# Towards Open-Domain Topic Classification

Hantian Ding1, Jinrui Yang1,2, Yuqian Deng1, Hongming Zhang1, Dan Roth1
1University of Pennsylvania, 2University of Melbourne
{hantian2, jinruiy, yuqiand, hzhangal<EMAIL_ADDRESS>

###### Abstract

We introduce an open-domain topic classification system that accepts user-
defined taxonomy in real time. Users will be able to classify a text snippet
with respect to any candidate labels they want, and get instant response from
our web interface. To obtain such flexibility, we build the backend model in a
zero-shot way. By training on a new dataset constructed from Wikipedia, our
label-aware text classifier can effectively utilize implicit knowledge in the
pretrained language model to handle labels it has never seen before. We
evaluate our model across four datasets from various domains with different
label sets. Experiments show that the model significantly improves over
existing zero-shot baselines in open-domain scenarios, and performs
competitively with weakly-supervised models trained on in-domain
data.111Interactive online demo at
https://cogcomp.seas.upenn.edu/page/demo_view/ZeroShotTC222Code and data
available at http://cogcomp.org/page/publication_view/980

##  1 Introduction

Text classification is a fundamental natural language processing problem, with
one of its major applications in topic labeling Lang (1995); Wang and Manning
(2012). Over the past decades, supervised classification models have achieved
great success in closed-domain tasks with large-scale annotated datasets Zhang
et al. (2015); Tang et al. (2015); Yang et al. (2016). However, they are no
longer effective in open-domain scenarios where the taxonomy is unbounded.
Retraining the model for every new label set often incurs prohibitively high
cost in the sense of both annotation and computation. By contrast, having one
classifier that is flexible with unlimited labels can save such tremendous
efforts while keeping the solution simple. Therefore, in this work, we build a
system for open-domain topic classification that can classify a given text
snippet into any categories defined by users.

At the core of our system is a zero-shot text classification model. While
supervised models are typically insensitive to class names, a zero-shot model
is usually label-aware, meaning that it can understand label semantics
directly from the name or definition of the label, without accessing any
annotated examples. Our model TE-Wiki combines a Textual Entailment (TE)
formulation with Wikipedia finetuning. Specifically, we construct a new
dataset that contains three million article-category pairs from Wikipedia’s
subcategory graph, and finetune a pretrained language model (e.g. BERT) to
predict the entailment relations between articles and their associated
categories. We simulate the diversity in open-domain classification with the
wide coverage of Wikipedia, while preserving label-awareness through an
entailment framework.

In our benchmarking experiments, TE-Wiki outperforms all previous zero-shot
methods on four benchmarks from different domains. It also shows competitive
performance against weakly-supervised models trained on in-domain data. By
learning from Wikipedia, our method does not require any data that is
specifically collected from the evaluation domains. On the other hand, since
our model is label-aware, it can flexibly classify text pieces into any labels
outside Wikipedia.

Finally, we compare our system against humans for further insights. We show
that even humans are sometimes confused by ambiguous labels through a
crowdsourcing study, which explains the performance gap between open-domain
and supervised classification. The gap is reduced significantly when label
meanings are clear and well aligned with the semantics of text. We also use an
example to illustrate the negative effect of a bad label name. Through the
analysis, we demonstrate the importance of choosing proper label names in
open-domain classification.

Figure 1: An overview of our open-domain topic classification system. Users
can choose multiple models (top), and define their own text input and
candidate labels (middle). Prediction results from different models are
displayed in the bar chart and the table (bottom).

##  2 Related Work

Open-domain zero-shot text classification was first studied in the NLP domain
in Chang et al. (2008) (under the name “dataless classification") as a method
that classifies solely based on a general knowledge source and does not
require any in-domain data, whether labeled or not. It was proposed to embed
both the text and labels into the same semantic space, via Explicit Semantic
Analysis, or ESA Gabrilovich and Markovitch (2007), and pick the label with
the highest relevance score. This idea was further extended to hierarchical
Song and Roth (2014) and cross-lingual Song et al. (2016) text classification.
Later on, Yin et al. (2019) called this protocol “label fully unseen" and
proposed an entailment approach to transfer knowledge from textual entailment
to text classification. It formulates an $n$-class classification problem as
$n$ binary entailment problems by converting labels into hypotheses and the
text into the premise, and selects the premise-hypothesis pair with highest
entailment score. More recently, another concurrent work Chu et al. (2021)
proposed to explore resources from Wikipedia for zero-shot text
classification, but with a different formulation.

There are many other methods that also require less labeling than supervised
classification, though in slightly different settings. For example, previous
works have explored to generalize from a set of known classes (with
annotation) to unknown classes (without annotation) using word embeddings of
label names Pushp and Srivastava (2017); Xia et al. (2018); Liu et al. (2019),
class correlation on knowledge graphs Rios and Kavuluru (2018); Zhang et al.
(2019), or joint embeddings of documents and labels Nam et al. (2016).
Besides, Weakly supervised approaches Mekala and Shang (2020); Meng et al.
(2020) learn from an unlabeled, but in-domain training set. Given a set of
pre-defined labels, a label-aware knowledge mining step is first applied to
find class-specific indicators from the corpus, followed by another self
training step to further enhance the model by propagating the knowledge to the
whole corpus. However, none of these approaches are suitable for building an
open-domain classification system. They either require domain-specific
annotation or knowing test labels beforehand.

##  3 System Description

Figure 2: An overview of our proposed TE-Wiki. Left: the data collection
process. For each of the top-level categories, we run DFS to find its
descendant categories as well as their member articles. These articles are
paired with the root category for model input. Right: the model architecture.
We use BERT for sequence classification. The article text is concatenated with
the category name to feed into a BERT encoder. The classification head takes
the output embedding of the "[CLS]" token to classify the input text-category
pair.

We present details about our open-domain topic classification system, starting
with an overview of our web interface, followed by the backend model.

###  3.1 User Interface

Figure 1 is a snapshot of our online demo. The system is supported by multiple
backend models for test and comparison. Among them, “Bert-Wiki", corresponding
to TE-Wiki in this paper, is the best-performing one in our evaluation. After
selecting the model(s), users can create their own taxonomy in the “Labels"
column, and input the text snippet. The system will then classify the text
with the user-defined taxonomy. Results are presented in two formats: a bar
chart and a ranking table. The table on the right provides a clear view of
rankings by each model, while the bar chart on the left is useful to compare
the scale of the scores from different models for different labels. These
scores, ranging from 0 to 1, are probabilities of the label being relevant to
the text, which we will explain further in the next section.

Consider the example in Figure 1. The input text is most relevant to
lifestyle, somewhat relevant to technology, and irrelevant to children, which
aligns with the prediction of our “Bert-Wiki" model.

###  3.2 TE-Wiki

We now describe our best performing model TE-Wiki. Previous work Yin et al.
(2019) has demonstrated that an $n$-way classification problem can be
converted into $n$ binary entailment problems. Specifically, we can use the
text as the premise, and candidate labels as the hypotheses, to generate $n$
statements “[Text]  Entails  [$\textup{Label}_{i}$]" for $i\in[n]$. The
motivation is that classification is essentially a special kind of entailment.
Suppose we want to classify a document into 3 classes: politics, business,
sports. We can ask three binary questions: “Is it about politics?”, “Is it
about business?”, “Is it about sports?”. By doing so, the model is no longer
constrained to a fixed label set, as we can always ask more questions to
handle new labels.

With the above framework, it is straightforward to train a model on an
entailment dataset (e.g. MNLI Williams et al. (2018), FEVER Thorne et al.
(2018), RTE Dagan et al. (2005); Wang et al. (2019).) and use it for
classification. However, this may not be the optimal choice as topic
classification only focuses on high-level concepts, while textual entailment
has a much wider scope and involves many other aspects (e.g., see Dagan et al.
(2013)). Therefore, we propose to construct a new dataset from Wikipedia with
articles as premises, and categories as hypotheses. Our desired training pair
should meet the following two criteria:

  1. 1.

The hypothesis is consistent with the premise, i.e. the categorization is
correct.

  2. 2.

The hypothesis should be abstract and concise to reflect the high-level idea
of the premise, rather than focus on certain details.

Directly using all the categories associated with an article satisfies the
first criterion, but fails with the second, as some of them do not represent
the article well. For example, the page Bill Gates is assigned
Category:Cornell family, which is correct about the person but probably not a
suitable label for the whole article. To resolve the issue, we instead use
higher-level categories on Wikipedia’s subcategory graph to yield better
hypotheses.

