#  Spectroscopic analysis of hot, massive stars in large spectroscopic surveys
with de-idealised models

J. M. Bestenlehner1,2, T. Enßlin3, M. Bergemann2, P. A. Crowther1, M.
Greiner3, and M. Selig4
1Department of Physics & Astronomy, Hounsfield Road, University of Sheffield,
S3 7RH, UK
2Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg , Germany
3Max Planck Institute for Astrophysics, Karl-Schwarzschildstraße 1, 85748
Garching, Germany
4DBFZ Deutsches Biomasseforschungszentrum gemeinnützige GmbH, Torgauer Straße
116, 04347 Leipzig, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Upcoming large-scale spectroscopic surveys with e.g. WEAVE and 4MOST will
provide thousands of spectra of massive stars, which need to be analysed in an
efficient and homogeneous way. Usually, studies of massive stars are limited
to samples of a few hundred objects which pushes current spectroscopic
analysis tools to their limits because visual inspection is necessary to
verify the spectroscopic fit. Often uncertainties are only estimated rather
than derived and prior information cannot be incorporated without a Bayesian
approach. In addition, uncertainties of stellar atmospheres and radiative
transfer codes are not considered as a result of simplified, inaccurate or
incomplete/missing physics or, in short, idealised physical models.

Here, we address the question of “How to compare an idealised model of complex
objects to real data?” with an empirical Bayesian approach and maximum a
posterior approximations. We focus on application to large scale optical
spectroscopic studies of complex astrophysical objects like stars. More
specifically, we test and verify our methodology on samples of OB stars in 30
Doradus region of the Large Magellanic Clouds using a grid of FASTWIND model
atmospheres.

Our spectroscopic model de-idealisation analysis pipeline takes advantage of
the statistics that large samples provide by determining the model error to
account for the idealised stellar atmosphere models, which are included into
the error budget. The pipeline performs well over a wide parameter space and
derives robust stellar parameters with representative uncertainties.

###### keywords:

methods: data analysis; methods: statistical; techniques: spectroscopic;
atomic data; stars: massive; stars: fundamental parameters

††pubyear: 2023††pagerange:  Spectroscopic analysis of hot, massive stars in
large spectroscopic surveys with de-idealised models –A

##  1 Introduction

With the advent of large spectroscopic surveys using instruments such as WEAVE
(Jin et al., 2022) and 4MOST (de Jong et al., 2019) tens of thousands spectra
of massive stars ($\gtrsim 10M_{\odot}$) will be obtained, which will need to
be analysed in a homogeneous and efficient way (e.g. Cioni et al., 2019;
Bensby et al., 2019; Chiappini et al., 2019). Current pipelines of large
spectroscopic surveys are largely designed for FGK stars, which are either
data driven (e.g. Ness et al., 2015; Guiglion et al., 2020) or model driven
(e.g. Allende-Prieto & Apogee Team, 2015; Ting et al., 2019). Traditionally,
and still widely performed today, massive stars have been analysed by “eye”,
which limits the sample size to $<100$ massive stars. In addition, stellar
parameters as well as uncertainties are estimated rather than determined.
Larger sample of a couple of hundreds of stars are usually analysed with a
$\chi^{2}$-minimisation algorithm, where the final fit often needs to be
visually verified depending on the goodness-of-fit.

Multi-dimensional probability distribution functions are obtained depending on
the number of free parameters and uncertainties are then defined on confidence
intervals rather than Gaussian standard deviations. Those uncertainties can be
highly asymmetric and very large in the case of degenerated parameters. In the
massive star community there are 2 main flavours of $\chi^{2}$-minimisation
algorithms, grid based (e.g. Simón-Díaz et al., 2011; Castro et al., 2012;
Bestenlehner et al., 2014) and Genetic Algorithms on the basis of natural
selection (e.g. Mokiem et al., 2007; Brands et al., 2022). All those
algorithms use a pre-defined selection of spectral line regions for their
analysis.

Theoretical models of complex physical systems are necessarily idealisations.
The implied simplifications allow us to focus the view on the essential
physics, to keep the model computationally feasible, and to investigate
systems for which not all of their components are perfectly known. In contrast
to solar-like stars, massive stars have strong stellar winds, which influence
the structure of the stellar atmospheres (line blanketing) and a pseudo
photosphere at optical depth 2/3, which is located in the stellar winds. The
inclusion of line-driven winds into stellar atmosphere models requires the
assumption of spherical geometry in the co-moving frame of the star (expanding
atmosphere). In addition to effective temperature and surface gravity, mass-
loss rate, velocity law, terminal velocity, wind inhomogeneity and line
blanketing need to be included into the stellar atmosphere code. The stellar
atmosphere of massive stars significantly depart from local thermal
equilibrium (LTE) and therefore must be computed in fully non-LTE, which is
computational expensive (e.g Santolaya-Rey et al., 1997; Hillier & Miller,
1998; Gräfener et al., 2002). This limits state-of-the-art stellar atmosphere
codes for hot, massive stars to 1D.

When faced to real data of the actual system these models often perform
insufficiently when judged on a goodness-of-fit basis. The difference between
real and modelled data can easily exceed the error budget of the measurement.
The reason is that the idealized models do not capture all aspects of the real
systems, but they are still present on the data. To discriminate these missing
aspects from measurement errors or noise, we use the term model errors to
describe these imperfections of our theoretical description (Oberpriller &
Enßlin, 2018).

Often one tries to determine the model parameters from the data via a
likelihood based methodology such as $\chi^{2}$-minimisation, maximum
likelihood or Bayesian parameter estimation that uses the measurement
uncertainties as a metric in data space. The resulting parameter estimates can
be strongly distorted by the desire of the method to minimize all apparent
differences between predicted and real data, indifferently if these are due to
measurement or model errors.

Thus the model errors should be included into the error budget of the
parameter estimation. This would require that we have a model for the not yet
captured aspects of the system or at least for the model errors these produce.
To do this thoroughly we would need to undertake a case by case analysis of
the missing physics.

However, this would be quite impractical in cases of the spectroscopic
analysis of complex astrophysical objects. Instead, we want to construct a
plausible, but by no means perfect, effective description of the model errors.
In the construction of the de-idealisation model, we will follow a pragmatic
route, but try to indicate the assumptions, approximations and simplifications
made.

In Section 2 we introduce the methodology, which is used in our spectroscopic
analysis pipeline (Section 3). Using grids of stellar atmospheres (Section
3.3) the pipeline is applied to observational data (Section 3.4). The results
are discussed in Section 4. We close with a brief conclusion and outlook
(Section 5).

##  2 Method

###  2.1 Data model

We assume we have a set of objects (e.g. stars, galaxies, …: labelled by
$i\in\\{1,\,2,\,\ldots\mathtt{n}\\}$) with observable signals
$s^{(i)}=(s_{x}^{(i)})_{x}$ over some coordinate $x$, e.g. the emitted
spectral energy distribution $s^{(i)}=(s_{\lambda}^{(i)})_{\lambda}$ as a
function of the wavelength $\lambda$. These signals are measured with a
linearly responding instrument (response matrix $R$) with additive Gaussian
noise ($n$) according to the measurement equation

$d^{(i)}=R^{(i)}s^{(i)}+n^{(i)}.$ (1)

The individual elements of the data vector ($d$) for the $i$-th object are
then given by

$d_{j}^{(i)}=\int dx\,R_{j\,x}^{(i)}s_{x}^{(i)}+n^{(i)},$ (2)

where in our spectroscopic cases $R_{j\,x}^{(i)}$ is the $j$-th bandpass of
our $i$-th observation as a function of wavelength $x=\lambda$. Spectroscopic,
colour filter, and bolometric measurements can thereby be treated with the
same formalism and even combined into a single data vector and response
matrix. In addition, we do not require that all objects are observed in the
same way by keeping the response matrix dependent on the object index $i$. In
this way, the formalism permits to combine heterogeneous observations.

For the measurement noise $n^{(i)}$ of the $i$-th observation we use the
error-spectrum from the data reduction which is assumed for simplicity to be
Gaussian with zero mean and signal independent,

$\displaystyle\mathcal{P}(n^{(i)}|s^{(i)})$ $\displaystyle=$
$\displaystyle\mathcal{G}(n^{(i)},N^{(i)})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{|2\pi
N^{(i)}|}}\exp\left[-\frac{1}{2}n^{(i)\dagger}\left(N^{(i)}\right)^{-1}n^{(i)}\right]$

with assumed noise covariance $N^{(i)}=\langle
n^{(i)}n^{(i)\dagger}\rangle_{(n^{(i)}|s^{(i)})}$. The dagger denotes the
transposed and complex conjugated vector. The noise of the different
observations is assumed to be independent as well,
$\mathcal{P}(n|s)=\mathcal{G}(n,N)=\prod_{i}\mathcal{G}(n^{(i)},N^{(i)})$ with
$n=(n^{(i)})_{i}$ and $s=(s^{(i)})_{i}$ being the combined noise and signal
vectors.

###  2.2 Model errors

Now we assume that some idealised model for our objects exists that predicts a
specific theoretical signal $t^{[p]}$ given a set of unknown model parameters
$p$. These parameters should be physical quantities like surface gravity,
radius, effective temperature, stellar wind properties or chemical composition
of the object, so that well defined values $p^{(i)}$ exist for each
object111Counter example would be purely phenomenological parameters,
describing aspects of the data that contain observation dependent properties,
or such that only make sense within a specific object description methodology.
Although the here proposed approach might be applicable to such
phenomenological descriptions as well, we currently demand the physical
existence of the used parameters in order to be on epistemologically firm
grounds.. Those idealised models can be generated with stellar atmosphere and
radiative transfer codes. Knowing these parameters for each object is the
primary goal of the inference. In principle, the relation between parameters
and signal could be stochastic, but for simplicity we concentrate on
deterministic models. The de-idealisation model we develop should serve as an
effective description for the remaining stochasticity.

The idealized model captures hopefully the dominant properties of the system
but certainly not all aspects. Therefore the real signal $s^{(i)}$ of an
object will deviate by an unknown stochastic component $u^{(i)}$, the model
error, so that

$s^{(i)}=t^{[p_{i}]}+u^{(i)}.$ (4)

The aim of model de-idealization is to find an appropriate stochastic model
$\mathcal{P}(u|p)=\prod_{i}\mathcal{P}(u^{(i)}|p^{(i)})$ for the model errors.
With such, the likelihood becomes

$\mathcal{P}(d|p)=\int\mathcal{\\!D}u\,\mathcal{P}(d|s=t^{[p]}+u)\,\mathcal{P}(u|p),$
(5)

where $\int\\!\mathcal{D}u$ denotes a phase space integral for the model
errors.

In our case, we want to restrict ourselves to using the simplest possible
representation of the model uncertainties. This means that we take only the
first and second moments of $u^{(i)}$ into account, $v^{(i)}=\langle
u^{(i)}\rangle_{(u|p)}$ and
$U^{(i)}=\langle(u-v)^{(i)}(u-v)^{(i)\dagger}\rangle_{(u|p)}$, and assume the
fluctuations of different objects to be independent, $\langle
u^{(i)}u^{(j)\dagger}\rangle_{(u|p)}=v^{(i)}v^{(j)\dagger}+\delta_{ij}U^{(i)}$.
The probability distribution that represents mean and variance without further
information on higher order correlations naturally is a Gaussian with this
mean and variance. Among all possible probability distributions with given
mean and variance it has a maximal entropy (e.g. Jaynes & Bretthorst, 2003;
Caticha, 2008). By adopting a Gaussian for the model errors,

$\mathcal{P}(u|p)=\mathcal{G}(u-v,\,U),$ (6)

the least amount of spurious information is entered into the inference system
in case only $v$ and $U=\langle(u-v)\,(u-v)^{\dagger}\rangle_{(u|p)}$ are
considered.

This does not mean that the model error statistics is a Gaussian in reality.
It just means that higher order correlations are ignored for the time being.
Taking such higher order correlations into account would most certainly
improve the method, but is left for future work.

The Gaussianity of measurement noise and modelling error description permits
us to integrate Equation (5) analytically leading to

$\displaystyle\mathcal{P}(d|p,\,v,\,M)$ $\displaystyle=$
$\displaystyle\mathcal{G}(d-R(v+t^{[p]}),\,M),$ (7)

with $M=N+R\,U\,R^{\dagger}$ being the combined error covariance.

###  2.3 Implicit hyperprior

The de-idealisation model requires that the auxiliary parameters $v$ and $U$
are determined as well, or better marginalised over. This requires that we
specify our prior knowledge on these parameters, $\mathcal{P}(v,U|p)$, which
is a very problem specific task. Using such a hyperprior we could then derive
an auxiliary parameter marginalised estimator for our desired model parameters
$p$. A good, but numerically very expensive approach would be to sample over
the joint space of model and auxiliary parameters, $p$, $v$, and $U$, for
example using the Gibbs sampling method (e.g. Wandelt et al., 2004; Jasche et
al., 2010).

In order to have a generic, affordable and pragmatic method we introduce a
number of approximations and simplifications. The first is that we replace the
auxiliary parameter marginalisation by an estimation using the following
approximation

$\displaystyle\mathcal{P}(d,p)$ $\displaystyle=$
$\displaystyle\int\\!\mathcal{D}v\int\mathcal{D}U\,\mathcal{P}(d|p,\,v,\,U)\>\mathcal{P}(v,U|p)\,\mathcal{P}(p)$
(8) $\displaystyle\approx$
$\displaystyle\mathcal{P}(d|p,\,v^{\star},\,U^{\star})\,\mathcal{P}(p)$

with $v^{\star}$ and $U^{\star}$ being suitable estimates of the auxiliary
parameters and $\mathcal{P}(p)$ the parameter prior. Instead of constructing
these point estimators using the so far unspecified and problem specific
priors we propose to pragmatically specify them with an educated ad-hoc
construction.

