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# Analyzing Stylistic Variation across Different Political Regimes

Liviu P. Dinu, Ana Sabina Uban
Faculty of Mathematics and Computer Science,
Human Language Technologies Research Center,
University of Bucharest
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this article we propose a stylistic analysis of texts written across two
different periods, which differ not only temporally, but politically and
culturally: communism and democracy in Romania. We aim to analyze the
stylistic variation between texts written during these two periods, and
determine at what levels the variation is more apparent (if any): at the
stylistic level, at the topic level etc. We take a look at the stylistic
profile of these texts comparatively, by performing clustering and
classification experiments on the texts, using traditional authorship
attribution methods and features. To confirm the stylistic variation is indeed
an effect of the change in political and cultural environment, and not merely
reflective of a natural change in the author’s style with time, we look at
various stylistic metrics over time and show that the change in style between
the two periods is statistically significant. We also perform an analysis of
the variation in topic between the two epochs, to compare with the varation at
the style level. These analyses show that texts from the two periods can
indeed be distinguished, both from the point of view of style and from that of
semantic content (topic).

Keywords: authorship attribution, stylome classification, stylistic variation,
Romanian

##  1 Previous Work and Motivation

Automated authorship attribution has a long history (starting from the early
20th century [12]) and has since then been extensively studied and elaborated
upon. The problem of authorship identification is based on the assumption that
there are stylistic features that can help distinguish the real author of a
text from any other theoretical author. This said set of stylistic features
was recently defined as a linguistic fingerprint (or stylome), which can be
measured, is largely unconscious, and is constant [17]. One of the oldest
studies to propose an approach to this problem is on the issue of the
_Federalist Papers_ , in which [13] try to determine the real author of a few
of these papers which have disputed paternity. This work remains iconic in the
field, both for introducing a standard dataset and for proposing an effective
method for distinguishing between the authors’ styles, that is still relevant
to this day, based on function words frequencies. Many other types of features
have been proposed and successfully used in subsequent studies to determine
the author of a text. These types of features generally contrast with the
content words commonly used in text categorization by topic, and are said to
be used unconsciously and harder to control by the author. Such features are,
for example, grammatical structures [1], part-of-speech n-grams [7], lexical
richness [16], or even the more general feature of character n-grams [6, 5].
Having applications that go beyond finding the real authors of controversial
texts, ranging from plagiarism detection to forensics to security, stylometry
has widened its scope into other related subtopics such as author verification
(verifying whether a text was written by a certain author) [8], author
profiling (e.g. gender or age prediction), author diarization, or author
masking (given a document, paraphrase it so that the original style does not
match that of its original author anymore).

In this work we attempt to explore a slightly different issue, strongly
related with the stylistic fingerprint of an author, namely if an author
preserves his stylome across different time periods, and across political and
cultural environments. More specifically, we want to see if we can
discriminate between texts written by the same author under a communist
regime, as compared to under democracy.

For this analysis, we chose the works of Solomon Marcus (1925-2016), one of
the most prominent scientists and men of culture of modern Romania, and a very
prolific author whose work spans a large period of time, with a consistent
amount of work both before and after the fall of communism in Romania. As a
scientist, he was active in an impressive range of different fields, such as
mathematics, computer science, mathematical and computational linguistics,
semiotics etc, and published over 50 books translated in more than 10
languages, and about 400 research articles in specialized journals in almost
all European countries. He is one of the initiators of mathematical
linguistics [11] and of mathematical poetics [9], and has been a member of the
editorial board of tens of international scientific journals covering all his
domains of interest. As a man of culture, he has written an equally impressive
amount of texts, having as preferred topics mathematical education, culture,
science, children, etc. His wide interests and complex personality have left
deep imprints in the Romanian scientific and cultural world.

We chose these works and this period in Romanian history as a particularly
interesting study case for stylistic variation. Marcus’ essays [10] were
written over a period spanning almost half a century: 22 years of communist
regime and 27 years of democracy (after the fall of communism in Romania in
1989). The significant changes in Romania’s cultural life with the fall of
communism motivate such an analysis beyond a simple temporal analysis of
texts: in the pre-democracy period, the communist norms demanded an elaborated
writing style, whereas the list of officially approved topics was much more
limited. On the other hand, with the fall of communism, we expect that the
author would would be able to radically change his style to express his ideas
freely.

Previous work on the same corpus [2] focused on a lexical quantitative
analysis of Marcus’ works, uncovering some differences in word usage between
the communist and the democracy period. We continue this work by performing
more robust analyses using various metrics and classification experiments to
show that the texts written during the two periods are indeed distinguishable
with a reasonable accuracy.

##  2 The Corpus

For the purposes of the analysis presented here, we used a collection of
Solomon Marcus’ non-scientific texts. The whole collection of Marcus’ non-
scientific publications is available in print, in six volumes entitled "Răni
deschise" (_Open wounds_), of which we analyze the first four: two containing
texts written before the fall of communism, and the other two with texts
written under democracy. The first volume comprises 1256 pages of texts,
conferences and interviews, from 2002 to 2011 (and a few published before
1990). The second volume "Cultură sub dictatură" (_Culture under
dictatorship_) of 1088 pages and the third "Depun Mărturie" (_I testify_), of
668 pages, is a collection of texts published between 1967 and 1989. The
fourth volume "Dezmeticindu-ne" (_Awaking_), of 1030 pages, contains texts
from the period immediately following the Romanian Revolution (December 1989)
to the year 2011.

Before the texts could be used for automatic analysis, some pre-processing was
involved, including extracting the texts from PDF files into text form, and
splitting the corpus into individual essays. We also parsed the table of
contents in order to label each text with its year of publication.
Additionally, we excluded some texts that would interfere with the
experiments, such as texts in languages different than Romanian (most of the
texts are in Romanian), and interviews (which also contain lines of the
interviewer, that should not be considered when analysing text authored by
Solomon Marcus).