Input : Top-level category set $S$,

Wikipedia subcategory graph $\mathcal{G}$,

max search depth $r=2$;

Initialize $d(x,c)=\infty$ for any article $x\in X$ and $c\in S$. $M=\\{\\}$;

for  _$c$ in $S$_ do

$T$ = DFS$(c,\mathcal{G},r)$;

for  _$t\textup{ in }T\textup{.nodes}$_ do

for  _$x\textup{ in }t\textup{.articles}$_ do

$d(x,c)=\min\\{d(x,c),1+\textup{depth}(t)\\}$;

end for

end for

end for

for  _$x$ in $X$_ do

if  _$\min_{c\in S}d(x,c) <\infty$_ then

$P=\textup{argmin}_{c\in S}d(x,c)$;

for  _$c$ in $P$_ do

Add $(x,c,1)$ to $M$;

end for

Sample $c^{\prime}$ from $S-P$;

Add $(x,c^{\prime},-1)$ to $M$;

end if

end for

Output : $M$

Algorithm 1 Collect training data

The overview of TE-Wiki is illustrated in Figure 2. Specifically, we start
with a set of 700 top-level categories from Wikipedia’s overview
page333https://en.wikipedia.org/wiki/Wikipedia:Contents/Categories as roots.
For each of them, we run a depth-first search (DFS) to find its subcategories.
In our experiment, we set the max depth to 2 to ensure the subcategories found
are strongly affiliated with the root. We collect all member articles of
categories in the DFS tree, including both leaves and internal nodes, and pair
them with the root to construct positive examples. In case an article can be
reached from multiple root categories, we only pair it with the root(s) that
has the smallest tree distance to the article to ensure supervision quality.
Then for each article, we randomly choose a different category to construct a
negative example. While we have tried more sophisticated negative sampling
strategies with the aim to confuse the model, none of them makes a significant
improvement. Thus, we keep to this simple version. The final training set
$D=\\{(x_{i},c_{i},p_{i})_{i=1}^{n}\\}$ consists of 3-tuples such that $x_{i}$
is a Wikipedia article, $c_{i}$ is the corresponding high-level category name,
and $p_{i}\in\\{+1,-1\\}$ is the label. The procedure for constructing the
training set is summarized in Algorithm 1.

We then fine-tune the pre-trained BERT model Devlin et al. (2019) with the
collected dataset. Given a tuple $(x_{i},c_{i},p_{i})$, the concatenation of
$x_{i}$ and $c_{i}$ is passed to a BERT encoder, followed by a classification
head to predict whether the article $x_{i}$ belongs to the category $c_{i}$.
During test, (i) for the single-labeled case, we pick the label with the
highest predicted probability, (ii) for the multi-labeled case, we pick all
labels predicted as positive (i.e. probability $>0.5$). We do not use any
hypothesis template to convert label names into sentences as in Yin et al.
(2019), for consistency with training.

##  4 Evaluation

We evaluate all the backend models of our system on four classification
benchmarks to compare their performance. We also compare them against weakly-
supervised and supervised models to quantify how much we can achieve without
any domain-specific training data.

Dataset | #Classes | #samples
---|---|---
Yahoo Zhang et al. (2015) | 10 | 100,000
Situation Mayhew et al. (2019) | 12 | 3525
AG News Zhang et al. (2015) | 4 | 7,600
DBPedia Lehmann et al. (2015) | 14 | 70,000
Table 1: Dataset Statistics. Supervision Type | Methods | Yahoo | Situation | AG News | DBPedia
---|---|---|---|---|---
Zero-shot | Word2Vec Mikolov et al. (2013) | 35.7 | 15.6 | 71.1 | 69.7
ESA Chang et al. (2008) | 40.4 | 30.2 | 71.1 | 64.7
TE-MNLI Yin et al. (2019) | 37.9 | 15.4 | 68.8 | 55.3
TE-FEVER Yin et al. (2019) | 40.1 | 21.0 | 78.0 | 73.0
TE-RTE Yin et al. (2019) | 43.8 | 37.2 | 60.5 | 65.9
TE-Wiki | 57.3 | 41.7 | 79.6 | 90.2
Weakly-supervised | LOTClass Meng et al. (2020) | 54.7 | N/A | 86.4 | 91.1
Supervised | BERT Devlin et al. (2019) | 75.3 | 58.0 | 94.4 | 99.3
Table 2: Test results of all methods on four datasets. Compared with Word2Vec
and ESA, ESA-WikiCate is overall the best among the three embedding-based
methods. TE-WikiCate outperforms all other zero-shot methods across all four
datasets, and performs competitively against the weakly-supervised LOTClass.

###  4.1 Experiment setup

Datasets:  We summarize all test datasets in Table 1. For Yahoo! Answers, we
use the reorganized train/test split by Yin et al. (2019). All datasets are in
English. Among the four, Situation Typing is a multi-labeled dataset with
imbalanced classes, for which we report the weighted average of per-class F1
score. We refer readers to Yin et al. (2019) for the class distribution
statistics. The other three are single-labeled and class-balanced, and we
report the classification accuracy.

Models:  Apart from TE-Wiki, we run five zero-shot models for open-domain
evaluation, as well as a weakly-supervised and a supervised model for close-
domain comparison.

  * •

Word2Vec Mikolov et al. (2013): To measure cosine similarity between the
embedding vectors of text and label.

  * •

ESA Chang et al. (2008): Same as above, except using embeddings in Wikipedia
title space

  * •

TE-MNLI, TE-FEVER, TE-RTE Yin et al. (2019): Textual entailment models by
finetuning BERT on MNLI, FEVER, and RTE respectively.444In experiments, we
always use bert-base-uncased.

  * •

LOTClass Meng et al. (2020): A weakly-supervised method that learns label
information from unlabeled, but in-domain training data.

  * •

BERT Devlin et al. (2019): We finetune a supervised BERT on training data for
each dataset.

Implementation: We finetune the bert-base-uncased model on the Wikipedia
article-category dataset to train TE-Wiki. We removed 26 categories whose name
starts with "List of" from the 700 top-level categories, resulting in 674
categories as hypotheses and 1,367,784 articles as premises. The final
training set contains 3,387,028 article-category pairs. We set the max
sequence length to be 128 tokens and the training batch size to be 64. The
model is optimized with AdamW Loshchilov and Hutter (2019) with initial
learning rate as 5e-5. Since we do not have a development set in the zero-shot
setting, we train the model for 1500 steps to prevent overfitting. For all
zero-shot methods, we train once and evaluate the model on all test datasets.
For supervised and weakly-supervised methods, we train a different model for
each different dataset.

###  4.2 Result Analysis

The main results are presented in Table 2. We observe that TE-Wiki performs
the best among all zero-shot methods on all four datasets with different
labels and from different domains, demonstrating its effectiveness as an open-
domain classifier. It also performs closely with the weakly-supervised
LOTClass which is trained on in-domain data with known taxonomy, showing that
an open-domain zero-shot model can achieve similar accuracy as those domain-
specific classifiers. In particular, TE-Wiki outperforms LOTClass on Yahoo,
whose training set contains quite a few ambiguous examples. These examples can
have negative impact on self-training. On the other hand, our zero-shot model
does not rely on any domain-specific data, making it more robust against
imperfect data.

| Yahoo | Situation | AG News | DBPedia
---|---|---|---|---
TE-Wiki | 57.3 | 41.7 | 79.6 | 90.2
\- Overlapping categories | 59.4(+2.1) | 38.8(-2.9) | 79.7(+0.1) | 88.9(-1.3)
Removed training categories | 15 | 2 | 4 | 10
Overlapping test examples (%) | 100.0 | 24.0 | 100.0 | 50.0
Table 3: Performance before and after removing the overlapping categories, as
well as their difference. We also show the number of removed categories, and
the percentage of test documents that belong to the overlapping labels.

It is possible that some of the testing labels also appear in the Wikipedia
categories used for training.555Throughout this subsection, we use the word
”category” for the training set and ”label” for the testing sets. To ensure
the quality and fairness of our zero-shot evaluation, we remove the
overlapping categories from Wikipedia training data, and retrain the TE-Wiki
model for each test set. Specifically, we normalize labels and categories by
their lower-cased, lemmatized names, and perform a token-based matching. We
report in Table 3 the performance before and after deduplication.

We find that deduplication has little or even positive influence on
performance, which shows that TE-Wiki does not rely on seeing test labels
during training. In particular, the performance on Yahoo gets improved with
deduplication. We suspect that exact match between training and testing labels
can lead to overfitting, since the same label may have different meanings
under different context. Notice that this study is only for justifying our
zero-shot evaluation. For real-word applications, excluding overlapping
categories is neither necessary nor feasible as users do not know the test
labels beforehand in zero-shot scenarios.

###  4.3 Early stopping and knowledge transfer

To study the convergence of our model, we sample a small dev set of 1000
examples from Yahoo’s original validation set. During training, we find that
with 25 steps the TE-Wiki model already achieves a reasonably good performance
on the dev set. Further training for longer steps yields some, but not
significant gains. Since the model has only seen $25\times 64=1600$ examples
at that point, there is little chance for the model to acquire label specific
knowledge with such a small amount of data. Hence, we believe that during the
early steps, the model actually learns “what topic classification is about",
while the knowledge specific to different labels has already been implicitly
stored in the pretrained BERT encoder. The category prediction task takes a
minor role in transferring world knowledge. Rather, it teaches the model how
to use existing knowledge to make a good inference.

