2021 [1]Andrea Russo

1]Department of Physics and Astronomy ”E. Majorana”, University of Catania

# Network analysis on political election; populist vs social emergent
behaviour

*<EMAIL_ADDRESS>Vincenzo Miracula  Antonio Picone  [

###### Abstract

Social networks play an important role in people’s daily socialization,
particularly through social media platforms, which have become key channels
for communication and information dissemination. The digital ecosystem does
not only evolve communication on multi-network (like TV, social media, and
online newspapers) but also provides the social researcher with useful data to
explain social-complex dynamics. Our work focus on cultural dynamics-reactions
that occurred during the 2020 Emilia-Romagna elections” in Italy, where a
stronghold culture felt in danger of losing against the strong populism and
Euro-scepticism present in digital ecosystems. We would like to show how the
interaction between parts of the society, during cultural and/or political
shifting, can lead to or induce emerging behaviour from society, creating
groups that react against or improve the status quo. We developed a word-entry
network based on three different levels of participation: pro, con, and
neutral. We have analyzed the tweets collected (as text) with the word
embedding tools, to see, the most used words (which may suggest the main
topics) and the most related words among the various groups. We show how a
careful analysis of groups through networks, can give important information
about the current event.

###### keywords:

Political election, Network science, Computation Social Science, Social
dynamics, Complex system, Populism

##  1 Introduction

Millions of people interact and keep themselves informed by using the
Internet. Everyone - somehow - is part of these news systems and is changing
the communication tools. Blogs, social networks, and online newspapers are now
social relationship tools Bennato . However, in the 2020s, strong and weak
mutations appear, which modify the society and make it react in different
ways. Wars, pandemics, and cultural dynamics processes have drastically
changed the sociological ecosystem.

Our work focus on cultural dynamics-reactions that occurred during the 2020
“Emilia-Romagna elections” cairney2012complexity  in Italy, where a stronghold
culture perez2015persistence  felt in danger of losing against a strong
populism and Euro-scepticism present in digital ecosystems. This election was
also important for the national equilibrium of the national government,
because, the left party affiliated with the stronghold culture is in the
government itself. Losing the election in his most important region can also
induce losing the position in the national government (because they are no
longer powerful and important) and let the government fall. So, this election
was important not only in a regional context but also at national and
international levels.

Inside these national and political dynamics, we would like to show how the
interaction between parts of the society, during cultural and/or political
shifting, can lead to or induce emerging behaviour from society, creating
groups that react against or improve the status quo. The cultural and/or
political shifting, is related to the growth of the right-wing populist party,
in a left-wing stronghold area, while during the election campaign, the
populist party has made a series of populist actions in the physical and
digital ecosystem.

Those actions have obtained powerful mediatic attention, because some of them
were also related to the drug cartel of the city (asking people if they were
drug dealers), violating people’s private sphere and inducing hate against
someone (especially black or migrant), many of this action were also linked to
legal and security aspects.

The research question that we would like to answer is related if it is
possible to understand the reason that led to the creation of the emerging
group, and if it is possible via computer science to understand the dynamics
with the other groups (left and right wings party). We think that the emerging
group, called “Le Sardine” (The Sardines), arises due to the fear of losing in
that stronghold area, and to react against a political populist party that
grows in political support.

We aim at studying this social phenomenon by building a network to evaluate
the interaction between the stronghold culture and the arising populism
culture in Emilia-Romagna.

##  2 Methodology

It is known that in order to study any kind of phenomenon data are needed,
whatever the subject that is being considered is. With the information
gathered, we developed a word-entry network based on three different levels of
participation: pro, con, and neutral.

The data was then further processed using NetworkX and Gephi, the former being
one of the most widely used libraries for generating, manipulating, and
visualizing networks through Python, while the latter is an open source
program for analyzing and visualizing social networks.

###  2.1 Data Gathering

In order to get the data for this analysis we decided to collect a significant
amount of textual datamitchell2018web  over time and from a specific social
network: Twitter.

The reasons behind this choice are fairly easy to understand. Twitter is the
only social network from which you can easily get data, as it provides a
regular RESTful API, which is a kind of API (Application Program Interface)
that is designed to exchange data over the Internet. It requires an
authentication method, that was requested through the Twitter developer
portalmakice2009twitter .

In order to consume these APIs we used a library that can be used through
Python code: Tweepy. It allows developers to interact with the Twitter API in
an easier way than making raw HTTP requests, so it is handy to handle big
amounts of data.

There is another feature that was useful for our purpose, you can filter your
query while using the Twitter API, language, location and hashtags can be
chosen. We decided to search all the tweets containing the hashtags
#bolognanonsilega, #sardine, and #salvini.

As already mentioned, additional filters were applied to our search query, not
only we selected the above-mentioned keywords but also we specified a
geofence, and we collected all the tweets geotaggeddavis2011inferring  in a
range of 50 km from the city of Bologna, which means that all the tweets in a
radius of 50 km from the coordinates of the city center of Bologna were
scraped and saved in our database as we wanted to study this phenomenon in a
controlled way.