##  3 Methodology

In order to analyze the style of Marcus’ work and its variation, we perform a
series of classification and clustering experiments, as well as a temporal
analysis of the evolution of the author’s style. We look at various features
to capture both style and semantic content, with a focus on metrics that that
characterize the stylistic fingerprint of the author.

We perform various classification and clustering experiments, as well as
visualisations complementing them, for which we experiment first with simple
bag of words models. To capture the stylistic fingerprint of the author, we
resort to one of the most commonly accepted indicators of personal style of an
author, namely the distribution of functional words, such as pronouns,
determiners, prepositions, etc, which were successfully used in many previous
studies to solve authorship attribution problems. For comparison, we perform a
parallel experiment where we use as features only the content words, assuming
they are descriptive of the semantic content of the texts, as opposed to their
style.

Aside from these feature sets, we also use more sophisticated features in some
additional classification experiments: various stylistic metrics (such as
readability or lexical richness) to capture the stylistic variation, as well
as features resulted from topic modelling to capture the semantic content.
These features will be described in more detail in the chapters dedicated
specifically to the classification experiments.

###  3.1 Clustering Experiments

Processing the text involves extracting all function words from each essay,
and representing each text as a vector of frequencies of function words. We
use a curated list of 120 functional Romanian words [4]. On these vectors we
then apply similarity metrics in order to cluster texts based on their
stylistic profile. The distance we have chosen is rank distance [3], [14], an
ordinal distance which was successfully previously used in other problems of
authorship [5], [4].

For our purposes, we applied a hierarchical clustering algorithm, and plotted
the resulted dendrogram, as shown in Figure 1. Texts from each volume are
marked with the prefix _RD <volume_number>_. The intention is to see whether
texts from the same period are clustered together: in this case, we would
expect to see texts from volumes 1 and 4 (communism) appear close together,
and texts from volumes 2 and 3 (post-communism) appear in a separate group.

An additional processing step we performed before applying the clustering
algorithm was to group the essays into longer texts, while keeping them
separated into volumes (texts from different volumes/periods were not grouped
together). For each volume, batches of 10 essays were grouped toghether in 1
text. This results in a fewer number of texts, which can be more easily
plotted on a dendrogram, and additionally should contain less noise than the
very short original texts.

Figure 1: Style dendrogram: Hierarchical clustering of function word vectors

###  3.2 Dimensionality Reduction and Visualization

In addition to the clustering experiments, we perform an experiment where we
plot the essays in the 4 volumes in 2-D space, by initially applying
dimensionality reduction (principal component analysis) on the texts,
represented as function word frequency vectors, as described above. The
resulted plot is shown in Figure 2.

Figure 2: Style space: 2-D visualization of function word vectors

In a separate experiment, we take a brief look at the content profile of the
texts (as opposed to the stylistic one), by performing the same analysis on
content words instead of function words. We extract all content words from all
the (grouped) texts, and represent the texts as vectors of content words
frequencies. We then apply PCA on the resulted vector space, and plot the
2-dimensional representations of the texts, as shown in Figure 3. Although
this is still a very crude analysis, the result should provide an insight into
the content similarities and dissimilarities between the texts in the 4
volumes.

We also perform a version of the clustering experiments on the same vectors of
content words, with the same methodology that was applied in the stylistic
analysis: the rank distance similarity is applied on the word vectors, and the
distances are used in a hierarchical clustering algorithm, which results in
the "content" dendrogram in Figure 4.

Figure 3: Topic dendrogram: Hierarchical clustering on content word vectors
Figure 4: Topic space: 2-D visualization of content word vectors

###  3.3 Classification Experiments

In order to have a more conclusive result regarding the differences between
the texts written in the two epochs and the levels at which they occur, we
also performed some classification experiments where we try to predict for
each essay if it was written before or after the fall of communism, as well as
to which specific volume it pertains.

For classifying the texts we used an SVM classifier with a linear kernel, and
tested its performance in a series of leave-one-out experiments.

####  3.3.1 Features

We used various sets of features, in turn specific to style and to semantic
content, in order to conclusively characterize the nature of the evolution of
Marcus’ writing over time and across the two periods.

As stylistic features, we experimented with 2 setups: first a bag-of-words
model using only functional words; and secondly a set of stylistic markers:

  * •

Automated Readability Index, as defined in [15], computed as:

$ARI=4.71\frac{c}{w}+0.5\frac{w}{s}-21.43$

where _c_ , _w_ and _s_ represent, respectively, total number of characters,
words and sentences in the text

  * •

Lexical richness, computed as the number of unique lemmas divided by the total
number of tokens in the text:

$LR=\frac{number\ of\ unique\ lemmas}{total\ number\ of\ tokens}$

  * •

Average word length

  * •

Average sentence length

To measure the variation in semantic content, we performed two classification
experiments. In the first experiment we used a simple bag-of-words model, this
time using only content words as features. For the second experiment, we used
topic modelling to extract topics in each article, that we then used as
features in the classifier. To extract the topics, we used a Latent Dirichlet
Allocation model, configured to generate 3 latent topics for each document.

Results are shown in Table 1 and Table 2: in Table 1 we report the results of
trying to classify the texts into the two distinct periods, while Table 2
shows the results of classifying each text into the specific volume to which
it pertains.

###  3.4 Temporal variation of stylistic metrics

In order to show that the stylistic variation discovered in the experiments
described in the previous chapter can indeed be attributed to the change of
political regime, and not simply to the passing of time and the natural
gradual change in the author’s style, we also look at how the various
stylistic metrics vary over time at a finer level.

Figures 5 to 8 show the evolution of each stylistic metric described in the
previous chapter from year to year. To achieve this, we computed each metric
for every article in the corpus, then for each year we averaged the values
obtained for the articles written during that year. To reduce the effect of
noise and variation in number of articles for each year, we smooth the
obtained values by applying a moving average over the articles in each year
with a window of 10 years.