##  5 Importance of label names

| TE-Wiki | Human | Supervised
---|---|---|---
Yahoo | 58.9 | 64.8 | 77.4
Yahoo-5 | 82.1(+23.2) | 88.1(+23.3) | 89.4(+12.0)
AG News | 79.0 | 80.2 | 94.7
AG News-5 | 86.4(+7.4) | 90.8(+10.6) | 96.4(+1.7)
Table 4: Classification accuracy on crowdsourcing datasets. Yahoo-5 and AG
News-5 count only examples for which all five workers choose the same label.

Since zero-shot classifiers understand a label by its name, the quality of
label names can be a important performance bottleneck in designing open-domain
text classification systems. To study this, We conduct crowdsourcing surveys
on subsets of Yahoo and AG News. For each dataset, we randomly sample 1,000
documents while preserving class balance. Every document is independently
annotated by five workers. In the survey question, we only provide the
document to be classified and names of candidate labels, without giving
workers examples for each class. We consider an example to be correctly
classified by humans only if at least three workers choose the gold label.
Details about the survey are in Appendix.

We summarize the results in Table 4. Row 1&3 are classification accuracy on
the whole crowdsourcing datasets, and row 2&4 are on subsets of examples where
all 5 workers choose the same label. We observe that when including all
examples, both TE-Wiki and humans perform much worse than the supervised
method. The supervised approach has the advantage that it learns data-specific
features to resolve ambiguity among different classes. On the other hand,
humans only make judgements based on their understanding of the labels and a
stand-alone test document, and so does our zero-shot algorithm. Ideally, this
task should not be difficult for humans as long as the labels properly
describe the text topics. However, in some cases the labels could be ambiguous
and confusing. Figure 3 shows an example of a bad label name leading to a
mistake. The word “Reference" in the correct label actually means “quoting
other people’s words". However, it is hard for an ordinary person to
understand the meaning without any example as illustration. 4 out of 5
annotators instead chose “Entertainment & Music" due to the movie “Star Wars".
By contrast, the supervised model has no difficulty in making the correct
decision because it has seen plenty of quotation examples during training and
can easily capture the useful pattern like “Who said XXX". The main reason for
humans’ confusion here is that the label name does not directly reflect the
semantics of the text. A better description of the class should be provided
for classification without examples.

We also calculate the accuracy on examples where all 5 workers agree, as in
row 2&4 in Table 4. We believe the high inter-annotator agreement here
indicates a better alignment between the semantics of text and label. We find
a significant improvement of human performance on these less ambiguous cases.
The same happens to our zero-shot model, but the supervised method benefits
much less. Consequently, the performance gap between humans and the supervised
model is also getting closer, which demonstrates that ambiguous labels have a
strongly negative impact on classification. Therefore, we believe picking good
labels is crucial for open-domain topic classification.

Figure 3: An example with a bad label name. Annotators are confused by the
word “Reference".

##  6 Conclusion

We introduce a system for open-domain topic classification. The system allows
users to define customized taxonomy and classify text with respect to that
taxonomy at real time, without changing the underlying model. To build a
powerful model, we propose to utilize Wikipedia articles and categories and
adopt an entailment framework for zero-shot learning. The resulting TE-Wiki
outperforms all existing zero-shot baselines in open-domain evaluations.
Finally, we demonstrate the importance of choosing proper label names in open-
domain topic classification through a crowdsourcing study.

## Acknowledgements

This work was supported by Contracts FA8750-19-2-1004 and FA8750-19-2-0201
with the US Defense Advanced Research Projects Agency (DARPA). Approved for
Public Release, Distribution Unlimited. The views expressed are those of the
authors and do not reflect the official policy or position of the Department
of Defense or the U.S. Government.

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##  Appendix A Crowdsourcing Setup

We conduct crowdsourcing annotations for 1000 documents sampled from the
Yahoo! Answers dataset and another 1000 from the AG News on Amazon Mechanical
Turk (AMTurk). Both crowdsourcing subsets preserve the class-balance as in the
original datasets. We avoid using long documents so that each document
contains no more than 512 characters. The 1000 samples are split into 40
assignments, each containing 25 examples. We request 5 AMTurk workers for
multiple-choice questions on each assignment. In order to ensure the response
quality, we use anchor examples and gold annotations from the original
datasets to filter out low-quality answers. Specifically, in each assignment
we insert two anchor examples that we believe are easy enough for workers to
choose the correct answer as long as they pay attention. We reject a
submission if a worker’s classification accuracy against gold annotations is
below 30%, or both anchor examples are wrongly classified. With a small
initial pilot, we estimate the average working time for labeling 25 examples
to be 22 minutes, and we set the pay rate to be $1.5 per assignment for each
valid submission. The overall cost is $300 for 200 valid submissions for each
dataset.


# Towards early-type eclipsing binaries as extragalactic milestones: III.
Physical properties of the O-type eclipsing binary OGLE LMC-ECL-21568 in a
quadruple system 111Based on observations collected at the European Southern
Observatory, Chile 222This paper includes data gathered with the 6.5m Magellan
Clay Telescope at Las Campanas Observatory, Chile.

Mónica Taormina Centrum Astronomiczne im. Mikołaja Kopernika PAN, Bartycka 18,
00-716 Warsaw, Poland  R.-P. Kudritzki  LMU München, Universitätssternwarte,
Scheinerstr. 1, 81679 München, Germany  Institute for Astronomy, University of
Hawaii at Manoa, Honolulu, HI 96822, USA  B. Pilecki Centrum Astronomiczne im.
Mikołaja Kopernika PAN, Bartycka 18, 00-716 Warsaw, Poland  G. Pietrzyński
Centrum Astronomiczne im. Mikołaja Kopernika PAN, Bartycka 18, 00-716 Warsaw,
Poland  I. B. Thompson  Carnegie Observatories, 813 Santa Barbara Street,
Pasadena, CA 91101-1292, USA  J. Puls  LMU München, Universitätssternwarte,
Scheinerstr. 1, 81679 München, Germany  M. Górski Centrum Astronomiczne im.
Mikołaja Kopernika PAN, Bartycka 18, 00-716 Warsaw, Poland  B. Zgirski
Universidad de Concepción, Departamento de Astronomía, Casilla 160-C,
Concepción, Chile  D. Graczyk Centrum Astronomiczne im. Mikołaja Kopernika
PAN, Rabiańska 8, 87-100 Toruń, Poland  W. Gieren  Universidad de Concepción,
Departamento de Astronomía, Casilla 160-C, Concepción, Chile  G. Hajdu Centrum
Astronomiczne im. Mikołaja Kopernika PAN, Bartycka 18, 00-716 Warsaw, Poland

(Accepted April 3, 2024)

###### Abstract

We present the results from a complex study of an eclipsing O-type binary
(Aa+Ab) with the orbital period $P_{A}=3.2254367$ days, that forms part of a
higher-order multiple system in a configuration (A+B)+C. We derived masses of
the Aa+Ab binary $M_{1}=19.02\pm 0.12\,M_{\odot}$, $M_{2}=17.50\pm
0.13\,M_{\odot}$, radii $R_{1}=7.70\pm 0.05\,R_{\odot}$, $R_{2}=6.64\pm
0.06\,R_{\odot}$, and temperatures $T_{1}=34250\pm 500$ K, $T_{2}=33750\pm
500$ K. From the analysis of radial velocities, we found a spectroscopic orbit
of A in the outer A+B system with $P_{A+B}=195.8$ days ($P_{A+B}/P_{A}\approx
61$). In the O-C analysis, we confirmed this orbit and found another component
orbiting the A+B system with $P_{AB+C}=2550$ days ($P_{AB+C}/P_{A+B}\approx
13$). From the total mass of the inner binary and its outer orbit, we
estimated the mass of the third object, $M_{B}\gtrsim 10.7M_{\odot}$. From the
light-travel time effect fit to the O-C data, we obtained the limit for the
mass of the fourth component, $M_{C}\gtrsim 7.3M_{\odot}$. These extra
components contribute to about 20% to 30% (increasing with wavelength) of the
total system light. From the comparison of model spectra with the multiband
photometry, we derived a distance modulus of 18.59 $\pm$ 0.06 mag, a reddening
of 0.16 $\pm$ 0.02 mag, and an $R_{V}$ of $3.2$. This work is part of our
ongoing project, which aims to calibrate the surface brightness-color relation
for early-type stars.

binaries: eclipsing — stars: early-type — stars: fundamental parameters —
Magellanic Clouds

††journal: ApJ††facilities: VLT:Kueyen (UVES), Magellan:Clay (MIKE and
MagE)††software: ESO Reflex (Freudling et al., 2013,
http://www.eso.org/sci/software/esoreflex/), PHOEBE (Prša & Zwitter, 2005,
http://phoebe-project.org/1.0),
RaveSpan (Pilecki et al., 2017,
https://users.camk.edu.pl/pilecki/ravespan/index.php), FASTWIND (Puls et al.,
2005; Rivero González et al., 2012)

##  1 Introduction

OGLE LMC-ECL-21568 (henceforth BLMC-03) is located in the region known as the
Lucke-Hodge 101 OB association (Lucke & Hodge, 1970) or NGC 2074 in the Large
Magellanic Cloud (LMC). It was included in photographic, photometric, and
spectroscopic studies of the region (e.g. Westerlund, 1961; Testor & Niemela,
1998) and observed in various photometric surveys of the LMC (e.g. Massey,
2002; Zaritsky et al., 2004).