We ended up with a dataset consisting of $\sim$2100 tweets, ranging from
12-11-2019 to 20-01-2020 and they were stored in CSV format (comma-separated
values). It is a text-based format where each column is separated by either a
comma, semicolon, or tab and can be imported into popular spreadsheet editing
software like Microsoft Excel. We then had all the data but the informative
value was missing for our research yet, as we aimed at understanding whatever
each tweet was against the leader of the populists, neutral or pro. We tried
to build a machine learning algorithm that is able to distinguish and
categorize these 3 categories but it did not give any relevant results.

The main reason for this failure is that a lot of tweets were either ironic or
they needed a background context to understand what they were referring to.
The only way to accomplish that - despite being more time-consuming - was
manually reading each tweet and tagging whenever each one of them was against,
neutral, or pro and we used - for the sake of simplicity - respectively 0, 1,
or 2.

###  2.2 Network & Community detection

Networks are defined as structures albert2002statistical  that represent the
group of objects and/or people, along with their relationship. Social networks
play an important role in people’s daily socialization lazer2009computational
, particularly through social media platforms, which have become key channels
for communication and information dissemination. It becomes increasingly
important to understand how information spreads within these networks. For
example, the speed at which information is shared, how users group themselves
by similar interests, how they interact with each other, and so on.

One way to study social networks is through the use of social network analysis
(SNA), which is a field of study that involves the use of statistical and
mathematical techniques to analyze and understand the relationships and
patterns within a network of individuals or organizations. SNA is often used
to identify key players and understand the dynamics of social, professional,
and communication networks. The aim of SNA is to understand a community by
mapping the relationships that connect them as a network and then try to draw
out key individuals, and groups within the network and/or associations between
the individuals. Twitter is a great platform for observing these behaviours.

In SNA, a network is typically represented as a graph, with nodes representing
the individual actors and edges representing the relationships between them. A
variety of different metrics (see Table 1) can be used to analyze these
networks, including measures of centrality (e.g., degree, betweenness),
measures of community structure (e.g., modularity), and measures of network
evolution albert2002statistical  (e.g., preferential attachment).

Statistic | Explanation
---|---
N. of nodes | N. of people in the network
N. of links | N. of relationships between people in the network
N. of components | N. of discrete groups in the network
Density | The proportion of all links that are actually present
Diameter | Size of the network
Mean avg distance | Avg n. of steps needed to go from one node to any other
Mean degree | Avg n. of links that pass through the nodes
Mean betweenness | Avg n. of unique paths that pass through the nodes
Table 1: Statistic and explanation

In these large social networks, especially linked to important issues such as
political elections, people group according to opinions by interacting with
some users. We will use Python’s implementation for the Louvain community
discovery algorithm. The algorithm is described in the work of Blondel et
al.blondel2008fast . The idea of the algorithm underlies the calculation of
modularity, as the metrics used to measure the quality of the division. The
modularity measure is a value in the interval [-1, 1] and measures the density
of links within communities compared to links between communities.

Controversial topics can be more engaging because they tend to arouse strong
emotions and opinions in people. When people encounter information or ideas
that challenge their beliefs or values, they may feel motivated to engage with
the content and express their opinions. This can lead to increased activity
and engagement on social media platforms, as people comment, share, and like
content related to the controversial topic.

In their work, Recuero et al.recuero2019using  conducted an SNA to identify
polarized political conversations on Twitter. To understand them, they used
several metrics, such as in-degree, out-degree, and modularity. According to
the authors, we use modularity as a measure of the strength of the division of
a network into distinct groups or communities.

Furthermore, controversial topics can generate a sense of urgency or
importance, as people may feel that the topic has immediate relevance or
consequences for themselves or others. This, too, can contribute to increased
involvement and participation.

It is worth noting that controversial topics can also generate negative or
harmful effects, such as misinformation and the spread of fake news. It is
important to approach controversial topics carefully and consider the
potential consequences of involvement in such content.

###  2.3 Word-embedding

Another tool used in this paper is word embedding. The word embedding makes it
possible to store both semantic and syntactic word information from an
unannotated corpus and constructs a vector space in which word vectors are
closer together if the words occur in the same linguistic contexts, i.e. if
they are recognized as semantically more similar (according to the hypothesis
of distributional semantics). In a stricter definition, word embedding is an
overall term for a set of modeling techniques in natural language processing
in which words or phrases of a vocabulary are mapped into vectors of real
numbers. We have analyzed the tweets collected (as text) with the word
embedding tools, to see, the most used words (which may suggest the main
topics) and the most related words among the various groups.

##  3 Results

The Emilia Romagna regional election background, was, indeed a test for the
national government and for the left-wing against right-wing national
leadership. The Emilia Romagna - given the election results R1  and as shown
in figure 1 \- is a stronghold region for the left-wing political party in
Italy, and for the first time, this region can be governed by the right
political party, with the former leader Matteo Salvini (Leader of Lega Nord).