The idea is to assume for a moment that a correct model parameter
classification $p^{(i)}$ for any object exists, which has later on to be
estimated self consistently with all the other estimates via iteration. The
difference of the signals reconstructed from the data $m^{(i)}=\langle
s^{(i)}\rangle_{(s^{(i)}|d^{(i)},\,p^{(i)})}$ and the one predicted from the
model $t^{[p^{(i)}]}$ plus the current guesses for $v^{\star}$,

$\text{$\delta$}^{(i)}=m^{(i)}-t^{[p^{(i)}]}-v^{(i)\star}$ (9)

can be analysed to provide information on $v$ and $U$. The signal
reconstruction can be done via a Wiener filter, since this is optimal in case
of a linear measurement and Gaussian noise model Equation (7). For the signal
difference this is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\delta^{(i)}$ $\displaystyle\\!\\!=$
$\displaystyle\\!\\!D^{(i)}R^{(i)\dagger}\left(N^{(i)}\right)^{-1}\\!\left[d^{(i)}-R^{(i)}\left(t^{[p^{(i)}]}+v^{(i)\star}\right)\right]\\!\\!,\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!D^{(i)}\\!\\!$ $\displaystyle=\\!\\!$
$\displaystyle\left[\left(U^{(i)\star}\right)^{-1}+R^{(i)\dagger}\left(N^{(i)}\right)^{-1}R^{(i)}\right]^{-1},$
(10)

where $D^{(i)}$ is the Wiener variance or uncertainty of the reconstruction.
For the current case we have used some guesses for $v^{\star}$ and $U^{\star}$
that need to be updated accordingly to the information contained in the
statistics of the signal differences $\delta^{(i)}$. To do so, we introduce a
suitable proximity measure in parameter space,
$\omega_{ii^{\prime}}=\mathrm{prox}(p^{(i)},p^{(i^{\prime})})$, that indicates
for an object $i$ how much another objects $i^{\prime}$ can be used to learn
about the model error statistics of $i$. A naive choice would be
$\omega_{ii^{\prime}}=1$ always, assuming that the model error statistics is
everywhere the same in the model parameter space. A more sophisticated method
would partition the parameter space into characteristic regions (e.g.
corresponding to the different known star and galaxy classes in our
spectroscopic example) and to set $\omega_{ii^{\prime}}=1$ or
$\omega_{ii^{\prime}}=0$ in case $i$ and $i^{\prime}$ belong to the same or
different classes. Even more sophisticated proximity weighting schemes can be
imagined with
$\omega_{ii^{\prime}}=1/(1+\mathrm{dist}(p^{(i)},p^{(i^{\prime})}))$ using
some distance measure in parameter space. However, in our case we group
objects together with respect to their main line diagnostics and analyse them
in the same batch.

Given such a scheme, the update operation for $v^{\star}$ and $U^{\star}$ are

$\displaystyle v^{(i)\star}$ $\displaystyle\rightarrow$ $\displaystyle
v^{(i)\star}+\sum_{i^{\prime}}\frac{\omega_{ii^{\prime}}}{\Omega_{i}}\delta^{(i^{\prime})},$
$\displaystyle U^{(i)\star}$ $\displaystyle=$
$\displaystyle\sum_{i^{\prime}}\frac{\omega_{ii^{\prime}}}{\Omega_{i}}\left(\text{$\delta$}^{(i^{\prime})}\text{$\delta$}^{(i^{\prime})\dagger}+D^{(i^{\prime})}\right)\mbox{
with}$ (11) $\displaystyle\Omega_{i}$ $\displaystyle=$
$\displaystyle\sum_{i^{\prime}}\omega_{ii^{\prime}}.$ (12)

The $v^{\star}$ update operation is a simple absorption of any systematic
difference into the mean component of the model error $v^{\star}$. For an
initial step it is better to set $v^{(i)\star}_{\mathrm{iteration\\#0}}=0$ as
it would absorb any differences between data and model, even though the model
might be not representative of the objects. However, we find that the the best
choice for $v$ is to represent non-stellar features, which are not part of the
model, like nebular lines, interstellar bands or telluric lines.

The $U^{\star}$ update incorporates the variance in the signal difference
reconstructions as well as their Wiener variances. The latter express the
level of missing fluctuations of the Wiener filter reconstruction. The
correction with the Wiener variance is done in analogy to the critical filter
methodology developed in (Enßlin & Frommert, 2011; Enßlin & Weig, 2010), where
a similar variance estimation was derived under a non-informative prior on the
power spectrum of a statistical homogeneous random process. For this study we
intuitively extended this to non-diagonal covariance structures. This means we
adopt implicitly a non-informative hyperprior for the model error statistics
as summarised by $v$ and $U$.

The logic behind this implicit prior is as follows. Assuming we would have
managed to specify an appropriate non-informative hyperprior for $v$ and $U$.
From this we would derive some recipe for our point estimates $v^{\star}$ and
$U^{\star}$ using some approximations. The resulting recipe should not
incorporate hidden spurious assumptions. The $v^{\star}$ and $U^{\star}$
estimates therefore can only be build from elements like
$\text{$\delta$}^{(i^{\prime})}$,
$\text{$\delta$}^{(i^{\prime})}\text{$\delta$}^{(i^{\prime})\dagger}$, and
$D^{(i^{\prime})}$, the latter being a summary of the former elements.
Requiring the estimators to be unbiased and to enclose the mentioned critical
filter as a limiting case, which is an unbiased power spectrum estimator, then
fixes the numerical coefficients in front of $\text{$\delta$}^{(i^{\prime})}$,
$\text{$\delta$}^{(i^{\prime})}\text{$\delta$}^{(i^{\prime})\dagger}$, and
$D^{(i^{\prime})}$ in Equation (11) to unity.

We admit that there are some frequentist elements in this derivation, since it
postulates an estimator and argues for its appropriateness using bias
arguments. We hope that it will be replaced by a more stringent calculation
once a suitable non-informative hyperprior has been specified. For the time
being, we use it in iteration with model parameter estimation.

###  2.4 Method summary

The combined de-idealized model parameter estimation method comprises the
following steps:

  1. 1.

Specify the weighting scheme
$\omega_{ii^{\prime}}=\mathrm{prox}(p^{(i)},p^{(i^{\prime})})$ that determines
how similar the model parameters of two objects have to be so that their model
error statistics can be assumed to be similar.

  2. 2.

Adopt some initial guess or default values for the model parameters $p^{(i)}$
and mode error parameters $v^{(i)\star}$ and $U^{(i)\star}$. A naive choice
could be $p^{(i)}=p$ for some plausible central $p$ within the physically
permitted parameter space, $v^{(i)\star}=0$, and
$U^{(i)\star}=\sum_{i^{\prime}}\left(R^{(i^{\prime})\dagger}d^{(i)}-t^{[p]}\right)\left(R^{(i^{\prime})\dagger}d^{(i)}-t^{[p]}\right)^{\dagger}$.

  3. 3.

Calculate $\delta^{(i)}$ and $D^{(i)}$ according to Equation (10) for all
objects.

  4. 4.

Update $v^{(i)\star}$ and $U^{(i)\star}$ according to Equation (11) for all
objects.

  5. 5.

Update the model parameters $p^{(i)}$ of all objects using the combined
likelihoods Equation (7) that incorporates measurement and model errors.

  6. 6.

Repeat steps 3-5 until convergence.

The resulting estimate will be similar to a joint maximum a posteriori
estimation of the model and model error parameters which are known to perform
worse than a properly marginalised with respect to the model error parameter
posterior mean estimator. However, given the large number of degrees of
freedom in the signal space (e.g. a highly resolved emission spectrum of a
star or galaxy), such optimal estimators can be extremely expensive
computationally. The proposed method might therefore be favourable in many
circumstances, despite its approximative and partly ad hoc nature.

In order to be explicit about the assumptions and approximations adopted we
provide an overview below. This list should help to judge the range of
applicability and to find possible improvements of the proposed method. In
particular we assume

  1. 1.

that the measurement noise is independent for the different objects and
independent of their signals, it has Gaussian statistics with known
covariance.

  2. 2.

a linear and known measurement response.

  3. 3.

that the model error knowledge can be approximated by a multivariate Gaussian
in signal space.

  4. 4.

that regions in parameter space exist and are known which have similar model
error statistics as parametrized by a mean and variance.

  5. 5.

no prior knowledge on the values of this model error mean and variances.

  6. 6.

that an iterated point estimate of all involved parameters leads to a
reasonable approximative solution of the joint inference problem.

##  3 Spectroscopic analysis pipeline

The pipeline has been developed using python3 with commonly used libraries
such as numpy, scipy, pandas, multiprocessing and ctypes plus astropy.io to
read fits files. Using commonly and maintained libraries will ensure that the
pipeline is easy to maintain and should be usable over a long period of time.
The following section outlines the required pre-processing steps (§ 3.1),
brief overview of the pipeline implementation (§ 3.2) and, description of the
grid of synthetic model spectra (§ 3.3) and the observational data to verify
the pipeline (§ 3.4).

###  3.1 Pre-processing

The pipeline requires that all observational data are read in at the beginning
as the spectra are required to be analysed all at once to determine
iteratively the stellar parameters and model uncertainties (§ 2.4). After a
spectrum is loaded it is corrected for the potential radial velocity shift,
transformed to the wavelength sampling of the synthetic spectra grid (§ 3.3)
and decomposed into principal components using the decomposition matrix
calculated from the synthetic spectra to reduce the memory usage and speed up
the analysis. This is essential, when analysing sample of spectra of more than
a few hundred sources.

The decomposed grid of synthetic spectra is loaded into shared-memory for
parallelisation purposes. The spectra are pre-convolved with combinations of
varying broadening parameters of projected rotational velocity ($\varv_{\rm
eq}\sin i$) and macro-turbulent velocity ($\varv_{\rm mac}$). The synthetic
grid preparation is also a pre-processing step, which is laid out in § 3.3. If
the sample is small and/or sufficient random access memory of the computing
system is available, the grid can be convolved with the star specific
broadening parameters. Even though the convolution is applied utilising the
fast Fourier transformation library of scipy, this could increase the pre-
processing time scale up to a few hours per star depending on the size of the
synthetic spectra grid ($\gg 100\,000$).

On a standard Desktop computer 1000 spectra can be analysed in less than half
a day. Larger set of samples we would advise to sort them into groups of
similar objects, which will also lead to more representative model error at a
specific parameter space when testing implemented physics or verifying
assumptions in the theoretical model.

###  3.2 Analysis pipeline

Figure 1: Probability heat map of surface gravity vs. effective temperature
for VFTS-072 (left) and VFTS-076 (right). Contours indicate 2D standard-
deviational ellipse confidence-intervals of 39.4%, 86.5% and 98.9% (e.g. Wang
et al., 2015). Spectroscopic fits of those stars are shown in Fig. 3.

After pre-processing the observational data including observational error
spectra and loading the synthetic grid of model spectra the pipeline is set up
according to Equation (1) with model de-idealisation Equation (4) assuming a
Gaussian noise model for the observational error (2.1). We are interested in
the posterior distribution of the signal given the data ($\mathcal{P}(s|d)$)
and apply the Bayes theorem

$\displaystyle\mathcal{P}(s|d)=\frac{\mathcal{P}(d|s)\mathcal{P}(s)}{\mathcal{P}(d)}$
(13)

with likelihood $\mathcal{P}(d|s)$, prior $\mathcal{P}(s)$ and evidence
$\mathcal{P}(d)$ to use, first, the likelihood $\mathcal{P}(d|s)$ and, second,
apply the Wiener filter (§ 2.3) to reconstruct the signal ($P(d|s)\to
P(s|d)$). We modified the likelihood as described in § 2.3
($\mathcal{P}(d|s)\Rightarrow\mathcal{P}(d|p)$), probability of the data $d$
given the stellar parameters $p$.

The best model is determined from a $\chi^{2}$-minimisation Ansatz with model
error variance matrix $U$ (Equation 7) to maximise the modified likelihood
$\mathcal{P}(d|p)$:

$\displaystyle\chi^{2}=\left(d^{(i)}-\mathrm{R^{(i)}}s^{(i)}\right)^{\mathrm{T}}\mathrm{N^{(i)}}^{-1}\left(d^{(i)}-\mathrm{R^{(i)}}s^{(i)}\right).$
(14)

For the mean and model error variance matrix $v$ and $U$, we find that it is
better to set $v=0$ and multiply $U$ by a factor
$\alpha=[10^{-5},0.35,0.7,1.0,1.0]$, which increases after each iteration, to
avoid that bad spectroscopic fits have significant impact on $v$ and $U$ and
therefore the determination of the best model. $v$ could be also set equal to
non-stellar features such as telluric bands, diffuse interstellar bands and
prominent interstellar lines like Ca H and K. However, interstellar
contribution can significantly vary with the line of side while telluric bands
change with atmospheric conditions and airmass. In cases, where the none
stellar features vary on a star by star basis, those contribution can be
combined with the observational error to give less weight to those spectral
regions.

$\omega_{ii^{\prime}}$ (Equation 11) can contain any prior information
$\mathcal{P}(p)$, e.g. parameter space of similar objects, stellar structure
models or population synthesis predictions. In the current implementation we
use a flat prior ($\omega_{ii^{\prime}}=1$).

The pipeline returns a multi-dimensional posterior distribution function for
each star while $U$ is the same for all sources analysed in one batch.
Parameters and their uncertainties are determine by defining confidence
intervals (Fig. 1). To increase the accuracy the posterior distribution
function can be multiplied by an appropriate prior. More details on the
implementation can be found in the source code of the pipeline.

###  3.3 Stellar atmosphere grid

The grid of synthetic model spectra was computed with the non-LTE stellar
atmosphere and radiative transfer code FASTWIND v10.6 (Santolaya-Rey et al.,
1997; Puls et al., 2005; Rivero González et al., 2012) including H, He, C, N,
O and Si as explicit elements. The FASTWIND LINES-list and FORMAL_INPUT file
is well tested and verified in the wavelength range from 4000 to 7000 Å. In
the FORMAL_INPUT file we included H , He i-ii, C ii-iv, N ii-v, O ii-v and Si
ii-iv in the wavelength range from $\lambda$3500 to 10000. On the basis of the
Vienna Atomic Line Database database222e.g.
http://vald.astro.uu.se/~vald/php/vald.php (VALD) version 3 and NIST
database333https://www.nist.gov/pml/atomic-spectra-database we added the
following lines to the FASTWIND LINES-list: C ii $\lambda$6784, C iii
$\lambda$7703 and $\lambda$9701-05-16, C iv $\lambda$4647 and
$\lambda$6592-93, N iii $\lambda$5321-27-52, $\lambda$3935-39 and
$\lambda$9402-24, N iv $\lambda$3748, $\lambda$5737, $\lambda$5776-85,
$\lambda$6212-15-29, $\lambda$7103-09-11-23-27-29, $\lambda$7425 and
$\lambda$9182-223, O iii $\lambda$3703, $\lambda$3707-15,
$\lambda$3755-57-60-74-91, O iv $\lambda$3560-63, $\lambda$3729-37,
$\lambda$7032-54, $\lambda$5769 and $\lambda$9454-88-92, O v $\lambda$5114 and
$\lambda$6500, and Si ii $\lambda$9413\. A full list of included spectral
lines can be found in the Appendix 1.

Some lines are located in the region of telluric bands, but could be of great
value, if a careful telluric correction has been performed. With data from
4MOST (de Jong et al., 2019) and in particular 4MOST/1001MC (Cioni et al.,
2019) we are going to verify the FASTWIND LINES-list beyond the well tested
wavelength range utilising the pipeline of this study.

Figure 2: $T_{\rm eff}-\log g$ plane of the computed grid of converged stellar
atmospheres. The high temperature low surface gravity regime (upper left area)
is empty as those models are unstable due to the Eddington limit ($\Gamma_{\rm
e}\approx 0.7$). Low temperature and high surface gravity models would be
better calculated with a plane-parallel code without stellar winds.