In order to confirm the variation in style is more pronounced around the year
1990 (that marks the beginning of democracy in Romania), we performed a series
of experiments in which we attempted temporal splits of the article set into
two periods using each year as a separator, and computed the statistical
significance of the difference shown at the level of each metric for each of
these splits. We assume that the more significant the split across a certain
year is, the more radical the author’s change in style around that year, and
compare the effect of the particular year that marked the beginning of
democracy in Romania to all of the other years during which Marcus wrote his
essays. We plot, in addition to the actual values of the stylistic metrics for
each year, the p-values corresponding to the statistical significance of the
difference for each metric between the articles written before and after each
year. The vertical line over the year 1990 splits the plot into two parts,
corresponding to the period before respectively after the fall of communism.

Figure 5: Evolution of readability over time Figure 6: Evolution of lexical
richness over time Figure 7: Evolution of average word length over time Figure
8: Evolution of average sentence length over time

##  4 Results and Analysis

The clustering experiments, as well as the 2-dimensional visualizations, show
a clear distinction between the works of the two periods. The style dendrogram
in Figure 1 shows that texts tend to be grouped according to the volume they
belong to, but moreover, the volumes belonging to each period tend to be
grouped together. It is noticeable that texts belonging to the first and
fourth volumes, written during the post-communist epoch (marked "RD1" and
"RD4"), are grouped together on the lower branch of the dendrogram, and the
same happens with texts in volumes 2 and 3 ("RD2" and "RD3"), written during
communism, which are grouped on the upper branch.

The same tendency is visible in the 2-dimensional vector space plot of the
texts. It is interesting that the separation between the pre-communist texts
(marked with green and blue points) and communist texts (marked with red and
yellow points) seems even more prominent than the separation between each of
the four volumes - indicating that the main variable contributing to style
differences is indeed the period when the text was written.

A better, more detailed understanding of how exactly Marcus’ style evolves can
be gained from the plots showing the variation in style from year to year.
Overall, after the change of regime in 1990, we can see a slight drop in all
of the stylistic features, especially in word length and lexical richness.
Most of the stylistic metrics (especially lexical richness and readability)
show a more pronounced change of style right after the year 1990 - the year of
the fall of the communist regime in Romania - suggesting that the change of
regime did have an effect of its own on the style of the author. An even more
interesting phenomenon can be observed by looking at the variation of the
p-values for splitting the article set into two arbitrary periods (before and
after each year). Except from average sentence length, for which the most
drastic change seems to happen around the year 1980, for all of the other
stylistic metrics splitting the articles set into texts written under
communism and under democracy is statistically significant (with p-values well
under 0.05 for the year 1990, and additionally having at least local minima
around this year).

We can conclude from these results that, independently on possible variation
at other levels (topic, etc), there is inarguable variation between the texts
at a stylistic level.

A similar but less obvious effect is noticeable in the content analysis. While
the vector space plot of the texts again shows a reasonably clear distinction
between the texts in the two periods, the cluster analysis is less conclusive.

Feature set | Precision
---|---
function words (style) | 75.80%
stylistic metrics (style) | 73.12%
content words (semantic content) | 74.46%
topic modelling (semantic content) | 71.24%
Table 1: Results of classifying texts into periods Feature set | Precision
---|---
function words | 61.29%
content words | 84.13%
Table 2: Results of classifying texts into volumes

The results of the classification experiments very clearly show that essays
can be easily separated into the two classes corresponding to communism and
democracy, both when using function words, stylistic metrics, and semantic
features (content words, topics) as a feature set. A slightly better accuracy
is obtained for stylistic classification. The stylistic features, in
comparison to the content words, seem to be more successful at predicting the
period rather than the specific volume, indicating that style may indeed be
the aspect of the texts that varies the most between the two periods and
political regimes. Analyses of the author’s change in style, as well as
statistical significance tests of the change in various stylistic metrics
across the two periods, further confirm that the change of political regime
does have its own effect on the author’s style.

We can conclude, given all of these results, that during the two periods,
corresponding to two different political regimes, Marcus produced works which
are clearly distinct both stylistically and from the point of view of the
topics discussed. It is noteworthy that the classifier manages not only to
classify the texts into the two periods in which they were written, but is
also successful at predicting the exact volume to which each essay pertains.
This suggests there may be other (volume-specific) factors contributing to the
distinctiveness of the essays in each class, which it may be interesting to
further explore.

##  5 Conclusions and Future Directions

We have proposed in this paper an analysis of the stylistic variation of the
works of Solomon Marcus between two distinct historical periods, and have
presented results that show the separation between the two periods is indeed
clear at the level of the author’s style. The clustering approach based on the
preference of the author regarding the functional words shows that the
functional words were unconsciously used by author in a different way in
communism than in democracy period, and we can use them to discriminate
between the communist and post-communist texts of the given author. We further
confirm in several classification experiments that the texts can be
automatically separated into the two periods, and are distinguishable both at
the level of style and topic. We show the contribution of the change in
political regime to the stylistic variation by looking at how the author’s
style gradually changes over time, and by performing statistical significance
tests on various stylistic metrics, comparing the two periods.

In future work, it would be very interesting to examine more closely how this
variation is manifested. From the stylistic point of view: what are the exact
changes that occur with the passing of time and the change of historical and
cultural context? From the point of view of topic modelling, what are the
specific changes in the topic of the text corresponding to the two periods? Do
these changes match our intuitions about writing in the communist as compared
to post-communist era?

Finally, continuing this analysis on other authors with works in both
communism and post-communism, or, more generally, across various political
regimes, would be interesting to confirm the phenomenon is not unique to this
author. Possibly extending this to other authors in different cultural
contexts could lead to an interesting analysis on the external factors that
influence style.

## Acknowledgement

Liviu P. Dinu is supported by UEFISCDI, project number 53BG/2016.