In the first paper of a series, regarding the program to find eclipsing
massive binaries in the LMC, Massey et al. (2012) presented a spectroscopic
follow-up of BLMC-03 (under the ID [M2002] LMC 172231 from the catalog of
Massey 2002) and classified it spectroscopically as O9V+O9.5V. They measured
the radial velocities of the components and obtained an orbital solution.
Combining these data with their new V-band photometry Massey et al. (2012)
provided the first characterization of the system. They found the binary’s
light and radial velocity curves to be consistent with a circular orbit and
assumed zero eccentricity for the rest of the study. We note that in their
orbital solution, radial velocities show relatively high scatter compared to
quoted errors. Moreover, the residuals for the two components are correlated,
indicating a possible unaccounted additional variability in the system.
Finally, they found almost identical masses for the components, $17.5\pm 0.3$
$M\odot$ and $17.6\pm 0.3$ $M\odot$, and stellar radii of $7.0\pm 0.3$
$R\odot$ and $6.5\pm 0.3$ $R\odot$ for the primary and secondary star,
respectively.

In our ongoing study of a carefully selected sample of early-type binaries in
the LMC, we aim to obtain accurate and precise physical parameters of their
components to calibrate the surface brightness-color (SBC) relation for hot
stars. We have already analyzed two early-type systems in this galaxy
(Taormina et al. 2019 \- Paper I, Taormina et al. 2020 \- Paper II), and
BLMC-03 is the next one on our list of the well-detached systems in the LMC.
As we need the highest accuracy and precision possible and a light curve
solution in the K-band, we decided to reanalyze this system using the
currently available photometric data in visual and near-infrared passbands
together with our new high-resolution spectra.

The paper is organized as follows. In Section 2 we present the data used in
the study, in Section 3, we describe the analysis and its direct results, and
in Section 3.4 we present a detailed spectroscopic analysis. In Section 4, we
derive the final physical parameters of the binary components and orbital
properties of the system. Our conclusions are presented in Section 5.

Table 1: General information about the system parameter | unit | value
---|---|---
our ID |  | BLMC-03
OGLE ID |  | LMC-ECL-21568
$\alpha_{2000}$ | [hh:mm:ss.ss] | 05:38:58.08
$\delta_{2000}$ | [$\pm$dd:mm:ss.s] | -69:30:11.3
orbital period | [days] | 3.225450
$I_{\mathrm{C}}$ \- band | [mag] | 14.212
$V$ \- band | [mag] | 14.125
$V-I_{\mathrm{C}}$ | [mag] | -0.087
spectral type |  | O9V+O9.5V aaMassey et al. (2012)

##  2 Observational Data

###  2.1 Photometry

As in our previous works (Paper I and II), we made use of the photometric data
from the published catalogs of eclipsing binary stars in the LMC (Graczyk et
al., 2011; Pawlak, 2016), based on phases 3 and 4 of the Optical Gravitational
Lensing Experiment survey (OGLE, Udalski, 2003; Udalski et al., 2015). We used
in total 436 $I_{\mathrm{C}}$-band observations from the OGLE-III and 383 from
the OGLE-IV survey as well as 45 and 72 V-band measurements from these two
OGLE surveys, respectively. To have a better time coverage, we have also
included in our analysis observations from the EROS-2 survey (Tisserand et
al., 2007), using 244 data points in the EROS R-band, which is almost
identical to the Johnson-Cousins $I_{C}$ filter. The light curves have been
cleaned of outliers and any significant long-term brightness trends.

Near-infrared K-band data is essential for our goal of improving the
calibration of the SBC relation. For that reason, in the analysis of BLMC-03
we used 14 measurements from the VISTA Magellanic Clouds IR photometric survey
(VMC, Cioni et al., 2011) together with 12 observations that we collected
using the SOFI imaging camera on the 3.58m ESO New Technology Telescope (NTT,
Moorwood et al., 1998) at the La Silla Observatory, Chile. The latter were
obtained using the Large Field setup with a field of view of 4.9’ x 4.9’ at a
scale of 0.288” per pixel. The gain and readout noise were 5.4e/ADU and 0.4e,
respectively. Deep J- and Ks-band observations of our target field were
acquired under good seeing conditions during several nights between December
2015 and December 2018. To account for the frequent sky-level variations in
the infrared spectral region, especially in the Ks-band, the observations were
performed with a dithering technique. For the Ks and J-band observations, we
took over 10 exposures333the exact number depended on the observing conditions
at a given pointing and moved the telescope to another position selected
randomly within a 25” x 25” square. Between 15 and 25 such dithering positions
were obtained for the Ks-band, and between 11 and 15 for the J-band.

To reduce the data we followed the procedure described in Pietrzyński & Gieren
(2002). The sky was subtracted from the images with a two-step process
implying the masking of the stars with the xdimsum IRAF package. The
individual images for each field and filter were then flat-fielded and stacked
into a final composite image. To obtain the light curve, the coordinates of
all stars on separate images were crossmatched using DAOMATCH and DAOMASTER
(Stetson, 1994) programs. The differential brightness of the eclipsing system
was calculated by comparison with the selected sample of 13 comparison stars
in the J-band, and 9 comparison stars in the Ks-band in each field. To
estimate the photometric accuracy of this procedure, we measured the rms of
brightness of a few random bright stars in the field (presumably constant
stars), and obtained rms of 0.008 mag for the J-band and 0.006 mag for the Ks-
band. For 6 photometric nights, we observed a significant number (8-12) of
photometric standard stars from the United Kingdom Infrared Telescope (UKIRT)
system (Hawarden et al. 2001) at a variety of air masses and spanning a broad
range in colors. It allowed us to calibrate brightness on the UKIRT standard
system, with the calibration uncertainty of 0.012 mag in the J-band, and 0.013
mag in the Ks-band.

In Fig. 1, we show the neighborhood of BLMC-03 as seen on one of the SOFI Ks-
band frames. The system can be easily resolved from all the other identified
nearby stars.

Figure 1:  A part of a SOFI $K_{\mathrm{S}}$-band frame showing a neighborhood
of the BLMC-03 system, which is marked with a red circle.

Unfortunately, because of randomly appearing patterns in one quadrant of the
camera, some of our measurements had to be discarded. However, joining the
SOFI data with the VMC Ks photometry, we have both eclipses covered and the
flat parts of the light curves (out of the eclipses) well-defined. As far as
we know, our final light curve with 26 data points is the best-covered Ks-band
light curve for an extra-galactic eclipsing binary system.

###  2.2 Spectroscopy

We acquired high-resolution optical echelle spectra of BLMC-03 using the UVES
spectrograph (Dekker et al., 2000) mounted on the Very Large Telescope (VLT)
at Paranal Observatory and the MIKE spectrograph (Bernstein et al., 2003) on
the Magellan Clay Telescope at the Las Campanas Observatory (LCO), both in
Chile. The setup of the instruments and the reduction procedure are the same
as described in Paper I. A total of 17 high-resolution spectra (9 UVES + 8
MIKE) were obtained with a typical signal-to-noise ratio near $4000$ Å of
about 50-60 for UVES and 30-40 for MIKE.

To increase the time coverage of observations, we used in addition 7 archival
spectra collected with the Magellan Echellette (MagE) spectrograph, also
mounted on the Clay telescope at the LCO. These data were used previously by
Massey et al. (2012), where a detailed description of the instrument setup,
reduction, and calibration can be found. Thus in total 24 spectra were used
for the determination of radial velocities (RVs) in a homogeneous way and in
the main analysis of the binary system.

We note that Massey et al. (2012) provide 4 more velocity determinations,
which come from the analysis of their spectra acquired with the Inamori-
Magellan Areal Camera and Spectrograph (IMACS) instrument on the Baade
Telescope at the LCO. We took these data into account in the analysis of the
outer (A+B) orbit but did not use them to obtain the final results (see
section 3.1.1 for details).

##  3 Analysis

The analysis of BLMC-03 is similar to those performed in our previous papers
(I and II) with the difference of taking into account the presence of other
bodies (see Section 3.1.1 and 3.3 below). Similarly to the study of BLMC-02
described in Paper II we also used NLTE spectral analysis to obtain
temperatures of the components and an independent estimate of their luminosity
ratio.