Figure 1: Election between 1970-2005 Figure 2: Election between 2005-2020

The event became a very important one, both by the amount of activity in
social networks giuffrida2021analyzing  russo2022entropy , but also for the
political and cultural left-wing political party R1 , because, as shown in
figure 2, since the election of 2005 the right-wing party has continuously
reduced the distance between both. So, if the left party lost, the election
would certificate not only the leadership of Matteo Salvini as the leader with
the most agreement throughout Italy, but also in the regional institutions of
Emilia Romagna (which have always dealt with left-wing culture) and also in
national institutions.

The data analysis shows that 50% of the collected posts refer to the new local
movement (known as 6000sardine), but the network analysis highlights that the
major and important hub is instead the leader of the populism and Euro-
skepticism party. This contrast shows the reaction of social dynamics to
populism propaganda during the election. The 6000sardine movement has
preferred to react to opposing political propaganda (given the various
populist events) than to voicing the presence of the stronghold culture in the
Emilia-Romagna area. This means that the 6000sardine wasn’t a stronghold
culture offshoot, born to protect themselves, but a social reaction against
the populism and Euro-skepticism presence in a digital environment.

Figure 3: Populist and Sardine Network

###  3.1 Network results

The first result of this paper is related to the networks themselves, shown in
Figure 3. The statistics of all networks are shown in Table 2. The names of
the nodes in the pictures are in proportion to each other. In this case, nodes
with more links will have larger names given their importance.

The network shows different characteristics: 1) For the colors for each
community, we chose blue for the Populists, red for the Sardine Movement,
yellow for M5s and journalists (the algorithm could not distinguish between
them), while the other colors are various scattered communities; 2) Networks
have 14 different communities (divided into populist, neutral, and pro), and
10 different hubs;
Of the 10 hubs (only selected if a minimum number of 30 likes were present and
connected to the others through citations and replies), 6 of them belonged to
the populist party, and 4 were against it.

Figure 4: Populist party Figure 5: Sardine movement Statistic | Both | Populist | Sardine
---|---|---|---
N. of nodes | 926 | 403 | 647
N. of links | 1104 | 422 | 749
N. of community | 34 | 12 | 20
Modularity | 0.533 | 0.636 | 0.58
Density | 0.0026 | 0.05 | 0.035
Mean degree | 0.015 | 0.025 | 0.02
Mean betweenness | 0.012 | 0.024 | 0.4
Table 2: Networks statistics

The main hub of the populist network is the leader of the populist party
(Matteo Salvini), as shown in Figure 4.

The same network also shows a few hubs close to the leader. This behaviour is
uncommon during the election because usually most of the attention should be
focused on the candidate (Lucia Borgonzoni), but she has a marginal role. Most
of the attention was focused on the leader of the populist party. We may find
an explanation for that in the leader’s various populist actions during the
election campaign, where he walked into an alleged drug dealer’s house asking
if he was dealing drugs spaccia , or when he exploited a criminal case
involving minors Bibbiano .

For these reasons, the Sardine Movement emerged, namely to counter the
populist phenomenon. In fact, as Figure 5 shows, the main hub of the network
is exactly the leader of the Populist group, Matteo Salvini. This can be
explained by what was said earlier, namely, the Sardine Movement was born as
an Anti-Populist movement, and not as an aid to leftist parties. In fact, the
network characteristics show very few hubs, as they are all focused on the
populist leader (the main one). Even in this case, there is no focus on the
candidate (leftist) during the election.

These network dynamics mark that the populist party has fought the stronghold
culture and that at the same time has also raised an attack against the
6000sardine movement. Both networks show a Barabasi Complex network
albert2002statistical  for populist and sardine networks.

###  3.2 Word-embedding Result

The second result is related to word embedding networks. The right-wing leader
party Salvini was the principal hub in both networks; this evidences (again)
that Salvini’s role-action was the main topic during the election campaign,
and not Emilia Romagna itself, with their respective political parties,
leaders, and political programs/projects. The word embedding score and main
topic/words used, both for Sardine and populist, is shown in Table 3.

Words | Sardine movement | Populist party | all
---|---|---|---
Salvini | 15 | 2 | 19
Citofono (intercom) | 10 | - | -
Bambola-gonfiabile (sex-doll) | 10 | - | -
Piazza | 5 | - | 5
Sardina | - | - | 5
Stato | - | 3 | 5
PD |  | 4 | -
Italia | - | - | 4
Sardine | - | 2 | -
Governo | - | 2 | 3
Gente | - | 2 | -
Lavoro | - | - | 3
Emilia | - | - | 3
Sinistra | - | - | 3
Table 3: Word-embedding Sardine movement

##  4 Conclusion

The present paper draws attention to how computational social science and
network science can explain both the complex dynamics of controversial and
challenging topics. The digital ecosystem does not only evolve communication
on multinetwork (like TV, social media, and newspapers) but also provides the
social researcher with useful data to explain social-complex dynamics.