The grid covers the parameter space for the effective temperature from $T_{\rm
eff}=17\,800$ K ($\log T_{\rm eff}=4.25$) to 56 200 K ($\log T_{\rm
eff}=4.75$), surface gravity $\log g/(\mathrm{g~{}cm^{-2}})=2.5$ to 4.5,
transformed mass-loss rate (e.g. Bestenlehner et al., 2014) from
$\log\dot{M}_{\rm t}/(\mathrm{M_{\odot}~{}yr^{-1}})=-6.5$ to $-5.0$ assuming a
constant radius and helium abundances by number from $Y=0.07$ to 0.15. 3
combination of CNO abundances representing LMC baseline abundances plus semi
and fully-processed CNO composition due to the CNO-cycle according to 60
$M_{\odot}$ evolutionary track by Brott et al. (2011). Figure 2 shows the
parameter space of the grid with respect to $\log g$ and $T_{\rm eff}$.

The high temperature, low surface gravity regime (upper left area) is
unpopulated as those models are unstable as they exceed the Eddington limit at
an Eddington parameter $\Gamma_{\rm e}\approx 0.7$ considering only the
electron opacity $\chi_{\rm e}$. Low temperature and high surface gravity
models can be computed with FASTWIND, $T_{\rm eff}$ between 17 800 K ($\log
T_{\rm eff}=4.25$) and 21 400 K ($\log T_{\rm eff}=4.33$) and $\log g>4.0$,
but a significantly larger number of depth points or high mass-loss rates
would be required to make them converge. The computational timescale exceeds 1
day in contrast to less than an hour. However, enhanced mass-loss rates are
only observed, if the star is to the proximity of the Eddington limit
(Bestenlehner et al., 2014). Therefore, those low temperature and high surface
gravity stellar atmosphere models are better calculated with a plane-parallel
geometry code without stellar winds (e.g. TLUSTY Hubeny & Lanz, 1995; Lanz &
Hubeny, 2007).

The clumping factor was set to $f_{\textrm{cl}}=1$, i.e. a homogeneous stellar
wind is adopted. We assumed a wind acceleration parameter of $\beta=1.0$ and a
fixed micro turbulence velocity of $10$ km/s. The terminal velocity was
calculated based on $\log g$ and stellar radius of the model using the escape-
terminal velocity relation of $\varv_{\rm esc}/\varv_{\infty}=2.6$ for models
hotter than 21 000 K and $\varv_{\rm esc}/\varv_{\infty}=1.3$ for cooler
models (Lamers et al., 1995). In total we computed of the order $\lesssim
150\,000$ stellar atmospheres. For around $\sim 20\%$ of those models the
atmospheric structure, ionisation balance and/or radiation field failed to
converge properly leading to negative fluxes, discontinuities or even failed
when the spectral lines were synthesised.

The grid was then convolved with a macro-turbulent velocity of $\varv_{\rm
mac}=20$ km/s and varying projected rotational velocity $\varv\sin
i=[0,20,50,100,150,200,250,300,400]$ km/s assuming $\varv_{\rm eq}\sin i$ is
the dominant broadening mechanism, which is a reasonable assumption given the
spectral resolution of the observational data (§ 3.4) and that typical
$\varv_{\rm mac}$ are of the order of a few 10s km/s. This results in a grid
of $\lesssim 1\,100\,000$ synthetic spectra, which has been used to compute
the decomposition matrix to decomposed the grid into its principal components
reducing the size by a factor $\sim 200$. The decomposition matrix is also
used to decompose the observational data (§ 3.1).

###  3.4 Observational data

To validate the methodology of the pipeline we used the VLT/FLAMES data of
VLT/FLAMES Tarantula survey (VFTS Evans et al., 2011) in the traditional blue-
optical wavelength regime with the LR02 ($\lambda 3964-4567$,
$\lambda/\Delta\lambda=6000$), LR03 ($\lambda 4501-5078$,
$\lambda/\Delta\lambda=7500$) and HR15N ($\lambda 6470-6790$,
$\lambda/\Delta\lambda=19\,000$) gratings. We selected 240 O-type stars
employing the same data and normalisation as used in Bestenlehner et al.
(2014); Sabín-Sanjulián et al. (2014, 2017); Ramírez-Agudelo et al. (2017) to
avoid the introduction of biases.

The second data set we used are the VLT/MUSE observation of $\sim 250$ OB
stars from Castro et al. (2018). The VLT/MUSE data cover the wavelength range
between 4600 to 9300 Å at spectral resolution of 2000 to 4000. The
normalisation of the spectra was fully automated simulating the work-flow of
the spectroscopic analysis of large datasets. 35 stars are in common with
VFTS, which will allow us to test the reliability of line diagnostics towards
the red of $\mathrm{H}_{\alpha}$ (§ 3.3) and the automated normalisation
routine.

##  4 Results and discussion

###  4.1 VLT/FLAMES: Evans et al. (2011)

Figure 3: Left, spectroscopic fit of an fast rotating early O2 V-III(n)((f*))
star VFTS-072 and, right, a late O9.2 III star VFTS-076 (right). Blue solid
line is the observation, red solid line the synthetic spectrum and the grey
shaded area is the square-root of the diagonal elements of the model-error
uncertainty-matrix calculated by the pipeline.

We analysed all 240 O-type stars including stars with low SNR and/or strong
nebular contamination (e.g. Fig. 10) while Bestenlehner et al. (2014); Sabín-
Sanjulián et al. (2014, 2017); Ramírez-Agudelo et al. (2017) provided reliable
results for 173 out of 240 sources. In Fig. 3 we show the spectroscopic fits
of a representative early and late O star (VFTS-072 and 076). The shaded error
area in Fig. 3 reveal where a general mismatch between model and observations
occurs and is the square-root of the diagonal elements of the model-error
uncertainty-matrix. In particular the line centres of the Balmer and He i-ii
lines seemed to be poorly fitted as a results of nebular lines, inaccurate
determined line-broadening, insufficient grid resolution and range for helium
abundances, fixed micro-turbulent velocity or shape of line profiles. We are
also able to locate spectra lines, which are potentially not included into the
FASTWIND LINES-list or require improved atomic data (e.g. Si iii
$\lambda$4552.6, 4567.8 and 4574.7). Overall the spectroscopic fit is good for
the synthetic spectra (red solid line) to the observations (blue solid line).

Figure 1 visualises the probabilities in the $\log g-T_{\rm eff}$ plane.
VFTS-072 (left panel) shows within 2-sigma a dual solution ($\sim$49 000 and
$\sim$46 000 K). At $\sim 45\,000$ K the He i disappear, but the N iv and v
lines sufficiently contributed to the $\chi^{2}$ so that the correct solutions
around $49\,000$ K has also the highest probabilities. By looking at the
3-sigma contour we notice a degeneracy between $\log g-T_{\rm eff}$ due to the
proximity of VFTS-072 to the Eddington limit. In contrast, the heat map of
VFTS-076 (right panel) is well centred on a specific $\log g-T_{\rm eff}$
region. However, a slightly higher surface gravity could be probable within
2-sigma, which is the results of a degeneracy between surface gravity and
mass-loss rates. High mass-loss rates fill in the wings of the Balmer lines
mimicking a lower surface gravity.

####  4.1.1 Model-error uncertainty-matrix

Figure 4: Model uncertainties extracted from model-error matrix as a function
of wavelength for $\mathrm{H}_{\alpha}$ (solid red), He i $\lambda 4471$
(black dashed) and He ii $\lambda 4686$ (solid blue). For better visualisation
uncertainties for He i $\lambda 4471$ and He ii $\lambda 4686$ are multiplied
a factor 25 and 10, respectively.

The model-error uncertainty-matrix is symmetric ($U^{\rm T}=U$) and shows
correlation between wavelength or pixel regions. An example is given in the
Appendix (Table 2), where we reduced the rank of the matrix from
11840$\times$11840 to 37$\times$37 for visualisation purposes. The strongest
correlations are between the Balmer lines, which are the most prominent lines
in O-type stars. On the other hand He i is present in mid to late O stars,
while He ii lines are only strong in early O stars. Therefore, to visualise
the model-error matrix and its correlations we plot in Fig. 4 the model
uncertainties for wavelengths of $\mathrm{H}_{\alpha}$, He i $\lambda 4471$
and He ii $\lambda 4686$.

$\mathrm{H}_{\alpha}$ and He ii $\lambda 4686$ are anti-correlated with each
other, although we amplified the uncertainties for He ii $\lambda 4686$ by a
factor of 10. Wavelength regions of Balmer lines are a blend of hydrogen and
He ii lines. With increasing temperature the He ii lines are stronger while
the Balmer lines become weaker. In addition, the line strength between
hydrogen and helium also determines their abundances, e.g. overall stronger
helium lines with respect to hydrogen lines mean lower hydrogen abundances.

He i $\lambda 4471$ (amplified by a factor of 25) is correlated with He ii
$\lambda 4686$ for helium lines, but anti-correlated for $\mathrm{H}_{\delta}$
and higher order Balmer lines following the trend of $\mathrm{H}_{\alpha}$. He
i $\lambda 4471$ show stronger correlations with Si iii and C iii, which are
only present in late O stars, where He i lines are strongest as well. Under
this supposition we would expect that we observe a strong correlation between
He ii and the higher ionised N iv and v, which seems not to be the case. A
reason might be that the number of early O stars is too small due to the
stellar mass-function to significantly contribute to the model-error ($<5$%).
Grouping similar objects together is advisable when testing model assumptions
in stellar atmosphere codes.

####  4.1.2 Challenges

The examples shown in Fig. 3 show low and modest nebular contamination and the
pipeline derives results in good agreement with VFTS. However, the pipeline
has difficulties when spectra show strong nebular lines. In the case of
VFTS-142 (Fig. 10) the temperature is still reasonably well reproduced, but
the surface gravity is by $\sim$0.3 dex too low. If the spectra is dominated
by nebular lines, the pipelines will fail, e.g. VFTS 410 (Fig. 11). Often
nebular lines are clipped, but in the case of VFTS-410 only few diagnostic
lines would remain. Even though we perform a single star analysis, for double-
lined spectroscopic binaries (SB2s) the pipeline is able to fairly fit the
primary component, but struggles with the mass-loss rate and helium abundances
due to the contribution of the colliding wind region of VFTS-527 (Fig. 12,
Taylor et al., 2011).

Figure 5: Effective temperatures (left) and surface gravities (right)
determined by the pipeline vs. the results from Bestenlehner et al. (2014);
Sabín-Sanjulián et al. (2014, 2017); Ramírez-Agudelo et al. (2017).

The goodness-of-fit is usually evaluated by calculating the reduced-chi-square
(RCS), which uses in our case the diagonal of the error co-variance matrix.
Due to the nebular contamination and diffuse interstellar bands none of our
fits had a RCS close to 1. To visualise how well the pipeline performs we
compare our results versus tailored analysis of VFTS targets in Fig. 5. Our
results agree well with Bestenlehner et al. (2014); Sabín-Sanjulián et al.
(2014, 2017); Ramírez-Agudelo et al. (2017) for fits with RCS $<100$.
Effective temperatures show a tighter relation than the surface gravity. The
determination of surface gravity is based on the wings of the Balmer lines,
which is influenced by the line broadening and therefore how well $\varv_{\rm
mac}$ and $\varv\sin i$ are determined. If the spectroscopic fit is poor, we
derive systematically lower temperatures and surface gravities. Low
temperature and gravity models have $\mathrm{H}_{\alpha}$ in emission to
somehow fit the none-stellar $\mathrm{H}_{\alpha}$ nebular line while higher
order Balmer lines remain in absorption. Overall there is good agreement
considering that our analysis took less than 30 min while the VFTS analysis
involved 3 PhD theses.

Looking at the error bars the pipeline obtains systematically larger
uncertainties in part due to the inclusion of the model error but mostly as a
result of the interpretation of the 4D posterior distribution function (PDF),
which includes the degeneracies between $T_{\rm eff}$, $\log g$, $\dot{M}_{\rm
t}$ and $Y$. The derived errors might be larger, but they are a complete
representation of the true uncertainties. A representative prior could
increase the accuracy, but might introduce additional biases (Bestenlehner et
al., 2020). Temperature uncertainties are systematically larger in the region
around 45,000 K as a result of the weakness of He i lines and the ionisation
balance is not based on He i-ii, but on the metal ions N iii-iv-v. Nitrogen
lines in early O star are relatively weak compared to He lines and therefore
contributed little to the global $\chi^{2}$ without specific weighting of
spectral lines. This can lead to an overall lower RCS, but inaccurate
temperature determination (red outliers in Fig.5). A similar behaviours occurs
in the transition from late O to early B stars due to the weakness of He ii
lines, where the main temperature diagnostics are Si iii and iv lines. So a
careful weighting scheme should be a very promising way for optimising the
pipeline by increasing the accuracy while at the same time reducing the
degeneracies between parameters.

###  4.2 VLT/MUSE: Castro et al. (2018)

Figure 6: Spectroscopic fit of an O2 If (Mk42, left) and a B2 Ib (VFTS-417,
right). Blue solid line is the observation, red solid line the synthetic
spectrum and the grey shaded area is the square-root of the diagonal elements
of the covariant-matrix calculated by the pipeline. In the left panel the
newly added N iv mulitplet at $\lambda 7103-29$ is able to reproduce the
observed line.

Figure 6 shows the spectroscopic fit of an Of supergiant and a B supergiant
with $\Delta T_{\rm eff}\approx 25\,000$ K and $\Delta\log\dot{M}\gtrsim 2.5$
dex. This highlights that stars covering a large spectral type range can be
successfully and reliably analysed with a single pipeline set up at the same
time. However, both stars would not be considered as similar, which has
implication on the model error $U$. Such a model error is averaged over a wide
parameter space and is probably not very helpful, when testing specific
physics (e.g. stellar wind physics or atomic data) in the model. Similar to
the VFTS data the pipeline performs not well for low signal to noise spectra
(S/N $\lesssim 10$ to 15), spectra with strong nebular lines and spectroscopic
binaries/multiples.

Figure 7: Effective temperatures (left) and surface gravities (right)
determined by the pipeline vs. the results from Castro et al. (2021).

In Fig. 7 we compare our results with those from Castro et al. (2021) which is
based on the ionisation balance of selected HeI and HeII and the wing of
$H_{\mathrm{\beta}}$. In contrast, we used all H, He plus CNO and Si metal
lines available in the VLT/MUSE wavelength range. The left panel compares
effective temperatures, which shows a large scatter, but mostly agrees within
their large uncertainties. Above 45 000 K, when He i disappears or weakly
present in the spectra, the temperature needs to be derive based on the
ionisation balance of metal lines. In the wavelength range of MUSE we have C
iii-iv and N iii-iv. While C iv and N iv are located in relative clean areas
of the spectra, the C iii and N iii are often found in the range of telluric
bands or near the Paschen jump, where we have issues with the normalisation
(Fig. 6). B type stars have per definition no He ii present in their spectra
and the temperature is based in the case of early B stars on the ionisation
balance of Si iii-iv lines. There is a reasonable number of lines in the MUSE
range, but the temperature determination suffers with the presence of nebular
lines or low SNR spectra.