Following the typical process for eclipsing binaries, we model the light and
radial velocity curves to characterize the system and obtain the physical
parameters of its components. Before the main analysis described in detail
below, preliminary fits to the available data were done to obtain an
approximate characterization of the system and the stars and identify possible
complications. The main analysis consists of the determination of radial
velocities of the primary and secondary components (free of higher-level
orbital motion) and simultaneous modeling of the light and RV curves corrected
for the light-travel time effect. This is followed by the NLTE spectral
analysis. Finally, from the combined results, the physical parameters of the
components are derived.

###  3.1 Radial velocities and orbital solution

We started our analysis with the determination of radial velocities of the
primary and secondary components of BLMC-03. For that, we applied the
Broadening Function technique (BF; Rucinski, 1992, 1999) to the reduced one-
dimensional spectra using the RaveSpan code (Pilecki et al., 2017). As a
template spectrum, we used a spectrum of a standard star from the ESO Data
Archive, that was selected to match the spectral type of the components.

Once we had extracted the radial velocities from the spectra, we modeled the
RV curve taking into account the orbital elements of a spectroscopic binary:
period ($P_{A}$), the orbital semi-amplitudes ($K_{A}$), the velocity of the
center-of-mass ($V_{A}$), the reference time ($T_{0,A}$), the orbital
eccentricity ($e_{A}$) and the argument of periastron passage of the primary
($\omega_{A}$). We fitted a model to the observed RV curve of each component
of the system and obtained a preliminary orbital solution.

####  3.1.1 Third body

In the preliminary RV curve analysis, we detected systematic and correlated
changes in the residual RVs for the primary and secondary components with a
period of about 200 days. This was a clear indication of the inner (Aa+Ab)
system orbiting the common center of mass (COM) of the higher-order system. As
from the light curve analysis, we also obtained significant values for the
third light444The parameter known as a ”third light” in binary modeling
describes any additional light that does not come from the components of the
eclipsing binary system. in all considered bands, we concluded that the system
is at least a triple, and included the presence of a third body in the RV
analysis. In the final model of the Aa+Ab binary, we thus included additional
parameters that describe the orbital motion of this binary around the A+B
system COM. We then fitted this model to the measured velocities of the
primary and secondary components.

Figure 2: Radial velocity of the inner (Aa-Ab) system center-of-mass and the
range of possible velocities of the tertiary component (assuming it is single
for the superior limit). Velocities of the primary and secondary components
are residuals after the subtraction of RVs from the inner binary model.

To increase the number of data points, we considered adding to the analysis
radial velocities measured by Massey et al. (2012) from their spectra obtained
with IMACS as mentioned in Section 2.2. These radial velocities were measured
using a low number of spectral lines: 1, 3, 4, or 6 depending on the spectrum
(see table 6 of Massey et al., 2012). We considered the measurement using just
one line as not reliable and indeed a quick check showed the difference
between the primary and secondary velocities to be about 50 km s-1 too low.
The uncertainties for them are also not provided. We thus used in the analysis
only measurements based on at least 3 lines, with determined uncertainties.
After doing the fit, we noted however that these literature RVs, although
being marginally consistent within error bars, are all lower than the best
model by about 15-20 km s-1. Moreover, this shift is also seen when comparing
one of these points with our two measurements done for spectra taken 5 days
later. As the data we used for our RV measurements dominate in number and
quality the exclusion of these external measurements leads to practically the
same results regarding the A+B system (amplitude change of 0.3$\sigma$), and
more importantly, negligible changes in the parameters of the inner one (below
0.2$\sigma$). Taking all of this into account and to maintain the homogeneity
of measurements we decided to use, here and in the rest of the analysis, only
RV measurements determined by us (using uniformly the BF method) from the
UVES, MIKE, and MagE spectra. The final orbital solution for the outer system
is shown in Fig. 2 and the properties of its orbit are given in Table 2.

Table 2: Orbital elements of the outer A+B orbit of BLMC-03

parameter | unit | value
---|---|---
$P_{B}$ | [days] | 195.78 $\pm$ 0.13
$T_{0,B}$ | [days] | 8038.3 $\pm$ 1.0
$e_{B}$ |  | 0.086 $\pm$ 0.024
$\omega_{B}$ | [deg] | 123 $\pm$ 15
$K_{A}$ | [km s-1] | 30 $\pm$ 0.6
$K_{B,min}$ | [km s-1] | 69∗
$K_{B,max}$ | [km s-1] | 103

**footnotetext: Assuming the third object is a single star.

The inclusion of the third body in the orbital solution has decreased the
scatter (rms) of residuals from about 22 to 4 km s-1. The orbital period of
the outer system converged firmly at $P_{A+B}=195.78\pm 0.13$ days, resulting
in the period ratio between the outer and inner system of
$P_{A+B}/P_{A}\approx 61$. A mass estimate for the tertiary will be derived
later, once we have the final masses of the inner components. The main result
from this part of the analysis is radial velocities of the primary and
secondary components free of additional orbital movement around the outer A+B
system COM and a preliminary orbital solution, which will be used as an input
in the final modeling.

####  3.1.2 Stellar rotation

From the BF method, the projected rotation velocities of the components of the
system ($v_{1,2}\sin i$) were also obtained. The broadening function response
for one of the spectra is presented in Fig 3. We have determined $v_{1,2}\sin
i$ for various spectra taken close to the orbital quadratures, where the
profiles are well separated, and taken an average. To obtain the true rotation
velocities we assume the inclination of the rotation axes to be the same as
the orbital inclination and correct measured velocities for the rotation
velocity of the template star. We did not correct for instrumental broadening
because it is negligible compared with the measured velocities. The rotation
velocities, obtained as described, are about 70 km s-1 and 80 km s-1 for the
primary and secondary components, respectively.

The final values are later used in the modeling in the Phoebe code (Prša &
Zwitter, 2005) as a synchronization parameter ($F$), defined as the ratio
between the rotational and orbital angular velocity. This information is
important for the modeling, as rotation changes the shape of stars, which
influences the light (mostly) and radial velocity (to a lower extent) curves.
For both components measured rotational velocities are sub-synchronous
($F_{1}$=0.61, $F_{2}$=0.75). This means that either their rotation is slower
than the angular orbital motion or their rotation axes have inclinations lower
than the orbital one. Note that we do not compare our rotation velocities with
a pseudo-synchronous velocity (defined by the orbital angular velocity at
periastron) as the difference between the synchronous and pseudo-synchronous
velocity for very low eccentricity of BLMC-03 is insignificant.

Figure 3: An example of Broadening Function profile for one of the analyzed
spectra. Apart from radial velocities, rotational velocities are measured,
which are lower than typical for early-type stars.

###  3.2 Light and RV curve modeling

The rest of the analysis closely resembles the one for BLMC-02 described in
Paper I and II, where a detailed description is provided. In short, we have
used a Python wrapper to the Phoebe 1 code to obtain a simultaneous solution
to the observed light and radial velocity curves. For that, we used a Markov
chain Monte Carlo analysis which provides also reliable uncertainties and lets
us find correlations between fitted parameters. The initial parameter values
were taken from the aforementioned preliminary low-resolution analysis. A
slight difference from the standard analysis was the usage of radial
velocities of the components corrected for the orbital motion around the outer
system’s center of mass (in comparison to using the velocities directly).

As typical for well-detached, non-totally eclipsing systems we found a slight
degeneracy between the parameters of the component stars. This can be seen,
for example, in the correlation between the radius of the primary ($R_{1}$)
and the luminosity ratio ($L_{2}/L_{1}$) in Fig. 4 (top). The solution is
well-defined (with small, symmetric uncertainties) within 1$\sigma$ but
strongly asymmetric for 3-$\sigma$ uncertainties as seen in the $\chi^{2}$
plot for $R_{1}$ in the bottom panel. However, we note that along the
$L_{2}/L_{1}$ \- $R_{1}$ relationship (top panel) the flux ratio $F_{2}/F_{1}$
= $(R_{1}/R_{2})^{2}(L_{2}/L_{1})$ remains constant. We measure $F_{2}/F_{1}$
= 0.971$\pm$0.002 in the V-band and 0.968$\pm$0.004 in both Ic and Ks bands.
This flux ratio will be used to constrain the effective temperature difference
between the primary and secondary components in the non-LTE spectral analysis
described in Section 3.4, which in turn will be used to obtain the final, non-
degenerate solution.

Figure 4: Top: Results from the Markov chain Monte Carlo analysis showing a
correlation between the radius of the primary and the luminosity ratio.
Bottom: The $\chi^{2}$ values vs. radius of the primary for the models
calculated in the analysis. Figure 5: $I_{\mathrm{C}}$-band light curve for
BLMC-03 not taking into account the presence of the 4th component. The best
model fit is shown as a red line. A high residual scatter can be seen around
eclipses.