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# Modelling the Light Curves of Transiting Exomoons: a Zero-order Photodynamic
Agent Added to the Transit and Light Curve Modeller

Szilárd Kálmán,1,2,3 Szilárd Csizmadia,4,5 Attila Simon E.,6,7 Kristine W. F.
Lam,4 Adrien Deline,8 Jan-Vincent Harre,4 and Gyula Szabó M.3,9
1 HUN-REN, Research Centre for Astronomy and Earth Sciences, Konkoly
Observatory, MTA Centre of Excellence,
H-1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary
2ELTE Eötvös Loránd University Doctoral School of Physics, Pázmány Péter
sétány 1/A, Budapest, H-1117, Hungary
3HUN-REN–ELTE Exoplanet Systems Research Group, Szent Imre h. u. 112.,
Szombathely, H-9700, Hungary
4Deutsches Zentrum für Luft- und Raumfahrt, Institute of Planetary Research,
Rutherfordstrasse 2, D-12489 Berlin, Germany
5HUN-REN–SZTE Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út Kt.
766, Hungary
6Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012
Bern, Switzerland
7Center for Space and Habitability, University of Bern, Gesellschaftsstrasse
6, 3012 Bern, Switzerland
8Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290
Versoix, Switzerland
9ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, Szent Imre
h. u. 112., H-9700, Hungary E-mail<EMAIL_ADDRESS>(SzK)

(Accepted 2023 November 07. Received 2023 November 07; in original form 2023
August 31)

###### Abstract

Despite the ever-growing number of exoplanets discovered and the extensive
analyses carried out to find their potential satellites, only two exomoon
candidates, Kepler-1625b-i and Kepler-1708 b-i, have been discovered to date.
A considerable amount of effort has been invested in the development of
algorithms for modelling, searching, and detecting exomoons in exoplanetary
light curves. In this work, we incorporate moon handling capabilities into the
state-of-the-art and publicly available code, the Transit and Light Curve
Modeller (TLCM). The code is designed for the analysis of transiting exoplanet
systems with the inclusion of a wavelet-based noise handling algorithm. Here
we present an updated version of TLCM that is capable of modelling a coplanar
planet-moon system on an elliptical orbit around its host, accounting for
mutual eclipses between the two bodies (and neglecting perturbative effects) –
a so-called photodynamic model. The key benefit of this framework is the
ability for a joint analysis of multiple planet–moon transits. We demonstrate
the necessity of this software on a case study of Kepler-1625b. Similarly to
prior works, we conclude that there is no firm evidence of an exomoon in that
system, by showing that temporally correlated noise can mimic apparent lunar
transits.

###### keywords:

celestial mechanics – techniques: photometric – planets and satellites:
detection – planets and satellites: individual: Kepler-1625b

††pubyear: 2023††pagerange: Modelling the Light Curves of Transiting Exomoons:
a Zero-order Photodynamic Agent Added to the Transit and Light Curve
Modeller–Modelling the Light Curves of Transiting Exomoons: a Zero-order
Photodynamic Agent Added to the Transit and Light Curve Modeller

##  1 Introduction

Since the discovery of the first extrasolar planet orbiting main sequence
stars (e.g. Mayor & Queloz, 1995), the field of exoplanet research grew
rapidly and became increasingly important in astronomy. Based on our
understanding of the Solar System, the quest to find satellites of these
planets (called ‘exomoons’) began to expand shortly thereafter (e.g.
Sartoretti & Schneider, 1999). Although there are $>5000$ known exoplanets to
date, only two exomoon-candidates have been identified (Kepler-1625b-i;
Teachey & Kipping, 2018) and (Kepler-1708 b-i; Kipping et al., 2022). Heller
et al. (2019) offer an alternative no-moon explanation for the signal observed
by Teachey & Kipping (2018), involving the presence of time-correlated noise
and the properties of Bayesian inference. In Kreidberg et al. (2019), the
authors also argue that the there is no conclusive evidence for an exomoon in
the transit light curves of Kepler-1625b. One can therefore argue that the
transiting exomoon candidate proposed by Teachey & Kipping (2018) is the
result of temporally variable noise mimicking the transit of a moon. There are
two distinct sources of this noise type (also commonly called red noise):
astrophysical and instrumental. The former of these include the presence of
stellar pulsations, flares, granulation, and spots, which may produce light
variations similar to the ones expected from transits of solid bodies (e.g.
Silva-Valio et al., 2010). The latter group includes cosmic rays causing
excess flux, temperature variations of the detectors/telescope assembly,
pointing drifts etc (see e.g. Csizmadia et al., 2023, for more discussion on
this topic). Time-correlated noise can manifest on a wide range of timescales.
The long-term trends can easily be accounted for by linear or parabolic
baselines (Teachey & Kipping, 2018). Kreidberg et al. (2019) suggests that
instrumental noise sources (of shorter time-scales) may be behind the apparent
exomoon transit.

Furthermore, Fox & Wiegert (2021) found six exomoon candidates (KOIs-268.01,
303.01, 1888.01, 1925.01, 2728.01, 3320.01) based on Transit Timing Variations
(TTVs). Kipping (2020) disproved these specific exomoon candidates from
observational aspects, while Quarles et al. (2020) suggest that the systems
mentioned above could not be stable from a dynamical perspective. The field of
exomoon research is intertwined with current and upcoming space-based
telescopes, such as CHEOPS (Simon et al., 2015; Benz et al., 2021; Ehrenreich
et al., 2023), PLATO (Rauer et al., 2014) and Ariel (Tinetti et al., 2021;
Szabó et al., 2022), the number of potentially discoverable exomoons is
therefore expected to rise.

In recent decades, a number of methods were proposed to detect and
characterise exomoons. These include, but are not limited to, the analysis of
TTVs (Simon et al., 2007; Kipping, 2009), the Rossiter-McLaughlin effect
(Simon et al., 2010), direct modelling of the transits (e.g. Kipping, 2011),
finding radio signatures caused by the interaction of the moon and its host
planet (e.g. Noyola et al., 2014, 2016), spectrophotometry (Jäger & Szabó,
2021) and direct imaging (Kleisioti et al., 2023). As expressed above, some of
these methods were used to identify exomoon candidates. The long-term
stability of the planet-moon systems is also favoured from an observational
point of view. A considerable effort was made to assess the dynamics of the
formation and evolution of such systems in general (Domingos et al., 2006;
Spalding et al., 2016; Alvarado-Montes et al., 2017; Sucerquia et al., 2019),
and the satellites of Kepler-1625b/Kepler-1708b (Moraes & Vieira Neto, 2020;
Moraes et al., 2023). The topic of habitability is also often discussed in
relation to the exomoons (e.g. Kipping et al., 2009; Kaltenegger, 2010; Dobos
et al., 2022).