In our work, we have shown how by analyzing political groups during important
events, it is possible to show some characteristics of the social and
political dynamics of various groups. For example, during the political
election in Emilia Romagna, a large presence was shown both in social and also
in more classical media, given the various behaviours of the leader Matteo
Salvini, which triggered social mobility. Our work was mainly based on the
analysis of networks, and words used during the election campaign, showing how
a careful analysis of groups through networks, can give important information
about the current event. In fact, none of the candidates was ever a main hub
in the networks, while much more attention was paid (in both networks) to the
actions of the populist leader.

##  5 Declarations

The authors declare no conflict of interest.
The authors disclosed receipt of the following financial support for the
research, authorship, and/ or publication of this article: This project has
not received funding from the University of Catania.

##  6 Authors contribution

All Authors have contributed equally.

## References

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# Characterisation of the eclipsing post-common-envelope binary TIC 60040774

R. Priyatikanto,1 C. Knigge,2 S. Scaringi,3 J. Brink4,5 and D.A.H.
Buckley4,5,6
1Research Center for Space, National Research and Innovation Agency, Bandung
40132, Indonesia
2School of Physics and Astronomy, University of Southampton, Highfield,
Southampton SO17 1BJ, United Kingdom
3Centre for Extragalactic Astronomy, Department of Physics, Durham University,
DH1 3LE, United Kingdom
4South African Astronomical Observatory, PO Box 9, Observatory, 7935, Cape
Town, South Africa
5Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
6Department of Physics, University of the Free State, PO Box 339, Bloemfontein
9300, South Africa
E-mail<EMAIL_ADDRESS>(RP)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Binaries with a white dwarf primary and a main sequence secondary can be used
to test our understanding of both single and binary star evolution. A small
fraction of such systems experienced a common-envelope phase from which they
emerged with a relatively short orbital period. Here, we present the
characterisation of an eclipsing post-common-envelope binary of this kind, TIC
60040774, based on the light curve provided by the Transiting Exoplanet Survey
Satellite (TESS), multi-band photometry collated from the virtual observatory,
and spectroscopic data obtained the Southern African Large Telescope (SALT).
With an orbital period of $0.404807\pm 0.000149$ days, this system consists of
a young white dwarf paired with an M6.5 dwarf companion. We estimate the
masses of the primary and secondary to be $0.598\pm 0.029$ M⊙ and $0.107\pm
0.020$ M⊙, while the effective temperatures are $14050\pm 360$ K and $2759\pm
50$ K, respectively. The eclipse ingress and egress profile is shallower than
expected from a simple geometric model such that more precise high-cadence
photometry is required to understand the nature of this system. Given the
similarity of TIC 60040774 to systems like GK Vir and NN Ser, it will be worth
tracking its eclipse times to check for the presence of one or more
circumbinary planets.

###### keywords:

binaries: eclipsing – binaries: close – stars: individual: TIC 60040774 –
white dwarfs

††pubyear: 2022††pagerange: Characterisation of the eclipsing post-common-
envelope binary TIC 60040774–References

##  1 Introduction

Binary systems can provide important insights into stellar structure and
evolution. Binaries, and especially eclipsing binaries, allow us to determine
fundamental stellar parameters (such as masses and radii) and to study their
evolution as a function of stellar age(Andersen, 1991; Torres et al., 2010).
Binary stars are therefore central to our understanding of stellar evolution
(Stancliffe et al., 2015; Claret, 2018). Eclipsing binaries also provide
opportunities to determine distances on extra-galactic scales with great
precision (Bonanos et al., 2006; Vilardell et al., 2010; Pietrzyński et al.,
2013), helping to anchor the cosmological distance ladder.

Binaries are not rare amongst stars in the Milky Way. Moe & Di Stefano (2017)
estimate that almost all massive stars tend to be in binary or multiple
systems, with a multiplicity fraction of 90%. This fraction decreases roughly
linearly with the logarithm of the primary mass, so, for example, still
${\gtrsim}40\%$ of solar-like stars belong to multiple systems. Close binaries
are also quite common. For example, ${\sim}15\%$ of solar-like stars become
the primary of close binary systems with separation of less than ${\sim}13$ AU
Moe & Di Stefano (2017). These systems will become interacting binaries where
the evolution of the components are affected by mass transfer either through
Roche-lobe overflow or a common-envelope phase. These binary evolution
channels give many exotic objects, such as blue stragglers (Perets & Fabrycky,
2009), cataclysmic variables (Knigge et al., 2011) and AM CVn stars (e.g.,
Nelemans et al., 2004). Common-envelope evolution, in particular, also
produces a significant population of close, but detached white-dwarf/main-
sequence (WDMS) binaries.

Rebassa-Mansergas et al. (2021) recently estimated a WDMS space density of
$3.7\times 10^{-4}$ pc-3, implying the presence of $\simeq 1500$ such systems
within a distance of $100$ pc. Approximately a quarter of WDMS are close
enough that they must be post-common-envelope binaries (PCEBs), and $\simeq
10\%$ of these PCEBs exhibit eclipses (Rebassa-Mansergas et al., 2016; Parsons
et al., 2013, 2015).