###  3.3 O-C analysis

Looking at the light curve models obtained in the previous section we found a
significant residual scatter around eclipses (see Fig. 5). To find out the
cause of this scatter and trying to independently check the orbital parameters
of the third body (B), we performed the O-C analysis similar to that described
in Pilecki et al. (2021) using the long time-span light curves from the EROS-2
and OGLE projects. The strongest variation seen in the O-C diagram (see Fig.
6) indeed looked like caused by the orbital motion but to our surprise, its
period was determined as 2550 $\pm$ 30 days. The shorter-period variability
previously detected in RVs was also found with a period of $195.3\pm 0.5$ days
but with much lower amplitude, consistent with the spectroscopic orbit
determined in Section 3.1.1 ($K_{A,O-C}=26\pm 3$ compared to $K_{A,RV}=30.0\pm
0.6$). This variability is best seen in the dataset of OGLE-IV (HJD $>2455000$
days) which has the highest density and quality of measurements. Both
variabilities can be described by the light-travel time effect (LTTE), as is
shown by the red line fit in Fig. 6. Apart from that, no other secular,
cyclic, or erratic changes were detected.

With these results, we could not only confirm the existence and the orbital
period of the third component found from the analysis of radial velocities but
also the presence of a fourth component. The system is thus at least a
quadruple with the configuration (Aa+Ab)+B+C. Because of a wider orbit, the
influence of the 4th body (C) on the phase shift of the eclipsing variability
of the A system is much stronger than that of the 3rd body (B). To account for
this periodic phase shift we subtracted it from the light and radial velocity
curves by adjusting the measurement time by the value obtained from the LTTE
fit. We then modeled these corrected data and obtained new model fits. As can
be seen in Fig. 7, the extra scatter in the eclipses disappeared almost
completely. As the depths and shape of the eclipses are now better defined,
some parameters, like the radius, temperature ratio, and the third light
parameter changed slightly. The final fitted values are close to the ones
determined before (within uncertainties) but are more precise. The degeneracy
also decreased. The final radial velocity curve is shown in Fig. 8.

Figure 6: O-C diagram showing the phase shifts between the data from around
the given date and the average model. The influence of two extra components
can be seen. The red line is the fit to the LTTE induced by the 3rd and 4th
body.

Figure 7: $I_{\mathrm{C}}$\- and $K_{\mathrm{S}}$-band light curves for
BLMC-03 corrected for the gravitational influence of the 4th component. The
residual scatter around eclipses disappeared almost completely. In the right
panel, VMC and SOFI data are represented by points and crosses, respectively.
Figure 8: Orbital solution for the inner binary system. Measured radial
velocities for the primary (blue) and secondary (red) are shown with residuals
in the upper panel.

###  3.4 Spectral analysis and reddening determination

The purpose of the non-LTE spectral analysis of the observed binary is
primarily to determine the effective temperatures of the components and
interstellar reddening and extinction. This will enable us to discuss the
physical nature and evolutionary status of the components and to constrain
interstellar reddening and extinction, providing intrinsic colors and surface
brightnesses as well as an independent estimate of the distance. We also used
this analysis to independently find the best solution along the slight
degeneracy described in Section 3.2.

We proceed in a very similar way as in Paper II. We use the non-LTE model
atmosphere code FASTWIND (Puls et al., 2005; Rivero González et al., 2012) and
calculate normalized flux spectra F1,λ and F2,λ for the primary and secondary
star, respectively, and then combine those to a composite spectrum through

$F_{\lambda}=w_{1}F_{1,\lambda}(T_{1},log~{}g_{1},v_{1})+w_{2}F_{2,\lambda}(T_{2},log~{}g_{2},v_{2})+w_{3}.$
(1)

$T_{i}$, $log\,g_{i}$ and $v_{i}$ denote effective temperature, gravity, and
radial velocity of the primary and secondary. The weight $w_{3}$ accounts for
the contribution of the additional light (the third light parameter in the
binary model) described in the previous sections and is defined as

$w_{3}={L_{3}\over L_{1}+L_{2}+L_{3}},$ (2)

where $L_{1}$ and $L_{2}$ are the V-band luminosities of the primary and
secondary, and $L_{3}$ is the total luminosity of all additional components.
From the photometry fits discussed above we obtain $w_{3}=0.19\pm 0.01$. The
weights $w_{1}$ and $w_{2}$ represent the continuum contributions by the
primary and secondary, respectively, and are given by

$w_{1}={L_{1}\over L_{1}+L_{2}+L_{3}}={1-w_{3}\over
1+(R_{2}/R_{1})^{2}(F_{2}/F_{1})}$ (3)

and

$w_{2}={L_{2}\over L_{1}+L_{2}+L_{3}}=1-w_{3}-w_{1}.$ (4)

We use the observed flux ratio given in section 3.2 for the calculation of
$w_{1}$ and $w_{2}$, together with the primary and secondary radii and stellar
gravities along the solutions seen in Fig. 4 as obtained from the light curve
and radial velocity curve modeling described in previous sections.

Figure 9: Fit of HeI lines with the best composite binary model spectrum
obtained from the $\chi^{2}$-analysis. The upper figure shows HeI 5048 (left)
and HeI 4143 (right) and the lower figure HeI 4713 (left) and HeI 4387
(right). The upper row in each figure corresponds to the superposition of
three UVES spectra around phase 0.75 and the lower row to three MIKE spectra
at phase 0.2. The line contributions from the primary and secondary are
indicated by (1) and (2), respectively.

Figure 10: Fit of HeII lines. The upper figure shows the MIKE spectrum of HeII
4541 (left) and the UVES spectrum of HeII 4200. The lower spectrum displays
fits of the UVES (upper row) and MIKE (lower row) spectra of HeII 5411.

In the first step, we determine the effective temperature difference $\Delta
T=T_{1}-T_{2}$ between primary and secondary. We use FASTWIND model SEDs to
calculate stellar V-band fluxes and compare their ratio with the observed
value. For the gravities from the binary model (as mentioned above) and with
temperatures between 30000 to 39000K the observed ratio leads to $\Delta
T=500\pm 100$ K. We then calculate a grid of composite spectra close to binary
quadrature with this temperature difference $\Delta T$ varying the primary
temperature between 31000 to 39000 K for each pair of radii and gravities. As
in Paper II, we adopt four values of the helium abundance N(He)/N(H), 0.08,
0.09, 0.1, and 0.15, two values for the microturbulence vturb = 10 and 15 km
s-1 and metallicity of [Z] = log Z/Z⊙ = -0.35 (Urbaneja et al., 2017). We
calculate normalized spectra with line profiles accounting for the observed
rotational velocities and the spectral resolution of the spectrographs by
convolving with the corresponding broadening profiles.

Figure 11: Probability distributions (blue circles) and Gaussian fits (red)
obtained from the Monte Carlo fit of observed colors as described in the text.
The red curves are Gaussian fits. Top left: N(E(B-V)), top right: N(RV),
bottom left: N(AV), bottom right: N(m-M).

In our analysis, we proceed exactly as in paper II. We select the same set of
HeI and HeII lines and compare model and observed spectra by calculating
$\chi^{2}$-values for all radii, temperatures, helium abundances, and
microturbulence values and use the minimum of $\chi^{2}$ combined with Monte
Carlo simulations to constrain the stellar parameters of the system. We obtain
$34250\pm 500$ and $33750\pm 500$ K for the effective temperatures of the
primary and secondary, respectively, together with a helium abundance
N(He)/N(H)=0.09$\pm 0.01$ and a microturbulence vturb = 10 km s-1. Our best
solution is obtained for the primary radius of $R_{1}=7.7R_{\odot}$ which
confirms the best solution from the binary modeling. Examples of the fits of
the helium lines with the final model are shown in Fig. 9 and Fig. 10.

####  3.4.1 Reddening and extinction

In the next step, we determine the interstellar reddening and extinction. This
is crucial for an independent estimate of distance and a measurement of
intrinsic stellar surface brightness and colors. For this purpose, we use
multiband photometry outside the eclipses, corrected for the contribution of
the additional components. In the Johnson-Cousins system, we have the
literature apparent magnitude B = 14.17$\pm$0.05 mag (Massey, 2002; Zaritsky
et al., 2004; Testor & Niemela, 1998), as well as V = 14.34$\pm$0.03 mag and
Ic = 14.46$\pm$0.03 mag from our light curve modeling (the system brightness
at phase 0.15). The applied corrections for the additional light were 0.18,
0.20, and 0.24 mag, respectively. Using our binary model we also derived
consistently out-of-eclipse _Gaia_ magnitudes from available light curves
(DR3, Gaia Collaboration, 2022) with GB = 14.22 $\pm$ 0.03 mag and GR = 14.38
$\pm$ 0.030 mag, where 3rd light corrections of 0.20 and 0.23 mag were
applied. From our modeling, we also derived the near-IR magnitudes of BLMC-03
in J (VMC data) and Ks-band (VMC+SOFI). We adopt J = 14.61$\pm$0.04 mag and Ks
= 14.70$\pm$0.05 mag with 3rd light corrections of 0.28 and 0.36 mag,
respectively.