Following the success of the transit method in the detection of exoplanets,
several algorithms have been put forward to model the light curve of a
transiting planet-moon system (e.g. Luger et al., 2017; Kipping, 2021; Saha &
Sengupta, 2022; Gordon & Agol, 2022). At the time of writing however, there is
only one open-source software available that is capable of fitting the light
curves of transiting planet-moon systems out-of-the-box (Hippke & Heller,
2022). Recently, time-correlated noise (either of stellar or instrumental
origin) has been demonstrated to mimic exomoon light curve components, and is
recognised to be a major source of false alarms (e.g. Ehrenreich et al.,
2023). Therefore, there is a clear need for algorithms with a combination of
noise-handling capabilities and the photodynamic effects. Red noise can arise
from instrumental and astrophysical sources, such as stellar spots,
microflares, pulsation etc. (see Csizmadia et al., 2023; Kálmán et al., 2023,
for more detailed discussions). To filter out the false-positive moon signals
that are consistent with noise level fluctuations, a light curve inversion
algorithm that accounts for red noise is needed.

In this paper, we introduce the photodynamic update of the Transit and Light
Curve Modeller (TLCM; Csizmadia, 2020), which uses the well-established
wavelet-based noise filtering algorithm of Carter & Winn (2009). Its benefits
in the precision and accuracy of exoplanet parameters in transit light curves
are well established (Csizmadia et al., 2023; Kálmán et al., 2023). The newly
integrated model also includes the ability to model mutual transits by the
planet and its satellite. In Section 2, we describe the calculation of the
position of the planet and its satellite. In Section 3, we provide example
light curves with arbitrary parameters and a specific case of Kepler-1625b-i
(Teachey & Kipping, 2018). In Section 4, we draw our conclusions.

##  2 Methods

Figure 1: Simulated light curve of an arbitrary transiting planet-moon system
(blue dots and solid red curve). Left. The moon remains hidden behind the
planet during the transit. Middle. A mutual transit event occurs, requiring
flux correction (solid blue curve). Right. The transit of the satellite occurs
after the transit of the exoplanet. Note that, for demonstrative purposes, the
sizes of both bodies are enlarged, and we assume no limb darkening of the host
star.

The TLCM software is designed to model a wide range of astrophysical effects
that can be observed in light curves of transiting exoplanets, including
transits, occultations, phase curve variability, ellipsoidal effect and
Doppler beaming (Csizmadia, 2020; Csizmadia et al., 2023). Special emphasis is
given to the handling of time-correlated noise: the wavelet-based noise
filtering of Carter & Winn (2009) is used to remove (or reduce) the amount of
red noise present in light curves (Csizmadia et al., 2023; Kálmán et al.,
2023). This noise filtering maximises the likelihood of the combination of the
noise and transit models, thus ‘forcing’ the signal into the shape with the
highest likelihood.

To add the transits of exomoons to TLCM, we make the following assumptions.
The common center of mass (CCM) of the planet-moon system has an elliptical
motion around the host star, and both of them are on circular orbits around
the CCM. Furthermore, the planet and its satellite orbit the CCM in the same
plane as the CCM orbits the star.

These assumptions help us avoid high degrees of degeneracy (Simon et al.,
2010; Hippke & Heller, 2022), and to limit the number of free parameters to
five to describe the moon (Mandel & Agol, 2002; Csizmadia, 2020). These are
the scaled semi-major axis of the moon, $a_{\rm Moon}/R_{\rm Planet}$, the
moon’s relative radius $R_{\rm Moon}/R_{\rm Planet}$, its relative mass,
$q=M_{\rm Moon}/M_{\rm Planet}$, its orbital period $P_{\rm Moon}$, and
$\Delta\varphi$ – a phase shift between the epoch of the CCM and the transit
of the satellite. The complete transit light curves are computed via numerical
integration of the two bodies, whose sky-projected positions are calculated
via the equations described below. The model presented here is also capable of
accounting for mutual eclipses of the planet and its satellite, as shown on
Fig. 1.

We describe these orbits by introducing two phase angles:

$\displaystyle\varphi_{\rm CMC}$ $\displaystyle=2\pi\frac{t-t_{0}}{P_{\rm
CMC}}$ (1) $\displaystyle\varphi_{\rm Moon}$
$\displaystyle=2\pi\frac{t-t_{0}}{P_{\rm Moon}},$ (2)

where $t_{0}$ is the epoch at which the CCM is collinear with the center of
the stellar disk and the observer, $P_{\rm CMC}$ and $P_{\rm Moon}$ are the
orbital periods of the CCM around the star and of the satellite around the
planet, respectively. The mean anomaly at $t_{0}$, denoted by $M_{0}$, is
calculated via Eqs. (12-15) of Csizmadia (2020). The mean anomaly can then be
written as $M=\varphi_{\rm CMC}+M_{0}$. Let $e_{\rm CMC}$ and $\omega_{\rm
CMC}$ denote the eccentricity and the argument of the periastron (measured
from the tangential plane of the sky) of the CCM, respectively. Using Kepler’s
equation, the eccentric anomaly, solved iteratively, can be written as $E_{\rm
CMC}=M+e_{\rm CMC}\sin E_{\rm CMC}$. The true anomaly, $\nu$, can then be
calculated as

$\tan\frac{\nu_{\rm CMC}}{2}=\sqrt{\frac{1+e_{\rm CMC}}{1-e_{\rm
CMC}}}\tan\frac{E_{\rm CMC}}{2}.$ (3)

In the orbital plane of the planet, the position of the common center of mass
at any point in time is