Eclipsing WDMS are essential objects for constraining the mass-radius
relationships for white dwarfs (WDs Parsons et al., 2017) and low-mass main-
sequence (MS) stars (Parsons et al., 2018). Moreover, multi-epoch observations
of some eclipsing PCEB systems reveal cyclic variations in the eclipse timings
that cannot be ascribed to magnetic activity via the Applegate mechanism
(Applegate, 1992). Examples here include NN Ser (Beuermann et al., 2010; Marsh
et al., 2014), RR Cae (Qian et al., 2012), DE CVn (Han et al., 2018), and GK
Vir Almeida et al. (2020). In these systems, the eclipse time variations are
therefore thought to be due to the presence of distant sub-stellar or
planetary companions. The existence of such companions to PCEBs raises obvious
questions for the theory of planetary formation (Bear & Soker, 2014; Völschow
et al., 2014). P-type or circumbinary planets that were formed from the same
cloud as the binary itself are expected to be ejected during the common-
envelope phase, due to the abrupt loss of gravitational binding energy. On the
other hand, S-type planets initially bound to one of the components component
are expected to be accreted by the primary, producing metal pollution
signatures in the white dwarf’s atmosphere (Koester et al., 2014).
Consequently, post-common-envelope planetary formation has been proposed to
explain the existence of planetary bodies orbiting systems like NN Ser
(Schleicher & Dreizler, 2014; Völschow et al., 2014). In order to test such
scenarios, it is vital to determine the frequency and statistical properties
of sub-stellar and planetary companions among a larger sample of eclipsing
PCEBs.

In this paper, we present the discovery and characterisation of a bright new
eclipsing PCEB. The system, TIC 60040774, is a high-proper-motion object that
lies close to the WD sequence in the colour-magnitude diagram and is located
fairly nearby, at a distance of $134$ pc. We show that TIC 60040774 is close
binary in which an M6.5V secondary eclipses a WD primary every $\simeq 0.4$
days. We also determine the binary parameters of the system and discuss its
properties in the context of other PCEBs.

##  2 Data

TIC 60040774 ($G=16.98$) is located ${\sim}15^{\circ}$ south of the Galactic
plane. Based on its Gaia EDR3 astrometry, it exhibits a relatively higher
proper motion of $\simeq 68$ mas yr-1 and is located at a distance of
$134.465$ parsecs (Gaia Collaboration et al., 2021). Some basic parameters of
TIC 60040774 are summarised in Table 1. According to its location in the
Hertzprung-Russell diagram (see Figure 1), Gentile Fusillo et al. (2021)
identified this object as a highly probable white dwarf with an estimated mass
of ${\sim}0.3$ M⊙ (assuming it to be a single WD). TIC 60040774 was observed
by the Transiting Exoplanet Survey Satellite (TESS, Ricker et al., 2015) with
two-minute cadence and identified as a target with significant transit by the
Science Processing Operations Center (SPOC) Pipeline (Caldwell et al., 2020).
However, the Data Validation pipeline, which fits limb-darkened transiting
planet models to the observed light curve (Twicken et al., 2018), excluded TIC
60040774 as a target of interest, since it assumed a solar-like primary and
predicted the secondary to be a non-planetary body with an estimated radius of
${\sim}0.8$ R⊙. Priyatikanto (2022) independently (re-)discovered TIC 60040774
as an eclipsing binary during a search for highly variable WDs in the TESS
data set. We then collected additional multi-wavelength photometric
measurements from the literature and obtained new spectroscopic observation
with the Southern African Large Telescope (SALT).

Figure 1: Left: the position of TIC 60040774 in the HR diagram plotted over the probable white dwarfs from Gentile Fusillo et al. (2021). Black lines are the cooling tracks of DA white dwarfs with masses from $0.2$ to $1.5$ solar masses. Right: TESS field-of-view around TIC 60040774 (cross) complemented with stars with $T_{\text{mag}}<16.5$. Table 1: Basic parameters of TIC 60040774 Other designations |  Gaia DR2 2943496741862496256
2MASS J06195643-1758186
---|---
Astrometry |  |
Right ascension | $\alpha_{2000}$ | $06^{\text{h}}\ 19^{\text{m}}\ 56.434^{\text{s}}$
Declination | $\delta_{2000}$ | $-17^{\circ}\ 58^{\prime}\ 18.621^{\prime\prime}$
Galactic longitude | $l$ | $225.796191^{\circ}$
Galactic latitude | $b$ | $-14.872777^{\circ}$
Proper motion | $\mu_{\alpha}\cos\delta$ |  $-23.306\pm 0.052$ mas/yr
| $\mu_{\delta}$ |  $-63.937\pm 0.057$ mas/yr
Parallax | $\pi$ |  $7.4369\pm 0.0611$ mas
Distance | $d$ |  $134.465\pm 1.105$ pc
Photometry |  |
Gaia EDR3 | $G$ | $16.97695\pm 0.00296$
| $B_{P}$ | $17.04543\pm 0.00596$
| $R_{P}$ | $16.79837\pm 0.00722$
TESS | $T_{\text{mag}}$ | $16.880\pm 0.009$
Pan-STARRS1 | $g$ | $16.949\pm 0.004$
| $r$ | $17.168\pm 0.003$
| $i$ | $17.208\pm 0.003$
| $z$ | $17.034\pm 0.006$
| $y$ | $16.797\pm 0.009$
2MASS | $J$ | $15.675\pm 0.054$
| $H$ | $15.220\pm 0.074$
| $Ks$ | $14.801\pm 0.099$
AllWISE | $W1$ | $14.737\pm 0.035$
| $W2$ | $14.611\pm 0.064$