The right panel of Fig. 7 compares surface gravities which show an even larger
scatter and uncertainties. Even though we utilised the Paschen lines as well,
there are 2 potential caveats, first, the normalisation near the Paschen jump
is not straightforward as the lines overlap and therefore no continuum,
second, the degree of overlapping depends not only on $\log g$ but also on the
line broadening due to the narrowness of the higher order Paschen lines, which
is only approximately determined during the fitting process. This results in a
degeneracy between $\log g$ and $\varv_{\rm eq}\sin i$. Surface gravities
cluster in the range of $\log g=3.5$ to 4.5, which is expected for a young
stellar population of 1 to 2 Myr largely consisting of dwarfs and giants in
the proximity of R136 (Bestenlehner et al., 2020).

Figure 8: Comparison of MUSE targets in common with VFTS: effective
temperatures (left) and surface gravities (right) determined by the pipeline
using VLT/MUSE data vs. the results from Bestenlehner et al. (2014); Sabín-
Sanjulián et al. (2014); McEvoy et al. (2015); Sabín-Sanjulián et al. (2017);
Ramírez-Agudelo et al. (2017) using VLT/FLAMES data.

To better quantify how reliable the analysis based on the VLT/MUSE data is, we
compare our results to VFTS. 35 VLT/MUSE targets are in common with VFTS
(Bestenlehner et al., 2014; Sabín-Sanjulián et al., 2014, 2017; Ramírez-
Agudelo et al., 2017) and the comparison is shown in Fig. 8. Uncertainties are
systematically larger. Effective temperatures agree within their 1$\sigma$
uncertainties for 26 out 35 stars (left panel) with differences largely as a
result of the cleanliness and quality of spectra. Surface gravities are in
consent for only half of the sample, which is expected due to the challenges
of the Paschen lines. Overall the agreement is reasonable considering the
difficulties in the analysing of the MUSE data.

Figure 9: Hertzsprung-Russell diagram of the analysed stars using the VLT/MUSE
data from Castro et al. (2018). Thin black lines are stellar evolutionary
tracks by Brott et al. (2011) and Köhler et al. (2015).

To place the stars into the Hertzprung-Russell diagram (HRD) we derived
bolometric luminosity on the basis of optical $UBV$ from Selman et al. (1999)
and near-infrared $JK_{\rm s}$ photometry from 2MASS ((2 Micron All Sky
Survey)) and Vista Magellanic Clouds survey (VMC Cioni et al., 2011) using the
methodology of Bestenlehner et al. (2020); Bestenlehner et al. (2022). The HRD
(Fig. 9) shows that most stars are populated near, and to the cool side of the
zero-age main-sequence (ZAMS). There are a couple of exceptions but their
uncertainties do not exclude a cooler location in agreement with the majority
of the sources. This can be improved by including a meaningful prior into the
analysis, e.g. based on evolutionary tracks, and could increase the accuracy
of the results, as we used only a flat prior ($\omega_{ii^{\prime}}=1$). For
example, only hydrogen deficient stars are found to be on the hot side of the
ZAMS, e.g. self-stripping through stellar winds or binary evolution. A prior
would give the star a higher probability to be found either on the hot or cool
side of the ZAMS depending on its helium composition. Stellar parameters can
be found in Table LABEL:at:muse_sp while individual spectroscopic fits for
visual inspection are in the supplementary online material. Mass-loss rates,
and He, C, N, and O abundances are not included. Optical data can only provide
an upper limit for most stars in our sample (flat PDF towards lower mass-loss
rates) while the PDF for Helium is cut off at the primordial abundance of 25%
by mass. CNO abundances are too coarsely sampled and linked to the predicted
chemical evolution of 60 $M_{\odot}$ star (§ 3.3).

##  5 Conclusions and Outlook

Large spectroscopic surveys with WEAVE and 4MOST will observed 10 000s of
massive stars, which need to be analysed in a homogeneous and efficient way.
The pipeline presented in this work takes advantage and utilises the
information that large data sets provide by determining the model
uncertainties, which are included into the error budget. This methodology
could be also applied to galaxies or other domains like biology and geophysics
for which approximate and incomplete theoretical models exist as well.

The runtime of the pipeline scales exponentially with the number of spectra,
because all stars are analysed at the time and the error-model uncertainties-
matrix is iteratively updated (§ 2.4). However, once a converged error-model
uncertainties-matrix is obtained, we can limit the matrix operations to the
$\chi^{2}$-minimisation and switch to a star by star analyses. In this case we
are able to analyse 1 star in less than a second.

Xiang et al. (2022, The HotPayne) applied the FGK methodology of The Payne
(Ting et al., 2019) to OBA stars to derive 2 stellar labels/parameters
($T_{\rm eff}$ and $\log g$) plus 15 chemical abundances. They used the plane-
parallel LTE atmospheric model calculated with ATLAS12 (Kurucz, 1970, 1993,
2005) and were able to analyse 330 000 spectra. While $T_{\rm eff}$ and $\log
g$ are sensible for A and mid-late B dwarfs the derived chemical abundances
suffer from non-negligible systematics due to non-LTE effects. AB supergiants
require spherical geometry (stellar radius $\ngg$ scale height) including
stellar winds as these effects cannot be neglected. In hotter and more
luminous stars (early B and O stars) a non-LTE treatment is necessary
(mihalas1970) such that The HotPayne results for $T_{\rm eff}$ and $\log g$
are not reliable including stars with weak winds. Even the inclusion of a
model error will not improve much due to the fundamental missing physics in
the ATLAS12 models.

To make The HotPayne usable for OBA stars the underlying stellar models must
be replaced with models computed with more sophisticated and fully non-LTE
stellar atmosphere codes designed for hot, massive stars with stellar winds,
e.g. cmfgen (Hillier & Miller, 1998), fastwind (Santolaya-Rey et al., 1997;
Puls et al., 2005) or PoWR (Gräfener et al., 2012; Hamann & Gräfener, 2003).
The approach of the The HotPayne needs to be expanded: to include the model
uncertainties into the error budget (e.g. this work) to account for the
assumptions and parametrisations utilised in those complex stellar atmosphere
codes. Additional stellar labels/parameters need to be incorporated such as
mass-loss rate, velocity law, wind-inhomogeneity, terminal and wind-turbulent
velocity plus Helium abundances. The helium abundance increases due to the CNO
cycle, which in turn increases the mean molecular weight ($\mu$), impacts the
mass-luminosity relation ($L\propto\mu^{4}M^{3}$), electron density, and
therefore the structure and ionisation balance of the stellar atmosphere. When
analysing optical spectra the wind parameters can be merged into a wind
strength parameter $Q$ (Puls et al., 1996), transformed radius $R_{\rm t}$
(Schmutz et al., 1989; Gräfener et al., 2002; Hamann & Gräfener, 2004) or
transformed mass-loss rate $\dot{M}_{\rm t}$ (Bestenlehner et al., 2014, §
3.3).

The fully automated spectroscopic analysis tool of this work reduces the human
interaction to a minimum to cope with the amount of data. It is able to
process $\sim$ 250 stars in less than half an hour ($\sim$ 6 CPU hours)
delivering results comparable to Bestenlehner et al. (2014); Sabín-Sanjulián
et al. (2014, 2017); Ramírez-Agudelo et al. (2017) over a decade. Overall the
quality of the spectroscopic fits are good, but around 15% of the stars need
additional attention as a result of strong nebular contamination, low S/N,
multiplicity et cetera. The pipeline performances well over a wide parameter
space which is support by the spectroscopic analysis of 3 benchmarck stars by
several groups within the X-Shooter and ULLYSES collaboration (XShootU Vink et
al., 2023) of optical VLT/XShooter data (Sander et al. in preparation).

Weights of spectral lines could increase the accuracy, but need to be adjusted
depending on the parameter space that would then require human interaction.
Determining weights for features (spectral lines) is a typical machine
learning problem and often solved with neural networks (deep learning).
However, to really take advantage of our statistical approach and optimise the
pipeline we will require much larger data sets, which will be soon provided by
WEAVE and 4MOST. Future advances of our pipeline will be released on the
pipelines repository.

## Acknowledgements

JMB and PAC are supported by the Science and Technology Facilities Council
research grant ST/V000853/1 (PI. V. Dhillon). MB is supported through the Lise
Meitner grant from the Max Planck Society. We acknowledge support by the
Collaborative Research centre SFB 881 (projects A5, A10), Heidelberg
University, of the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation). This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant agreement No. 949173).

## Data Availability

The data underlying this article are available in the article and in its
supplementary material. The pipeline will be made publicly available after
acceptance of this manuscript. Spectroscopic data are available via the ESO
archive facility while grids of synthetic spectra can be requested from the
lead author.

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##  Appendix A Additional material