We then proceed exactly in the same way as in Paper II and use the composite
SED of our final model of primary and secondary to calculate reddened model
colors (B-V), (GB-V), (V-GR), (V-IC), (V-J), (V-Ks) in the same passbands as
observed assuming a grid of E(B-V) and RV = AV/E(B-V) values in a range from
0.0 to 0.3 mag and 1.0 to 7.0, respectively. From the $\chi^{2}$ comparison
between calculated and observed colors, we determine the best pair of E(B-V)
and RV at the minimum of $\chi^{2}$. In a Monte Carlo simulation, we modify
the observed colors assuming a Gaussian distribution of uncertainties and
construct probability distribution functions N(E(B-V)), N(RV), N(AV), and
N(m-M) for reddening, the ratio of extinction to reddening, extinction, and
distance modulus, respectively. Fig. 11 shows the result assuming the
monochromatic reddening law by Maíz Apellániz et al. (2014, 2017). We have
also applied the reddening laws by Cardelli et al. (1989), O’Donnell (1994),
and Fitzpatrick (1999) which yield very similar results. We obtain as average
values E(B-V) = 0.16 $\pm$ 0.02 mag, RV = 3.2 $\pm$ 0.8, AV = 0.51 $\pm$ 0.08
mag and a distance modulus m-M = 18.59 $\pm$ 0.06 mag.

Figure 12: Location of the binary components in the HRD compared to MESA
evolutionary tracks with ZAMS masses of 18 (green), 19 (blue), and 20 (pink)
M⊙.

####  3.4.2 Reddening from the sodium line

Similar to the systems analyzed in previous papers, we did an independent
determination of the reddening value through the measurement of the equivalent
widths of the sodium doublet lines. The fit is shown in Fig. 13. From the
calibration of Munari & Zwitter (1997) we then determined the reddening values
for the Milky Way and LMC components separately. for the Milky Way E(B-V) =
0.070, while for the LMC E(B-V) = 0.075. The total reddening is E(B-V) =
0.145, in good agreement with the value obtained from the spectral analysis.

Figure 13: A multi-component fit to the sodium doublet at 5890Å. Individual
components are marked with black thick vertical lines. For both the LMC and
MW, four components were identified. The red line shows the fit to the data
including extra lines, while the blue one marks the profiles of the sodium
lines only. Multiple spectra (lines of different colors) are fitted at once.
The resulting reddening E(B-V) is 0.145.

##  4 Results

To obtain the final solution we combine the results from the Monte Carlo
analysis of the light and RV curves corrected for the presence of extra
components with those from the NLTE spectroscopic analysis. The final results
are summarized in Table 3, where we give the physical parameters of the
components together with the orbital properties of the inner system. Note that
the temperature difference between the primary and secondary is determined
much better than their individual temperatures. From the binary model $\Delta
T=510\pm 40$, which is almost identical to the one obtained from the
spectroscopic analysis ($\Delta T=500\pm 100$).

Figure 14: Evolutionary constraints on the 3rd and 4th objects. We show MESA
evolutionary tracks with ZAMS masses of 7.4 (blue) and 11 (green) M⊙
corresponding to the minimum masses of the 4th and 3rd objects, respectively.
Absolute magnitudes in the V-band (upper left), IC (upper right), and Ks
(lower panel) are plotted versus effective temperature. The dashed part of the
tracks corresponds to the evolution towards the ZAMS. The horizontal dotted
red line corresponds to the de-reddened total absolute magnitude of the
combined third and fourth objects. The orange circles, pink squares, and cyan
stars on the tracks are defined and discussed in the text.

The distance modulus m-M = 18.59 $\pm$ 0.06 mag obtained through our spectral
analysis and de-reddening can be compared with the value of 18.45$\pm$0.02 mag
derived from the one percent precision distance to the LMC determined by
Pietrzyński et al. (2019) and a small geometrical correction accounting for
the location of the system within the LMC using the geometrical LMC model by
van der Marel & Kallivayalil (2014). Being 2 $\sigma$ larger, it just agrees
within the error margins. It is interesting to note that for BLMC-02 analyzed
in Paper II we also found a spectroscopic distance modulus 0.05 mag larger.
This may hint at a small systematic effect caused by the use of our non-LTE
model atmospheres, but we will need the analysis of additional objects to
investigate this further.

With our determination of effective temperatures and luminosities, we can
discuss the evolutionary status of the O-star binary system in the
Hertzsprung-Russell diagram. In Fig. 12 we show MESA evolutionary tracks (Choi
et al., 2016; Dotter, 2016) for 18, 19, 20 M⊙ and the location of the primary
and secondary in the HRD. From the MESA grid of evolutionary models, we have
selected tracks with a metallicity [Z] = -0.5 and without rotation. We have
selected the latter because the rotational velocities are relatively small
compared to the alternative in the MESA grid with $v_{rot}/v_{crit}=0.4$).
According to the tracks, the binary components have a consistent age of 6.3
million years. The tracks indicate masses of 20 and 18 M⊙, slightly higher
than obtained with our analysis but at the margin of the uncertainties.

Table 3: Physical parameters and other properties of the inner binary (A) Parameter [unit] | Primary (Aa) | Secondary (Ab)
---|---|---
mass [$M_{\odot}$] | 19.02 $\pm$ 0.12 | 17.50 $\pm$ 0.13
radius [$R_{\odot}$] | 7.70 $\pm$ 0.05 | 6.64 $\pm$ 0.06
$\log g$ [cgs] | 3.944 $\pm$ 0.005 | 4.035 $\pm$ 0.007
temperature [K] | 34250 $\pm$ 500 | 33750 $\pm$ 500
$(T_{2}-T_{1})_{spec}$ [K] | $500\pm 100$
$(T_{2}/T_{1})_{phot}$ | $0.9851\pm 0.0011$
$\log L$ [$L_{\odot}$] | 4.854 $\pm$ 0.010 | 4.700 $\pm$ 0.003
$j_{21,V}=F_{2}/F_{1}(V)$ | 0.971 $\pm$ 0.002
$V$ [mag] | 14.963 $\pm$ 0.021 | 15.285 $\pm$ 0.025
$I_{\mathrm{C}}$ [mag] | 15.086 $\pm$ 0.021 | 15.413 $\pm$ 0.024
$K_{\mathrm{S}}$ [mag] | 15.316 $\pm$ 0.029 | 15.644 $\pm$ 0.029
$v_{rot}$ [$km\,s^{-1}$] | 70 $\pm$ 7 | 80 $\pm$ 8
E(B-V) [mag] | 0.16 $\pm$ 0.02
orbital period, $P$ [days] | 3.2254367 $\pm$ 0.0000003
semi-major axis [$R_{\odot}$] | 30.48 $\pm$ 0.07
orbital inclination, $i$ | 87.43 $\pm$ 0.21
eccentricity, $e$ | 0.0006 $\pm$ 0.0003
argument of periastron, $\omega$ [rad] | 0.3 $\pm$ 1.0
apsidal motion, $\dot{\omega}$ [rad/d] | -0.005 $\pm$ 0.004
orbital semiamplitude, $K$ [km s-1] | 228.5 $\pm$ 1.3 | 248.7 $\pm$ 1.1
systemic velocity, $\gamma$ [km s-1] | 280.7 $\pm$ 0.4
mass ratio, $q=m_{2}/m_{1}$ | 0.919 $\pm$ 0.007

Having determined the masses of the inner system components we can also
estimate them for the additional components. Assuming an orbital inclination
of 90 degrees (outer orbit seen edge-on) from the spectroscopic orbit of the A
system, the minimum mass of the tertiary is estimated to be about $M_{B}\sim
10.7M_{\odot}$. With this limit at hand, and having the orbital semiamplitude
of the A+B system derived from the detected LTTE ($K_{AB}=8.6\pm 0.4$), we
obtain the minimum mass of the fourth component of $M_{C}\sim 7.3M_{\odot}$.
Moreover, as we know that these objects provide about 20% of the total system
light and are not visible in the spectra, the individual components should be
less massive than the primary and secondary components, individually. This
puts a conservative upper limit to the mass of about $17.5M_{\odot}$ (the mass
of the secondary) for a single star. From the fact that the more inclined the
outer orbit is the more massive tertiary we would obtain, we can also
constrain the inclination of the A+B system, $i_{A+B}\gtrsim 42^{\circ}$,
assuming B is a single star. With the same assumption, we expect $M_{B}$ to be
close to the lower limit and to have an inclination similar to that of the
inner system, although the orbits do not necessarily have to be co-planar.