$\displaystyle x_{\rm CMC}$ $\displaystyle=\frac{a_{\rm
CMC}}{R_{\star}}\cos\left(\nu_{\rm CMC}+\omega_{\rm
CMC}\right)\frac{1-e^{2}_{\rm CMC}}{1+e\cos\left(\nu_{\rm CMC}\right)}$ (4)
$\displaystyle y_{\rm CMC}$ $\displaystyle=\frac{a_{\rm
CMC}}{R_{\star}}\sin\left(\nu_{\rm CMC}+\omega_{\rm
CMC}\right)\frac{1-e^{2}_{\rm CMC}}{1+e\cos\left(\nu_{\rm CMC}\right)}$ (5)
$\displaystyle z_{\rm CMC}$ $\displaystyle=0,$ (6)

where $a_{\rm CMC}$ is the semi-major axis of the CCM, and $R_{\star}$ is the
stellar radius. With the introduction of the moon-to-planet mass ratio,
$q=M_{\rm Moon}/M_{\rm Planet}$, and the semi-major axis of the orbit of the
moon, $a_{\rm Moon}$, the position of the planet in the orbital plane can be
written as

$\displaystyle x_{\rm Planet}$ $\displaystyle=x_{\rm CMC}-\frac{a_{\rm
Moon}}{R_{\rm Planet}}\frac{R_{\rm
Planet}}{R_{\star}}\frac{q}{1+q}\cos\left(\varphi_{\rm
Moon}+\Delta_{\varphi}\right)$ (7) $\displaystyle y_{\rm Planet}$
$\displaystyle=y_{\rm CMC}-\frac{a_{\rm Moon}}{R_{\rm Planet}}\frac{R_{\rm
Planet}}{R_{\star}}\frac{q}{1+q}\sin\left(\varphi_{\rm
Moon}+\Delta_{\varphi}\right)$ (8) $\displaystyle z_{\rm Planet}$
$\displaystyle=0.$ (9)

The position of the moon in the orbital planet of the planet is given by

$\displaystyle x_{\rm Moon}$ $\displaystyle=x_{\rm CMC}+\frac{a_{\rm
Moon}}{R_{\rm Planet}}\frac{R_{\rm
Planet}}{R_{\star}}\frac{1}{1+q}\cos\left(\varphi_{\rm
Moon}+\Delta_{\varphi}\right)$ (10) $\displaystyle y_{\rm Moon}$
$\displaystyle=y_{\rm CMC}+\frac{a_{\rm Moon}}{R_{\rm Planet}}\frac{R_{\rm
Planet}}{R_{\star}}\frac{1}{1+q}\sin\left(\varphi_{\rm
Moon}+\Delta_{\varphi}\right)$ (11) $\displaystyle z_{\rm Moon}$
$\displaystyle=0.$ (12)

In the equations above, $\Delta\varphi$ is the phase shift of the epoch of the
satellite, and $R_{\rm Planet}$ is the planetary radius. The sky-projected
position of the exomoon and the exoplanet are then calculated from Eqs. (7 –
12) via the equations described in Sect. 2.2. of Csizmadia (2020).

From the known sky-projected positions, the limb-darkened transits of the two
bodies are calculated via the commonly used numerical integration tools to
account for the light blocked by the planet and the moon. The resulting light
curve therefore consists a ‘planet component’ and a ‘moon component’
superimposed on each other (Simon et al., 2007). Owing to the assumed
coplanarity, the projected orbital trajectories of the planet and the moon
will only coincide if the inclination of the orbital plane of the CCM (with
respect to the line of sight) is $90^{\circ}$. This is highligted on Fig. 2.

Furthermore, there is a check for the occurrence of mutual transits (i.e. the
moon passing in front of, or behind the planet). If such an event is detected,
a correction is applied so that the mutually eclipsed area is calculated only
once. A mutual transit event can occur if and only if all of the following
conditions are fulfilled:

$\displaystyle\delta_{\mathrm{Planet-Moon}}$ $\displaystyle\leq
R_{\mathrm{Planet}}+R_{\mathrm{Moon}}$ (13)
$\displaystyle\delta_{\star\mathrm{-Planet}}$ $\displaystyle\leq
R_{\star}+R_{\mathrm{Planet}}$ (14)
$\displaystyle\delta_{\star\mathrm{-Moon}}$ $\displaystyle\leq
R_{\star}+R_{\mathrm{Moon}}$ (15)

Here $\delta$ denotes the sky-projected mutual distance of the different
objects. When all three conditions are simultaneously fulfilled, the arcs of
the circular boundaries between the star, planet and moon are determined to
calculate the mutually eclipsed area. This is carried out in a numerical way.
A 101x101 pixel image is centered around the moon and planet and we check
which points are inside the sky-projected images of the star, planet and moon.
The mutually eclipsed area is enclosed by either a full circle (in the limit
if the moon is completely behind or in front of the planet, or, conversely if
the planet is smaller than the moon), or by two or three arcs (see Fig. 2).The
user can choose between $6\times 6$, $10\times 10$, $20\times 20$, $48\times
48$ or $96\times 96$ integration points which are distributed over the
mutually eclipsed area according to the Gauss-Legendre quadrature rules.
According to our experiences, a $6\times 6$ grid is enough to get a few ppm
precision. The intensity behind the mutually eclipsed area is then calculated
with the Gauss-Legendre quadrature and the limb darkening is taken into
account. This correction flux is added accordingly to the light curve, because
the simple transit events of the moon and the planet would result in a
duplicated subtraction of the flux behind the mutually eclipsed area. An
illustration of how this works is given by Figs. 1 and 2.

We note that the current implementation of the photodynamic model is highly
simplified, as the orbital plane of the moon is considered to coincide with
the one of the CCM. Furthermore, the two bodies are also considered to be on
Keplerian orbits (hence the term ‘zero order’). Future updates will focus on
the inclusion of the possibility of misaligned orbital plane for the
satellite, as well as the inclusion of various perturbative effects, which may
lead to variations in periodicity, transit depth and duration etc. (see e.g.
Borkovits et al., 2022, for a more detalied discussion on these effects in
triple star systems).