Table 1: Lines synthesised when calculation the formal integral. Wavelength ranges are multiplets with diverging central wavelengths. Ion | Wavelength | Ion | Wavelength | Ion | Wavelength | Ion | Wavelength | Ion | Wavelength | Ion | Wavelength
---|---|---|---|---|---|---|---|---|---|---|---
| [Å] |  | [Å] |  | [Å] |  | [Å] |  | [Å] |  | [Å]
H i | $3835.4$ | He i | $3888.6$ | C ii | $3919.0-3920.7$ | N ii | $3995.9$ | O ii | $3945.0-3954.4$ | Si ii | $3853.7-3862.6$
H i | $3889.1$ | He i | $3964.7$ | C ii | $4267.0-4267.3$ | N ii | $4447.0$ | O ii | $4069.6-4075.9$ | Si ii | $4128.1-4130.9$
H i | $3970.1$ | He i | $4009.3$ | C ii | $4637.6-4639.1$ | N ii | $4530.4$ | O ii | $4317.1-4366.9$ | Si ii | $5041.0-5056.3$
H i | $4101.7$ | He i | $4026.2$ | C ii | $5132.9-5151.1$ | N ii | $4552.5$ | O ii | $4414.9-4452.4$ | Si ii | $5957.6-5978.9$
H i | $4340.5$ | He i | $4120.8$ | C ii | $5648.1-5662.5$ | N ii | $4601.5-4643.1$ | O ii | $4638.9-4676.2$ | Si ii | $6347.1-6371.4$
H i | $4861.4$ | He i | $4143.8$ | C ii | $6151.3-6151.5$ | N ii | $5005.2$ | O ii | $4890.9-4906.8$ | Si ii | $9412.7-9412.8$
H i | $6562.8$ | He i | $4387.9$ | C ii | $6461.9$ | N ii | $5007.3$ | O ii | $4941.1-4943.0$ | Si iii | $3791.4-3806.8$
H i | $8392.2$ | He i | $4471.5$ | C ii | $6578.1-6582.9$ | N ii | $5045.1$ | O iii | $3703.4$ | Si iii | $4552.6-4574.8$
H i | $8413.1$ | He i | $4713.1$ | C ii | $6783.9$ | N ii | $5666.6-5710.8$ | O iii | $3707.3-3715.1$ | Si iii | $4716.7$
H i | $8437.8$ | He i | $4921.9$ | C iii | $4056.1$ | N ii | $5931.9-5941.7$ | O iii | $3754.7-3791.3$ | Si iii | $4813.3-4829.0$
H i | $8467.0$ | He i | $5015.7$ | C iii | $4068.9-4070.3$ | N ii | $6482.1$ | O iii | $3961.6$ | Si iii | $5739.7$
H i | $8502.3$ | He i | $5047.7$ | C iii | $4152.5-4162.9$ | N ii | $6610.6$ | O iii | $4072.6-4089.3$ | Si iii | $7461.9-7466.3$
H i | $8545.2$ | He i | $5875.6$ | C iii | $4186.9$ | N iii | $3934.5-3938.5$ | O iii | $4366.5-4375.9$ | Si iii | $8262.6-8271.9$
H i | $8598.2$ | He i | $6678.2$ | C iii | $4647.4-4651.5$ | N iii | $3998.6-4003.6$ | O iii | $4799.8$ | Si iii | $9799.9$
H i | $8664.8$ | He i | $7065.2$ | C iii | $4663.6-4665.9$ | N iii | $4097.4-4103.4$ | O iii | $5268.3$ | Si iv | $4088.9-4116.1$
H i | $8750.3$ | He i | $7281.4$ | C iii | $5249.1$ | N iii | $4195.8-4200.1$ | O iii | $5508.2$ | Si iv | $4212.4$
H i | $8862.6$ | He ii | $3796.3$ | C iii | $5253.6-5272.5$ | N iii | $4332.9-4345.7$ | O iii | $5592.3$ | Si iv | $4950.1$
H i | $9014.7$ | He ii | $3813.5$ | C iii | $5695.9$ | N iii | $4379.1$ | O iii | $7711.0$ | Si iv | $6667.6-6701.3$
H i | $9228.8$ | He ii | $3833.8$ | C iii | $5826.4$ | N iii | $4510.9-4547.3$ | O iv | $3560.4-3563.3$ | Si iv | $7047.9-7068.4$
H i | $9545.7$ | He ii | $3858.1$ | C iii | $6731.0-6744.3$ | N iii | $4527.9-4546.3$ | O iv | $3729.0-3736.9$ | Si iv | $8957.3$
|  | He ii | $3887.5$ | C iii | $7707.4$ | N iii | $4634.1-4641.9$ | O iv | $3995.1$ | Si iv | $9018.1$
|  | He ii | $3923.5$ | C iii | $8500.3$ | N iii | $4858.7-4867.2$ | O iv | $4654.1$ |  |
|  | He ii | $3968.4$ | C iii | $9701.1-9715.1$ | N iii | $5320.8-5352.5$ | O iv | $4813.2$ |  |
|  | He ii | $4025.6$ | C iv | $4646.6-4647.0$ | N iii | $6445.3-6487.8$ | O iv | $7004.1$ |  |
|  | He ii | $4100.1$ | C iv | $5016.6-5018.4$ | N iii | $9402.5-9424.5$ | O iv | $7032.3-7053.6$ |  |
|  | He ii | $4199.8$ | C iv | $5801.3-5812.0$ | N iv | $3747.5$ | O iv | $9453.9-0402.4$ |  |
|  | He ii | $4338.7$ | C iv | $6591.5-6592.6$ | N iv | $4057.8$ | O v | $5114.1$ |  |
|  | He ii | $4541.6$ |  |  | N iv | $5200.4-5205.1$ | O v | $6500.2$ |  |
|  | He ii | $4685.7$ |  |  | N iv | $5736.9$ |  |  |  |
|  | He ii | $4859.3$ |  |  | N iv | $5776.3-5784.8$ |  |  |  |
|  | He ii | $5411.5$ |  |  | N iv | $6212.4-6219.9$ |  |  |  |
|  | He ii | $6074.2$ |  |  | N iv | $7103.2-7129.2$ |  |  |  |
|  | He ii | $6118.3$ |  |  | N iv | $7425.3$ |  |  |  |
|  | He ii | $6170.7$ |  |  | N iv | $9182.2-9223.0$ |  |  |  |
|  | He ii | $6233.8$ |  |  | N v | $4603.7-4620.0$ |  |  |  |
|  | He ii | $6310.9$ |  |  |  |  |  |  |  |
|  | He ii | $6406.4$ |  |  |  |  |  |  |  |
|  | He ii | $6527.1$ |  |  |  |  |  |  |  |
|  | He ii | $6560.1$ |  |  |  |  |  |  |  |
|  | He ii | $6683.2$ |  |  |  |  |  |  |  |
|  | He ii | $6890.9$ |  |  |  |  |  |  |  |
|  | He ii | $7177.5$ |  |  |  |  |  |  |  |
|  | He ii | $7592.8$ |  |  |  |  |  |  |  |
|  | He ii | $8236.8$ |  |  |  |  |  |  |  |
|  | He ii | $9011.2$ |  |  |  |  |  |  |  |
|  | He ii | $9108.6$ |  |  |  |  |  |  |  |
|  | He ii | $9225.3$ |  |  |  |  |  |  |  |
|  | He ii | $9344.9$ |  |  |  |  |  |  |  |
|  | He ii | $9367.1$ |  |  |  |  |  |  |  |
|  | He ii | $9542.1$ |  |  |  |  |  |  |  |
Figure 10: VFTS 142: moderate nebular contamination. Blue solid line is the observation, red solid line the synthetic spectrum and the grey shaded area is the square-root of the diagonal elements of the covariant-matrix calculated by the pipeline. Effective temperature is well reproduce while the surface gravity is 0.2 to 0.3 dex too low. Figure 11: VFTS 410: strong nebular contamination. Blue solid line is the observation, red solid line the synthetic spectrum and the grey shaded area is the square-root of the diagonal elements of the covariant-matrix calculated by the pipeline. The pipeline is unable to reproduce the stellar parameters. Figure 12: VFTS 527: double-lined spectroscopic binary VFTS-527 (Taylor et al., 2011). Blue solid line is the observation, red solid line the synthetic spectrum and the grey shaded area is the square-root of the diagonal elements of the covariant-matrix calculated by the pipeline. Effective temperature and surface gravity of the primary are reproduced, but mass-loss rate is too high while helium abundance is too low due to the contribution of the colliding wind region largely effecting $\mathrm{H}_{\alpha}$ and $\mathrm{H}_{\beta}$. Table 2: Error-model uncertainties-matrix based on the VFTS analysis (§ 4.1). The rank of the matrix has been reduced by merging elements together for visualisation purposes. The Balmer lines show the strongest correlations have been highlighted. |  |  |  |  | Hδ |  |  |  |  |  | Hγ |  |  |  |  |  |  |  |  |  |  |  |  | Hβ |  |  |  |  |  |  | Hα |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\lambda$ [Å] | 3987 | 4016 | 4045 | 4072 |  4089 | 4115 | 4136 | 4160 | 4192 | 4233 | 4310 |  4325 | 4346 | 4378 | 4424 | 4474 | 4511 | 4532 | 4559 | 4613 | 4638 | 4651 | 4669 | 4717 | 4792 |  4824 | 4875 | 4914 | 4987 | 5008 | 5025 | 6447 | 6472 |  6520 | 6582 | 6643 | 6728
3987 | 0.03 | 0.02 | 0.01 | -0.0 | 0.01 | 0.02 | 0.02 | 0.0 | -0.01 | 0.0 | 0.01 | 0.0 | 0.01 | 0.01 | 0.0 | 0.03 | 0.0 | -0.01 | 0.02 | 0.02 | 0.05 | -0.07 | -0.02 | 0.01 | 0.0 | 0.02 | 0.0 | 0.01 | 0.0 | 0.03 | 0.0 | 0.01 | 0.01 | 0.06 | -0.13 | 0.03 | -0.0
4016 | 0.02 | 0.02 | 0.01 | -0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.0 | 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | -0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.04 | 0.01 | 0.01 | 0.03 | 0.02 | 0.02 | 0.02 | -0.04 | 0.17 | 0.0 | 0.01
4045 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.0 | -0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.02 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | -0.03 | 0.13 | -0.0 | 0.01
4072 | -0.0 | -0.0 | -0.01 | 0.03 | -0.03 | -0.01 | 0.0 | -0.0 | -0.01 | -0.0 | 0.0 | -0.0 | -0.01 | -0.0 | 0.01 | 0.0 | -0.01 | -0.02 | 0.01 | 0.0 | 0.0 | -0.03 | -0.0 | -0.01 | -0.0 | -0.0 | 0.02 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.0 | -0.01 | 0.1 | -0.01 | -0.0
4089 | 0.01 | 0.0 | 0.0 | -0.03 | 0.03 | 0.02 | -0.0 | -0.0 | 0.0 | -0.0 | -0.01 | 0.0 | 0.01 | 0.0 | -0.01 | 0.0 | 0.01 | 0.01 | -0.01 | -0.01 | 0.0 | 0.01 | -0.01 | 0.0 | -0.0 | 0.01 | -0.04 | -0.0 | -0.01 | -0.01 | -0.01 | -0.01 | 0.0 | 0.05 | -0.25 | 0.03 | -0.01
4115 | 0.02 | 0.01 | 0.0 | -0.01 | 0.02 | 0.02 | 0.01 | -0.01 | -0.01 | -0.01 | -0.0 | 0.01 | 0.01 | 0.0 | -0.01 | 0.01 | -0.0 | -0.01 | 0.01 | -0.0 | 0.03 | -0.06 | -0.03 | 0.0 | -0.01 | 0.03 | -0.04 | 0.0 | -0.01 | -0.0 | -0.01 | -0.01 | 0.0 | 0.1 | -0.34 | 0.04 | -0.01
4136 | 0.02 | 0.01 | 0.01 | 0.0 | -0.0 | 0.01 | 0.02 | 0.0 | -0.01 | 0.0 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.02 | 0.0 | -0.0 | 0.02 | 0.01 | 0.03 | -0.05 | -0.01 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.0 | 0.02 | 0.01 | 0.01 | 0.01 | 0.02 | -0.01 | 0.02 | 0.0
4160 | 0.0 | 0.01 | 0.01 | -0.0 | -0.0 | -0.01 | 0.0 | 0.02 | 0.02 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.02 | 0.0 | 0.01 | -0.01 | 0.04 | 0.03 | 0.01 | 0.01 | -0.03 | 0.05 | 0.01 | 0.02 | 0.03 | 0.02 | 0.02 | 0.01 | -0.1 | 0.35 | -0.02 | 0.02
4192 | -0.01 | 0.01 | 0.01 | -0.01 | 0.0 | -0.01 | -0.01 | 0.02 | 0.02 | 0.02 | 0.0 | -0.01 | -0.0 | 0.01 | 0.01 | -0.01 | 0.02 | 0.03 | -0.01 | 0.01 | -0.03 | 0.08 | 0.04 | 0.01 | 0.01 | -0.03 | 0.05 | 0.0 | 0.02 | 0.01 | 0.02 | 0.01 | 0.01 | -0.11 | 0.36 | -0.03 | 0.02
4233 | 0.0 | 0.01 | 0.01 | -0.0 | -0.0 | -0.01 | 0.0 | 0.02 | 0.02 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.02 | 0.0 | 0.01 | -0.01 | 0.04 | 0.03 | 0.01 | 0.02 | -0.03 | 0.06 | 0.01 | 0.02 | 0.03 | 0.02 | 0.02 | 0.01 | -0.1 | 0.37 | -0.02 | 0.02