We can use the additional light found in our light curve analysis of section
3.2 for an attempt to further constrain the nature of the 3rd and 4th objects.
With the magnitude corrections given above, we obtain V = 16.05$\pm$0.03 mag,
V - Ic = 0.06$\pm$0.02 mag and V - Ks = 0.35$\pm 0.04$ mag. With E(B-V) = 0.16
mag and RV = 3.2 (see previous section) we can correct for extinction and
calculate intrinsic colors and magnitude: V0 = 15.54$\pm$0.08 mag, (V - Ic)0 =
-0.15$\pm$0.10 mag and (V - Ks)0 = -0.10$\pm 0.10$ mag.

With a distance modulus of 18.45 (see above) we then obtain absolute
magnitudes MV = -2.91$\pm$0.08 mag, M${}_{I_{C}}$ = -2.76$\pm$0.06 mag and
M${}_{K_{s}}$ = -2.80$\pm$0.05 mag. This allows us to compare the joint
magnitudes of the objects with MESA evolutionary tracks of 7.4 and 11 M⊙,
corresponding to the minimum masses of the fourth and third object,
respectively. This is done in Fig. 14 where we show absolute magnitudes in V,
IC, and Ks versus effective temperature. The MESA tracks are from the same
evolutionary grid as used in Fig. 12. They include pre-main sequence evolution
towards the zero-age main sequence (ZAMS) and post-main sequence evolution
away from ZAMS. The observed absolute magnitudes of the combined third and
fourth objects are also shown in each panel.

Fig. 14 indicates that only a small range of effective temperature of the
tracks is compatible with the observed magnitudes. From the V-band we see
immediately that the effective temperature of the 3rd object (represented by
the track with $11M_{\odot}$) is tightly constrained to an evolutionary phase
very close to the ZAMS with log T${}_{eff}\geq$ 4.4. This also restricts the
evolutionary status of the 4th object (the track with $7.4M_{\odot}$), because
the combined brightness of both objects needs to match the observed magnitude
shown as the red dotted line. At the minimum brightness of the 3rd object, the
4th object must have a very similar magnitude as indicated by the orange
circles on the tracks. The maximum possible brightness of the 3rd object is
constrained through the minimum V-band magnitude of the 4th object on the
$7.4M_{\odot}$ track. The pink squares on each track define the corresponding
evolutionary phases and magnitudes. The range between the pink squares and the
orange circles on both tracks describes the evolutionary phase during which
the joint brightness of both objects matches the V-band observation. The cyan
stars provide an example within this range.

We then use exactly the same evolutionary phases represented by the orange
circles, pink squares, and cyan stars in the plot of M${}_{I_{C}}$ in Fig. 14.
Combining the corresponding magnitudes on the two tracks we find that they
also agree with the observation within the error margins. However, we
encounter a severe discrepancy with the observations for M${}_{K_{s}}$ as
shown in Fig. 14. The combined magnitudes of the orange circles, cyan stars,
and pink squares, respectively, are 0.6 to 0.75 magnitudes too faint. This may
indicate an IR excess of these objects, for instance, caused by a
circumstellar disk or an even more complex structure of the system with a
possible multiplicity of the B and C components.

In a similar analysis, we also found that evolutionary track for a star of 16
M⊙ would lie entirely above the red dotted line in the V-band panel of Fig. 14
(passing slightly below it for the I-band), i.e. the star at any evolutionary
stage would be brighter than the additional light we found. This puts a
stronger (than the mass of the secondary) mass limit of 16 M⊙ on the 3rd and
4th component (or individual stars they consist of in the case any of them is
a binary itself).

##  5 Conclusions

From our multi-method analysis, the BLMC-03 system resulted to be much more
complex than expected. We found strong evidence for the existence of two extra
components orbiting around the studied binary system. Each of these two extra
components may be a binary itself but we have no direct evidence for that,
apart from a small hint from the additional light in the Ks-band, which may
suggest more than 2 but cooler and redder stars in the system.

This (at least) quadruple system belongs to the less common group of 2+1+1
systems (2+2 are more frequent, see e.g. Tokovinin 2014). The period ratios
are $P_{A+B}/P_{A}=60.7$ and $P_{AB+C}/P_{A+B}=13$. The system is thus
relatively tight for a 3-tier hierarchical system.

One may question if such a complex system can be used for our final goal of
the calibration of the SBC relation. However, we have full control over the
additional light in the system and take into account the effects of additional
components on the measured quantities. As a result, the accuracy and precision
of determined parameters are comparable to those for our other systems. One
may also argue that it is better to have a binary as a part of a multiple
system with detected additional components than a binary with such components
not yet detected. This also points to the importance of high-quality data and
a detailed study of any extragalactic binary system that will be used in the
future for distance measurement.

##  6 Acknowledgments

We thank Dr. Nidia Morrell, Dr. Philip Massey and Dr. Kathryn Neugent for
sharing with us the MagE spectra of BLMC-03. We also thank Nidia Morrell for
acquiring a few additional recent MIKE spectra that helped constrain the outer
orbit. The research leading to these results has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme under grant agreement No 951549 (project
UniverScale) and from the National Science Center, Poland grant BEETHOVEN
UMO-2018/31/G/ST9/03050. RPK has been supported by the Munich Excellence
Cluster Origins funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC-2094 390783311.
M.T., B.P., and G.P. received support from the Munich Institute for Astro-,
Particle- and BioPhysics (MIAPbP) which is also funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC-2094 390783311. B.P. acknowledges funding from the
Polish National Science Center grant SONATA BIS 2020/38/E/ST9/00486. We also
acknowledge support from the Polish Ministry of Science and Higher Education
grant DIR-WSIB.92.2.2024. The research was based on data collected under the
Polish-French Marie Skłodowska-Curie and Pierre Curie Science Prize awarded by
the Foundation for Polish Science.

This research is based on observations collected at the Las Campanas
Observatory and the European Southern Observatory under ESO programs:
098.D-0263(A), 0100.D-0339(A), 0102.D-0469 and 102.D-0590 (B).

This research has made use of NASA’s Astrophysics Data System Service.

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> Proceedings of the
>  1st International Workshop on
>  Reading Music Systems
>
> 20th September, 2018
> Paris, France

Proceedings of the 1st International Workshop on Reading Music Systems, Paris,
2018

Edited by Jorge Calvo-Zaragoza, Jan Hajič jr. and Alexander Pacha

© The respective authors.

Licensed under a Creative Commons Attribution 4.0 International License (CC-
BY-4.0).

Logo made by Freepik from www.flaticon.com. Adapted by Alexander Pacha

## Organization

General Chairs

Jorge Calvo-Zaragoza |  | Universitat Politècnica de València, Spain
---|---|---
Jan Hajič jr. |  | Charles University, Czech Republic
Alexander Pacha |  | TU Wien, Austria

Local Organizer

Philippe Rigaux |  | Conservatoire national des arts et métiers, France
---|---|---

Program Committee

Ichiro Fujinaga |  | McGill University, Canada
---|---|---
Alicia Fornés |  | Computer Vision Center, Spain
Andreas Arzt |  | Johannes Kepler Universität, Austria
Horst Eidenberger |  | TU Wien, Austria
Jose M. Iñesta |  | University of Alicante, Spain
Ana Rebelo |  | INESC Tec, Portugal
Christopher Raphael |  | Indiana University, United States

Support

The organization of this workshop has been possible thanks to the support from
the Music Notation Information Retrieval (MuNIR) project.

## Preface

Dear colleagues,

it is our greatest pleasure to introduce the proceedings of the 1st
International Workshop on Music Reading Systems (WoRMS), an event that many of
us have been looking forward to for a long time. Considering that Optical
Music Recognition (OMR) has been subject to research for over 50 years, it is
high time for all active researchers in this field to gather in one place to
exchange ideas and build a stable community that will drive the research for
the years to come.

At the Graphics Recognition Workshop (GREC) 2017 in Kyoto, many researchers in
this field met in person for the first time. It was a unique opportunity that
created a wide range of collaborations and scientific exchange. In that
spirit, we decided to create a similar event, one that is dedicated entirely
to OMR and is attached to the annual International Society for Music
Information Retrieval Conference (ISMIR), which feels to us like a natural
choice for the venue, considering that OMR lives at the intersection between
music and technology and is mainly used by people that are interested in music
or music information retrieval.

It is an exciting time to be an active researcher in this field with
significant advances happening all around the globe, making it sometimes hard
to oversee all of them. We are very happy that the topics of the four sessions
span the entire bandwidth: Community, Applications and Interactive Systems,
Technical Solutions, and User Perspectives. The brief presentation format with
an open-ended discussion towards the end of the workshop will hopefully
trigger lively discussions, spawn inspiring ideas and help initiate
collaborations that will make this workshop a memorable event.

Jorge Calvo-Zaragoza, Jan Hajič jr. and Alexander Pacha

###### Contents

  1. Preface

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