Figure 2: Possible mutual eclipse cases with an illustration of the Gauss-
Legendre quadrature integration points (blue) distributed during the mutual
eclipses. The three discs represent, in decreasing order of diameter, the
star, the planet and the moon. The projected orbits of the planet and the moon
are shown with dashed black and magenta lines. The dashed blue line is used to
visualise the impact parameters of the two bodies. The limb darkening of the
stellar disc is generated via solar-like parameters according to the routines
of Harre & Heller (2021). The mutually eclipsed area (by the planet and the
moon) is outlined with solid red lines. Case A: the planet fully occults the
moon (or the moon totally transits the planet, i.e. the moon can be behind or
in front of the planet). Case B: The planet and the moon exhibit partial
mutual eclipse but both totally eclipses the star. Case C: During the partial
eclipses of the planet and/or the moon, three arcs may be needed to find the
mutually eclipsed area. Case D: no mutual eclipse.

##  3 Light curve simulation

###  3.1 A schematic configuration

An example light curve, computed via the methods described in Sect. 2, is
shown on Figure 1. This system was set up with the following exaggerated,
arbitrary parameters: $a_{\rm CMC}/R_{\star}=50$, $a_{\rm Moon}/R_{\rm
Planet}=10.2$, $\Delta\varphi=0.4$, $R_{\rm Moon}/R_{\rm Planet}=0.53$,
$P_{\rm Moon}=2.5789$ days, $R_{\rm Planet}/R_{\star}=0.51$, $q=0.3$, $e_{\rm
CMC}=0.35$, $\omega_{\rm CMC}=124^{\circ}$, and $P_{\rm CMC}=21.5089$ days.
For the purposes of this demonstration, we assumed uniform surface brightness
of the stellar disk (i.e. no limb darkening) and we enlarged the mass and
radius of the moon. Furthermore, we added a synthetic noise model with
autocorrelated effects, described by an Autoregressive Integrated Moving
Average (ARIMA) model (e.g. Wilson, 2016). An $ARIMA(p,d,q)$ process consists
of an autogregressive model of the order $p$ and a moving average model of the
order $q$, with a $d$-order integration. In time series analysis, integration
can be thought of as the inverse of differencing, a smoothing operation that
generates from the time series $y(t)$ the series $y^{\prime}(t)$ by
representing the change between each consecutive element of $y(t)$, thus:

$y^{\prime}_{i}(t)=y_{i}(t)-y_{i-1}(t).$ (16)

The autoregressive and moving average processes can be represented as

$\displaystyle y_{i}(t)$
$\displaystyle=c_{1}+\sum_{k=1}^{p}\phi_{k}y_{i-k}(t)+\varepsilon_{i}$ (17)
$\displaystyle y_{i}(t)$
$\displaystyle=c_{2}+\varepsilon_{i}+\sum_{k=1}^{q}\theta_{k}\varepsilon_{i-k}.$
(18)

Here $c_{1,2}$ are constants, $\phi_{k}$ and $\theta_{k}$ are the coefficients
of the two processes, while the noise terms $\varepsilon$ are drawn from a
Gaussian distribution with zero mean (i.e. white noise). In our case (Figure
1), we set up the model with the parameters
$ARIMA([1,-0.5],[1],[0.5,-1])$111In this case, $p=q=2$ and $d=1$. The
coefficients $\theta_{1,2}$ and $\phi_{1,2}$ are expressed in the brackets..
Figure 1 clearly shows that the algorithm is capable of handling different
angular velocities for the planet and its satellite (manifesting in different
transit durations for the two objects).

###  3.2 A realistic configuration

We also explore a realistic planet-moon case. To that end, we adopted the
parameters of the planet and its candidate satellite from Teachey & Kipping
(2018), with arbitrary limb darkening parameters, while restricting the
orbital plane of the moon to coincide with that of the planet. The left and
middle columns in Fig. 3 show two transits with arbitrarily positioned moons.
In the rightmost column, we demonstrate a false-positive detection in a no-
moon case.

The top panels show the synthetic input transits with a white noise component
only (with a $200$ ppm point-to-point scatter, compared to the $\sim 270$ ppm
transit depth of the moon). The three columns present two systems with moons
(A and B) and a moonless case (C). The middle row shows the same input
signals, but with a red noise component added as well. To be realistic, here
we introduced the red noise via an $ARIMA([0.6819],[1],[-0.3603,-0.5747])$
process, cloning the noise of the residuals of the original measurement of
Teachey & Kipping (2018). The bottom row presents a comparison between the
“moon” solution and the synthetic planet+moon+red noise. We observe that in
cases A and B the presence of the moon is convincing in the synthetic
measurements even by visual inspection. On the other hand, in scenario (C)
where the moon is absent, we encounter a false positive detection due to the
similarity of the red-noise-induced signal to that of the moon’s.

One possible way to exclude false detections like this is to schedule
observations that are long enough so that a noise filtering algorithm (e.g.
the wavelet-filter of Carter & Winn, 2009) is able to discern the actual noise
properties in detail.

The true advantage of this filtering technique lies in the fact that it does
not require prior knowledge of the exact source or realisation of the time-
correlated noise, since it operates by constructing a time series model in a
wavelet basis. It is therefore efficient in removing the red noise down to the
time scales of minutes (Csizmadia et al., 2023). It is also important to
stress that the test carried out with the TLCM thus far (although they were
limited to transiting exoplanets) indicate that the radius of the transiting
object can be recovered with high precision and accuracy even at low S/N
(Csizmadia et al., 2023; Kálmán et al., 2023). This would also be one of the
most important parameters for a potential exomoon candidate.

Fig. 3 has two important lessons for us:

  * •

When chasing a moon, the measurement of the null signal (noise only) before
and after the transit has to be as long as or longer than the part of the
presumed signal itself, to enable the algorithms to distinguish between noise
and signal.

  * •

The threshold of a secure detection depends on the length of the noise
sampling as well as the actual geometrical configuration. This has to be
tested even at the observation planning phase, and has to be taken into
account in reasoning the feasibility of any kind of measurement in this field.