4310 | 0.01 | 0.01 | 0.0 | 0.0 | -0.01 | -0.0 | 0.01 | 0.01 | 0.0 | 0.01 | 0.0 | -0.0 | -0.0 | 0.01 | 0.01 | 0.01 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | -0.0 | 0.01 | 0.01 | 0.01 | -0.01 | 0.03 | 0.0 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | -0.03 | 0.14 | -0.01 | 0.01
4325 | 0.0 | -0.0 | -0.0 | -0.0 | 0.0 | 0.01 | 0.0 | -0.01 | -0.01 | -0.01 | -0.0 | 0.01 | 0.0 | -0.0 | -0.01 | 0.0 | -0.01 | -0.01 | 0.0 | -0.01 | 0.01 | -0.04 | -0.02 | -0.01 | -0.01 | 0.02 | -0.03 | -0.0 | -0.01 | -0.01 | -0.01 | -0.01 | -0.01 | 0.07 | -0.24 | 0.02 | -0.01
4346 | 0.01 | 0.01 | 0.0 | -0.01 | 0.01 | 0.01 | 0.01 | -0.0 | -0.0 | -0.0 | -0.0 | 0.0 | 0.01 | 0.0 | -0.0 | 0.01 | -0.0 | -0.0 | 0.0 | 0.0 | 0.01 | -0.02 | -0.01 | 0.0 | -0.0 | 0.01 | -0.01 | 0.0 | -0.0 | 0.0 | -0.0 | -0.0 | 0.0 | 0.04 | -0.13 | 0.02 | -0.0
4378 | 0.01 | 0.02 | 0.01 | -0.0 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | -0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.04 | 0.01 | 0.01 | 0.03 | 0.02 | 0.02 | 0.01 | -0.04 | 0.17 | -0.0 | 0.01
4424 | 0.0 | 0.01 | 0.01 | 0.01 | -0.01 | -0.01 | 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.01 | 0.02 | 0.02 | 0.01 | 0.02 | -0.0 | 0.03 | 0.03 | 0.01 | 0.02 | -0.03 | 0.06 | 0.01 | 0.02 | 0.03 | 0.03 | 0.02 | 0.01 | -0.1 | 0.4 | -0.03 | 0.02
4474 | 0.03 | 0.02 | 0.01 | 0.0 | 0.0 | 0.01 | 0.02 | 0.0 | -0.01 | 0.0 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.02 | 0.0 | -0.01 | 0.02 | 0.02 | 0.04 | -0.06 | -0.02 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.0 | 0.03 | 0.01 | 0.01 | 0.01 | 0.03 | -0.03 | 0.02 | 0.0
4511 | 0.0 | 0.01 | 0.01 | -0.01 | 0.01 | -0.0 | 0.0 | 0.02 | 0.02 | 0.02 | 0.0 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.03 | -0.0 | 0.01 | -0.01 | 0.05 | 0.03 | 0.01 | 0.02 | -0.03 | 0.05 | 0.01 | 0.02 | 0.03 | 0.02 | 0.02 | 0.01 | -0.09 | 0.31 | -0.02 | 0.02
4532 | -0.01 | 0.01 | 0.01 | -0.02 | 0.01 | -0.01 | -0.0 | 0.02 | 0.03 | 0.02 | 0.0 | -0.01 | -0.0 | 0.01 | 0.02 | -0.01 | 0.03 | 0.04 | -0.01 | 0.01 | -0.03 | 0.09 | 0.05 | 0.01 | 0.02 | -0.04 | 0.05 | 0.0 | 0.02 | 0.02 | 0.03 | 0.02 | 0.01 | -0.13 | 0.41 | -0.03 | 0.02
4559 | 0.02 | 0.01 | 0.0 | 0.01 | -0.01 | 0.01 | 0.02 | 0.0 | -0.01 | 0.0 | 0.01 | 0.0 | 0.0 | 0.01 | 0.01 | 0.02 | -0.0 | -0.01 | 0.02 | 0.01 | 0.04 | -0.06 | -0.02 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.0 | 0.02 | 0.0 | 0.01 | 0.01 | 0.03 | -0.03 | 0.01 | -0.0
4613 | 0.02 | 0.02 | 0.01 | 0.0 | -0.01 | -0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.02 | 0.02 | 0.02 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | -0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.05 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.02 | -0.05 | 0.24 | -0.01 | 0.01
4638 | 0.05 | 0.02 | 0.01 | 0.0 | 0.0 | 0.03 | 0.03 | -0.01 | -0.03 | -0.01 | 0.01 | 0.01 | 0.01 | 0.02 | -0.0 | 0.04 | -0.01 | -0.03 | 0.04 | 0.02 | 0.09 | -0.14 | -0.06 | 0.01 | -0.0 | 0.05 | -0.02 | 0.01 | -0.01 | 0.03 | -0.01 | 0.01 | 0.01 | 0.15 | -0.37 | 0.06 | -0.01
4651 | -0.07 | -0.01 | 0.01 | -0.03 | 0.01 | -0.06 | -0.05 | 0.04 | 0.08 | 0.04 | -0.0 | -0.04 | -0.02 | -0.01 | 0.03 | -0.06 | 0.05 | 0.09 | -0.06 | -0.01 | -0.14 | 0.32 | 0.14 | 0.01 | 0.03 | -0.12 | 0.11 | -0.01 | 0.04 | -0.0 | 0.05 | 0.02 | 0.0 | -0.38 | 1.1 | -0.12 | 0.06
4669 | -0.02 | 0.01 | 0.01 | -0.0 | -0.01 | -0.03 | -0.01 | 0.03 | 0.04 | 0.03 | 0.01 | -0.02 | -0.01 | 0.01 | 0.03 | -0.02 | 0.03 | 0.05 | -0.02 | 0.01 | -0.06 | 0.14 | 0.08 | 0.02 | 0.03 | -0.07 | 0.1 | 0.0 | 0.03 | 0.02 | 0.04 | 0.03 | 0.01 | -0.23 | 0.75 | -0.07 | 0.04
4717 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | -0.01 | 0.04 | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.01 | -0.05 | 0.21 | -0.01 | 0.01
4792 | 0.0 | 0.01 | 0.01 | -0.0 | -0.0 | -0.01 | 0.0 | 0.01 | 0.01 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.02 | 0.0 | 0.01 | -0.0 | 0.03 | 0.03 | 0.01 | 0.01 | -0.02 | 0.05 | 0.01 | 0.01 | 0.03 | 0.02 | 0.02 | 0.01 | -0.09 | 0.31 | -0.02 | 0.02
4824 | 0.02 | -0.01 | -0.01 | -0.0 | 0.01 | 0.03 | 0.01 | -0.03 | -0.03 | -0.03 | -0.01 | 0.02 | 0.01 | -0.01 | -0.03 | 0.01 | -0.03 | -0.04 | 0.01 | -0.01 | 0.05 | -0.12 | -0.07 | -0.01 | -0.02 | 0.06 | -0.09 | -0.0 | -0.03 | -0.02 | -0.04 | -0.02 | -0.01 | 0.21 | -0.69 | 0.06 | -0.03
4875 | 0.0 | 0.04 | 0.02 | 0.02 | -0.04 | -0.04 | 0.01 | 0.05 | 0.05 | 0.06 | 0.03 | -0.03 | -0.01 | 0.04 | 0.06 | 0.01 | 0.05 | 0.05 | 0.01 | 0.05 | -0.02 | 0.11 | 0.1 | 0.04 | 0.05 | -0.09 | 0.2 | 0.02 | 0.05 | 0.09 | 0.08 | 0.06 | 0.04 | -0.33 | 1.25 | -0.08 | 0.06
4914 | 0.01 | 0.01 | 0.01 | 0.0 | -0.0 | 0.0 | 0.01 | 0.01 | 0.0 | 0.01 | 0.0 | -0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.01 | 0.01 | -0.0 | 0.02 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | -0.02 | 0.11 | 0.0 | 0.01
4987 | 0.0 | 0.01 | 0.01 | 0.0 | -0.01 | -0.01 | 0.0 | 0.02 | 0.02 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.02 | 0.0 | 0.01 | -0.01 | 0.04 | 0.03 | 0.01 | 0.01 | -0.03 | 0.05 | 0.01 | 0.02 | 0.03 | 0.02 | 0.02 | 0.01 | -0.09 | 0.35 | -0.02 | 0.02
5008 | 0.03 | 0.03 | 0.02 | 0.0 | -0.01 | -0.0 | 0.02 | 0.03 | 0.01 | 0.03 | 0.02 | -0.01 | 0.0 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.02 | 0.04 | 0.03 | -0.0 | 0.02 | 0.02 | 0.03 | -0.02 | 0.09 | 0.02 | 0.03 | 0.06 | 0.04 | 0.04 | 0.03 | -0.1 | 0.46 | -0.02 | 0.03
5025 | 0.0 | 0.02 | 0.01 | 0.0 | -0.01 | -0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.01 | -0.01 | -0.0 | 0.02 | 0.03 | 0.01 | 0.02 | 0.03 | 0.0 | 0.02 | -0.01 | 0.05 | 0.04 | 0.02 | 0.02 | -0.04 | 0.08 | 0.01 | 0.02 | 0.04 | 0.03 | 0.03 | 0.02 | -0.14 | 0.51 | -0.03 | 0.03
6447 | 0.01 | 0.02 | 0.01 | 0.0 | -0.01 | -0.01 | 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | -0.01 | -0.0 | 0.02 | 0.02 | 0.01 | 0.02 | 0.02 | 0.01 | 0.02 | 0.01 | 0.02 | 0.03 | 0.02 | 0.02 | -0.02 | 0.06 | 0.01 | 0.02 | 0.04 | 0.03 | 0.02 | 0.02 | -0.09 | 0.36 | -0.02 | 0.02
6472 | 0.01 | 0.02 | 0.01 | -0.0 | 0.0 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | -0.01 | 0.0 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.01 | 0.0 | 0.01 | 0.01 | 0.01 | -0.01 | 0.04 | 0.01 | 0.01 | 0.03 | 0.02 | 0.02 | 0.01 | -0.05 | 0.2 | -0.0 | 0.01
6520 | 0.06 | -0.04 | -0.03 | -0.01 | 0.05 | 0.1 | 0.02 | -0.1 | -0.11 | -0.1 | -0.03 | 0.07 | 0.04 | -0.04 | -0.1 | 0.03 | -0.09 | -0.13 | 0.03 | -0.05 | 0.15 | -0.38 | -0.23 | -0.05 | -0.09 | 0.21 | -0.33 | -0.02 | -0.09 | -0.1 | -0.14 | -0.09 | -0.05 | 0.73 | -2.46 | 0.21 | -0.12
6582 | -0.13 | 0.17 | 0.13 | 0.1 | -0.25 | -0.34 | -0.01 | 0.35 | 0.36 | 0.37 | 0.14 | -0.24 | -0.13 | 0.17 | 0.4 | -0.03 | 0.31 | 0.41 | -0.03 | 0.24 | -0.37 | 1.1 | 0.75 | 0.21 | 0.31 | -0.69 | 1.25 | 0.11 | 0.35 | 0.46 | 0.51 | 0.36 | 0.2 | -2.46 | 8.74 | -0.7 | 0.41
6643 | 0.03 | 0.0 | -0.0 | -0.01 | 0.03 | 0.04 | 0.02 | -0.02 | -0.03 | -0.02 | -0.01 | 0.02 | 0.02 | -0.0 | -0.03 | 0.02 | -0.02 | -0.03 | 0.01 | -0.01 | 0.06 | -0.12 | -0.07 | -0.01 | -0.02 | 0.06 | -0.08 | 0.0 | -0.02 | -0.02 | -0.03 | -0.02 | -0.0 | 0.21 | -0.7 | 0.07 | -0.03
6728 | -0.0 | 0.01 | 0.01 | -0.0 | -0.01 | -0.01 | 0.0 | 0.02 | 0.02 | 0.02 | 0.01 | -0.01 | -0.0 | 0.01 | 0.02 | 0.0 | 0.02 | 0.02 | -0.0 | 0.01 | -0.01 | 0.06 | 0.04 | 0.01 | 0.02 | -0.03 | 0.06 | 0.01 | 0.02 | 0.03 | 0.03 | 0.02 | 0.01 | -0.12 | 0.41 | -0.03 | 0.02
Table 3: Stellar parameters of the VLT/MUSE sources (Castro et al., 2018), which must be verified before using them for scientific purposes (e.g. plots in the supplementary online material). A very basic automated quality control has been performed, comments column: grid boundary, derived stellar parameters are located at the edge of the $T_{\rm eff}-\log g$ parameter space of the stellar atmosphere grid; bad fit, large miss match between He i and ii lines; no photometry, insufficient photometric data were available to derive bolometric luminosities. ID | $T_{\rm eff}$ | $\log g$ | $\log L$ | comments
---|---|---|---|---
(Castro et al., 2018) | [kK] | [g/cm2] | [$L/L_{\odot}$] |
178 | $21.22^{+0.96}_{-0.65}$ | $2.99^{+0.25}_{-0.21}$ | – | no photometry
195 | $37.63^{+0.85}_{-0.81}$ | $3.84^{+0.12}_{-0.11}$ | $4.91^{+0.04}_{-0.04}$ |
225 | $25.9^{+1.5}$ | $2.62^{+0.07}_{-0.02}$ | $3.64^{+0.14}_{-0.12}$ | grid boundary, bad fit
249 | $26.64^{+0.54}_{-0.46}$ | $3.6^{+0.24}_{-0.25}$ | $4.01^{+0.59}_{-0.59}$ |
276 | $37.67^{+1.08}_{-0.81}$ | $3.71^{+0.1}_{-0.08}$ | $5.23^{+0.06}_{-0.05}$ |
290 | $36.75^{+0.77}_{-0.69}$ | $3.81^{+0.1}_{-0.11}$ | $5.34^{+0.04}_{-0.04}$ |
322 | $21.61^{+5.27}_{-0.73}$ | $3.81^{+0.25}_{-0.16}$ | $3.46^{+0.36}_{-0.27}$ |
367 | $22.95^{+1.31}_{-1.31}$ | $3.82^{+0.08}_{-0.22}$ | $3.84^{+0.14}_{-0.14}$ |
375 | $34.6^{+0.65}_{-1.42}$ | $3.4^{+0.08}_{-0.15}$ | $4.73^{+0.03}_{-0.07}$ |
410 | $32.52^{+0.77}_{-1.23}$ | $3.19^{+0.1}_{-0.14}$ | $4.2^{+0.05}_{-0.07}$ |
466 | $36.33^{+6.11}_{-6.23}$ | $3.09^{+0.32}_{-0.37}$ | $4.97^{+0.22}_{-0.23}$ | bad fit
489 | $21.41^{+0.81}_{-0.58}$ | $3.8^{+0.04}_{-0.06}$ | $3.41^{+0.22}_{-0.22}$ |
501 | $33.44^{+2.46}_{-1.42}$ | $3.41^{+0.33}_{-0.19}$ | $4.29^{+0.12}_{-0.08}$ |
520 | $20.53^{+0.46}_{-0.46}$ | $2.59^{+0.09}_{-0.09}$ | $3.57^{+0.13}_{-0.13}$ |
543 | $35.64^{+2.61}_{-1.88}$ | $3.7^{+0.41}_{-0.27}$ | $4.48^{+0.11}_{-0.08}$ |
549 | $23.14^{+3.34}_{-1.54}$ | $3.9^{+0.11}_{-0.09}$ | $4.19^{+0.2}_{-0.1}$ |
605 | $21.26^{+4.88}_{-0.5}$ | $3.8^{+0.23}_{-0.08}$ | $3.68^{+0.3}_{-0.13}$ |
689 | $43.55^{+2.31}_{-2.04}$ | $3.71^{+0.1}_{-0.07}$ | $4.92^{+0.08}_{-0.08}$ |