###  2.1 SALT spectra

The Robert Stobie Spectrograph (RSS, Burgh et al., 2003), operated at the
prime focus of SALT, was used to acquire optical spectra of TIC 60040774 on
2021-11-14 22:48 UT and 2021-11-16 22:19 UT. The observations were taken in
long-slit spectroscopy mode and used the PG0900 grating, yielding a resolution
of $R=\lambda/\Delta\lambda\approx 1065$ at the central wavelength of $6050$
Å. The blue and red segments of the optical spectrum were acquired separately.
For the blue segment ($3900$ to $7000$ Å), a UV blocking filter was used,
while the red segment ($6000$ to $9000$ Å) was obtained with a blue blocking
filter. Exposure times of $1200$ s yielded sufficient signal-to-noise for our
purposes. PySALT (Crawford et al., 2010) was used to reduce and calibrate the
spectra.

###  2.2 Multiband photometry

The characterisation of the faint, red secondary component requires access to
wavelengths beyond the optical range. Fortunately, TIC 60040774 was observed
by several survey missions, allowing the construction of a broad-band Spectral
Energy Distribution (SED) that extends well into the infrared region. We used
the Virtual Observatory SED Analyzer (VOSA111http://svo2.cab.inta-
csic.es/theory/vosa/, Bayo et al., 2008) to access and analyse these
photometric data sets, specifically the Gaia Early Data Release 3 (Gaia EDR3,
Gaia Collaboration et al., 2021), Panoramic Survey Telescope and Rapid
Response System Data Release 1 (Pan-STARRS1, Chambers et al., 2016), Two
Micron All Sky Survey catalogue of point sources (2MASS, Skrutskie et al.,
2006), and the AllWISE source catalogue (Cutri et al., 2021) from the Wide-
field Infrared Survey Explorer. The observed magnitudes of TIC 60040774 from
these catalogues are summarised in Table 1. Using specific zero points for
selected photometric filters, those magnitudes were transformed into flux
densities and then compared to the synthetic SEDs generated from selected
models. The synthetic SEDs were normalised by adopting the distance of
$d=134.465\pm 1.105$ derived from Gaia EDR3 parallax (Gaia Collaboration et
al., 2021). At this distance, the Galactic extinction in the direction of TIC
60040774 is negligible (Green et al., 2019)222http://argonaut.skymaps.info/.

###  2.3 TESS light curve

TIC 60040774 was observed by TESS in Sector 33 from 17 December 2020 to 13
January 2021. TESS uses four cameras comprising 16 CCDs to monitor the
variability of the sky in sectors that change every 27.4 days (Ricker et al.,
2015). Each sector covers an area of $24^{\circ}\times 90^{\circ}$ with $21"$
pixel resolution and 2 s exposure time. Full-frame images are not stored at
that cadence, however. Instead, different cadences are adopted, depending on
the target priorities. There are 20 s cadence data products for approximately
1000 target stars for astroseismology, 120 s cadence data products for planet
search priority targets, and 30 min cadence full-frame images for general-
purpose photometry. Within this hierarchy, TIC 60040774 was observed with 120
s cadence. The data was processed using the TESS Science Processing Operations
Center pipeline to produce calibrated target pixels, simple aperture
photometry flux time series (SAP), and pre-search data conditioning corrected
flux time series (PDCSAP) (Caldwell et al., 2020). The light curve for TIC
60040774 is available at the Mikulski Archive for Space Telescopes (MAST) and
easily accessed and processed using Lightkurve Python package (Lightkurve
Collaboration et al., 2018). Some analyses were also performed using
lightkurve package, including the recovery of the eclipse period through the
construction a box least square (BLS) periodogram (Kovács et al., 2002), which
is suitable for analysing periodic transit-like events. Simple light curve
models with periods ranging from $0.3$ to $0.5$ days and transit/eclipse
durations from $1.44$ to $14.4$ minutes were evaluated to construct the
periodogram shown in Figure 2). This yields an optimal period estimate of
$P=0.404807$ days.

Figure 2: Top: BLS periodogram of TIC 60040774 shows prominent periodicity at
$P=0.404807$ days. Bottom: folded light curve shows primary eclipse with very
short duration. Normalised fluxes from PDCSAP are represented by grey dots
with typical error of $0.193$. To suppress the scatter, the light curve is
binned (shown as black line). There is no significant difference between even
and odd eclipses as indicated by the mean and standard deviation of fluxes
during the eclipses.