We note however that the possibility of dense rings around Kepler-1625b, or
even the as yet unconfirmed Kepler-1625b-i (Sucerquia et al., 2022) can not be
excluded, although exploring that possibility is beyond the scope of this
paper and will be the subject of a subsequent one.

Figure 3: Demonstration of time-correlated noise mimicing transits of an
exomoon. In cases A (left column) and B (middle column), the transits of the
moon are present in the synthetic photometry (points), case C (right column)
presents the transit of an exoplanet only but leads to false detection because
of the correlated noise in combination with short observations. The top row
represents the ideal case where only white noise is present on the light
curve. The middle and bottom rows also have an ARIMA noise model superimposed
upon them. The synthetic photometry in the bottom row is the same as in the
middle row, but the solid red line represents a planet-and-moon scenario. We
emphasise that these figures are not results of light curve fitting.

##  4 Discussion and summary

Motivated primarily by the lack of confirmed exomoons and the work of Heller
et al. (2019), we developed a complex photodynamical model, capable of
describing a planet–moon system on an elliptical orbit around a host star in
the presence of red noise. We integrated this model into the TLCM code
(Csizmadia, 2020). One of the primary features of the TLCM algorithm is the
ability to handle time-correlated noise via the wavelet-based routines of
Carter & Winn (2009).

It is therefore expected that by combining this noise handling method and the
photodynamical model presented here, we will be able to ($i$) reject false
positive exomoon candidates (Heller et al., 2019), ($ii$) uncover exomoons
‘hidden’ by red noise, and ($iii$) characterise real exomoons with
sufficiently high precision and accuracy, as it is commonly done for
exoplanets (Csizmadia et al., 2023).

The results shown in the paper can be summarised as follows:

  * •

The Exomoon Agent embedded in the TLCM algorithm is introduced. With this
development, users can now search for planet+moon solutions in addition to
planet-only cases, and will be able to chose the best model family based on
the e.g. Akaike and Bayesian Information Criteria.

  * •

As a case study, we investigated possible transits of a planet–moon system
with parameters taken from Teachey & Kipping (2018). We added correlated noise
to the simulations using ARIMA clones of the actual residual of the
observations.

  * •

We observed that correlated noise components can mimic the characteristic
pattern of exomoons, leading to false positive detections. Our results are in
accordance with Heller et al. (2019).

  * •

The single way to avoid these false detections is to take long enough
observations. They must include the pre- and post-transit areas of null signal
to teach the algorithm the noise properties in details, that can lead to a
reliable mitigation. The threshold of a suspected detection depends on the
sampling of the observed data, which must be calibrated by accounting for the
length of the ‘red noise only’ (or out-of-transit) light curves.

  * •

With the currently available instruments (including those that are in advanced
stages of development such as Ariel), we will likely never be able to claim
with full confidence that an exomoon has been detected by modelling a single
transit. This is partly because the proper separation of time-correlated noise
of astrophysical and instrumental origin itself can be challenging in and of
itself, even when the instrumental effects are well understood as in the case
of CHEOPS (Hoyer et al., 2020). In order to separate the temporally correlated
noise from the signal of a transit, the desirable strategy of observing
exomoons with HST, JWST or even Ariel would have to rely on multiple transits,
spanning years (due to the relatively long orbital periods of the exomoon-
hosting planets). This would, ideally, allow observations of the target system
in multiple configurations (Fig. 2), thus providing a deeper understanding of
the probability of false alarm. A strong claim to the presence of an exomoon
would have to be the change in occurrence of its transit from observation to
observation (relative to the transit of its host). Ideally, TTVs would also
have to be observed, and a thorough follow-up would be needed.

  * •

While an understanding of systematic noise sources is beneficial, the wavelet-
based noise handling of TLCM can reasonably be expected to allow us do
distinguish between an actual moon and a false detection. This has been the
inspiration behind the Exomoon Agent upgrade of TLCM.

The software is available publicly for further tests and uses. The thorough
validation of the parameter-retrieval capabilities is beyond the scope of this
Letter, and it remains in the user’s responsibility to look for the stability
of solutions in the actual modelling. In an upcoming publication, we will test
the parameter stabilities in simulated data sets and data from real
measurements. It is expected however, that the Exomoon Agent can be used to
reanalyse existing data.

We expect that the current chase for exomoons will remain with throughout the
operations of the current space telescopes. In the search for ever smaller
satellites, orbiting planets whose hosts are ever farther away, there will be
a clear need for data from larger and more sensitive future space-born
telescopes, such as the Large UV Optical Infrared Surveyor (LUVOIR; The LUVOIR
Team, 2019).

## Acknowledgements

SzK and GyMSz acknowledge the PRODEX Experiment Agreement No. 4000137122
between the ELTE Eötvös Loránd University and the European Space Agency
(ESA-D/SCI-LE-2021-0025). Project no. C1746651 has been implemented with the
support provided by the Ministry of Culture and Innovation of Hungary from the
National Research, Development and Innovation Fund, financed under the
NVKDP-2021 funding scheme. SzCs gratefully acknowledges the European Space
Agency and the PLATO Mission Consortium, whose outstanding efforts have made
these results possible. We thank DFG Research Unit 2440: ’Matter Under
Planetary Interior Conditions: High Pressure, Planetary, and Plasma Physics’
for support. We also acknowledge support by DFG grants RA 714/14-1,2 within
the DFG Schwerpunkt SPP 1992: ’Exploring the Diversity of Extrasolar Planets’.
ASE acknowledges support from the Swiss Space Office through the ESA PRODEX
program. This work was supported by European Union’s Horizon Europe Framework
Programme under the Marie Skłodowska-Curie Actions grant agreement No.
101086149 (EXOWORLD). A.De. acknowledges financial support from the Swiss
National Science Foundation (SNSF) for project 200021_200726. We would like to
dedicate this paper to the memory of Prof. István Jankovics (1943-2023),
recognizing the profound influence he had on our scientific community and the
enduring inspiration he provides to current and future generations of
astronomers.

## Data Availability

No new data were generated or analysed in support of this research. The
Transit and Light Curve Modeller software is available at http://transits.hu/.

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