710 | $32.68^{+1.81}_{-1.19}$ | $3.21^{+0.15}_{-0.13}$ | $4.27^{+0.09}_{-0.07}$ |
739 | $20.8^{+1.23}_{-1.08}$ | $3.79^{+0.07}_{-0.18}$ | $3.74^{+0.12}_{-0.11}$ |
762 | $27.37^{+0.73}_{-0.69}$ | $2.72^{+0.05}_{-0.06}$ | $5.08^{+0.06}_{-0.06}$ |
774 | $39.83^{+0.65}_{-0.65}$ | $3.49^{+0.06}_{-0.06}$ | $5.49^{+0.03}_{-0.03}$ |
805 | $33.25^{+3.34}_{-1.69}$ | $3.48^{+0.45}_{-0.27}$ | $4.55^{+0.17}_{-0.13}$ |
813 | $40.82^{+0.73}_{-1.27}$ | $4.37^{+0.03}_{-0.67}$ | $4.95^{+0.04}_{-0.05}$ | bad fit
830 | $36.83^{+1.81}_{-1.23}$ | $3.88^{+0.22}_{-0.21}$ | $5.41^{+0.08}_{-0.06}$ |
835 | $18.72^{+0.73}_{-0.92}$ | $3.6^{+0.11}_{-0.18}$ | $3.43^{+0.2}_{-0.21}$ |
888 | $29.02^{+1.35}_{-2.34}$ | $4.08^{+0.05}_{-0.1}$ | $4.39^{+0.08}_{-0.12}$ |
898 | $26.95^{+4.11}_{-5.42}$ | $4.0^{+0.26}_{-0.21}$ | $3.78^{+0.29}_{-0.32}$ |
909 | $33.41^{+0.69}_{-1.04}$ | $3.5^{+0.09}_{-0.11}$ | $4.5^{+0.05}_{-0.06}$ |
911 | $33.33^{+1.27}_{-0.88}$ | $4.19^{+0.06}_{-0.16}$ | $4.9^{+0.1}_{-0.09}$ |
912 | $30.14^{+1.08}_{-1.04}$ | $4.09^{+0.05}_{-0.1}$ | $4.22^{+0.18}_{-0.18}$ |
920 | $17.8^{+0.5}$ | $3.69^{+0.08}_{-0.07}$ | $3.23^{+1.07}_{-1.07}$ |
935 | $28.91^{+1.23}_{-1.08}$ | $4.07^{+0.05}_{-0.09}$ | $4.83^{+0.41}_{-0.41}$ |
955 | $38.36^{+0.77}_{-0.85}$ | $4.28^{+0.06}_{-0.28}$ | $4.89^{+0.04}_{-0.05}$ |
960 | $25.26^{+4.65}_{-7.46}$ | $2.52^{+1.13}_{-0.52}$ | $3.76^{+0.35}_{-0.41}$ | bad fit
963 | $44.71^{+3.34}_{-2.54}$ | $3.82^{+0.15}_{-0.09}$ | $5.63^{+0.12}_{-0.09}$ |
1008 | $38.67^{+0.69}_{-0.77}$ | $4.0^{+0.16}_{-0.14}$ | $5.08^{+0.05}_{-0.05}$ |
1040 | $35.48^{+0.81}_{-0.58}$ | $3.93^{+0.25}_{-0.12}$ | $4.77^{+0.05}_{-0.04}$ |
1068 | $44.78^{+3.0}_{-1.27}$ | $3.7^{+0.13}_{-0.09}$ | $5.48^{+0.11}_{-0.06}$ |
1138 | $35.6^{+0.54}_{-0.62}$ | $4.13^{+0.11}_{-0.14}$ | $4.46^{+0.14}_{-0.14}$ |
1143 | $22.03^{+0.96}_{-1.19}$ | $3.84^{+0.1}_{-0.14}$ | $4.27^{+0.41}_{-0.41}$ |
1152 | $25.99^{+0.85}_{-1.81}$ | $3.7^{+0.2}_{-0.2}$ | $4.51^{+0.05}_{-0.11}$ |
1177 | $35.44^{+1.04}_{-1.15}$ | $3.61^{+0.16}_{-0.14}$ | $5.32^{+0.05}_{-0.05}$ |
1184 | $31.56^{+0.54}_{-0.46}$ | $3.3^{+0.06}_{-0.07}$ | $5.31^{+0.1}_{-0.1}$ |
1199 | $56.2_{-1.61}$ | $4.01^{+0.05}_{-0.07}$ | $6.33^{+0.5}_{-0.5}$ | grid boundary
1210 | $26.64^{+0.85}_{-0.38}$ | $4.0^{+0.04}_{-0.05}$ | $3.87^{+0.23}_{-0.22}$ |
1223 | $34.48^{+0.54}_{-0.81}$ | $3.35^{+0.09}_{-0.09}$ | $5.45^{+0.03}_{-0.04}$ |
1226 | $18.34^{+0.35}_{-0.54}$ | $3.62^{+0.09}_{-0.17}$ | $3.36^{+0.03}_{-0.05}$ |
1274 | $35.64^{+1.08}_{-1.11}$ | $3.6^{+0.13}_{-0.11}$ | $5.62^{+0.05}_{-0.05}$ |
1275 | $31.64^{+1.35}_{-1.04}$ | $3.2^{+0.27}_{-0.13}$ | $4.63^{+0.36}_{-0.36}$ |
1276 | $25.3^{+0.73}_{-2.31}$ | $3.83^{+0.18}_{-0.32}$ | $4.44^{+0.67}_{-0.67}$ |
1279 | $27.56^{+1.31}_{-0.85}$ | $2.72^{+0.06}_{-0.06}$ | $5.43^{+0.08}_{-0.06}$ |
1282 | $23.45^{+6.65}_{-3.46}$ | $4.05^{+0.14}_{-0.13}$ | $3.83^{+0.74}_{-0.73}$ |
1293 | $31.56^{+0.54}_{-0.46}$ | $3.3^{+0.06}_{-0.07}$ | $5.65^{+0.03}_{-0.03}$ | bad fit
1297 | $26.64^{+0.85}_{-0.46}$ | $3.99^{+0.05}_{-0.06}$ | $4.36^{+0.05}_{-0.03}$ |
1324 | $32.37^{+1.96}_{-1.54}$ | $4.17^{+0.06}_{-0.16}$ | $3.97^{+0.61}_{-0.61}$ |
1325 | $38.83^{+2.31}_{-0.85}$ | $3.71^{+0.13}_{-0.09}$ | $5.07^{+0.69}_{-0.69}$ |
1334 | $34.56^{+0.77}_{-0.62}$ | $3.6^{+0.1}_{-0.08}$ | $4.72^{+0.05}_{-0.05}$ |
1340 | $19.41^{+0.42}_{-0.38}$ | $3.61^{+0.08}_{-0.08}$ | $3.96^{+1.13}_{-1.13}$ | bad fit
1346 | $36.83^{+1.11}_{-0.88}$ | $3.99^{+0.17}_{-0.15}$ | $5.44^{+0.05}_{-0.04}$ |
1370 | $27.95^{+2.46}_{-3.69}$ | $4.09^{+0.06}_{-0.08}$ | $4.13^{+0.18}_{-0.22}$ |
1373 | $37.6^{+1.65}_{-1.23}$ | $3.69^{+0.15}_{-0.13}$ | $5.6^{+0.07}_{-0.05}$ |
1387 | $18.45^{+0.35}_{-0.46}$ | $3.68^{+0.04}_{-0.17}$ | $3.98^{+0.9}_{-0.9}$ |
1390 | $26.64^{+0.46}_{-3.38}$ | $3.9^{+0.1}_{-0.13}$ | $4.24^{+0.2}_{-0.25}$ |
1398 | $32.56^{+1.54}_{-1.08}$ | $3.43^{+0.25}_{-0.15}$ | $4.96^{+0.09}_{-0.07}$ |
1399 | $19.34^{+0.42}_{-0.38}$ | $3.29^{+0.17}_{-0.16}$ | $4.79^{+0.04}_{-0.03}$ |
1401 | $40.75^{+1.42}_{-2.08}$ | $3.9^{+0.19}_{-0.14}$ | $5.42^{+0.66}_{-0.66}$ |
1405 | $25.99^{+1.69}_{-0.85}$ | $2.62^{+0.11}_{-0.07}$ | $4.61^{+0.13}_{-0.1}$ |
1409 | $35.6^{+1.42}_{-1.19}$ | $3.8^{+0.46}_{-0.22}$ | $5.31^{+0.18}_{-0.17}$ |
1415 | $44.71^{+0.69}_{-2.92}$ | $3.59^{+0.06}_{-0.14}$ | $5.09^{+1.06}_{-1.06}$ |
1423 | $44.71^{+0.81}_{-1.04}$ | $3.59^{+0.07}_{-0.07}$ | $5.87^{+0.29}_{-0.29}$ |
1428 | $29.14^{+2.15}_{-1.42}$ | $4.08^{+0.07}_{-0.09}$ | $4.12^{+0.41}_{-0.4}$ |
1433 | $44.86^{+1.61}_{-1.0}$ | $3.7^{+0.07}_{-0.07}$ | $5.86^{+0.06}_{-0.04}$ |
1434 | $29.83^{+1.42}_{-1.5}$ | $4.1^{+0.05}_{-0.09}$ | $4.11^{+0.14}_{-0.14}$ |
1439 | $36.44^{+1.0}_{-1.27}$ | $3.76^{+0.19}_{-0.2}$ | $5.11^{+0.05}_{-0.06}$ |
1445 | $36.56^{+1.5}_{-1.35}$ | $4.28^{+0.06}_{-0.14}$ | $4.95^{+0.62}_{-0.62}$ |
1447 | $43.44^{+1.69}_{-1.73}$ | $4.4^{+0.06}_{-0.14}$ | $5.66^{+0.08}_{-0.08}$ |
1453 | $21.64^{+0.88}_{-1.0}$ | $3.81^{+0.04}_{-0.04}$ | $4.09^{+0.54}_{-0.54}$ |
1456 | $21.14^{+1.08}_{-3.34}$ | $3.26^{+0.39}_{-0.27}$ | $3.86^{+0.56}_{-0.56}$ |
1457 | $17.8^{+0.54}$ | $3.59^{+0.05}_{-0.05}$ | $3.67^{+0.05}_{-0.0}$ | grid boundary
1459 | $37.6^{+0.69}_{-0.77}$ | $3.4^{+0.07}_{-0.08}$ | $5.86^{+0.28}_{-0.28}$ |
1464 | $35.44^{+0.69}_{-0.73}$ | $4.07^{+0.16}_{-0.14}$ | $4.35^{+0.12}_{-0.12}$ |
1472 | $37.94^{+1.38}_{-1.65}$ | $3.82^{+0.29}_{-0.18}$ | $5.14^{+0.13}_{-0.13}$ |
1494 | $51.47^{+3.96}_{-13.68}$ | $3.9^{+0.15}_{-0.8}$ | $5.97^{+0.86}_{-0.86}$ |
1503 | $36.48^{+1.38}_{-1.27}$ | $4.28^{+0.05}_{-0.13}$ | $5.16^{+0.71}_{-0.71}$ |
1504 | $27.22^{+3.11}_{-1.69}$ | $3.98^{+0.12}_{-0.35}$ | $4.22^{+0.27}_{-0.25}$ |
1505 | $26.22^{+1.19}_{-2.0}$ | $3.98^{+0.05}_{-0.05}$ | $4.38^{+1.05}_{-1.05}$ |
1522 | $36.33^{+1.08}_{-1.38}$ | $4.27^{+0.05}_{-0.12}$ | $5.12^{+0.52}_{-0.52}$ |
1523 | $36.67^{+1.5}_{-0.96}$ | $4.24^{+0.09}_{-0.19}$ | $5.29^{+0.47}_{-0.47}$ |
1527 | $36.71^{+0.85}_{-0.65}$ | $4.0^{+0.17}_{-0.12}$ | $5.26^{+0.95}_{-0.95}$ |
1535 | $35.48^{+0.62}_{-0.73}$ | $4.06^{+0.17}_{-0.17}$ | $5.7^{+0.09}_{-0.09}$ |
1537 | $35.87^{+1.42}_{-1.0}$ | $3.77^{+0.23}_{-0.15}$ | $5.33^{+0.35}_{-0.35}$ |
1572 | $40.98^{+1.0}_{-1.88}$ | $4.38^{+0.04}_{-0.2}$ | $5.64^{+0.04}_{-0.08}$ |
1580 | $46.01^{+1.19}_{-3.31}$ | $3.81^{+0.08}_{-0.07}$ | $5.67^{+0.64}_{-0.64}$ |
1607 | $31.52^{+1.19}_{-0.85}$ | $3.5^{+0.17}_{-0.17}$ | $4.67^{+0.2}_{-0.19}$ |
1625 | $35.52^{+0.88}_{-6.84}$ | $3.72^{+0.36}_{-0.16}$ | $5.19^{+0.26}_{-0.33}$ |
1632 | $35.44^{+0.65}_{-0.96}$ | $3.69^{+0.1}_{-0.13}$ | $6.36^{+1.14}_{-1.14}$ |
1641 | $22.18^{+1.27}_{-1.11}$ | $3.51^{+0.28}_{-0.25}$ | $4.41^{+0.09}_{-0.08}$ |
1651 | $25.1^{+0.65}$ | $2.62^{+0.03}_{-0.02}$ | $4.24^{+0.08}_{-0.07}$ | grid boundary, bad fit
1682 | $29.02^{+0.5}_{-3.88}$ | $4.0^{+0.07}_{-0.12}$ | $3.96^{+0.7}_{-0.7}$ |
1689 | $30.72^{+0.65}_{-0.81}$ | $3.1^{+0.08}_{-0.1}$ | $5.78^{+0.04}_{-0.05}$ |
1699 | $39.79^{+0.96}_{-1.04}$ | $3.72^{+0.19}_{-0.14}$ | $5.22^{+0.05}_{-0.05}$ |
1700 | $36.06^{+1.19}_{-0.92}$ | $4.17^{+0.14}_{-0.29}$ | $5.52^{+0.05}_{-0.04}$ |
1714 | $42.25^{+1.73}_{-2.42}$ | $4.21^{+0.2}_{-0.26}$ | $5.39^{+0.14}_{-0.15}$ |
1725 | $27.4^{+0.41}_{-1.5}$ | $2.8^{+0.06}_{-0.13}$ | $4.82^{+0.99}_{-0.99}$ | bad fit
1749 | $51.82^{+2.04}_{-1.77}$ | $4.01^{+0.1}_{-0.08}$ | $6.16^{+0.07}_{-0.06}$ |
1763 | $36.63^{+1.58}_{-1.31}$ | $3.72^{+0.17}_{-0.14}$ | $5.6^{+0.59}_{-0.59}$ |
1764 | $23.03^{+1.88}_{-1.31}$ | $3.79^{+0.11}_{-0.26}$ | $3.88^{+0.14}_{-0.11}$ |
1768 | $26.64^{+1.23}_{-3.84}$ | $4.0^{+0.06}_{-0.1}$ | $4.3^{+0.18}_{-0.25}$ |
1778 | $30.87^{+1.42}_{-1.23}$ | $4.11^{+0.07}_{-0.1}$ | $5.49^{+0.16}_{-0.16}$ |
1826 | $37.63^{+1.15}_{-0.77}$ | $4.29^{+0.06}_{-0.24}$ | $4.38^{+0.21}_{-0.2}$ |
1827 | $39.79^{+0.85}_{-0.85}$ | $3.73^{+0.16}_{-0.09}$ | $5.62^{+0.04}_{-0.04}$ |
1847 | $18.53^{+0.58}_{-0.46}$ | $3.68^{+0.04}_{-0.15}$ | $2.82^{+1.24}_{-1.24}$ |
1857 | $37.37^{+0.77}_{-0.88}$ | $4.22^{+0.1}_{-0.11}$ | $6.23^{+0.22}_{-0.23}$ |
1858 | $25.95^{+1.38}_{-1.81}$ | $3.96^{+0.07}_{-0.07}$ | $4.87^{+0.16}_{-0.17}$ | bad fit
1870 | $36.48^{+0.73}_{-0.85}$ | $3.67^{+0.09}_{-0.11}$ | $5.01^{+0.54}_{-0.54}$ |
1890 | $38.52^{+0.77}_{-1.15}$ | $3.86^{+0.22}_{-0.15}$ | $5.42^{+0.05}_{-0.06}$ |
1894 | $31.48^{+0.73}_{-1.11}$ | $3.5^{+0.21}_{-0.15}$ | $4.47^{+0.05}_{-0.07}$ |
1901 | $33.33^{+2.04}_{-1.31}$ | $3.21^{+0.15}_{-0.11}$ | $4.73^{+0.76}_{-0.76}$ | bad fit
1912 | $38.71^{+2.38}_{-1.15}$ | $4.14^{+0.24}_{-0.22}$ | $6.12^{+0.21}_{-0.2}$ |
1920 | $36.63^{+1.42}_{-0.96}$ | $3.92^{+0.26}_{-0.16}$ | $5.15^{+0.91}_{-0.91}$ |
1941 | $27.22^{+0.62}_{-0.77}$ | $3.99^{+0.04}_{-0.2}$ | $4.87^{+0.08}_{-0.08}$ | bad fit
1951 | $31.6^{+0.65}_{-4.31}$ | $3.31^{+0.11}_{-0.12}$ | $5.02^{+0.05}_{-0.19}$ |
1952 | $21.22^{+2.31}_{-0.46}$ | $3.81^{+0.14}_{-0.06}$ | $4.08^{+0.45}_{-0.44}$ |
1956 | $51.55^{+0.85}_{-16.3}$ | $4.0^{+0.06}_{-1.08}$ | $5.59^{+0.83}_{-0.84}$ |
1968 | $21.84^{+1.0}_{-0.62}$ | $3.8^{+0.06}_{-0.11}$ | $4.48^{+0.14}_{-0.13}$ |
1969 | $35.48^{+1.96}_{-10.49}$ | $3.61^{+0.44}_{-0.2}$ | $4.88^{+0.67}_{-0.69}$ |
1973 | $25.1^{+20.8}$ | $2.6^{+1.13}$ | $3.86^{+0.72}_{-0.55}$ | grid boundary, bad fit
1974 | $37.33^{+0.69}_{-0.96}$ | $4.21^{+0.08}_{-0.19}$ | $5.45^{+0.04}_{-0.05}$ |
1977 | $21.99^{+0.88}_{-0.73}$ | $3.81^{+0.05}_{-0.08}$ | $3.49^{+0.29}_{-0.29}$ |
1979 | $37.67^{+1.23}_{-0.88}$ | $3.51^{+0.1}_{-0.08}$ | $5.43^{+0.06}_{-0.05}$ |