As depicted on the right panel of Figure 1, the object is surrounded by
several brighter stars that might contaminate the target flux within the
specified aperture. There are four stars with $T_{\text{mag}}<16.5$ within
$1^{\circ}$ radius from TIC 60040774, including TIC 60040779 and TIC 60040763,
with magnitudes of $12.7$ and $13.9$, respectively. These are located less
than 2 pixels from the target star. In a dense region, point spread function
(PSF) modelling might be useful to remove the contamination from the
neighboring stars (e.g. Nardiello et al., 2019). However, this is not required
in our case, since the neighbouring stars show no variability. The flux time
series data produced by the TESS-SPOC behaves very well, as indicated by the
linearity of TIC 60040774 and some neighboring stars in flux-magnitude space.
We confirmed that the PDCSAP flux is background-corrected and free from
contamination.

##  3 Characteristics of TIC 60040774

###  3.1 White dwarf parameters

Figure 3: Extracted and normalised Balmer lines from H$\alpha$ (bottom) to
H$\epsilon$ (top) are compared to the best-fit model.

We used the blue segment of the spectra to constrain the physical parameters
of the white dwarf. At wavelengths shorter than $7000$ Å, the white dwarf
component of TIC 60040774 is the dominant source of light. A set of DA-type
white dwarf synthetic spectra with pure hydrogen atmosphere (Koester, 2010)
were compared to the observed data using standard procedures (Bergeron et al.,
1992). Each model atmosphere is parameterised by a certain $T_{\text{eff{}}}$
and $\log(g)$. Balmer lines from H$\alpha$ to H$\epsilon$ were extracted and
normalised separately. To find the normalisation factor, we fitted a pseudo-
Gaussian function to each line profile. DA-type white dwarfs with effective
temperatures of $9000\text{ K}\lesssim T_{\text{eff}}\lesssim 16000\text{ K}$
show very strong Balmer lines which are well matched to pseudo-Gaussian
functions (Bergeron et al., 1995). We implemented the same extraction and
normalisation to the synthetic spectra prior to the fitting. Additionally,
line profiles from the models were convolved with a Gaussian kernel with FWHM
of $2$ Å in order to match the instrumental profile. After some experiments
with a coarser parameter grid, $T_{\text{eff{}}}$ models ranging from $13000$
to $15000$ K with $250$ K resolution and $\log(g)$ ranging from $7.00$ to
$9.00$ with $0.25$ resolution were fitted to the extracted line profiles. The
$\chi^{2}$ was used as the cost function to find the best-fit model and obtain
the optimum model parameters. Subsequently, more physical parameters such as
mass, radius, and the cooling age of the white dwarf can be estimated using
theoretical models (Tremblay et al., 2013).

The best fit to the Balmer lines ($H\alpha$ to $H\epsilon$) yields
$T_{\text{eff{,A}}}=14050\pm 370$ K and $\log(g_{A})=7.98\pm 0.07$ (in cgs).
Figure 3 displays the extracted line profiles and best fit model. The
contamination from the red companion is negligible even at $H\alpha$
wavelength, such that the solution obtained using all available Balmer lines
is reliable. Additionally, the deviation between the 1D model and the more
realistic 3D atmospheric model at the quoted temperature and gravity is
minimal (Tremblay et al., 2013). According to the cooling track model provided
by the Montreal group (Tremblay et al., 2011; Bédard et al., 2020), the
estimated mass of the white dwarf is $M_{A}=0.598\pm 0.029$ M⊙, with a
corresponding radius of $R_{A}=0.0131\pm 0.0005$ R⊕, while the cooling age is
$t_{\text{cool}}=243\pm 24$ Myr. The uncertainties for these derived
parameters were estimated by randomly sampling the models with possible sets
of $T_{\text{eff{}}}$ and $\log(g)$ following a distribution dictated by their
uncertainties. This solution is consistent with the mass and radius calculated
using Koester’s model, which was scaled by the observed magnitude in $g$,
$B_{P}$, and $G$ bands. Using bolometric luminosities derived from those
magnitudes, the obtained mass and radius are $0.63\pm 0.10$ M⊙ and $0.0133\pm
0.003$ R⊙ respectively. This adopts the bolometric correction table from Chen
et al. (2019) and standard formulae and solar quantities from Prša et al.
(2016a).

With a mass of $0.598$ M⊙ and $T_{\text{eff{}}}=14050$ K, the primary
component of TIC 60040774 can be regarded as a typical white dwarf (Kleinman
et al., 2013). It is far enough from the instability strip that variability
beyond the eclipse is not expected. As the first-order estimate of the system
age, the progenitor mass of the white dwarf can be estimated using initial-
final mass relations. Its main sequence lifetime could also be estimated using
reliable stellar evolution model. In this scheme, the white dwarf is assumed
to evolve without significant influence from its companion. Using the semi-
empirical relation from Catalán et al. (2008), the estimated initial mass of
TIC 60040774a is $1.78\pm 0.48$ M⊙. Stars with solar metallicities as those
inferred here spend $1.49$ Gyr on the main sequence and an additional $0.30$
Gyr during the post-main sequence phase before cooling as white dwarfs (Choi
et al., 2016). The use of the initial-final mass relationship overestimates
the age of TIC 60040774. This is because the mass transfer from a massive
progenitor, as well as common envelope evolution, were likely to occur in this
system. We will discuss the past evolution of TIC 60040774 Section 3.4.

