions are confirmed in this
investigation. This resulted in a reduced ionisation imbalance for the metal-
poor giants, a slight reduction in the disagreement between dwarf and giant
titanium abundances at lower metallicities, and a shift in the [Ti/Fe] trend
of Ti I to more closely match that of Ti II at low metallicities. This takes
observations further away from predictions by some GCE models of titanium
(Kobayashi et al. 2020, and references therein). Overall, this suggests that
the inclusion of the departure coefficient grid and correcting for non-LTE
effects results in more accurate Ti I abundances for giants at lower
metallicities, although this is dependent on the line selection.

Due to the small sample of metal-poor dwarfs and the relatively low accuracy
of the GALAH sample, the limitations of this analysis are apparent. In
Mallinson et al. (2022), metal-poor dwarfs were seen to have their imbalance
worsened by the inclusion of non-LTE, a conclusion that is reinforced by our
analysis of the Sneden et al. (2023) sample in this work. An investigation of
3D non-LTE effects for Ti in metal-poor stars may shed further light on the
problem. Additionally, the low number of lines and their quality —
particularly for Ti II — further limit the reliability of these results.

Finally, there were larger departures from LTE towards higher effective
temperatures and lower surface gravities, as well as lower metallicities.
However, at lower surface gravities of $\log{g}<3.5$, increasing the
temperature beyond $T_{\mathrm{eff}}\approx 5500$K reduced the departures from
LTE. Likewise, a lower metallicity can occasionally result in lower non-LTE
effects in all lines analysed, primarily at low temperatures and high surface
gravity. In summary, we find the non-LTE corrections reach up to $+0.41$ dex
at $T_{\mathrm{eff}}=5000$K, $\log{g}=1.0$, [Fe/H] $=-4.0$.

###### Acknowledgements.

AMA acknowledges support from the Swedish Research Council (VR 2020-03940).
This research was supported by computational resources provided by the
Australian Government through the National Computational Infrastructure (NCI)
under the National Computational Merit Allocation Scheme and the ANU Merit
Allocation Scheme (project y89). 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. We thank the PDC Center for High Performance
Computing, KTH Royal Institute of Technology, Sweden, for providing access to
computational resources and support. JWEM, KL, and KY acknowledge funds from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 852977) and funds from
Knut and Alice Wallenberg Foundation. We acknowledge Chris Sneden for the
sharing of his data and advice in its analysis.

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##  Appendix A Additional non-LTE effects and microturbulence sensitivity

This appendix contains additional figures where Figure 13 represents the non-
LTE effects for lines as in Fig. 12 but for Ti II, and the impact titanium
abundance has on non-LTE effects in benchmark stars. Figures 14-15 shows the
impact reduced equivalent width has on microturbulence sensitivity in GALAH
stars with their microturbulences replaced with 1 and 2kms-1. Lines above
$-4.9$ were removed due to the saturation effect and microturbulence
sensitivity, shown in orange.

Figure 13: Trend lines showing the non-LTE effects for unsaturated lines
($-6.0<W_{\mathrm{red},\lambda}=\mathrm{log(W/\lambda)}<-4.9$) of Ti II across
a range of stellar parameters. All models shown have $\xi_{\mathrm{t}}$ = 2
$\mathrm{km\,s^{-1}}$ and at least two lines from Table 1.  Figure 14:
Difference in abundance for GALAH stars with microturbulences replaced by
$\xi=1$ and $\xi=2$ kms-1 for Ti I. Figure 15: Difference in abundance for
GALAH stars with microturbulences replaced by $\xi=1$ and $\xi=2$ kms-1 for Ti
II.