1988 | $28.99^{+1.11}_{-1.11}$ | $4.07^{+0.05}_{-0.15}$ | $4.43^{+0.44}_{-0.44}$ | bad fit
1998 | $36.06^{+1.77}_{-1.58}$ | $3.81^{+0.47}_{-0.26}$ | $4.86^{+0.16}_{-0.16}$ |
2003 | $39.83^{+0.54}_{-0.58}$ | $3.71^{+0.07}_{-0.06}$ | $6.29^{+0.63}_{-0.63}$ |
2005 | $45.82^{+0.73}_{-15.64}$ | $3.69^{+0.05}_{-1.03}$ | $5.03^{+0.05}_{-0.37}$ |
2016 | $36.6^{+0.92}_{-0.96}$ | $3.82^{+0.19}_{-0.12}$ | $5.53^{+0.2}_{-0.2}$ |
2017 | $26.99^{+0.85}_{-0.62}$ | $4.01^{+0.04}_{-0.04}$ | $3.67^{+0.22}_{-0.22}$ |
2018 | $21.95^{+11.19}_{-0.77}$ | $3.84^{+0.42}_{-0.09}$ | $4.26^{+0.82}_{-0.8}$ |
2024 | $21.61^{+0.69}_{-0.54}$ | $3.78^{+0.05}_{-0.24}$ | $4.65^{+0.05}_{-0.04}$ |
2033 | $30.6^{+1.0}_{-0.92}$ | $4.1^{+0.05}_{-0.05}$ | $5.34^{+0.44}_{-0.44}$ |
2038 | $43.44^{+2.61}_{-10.26}$ | $3.51^{+0.08}_{-0.67}$ | $4.69^{+0.09}_{-0.29}$ | bad fit
2044 | $38.67^{+0.58}_{-0.96}$ | $4.09^{+0.23}_{-0.18}$ | $5.28^{+0.1}_{-0.1}$ |
2053 | $42.44^{+2.69}_{-1.38}$ | $3.6^{+0.13}_{-0.07}$ | $5.86^{+0.11}_{-0.08}$ |
2057 | $42.25^{+0.69}_{-0.85}$ | $3.49^{+0.06}_{-0.05}$ | $5.3^{+0.04}_{-0.04}$ | bad fit
2077 | $44.94^{+4.23}_{-4.19}$ | $3.91^{+0.12}_{-0.08}$ | $5.81^{+0.14}_{-0.14}$ |
2086 | $31.52^{+0.42}_{-1.04}$ | $3.49^{+0.07}_{-0.19}$ | $4.2^{+0.59}_{-0.59}$ |
2090 | $34.6^{+1.0}_{-0.65}$ | $4.19^{+0.06}_{-0.22}$ | – | no photometry
2097 | $26.6^{+1.19}_{-3.11}$ | $3.99^{+0.06}_{-0.35}$ | $4.33^{+0.07}_{-0.17}$ |
2098 | $21.8^{+0.81}_{-0.65}$ | $3.82^{+0.05}_{-0.09}$ | $4.18^{+0.65}_{-0.65}$ |
2102 | $44.75^{+0.69}_{-0.77}$ | $3.8^{+0.09}_{-0.09}$ | $6.48^{+0.21}_{-0.21}$ |
2112 | $46.01^{+4.38}_{-11.8}$ | $3.7^{+0.14}_{-0.77}$ | $5.7^{+0.14}_{-0.31}$ |
2114 | $32.52^{+0.62}_{-0.54}$ | $3.2^{+0.06}_{-0.07}$ | $4.3^{+0.3}_{-0.3}$ |
2128 | $39.98^{+1.19}_{-0.77}$ | $3.71^{+0.18}_{-0.1}$ | $5.51^{+0.53}_{-0.53}$ |
2130 | $34.52^{+0.54}_{-0.54}$ | $4.09^{+0.13}_{-0.13}$ | $5.32^{+0.16}_{-0.16}$ |
2140 | $18.18^{+0.27}_{-0.38}$ | $3.3^{+0.23}_{-0.14}$ | $3.26^{+0.3}_{-0.3}$ |
2147 | $26.64^{+1.61}_{-3.96}$ | $3.98^{+0.08}_{-0.41}$ | $3.85^{+0.5}_{-0.51}$ |
2159 | $21.37^{+0.96}_{-0.42}$ | $3.7^{+0.18}_{-0.19}$ | $3.88^{+0.6}_{-0.6}$ |
2165 | $35.6^{+1.42}_{-1.04}$ | $3.5^{+0.19}_{-0.12}$ | $5.85^{+0.08}_{-0.07}$ |
2174 | $22.3^{+0.62}_{-0.54}$ | $3.81^{+0.04}_{-0.06}$ | $3.91^{+0.29}_{-0.29}$ | bad fit
2176 | $18.49^{+0.5}_{-0.35}$ | $3.79^{+0.04}_{-0.12}$ | $3.97^{+0.54}_{-0.54}$ |
2190 | $28.22^{+0.58}_{-0.58}$ | $2.79^{+0.06}_{-0.06}$ | $5.49^{+0.12}_{-0.12}$ |
2193 | $36.63^{+0.81}_{-0.69}$ | $3.79^{+0.16}_{-0.15}$ | $5.79^{+0.77}_{-0.77}$ |
2231 | $38.48^{+1.58}_{-1.88}$ | $3.84^{+0.19}_{-0.13}$ | $5.41^{+0.11}_{-0.12}$ |
2233 | $32.45^{+3.27}_{-1.38}$ | $4.16^{+0.07}_{-0.23}$ | $4.88^{+0.15}_{-0.07}$ |
2245 | $37.75^{+1.35}_{-1.11}$ | $4.23^{+0.13}_{-0.16}$ | $5.16^{+0.06}_{-0.05}$ |
2256 | $31.37^{+1.81}_{-1.77}$ | $3.48^{+0.31}_{-0.27}$ | $4.87^{+0.1}_{-0.1}$ |
2270 | $29.06^{+1.77}_{-2.0}$ | $4.08^{+0.06}_{-0.06}$ | $4.75^{+0.09}_{-0.11}$ |
2301 | $28.99^{+1.42}_{-3.0}$ | $4.1^{+0.04}_{-0.07}$ | $5.04^{+0.08}_{-0.15}$ | bad fit
2302 | $17.8^{+0.58}$ | $3.69^{+0.03}_{-0.04}$ | $3.72^{+0.84}_{-0.84}$ | grid boundary, bad fit
2385 | $54.55^{+1.65}_{-4.42}$ | $4.5_{-0.14}$ | $5.7^{+0.05}_{-0.12}$ | grid boundary
2388 | $33.52^{+2.27}_{-5.11}$ | $3.2^{+0.15}_{-0.54}$ | $4.36^{+0.74}_{-0.74}$ |
2389 | $31.02^{+3.84}_{-0.88}$ | $4.1^{+0.11}_{-0.18}$ | $4.93^{+0.18}_{-0.05}$ |
2395 | $49.78^{+1.73}_{-3.08}$ | $4.19^{+0.22}_{-0.15}$ | $6.23^{+0.06}_{-0.1}$ |
2401 | $36.67^{+1.58}_{-1.77}$ | $4.28^{+0.06}_{-0.13}$ | $4.54^{+0.53}_{-0.53}$ |
2417 | $33.64^{+1.08}_{-0.88}$ | $3.24^{+0.13}_{-0.1}$ | $5.49^{+0.05}_{-0.04}$ |
2438 | $18.49^{+0.5}_{-0.31}$ | $3.69^{+0.05}_{-0.07}$ | $3.67^{+0.41}_{-0.41}$ |
2447 | $53.09^{+1.27}_{-9.88}$ | $4.5_{-0.45}$ | $6.04^{+0.04}_{-0.24}$ | grid boundary
2451 | $44.75^{+3.84}_{-0.96}$ | $4.0^{+0.19}_{-0.15}$ | $5.46^{+0.13}_{-0.04}$ |
2453 | $25.78^{+6.56}_{-0.38}$ | $2.6^{+0.31}_{0.0}$ | $3.68^{+0.39}_{-0.31}$ | bad fit
2454 | $33.37^{+1.11}_{-0.73}$ | $4.19^{+0.05}_{-0.1}$ | $4.9^{+0.24}_{-0.24}$ | bad fit
2498 | $50.13^{+1.19}_{-1.11}$ | $4.5_{-0.2}$ | $5.64^{+0.05}_{-0.05}$ | grid boundary
2506 | $21.14^{+0.65}_{-2.88}$ | $2.9^{+0.21}_{-0.27}$ | $3.36^{+0.21}_{-0.26}$ |
2511 | $26.95^{+0.88}_{-0.58}$ | $4.01^{+0.04}_{-0.04}$ | $4.18^{+0.5}_{-0.5}$ |
2521 | $27.18^{+0.62}_{-0.81}$ | $3.72^{+0.24}_{-0.26}$ | $4.72^{+0.04}_{-0.05}$ |
2543 | $24.8^{+1.69}_{-2.04}$ | $3.84^{+0.13}_{-0.3}$ | $4.14^{+0.17}_{-0.18}$ |
2545 | $38.98^{+7.03}_{-8.99}$ | $3.17^{+0.35}_{-0.51}$ | $4.99^{+0.24}_{-0.28}$ | bad fit
2565 | $35.6^{+1.23}_{-1.04}$ | $3.66^{+0.16}_{-0.13}$ | $5.09^{+0.06}_{-0.05}$ |
2570 | $37.6^{+2.04}_{-1.65}$ | $4.29^{+0.08}_{-0.46}$ | $5.3^{+0.09}_{-0.07}$ |
2572 | $32.25^{+3.31}_{-7.46}$ | $4.19^{+0.1}_{-0.28}$ | $4.45^{+0.29}_{-0.35}$ |
2607 | $28.91^{+0.92}_{-4.77}$ | $3.99^{+0.1}_{-0.54}$ | $4.37^{+0.2}_{-0.27}$ |
2610 | $28.64^{+0.81}_{-0.85}$ | $3.99^{+0.07}_{-0.36}$ | $4.53^{+0.05}_{-0.05}$ |
2653 | $32.6^{+0.73}_{-0.81}$ | $3.5^{+0.13}_{-0.11}$ | $5.73^{+0.04}_{-0.04}$ |
2665 | $34.67^{+1.65}_{-1.96}$ | $3.4^{+0.22}_{-0.22}$ | $4.82^{+0.08}_{-0.09}$ |
2681 | $24.41^{+1.65}_{-0.65}$ | $3.32^{+0.27}_{-0.2}$ | $4.08^{+0.11}_{-0.05}$ |
2703 | $30.41^{+1.11}_{-1.04}$ | $4.08^{+0.06}_{-0.28}$ | $5.07^{+0.17}_{-0.17}$ |
2718 | $33.52^{+0.5}_{-0.5}$ | $3.96^{+0.14}_{-0.12}$ | $4.85^{+0.05}_{-0.05}$ |
2735 | $28.37^{+1.69}_{-3.15}$ | $4.1^{+0.04}_{-0.05}$ | $4.14^{+0.42}_{-0.43}$ |
2748 | $38.52^{+0.73}_{-2.54}$ | $3.61^{+0.11}_{-0.11}$ | $5.55^{+0.05}_{-0.11}$ |
2760 | $39.71^{+0.88}_{-1.42}$ | $3.61^{+0.1}_{-0.11}$ | $5.64^{+0.04}_{-0.06}$ |
2763 | $40.02^{+3.27}_{-0.85}$ | $3.7^{+0.11}_{-0.1}$ | $5.17^{+0.12}_{-0.04}$ |
2771 | $21.49^{+0.73}_{-0.42}$ | $3.71^{+0.12}_{-0.29}$ | $4.15^{+0.11}_{-0.1}$ |
2776 | $32.56^{+1.54}_{-5.3}$ | $3.2^{+0.16}_{-0.56}$ | $5.04^{+0.18}_{-0.26}$ |
2780 | $37.56^{+0.69}_{-10.53}$ | $3.39^{+0.07}_{-0.83}$ | $5.41^{+0.04}_{-0.33}$ |
2782 | $36.52^{+0.73}_{-0.54}$ | $3.75^{+0.11}_{-0.09}$ | $4.82^{+0.04}_{-0.03}$ |
2789 | $37.98^{+1.04}_{-1.19}$ | $4.31^{+0.05}_{-0.11}$ | $5.31^{+0.05}_{-0.05}$ |
2804 | $24.26^{+0.46}_{-0.96}$ | $3.07^{+0.16}_{-0.15}$ | $4.49^{+0.1}_{-0.11}$ |
2813 | $22.34^{+0.65}_{-0.62}$ | $3.8^{+0.06}_{-0.11}$ | $3.94^{+0.05}_{-0.05}$ |
2815 | $27.52^{+1.5}_{-1.15}$ | $4.01^{+0.07}_{-0.17}$ | $4.37^{+0.41}_{-0.4}$ |
2819 | $40.82^{+0.81}_{-1.38}$ | $3.62^{+0.09}_{-0.08}$ | $5.84^{+0.05}_{-0.07}$ |
2846 | $33.29^{+1.19}_{-1.35}$ | $4.18^{+0.06}_{-0.44}$ | $4.36^{+0.95}_{-0.95}$ |
2848 | $21.45^{+0.77}_{-0.46}$ | $3.59^{+0.24}_{-0.26}$ | $3.81^{+0.09}_{-0.08}$ |
2861 | $28.06^{+1.73}_{-3.0}$ | $4.04^{+0.07}_{-0.14}$ | $4.13^{+0.33}_{-0.34}$ |
2876 | $35.44^{+0.69}_{-1.23}$ | $3.85^{+0.2}_{-0.23}$ | $5.1^{+0.05}_{-0.07}$ |
2884 | $38.02^{+0.96}_{-1.04}$ | $3.6^{+0.07}_{-0.07}$ | $5.14^{+0.06}_{-0.06}$ |
2893 | $47.47^{+2.11}_{-2.92}$ | $4.1^{+0.26}_{-0.16}$ | $5.68^{+0.1}_{-0.12}$ |
2897 | $37.44^{+1.31}_{-1.35}$ | $4.3^{+0.05}_{-0.31}$ | $5.14^{+0.06}_{-0.06}$ |
2901 | $53.09^{+1.58}_{-6.53}$ | $4.5_{-0.5}$ | $5.58^{+0.05}_{-0.17}$ | grid boundary
2911 | $38.67^{+1.42}_{-1.31}$ | $3.51^{+0.09}_{-0.07}$ | $5.47^{+0.61}_{-0.61}$ |
2913 | $36.48^{+0.92}_{-0.85}$ | $3.96^{+0.15}_{-0.14}$ | $5.39^{+0.07}_{-0.06}$ |
2928 | $22.03^{+10.76}_{-0.85}$ | $3.84^{+0.4}_{-0.11}$ | $4.15^{+0.51}_{-0.31}$ |
2929 | $35.67^{+0.96}_{-0.62}$ | $3.61^{+0.16}_{-0.09}$ | $4.04^{+0.6}_{-0.6}$ |
2939 | $26.64^{+1.5}_{-4.34}$ | $3.89^{+0.17}_{-0.3}$ | $4.29^{+0.67}_{-0.67}$ |
2944 | $33.25^{+2.34}_{-1.77}$ | $4.19^{+0.05}_{-0.1}$ | $4.68^{+1.29}_{-1.29}$ |
2945 | $36.6^{+0.73}_{-0.62}$ | $3.77^{+0.1}_{-0.11}$ | $4.92^{+0.04}_{-0.04}$ |
2946 | $42.09^{+4.5}_{-3.04}$ | $3.89^{+0.23}_{-0.15}$ | $5.58^{+0.15}_{-0.11}$ |
2966 | $23.07^{+4.11}_{-1.69}$ | $3.91^{+0.11}_{-0.47}$ | $3.21^{+1.18}_{-1.18}$ |
2977 | $36.67^{+0.92}_{-0.65}$ | $3.8^{+0.15}_{-0.1}$ | $5.27^{+1.28}_{-1.28}$ |
2981 | $38.02^{+1.04}_{-1.31}$ | $4.14^{+0.12}_{-0.18}$ | $5.47^{+0.05}_{-0.06}$ |
2985 | $18.61^{+0.69}_{-0.81}$ | $3.42^{+0.22}_{-0.21}$ | $3.98^{+0.73}_{-0.73}$ |
2987 | $48.44^{+3.61}_{-18.49}$ | $3.81^{+0.13}_{-1.15}$ | $5.6^{+0.62}_{-0.66}$ |
3007 | $29.87^{+2.34}_{-3.08}$ | $3.3^{+0.48}_{-0.42}$ | $4.8^{+0.12}_{-0.15}$ | bad fit
3010 | $44.67^{+2.88}_{-3.0}$ | $3.59^{+0.1}_{-0.15}$ | $4.89^{+0.1}_{-0.11}$ | bad fit
3018 | $30.45^{+2.96}_{-4.04}$ | $4.16^{+0.07}_{-0.07}$ | $4.29^{+0.81}_{-0.81}$ |
3027 | $37.71^{+3.88}_{-6.38}$ | $3.2^{+0.24}_{-0.38}$ | $4.59^{+0.15}_{-0.23}$ | bad fit
3030 | $43.4^{+1.46}_{-1.85}$ | $4.4^{+0.05}_{-0.24}$ | $5.41^{+0.07}_{-0.08}$ |
3032 | $25.06^{+1.85}_{-2.0}$ | $4.02^{+0.04}_{-0.05}$ | $3.71^{+0.28}_{-0.28}$ |
3034 | $35.79^{+1.04}_{-0.81}$ | $3.6^{+0.1}_{-0.08}$ | $5.25^{+0.08}_{-0.07}$ |
3043 | $35.56^{+1.0}_{-0.69}$ | $3.53^{+0.15}_{-0.09}$ | $5.94^{+0.05}_{-0.04}$ |
3062 | $32.52^{+0.54}_{-0.5}$ | $3.2^{+0.06}_{-0.06}$ | $5.68^{+0.5}_{-0.5}$ |
3081 | $37.67^{+0.58}_{-0.88}$ | $4.12^{+0.11}_{-0.22}$ | $5.36^{+0.03}_{-0.04}$ |
3106 | $33.33^{+0.73}_{-0.85}$ | $4.06^{+0.16}_{-0.19}$ | $4.61^{+0.04}_{-0.05}$ |
3129 | $22.18^{+4.23}_{-1.19}$ | $3.8^{+0.24}_{-0.1}$ | $3.44^{+0.29}_{-0.2}$ |
3149 | $22.34^{+0.65}_{-0.92}$ | $3.8^{+0.07}_{-0.1}$ | $4.06^{+0.51}_{-0.52}$ |
3151 | $22.18^{+0.69}_{-0.96}$ | $3.83^{+0.08}_{-0.17}$ | $3.78^{+0.97}_{-0.97}$ | bad fit