Figure 4: The best-fit model of white dwarf-M dwarf binary (blue) overlaid to
the optical spectrum from SALT/RSS (grey) and the observed SED (dark grey).
The relative errors (bottom panel) between the model and observation are below
$20\%$.

###  3.2 M dwarf parameters

The flux density of the secondary component ($F_{\lambda,2}$) was estimated by
subtracting the observed total flux ($F_{\lambda,\text{total}}$) with the
contribution from the white dwarf ($F_{\lambda,1}$) derived from the white
dwarf model with $T_{\text{eff{,A}}}=14050$ K and $\log(g_{A})=7.98$.
Interpolation over a rectangular mesh of $T_{\text{eff{}}}-\log(g)$ was done
using a bivariate cubic spline to yield precise synthetic SED for the white
dwarf. After that, the fluxes of the secondary component (TIC 60040774B) at
specified photometric bands were calculated using the following equation:

$F_{\lambda,2}=F_{\lambda,\text{total}}\left[1-10^{-0.4(m_{\lambda,1}-m_{\lambda,\text{total}})}\right],$
(1)

where $m_{\lambda}$ represents the apparent magnitude at a particular band.
The result is an SED with a peak at around $1\ \mu\text{m}$, implying that the
effective temperature of this star is around $2900$ K.

The SED can also be translated to the corrected magnitudes and colours of the
M dwarf. Applying the potometric classification scheme presented by Skrzypek
et al. (2015) in combination with the empirical relation between colors and
spectral types, we estimated that TIC 60040774B is an M6.5 dwarf with an
effective temperature of $2600-2900$ K. In this approach, we used the
polynomial functions to relate colour indices to the spectral types based on
Best et al. (2018) The $\chi^{2}$ minimisation method was based on the
prescription from Skrzypek et al. (2015). Colour indices were calculated using
magnitudes observed at $i$-band and longer wavelengths. At a shorter
wavelengths, the flux from the M dwarf is very low and thus uncertain. The
translation from spectral type to temperature is based on Rajpurohit et al.
(2013).

In more detail, the calculated SED from the secondary was then compared to the
collection of synthetic spectra from BT-Settl CIFIST (Caffau et al., 2011;
Allard et al., 2012) covering $2000\text{ K}\leq T_{\text{eff{}}}\leq
4000\text{ K}$ with $100$ K resolution and $2.5\leq\log(g)\leq 5.5$ with $0.5$
resolution. BT-Settle CIFIST provides excellent atmospheric models which are
compatible with the observed properties of very-low-mass stars and brown
dwarfs with solar metallicity. The optimum parameters were obtained through
$\chi^{2}$ minimisation using VOSA online service.

The SED analysis through VOSA resulted in the atmospheric parameters
$T_{\text{eff{,B}}}=2700\pm 150$ K and $\log(g_{B})=4.14\pm 0.97$, where the
mean values and the corresponding uncertainties are based on the profile of
smoothed $\chi^{2}$. The quoted uncertainties are associated with the
$\chi^{2}=\chi^{2}_{\text{min}}+\Delta\chi^{2}$, where $\Delta\chi^{2}=2.3$
for two-parameter-model fitting (e.g., Avni, 1976). Moreover, VOSA also
provided the scaling factor of $M_{d}=R^{2}/d^{2}=(5.54\pm 0.60)\times
10^{-22}$, which can be used to derive the radius of the object, which is
$R_{2}=0.140\pm 0.008$ R⊙. From the adopted surface gravity and radius, the
derived mass of the dwarf is $\log(M/M_{\odot})=-1.98\pm 0.98$. The estimated
radius is in good agreement with the one calculated using the semi-empirical
relation between radius and absolute magnitude in the $Ks$ band (calibrated
using nearby K7–M7 dwarfs (Mann et al., 2015)). Using the corrected magnitude
and the distance from Gaia EDR3, we got $M_{Ks}=9.26\pm 0.10$ and a radius of
$0.133\pm 0.004$. Additionally, based on the mass-luminosity relations from
Mann et al. (2019), we estimated that the secondary component has a mass of
$0.107\pm 0.020$ M⊙. Though the mass-luminosity relation from Mann et al.
(2019) is more accurate, especially for low-mass stars, the intrinsic scatter
in the relation becomes the major source of the quoted uncertainty. The latter
mass estimate is more plausible than the gravity-based estimate since the
uncertainty in the $\log(g)$ from the SED fitting is large.

Figure 4 shows the comparison between the observed SED with the model
containing $14050$ K white dwarf with $\log(g_{B})=7.98$ and the M dwarf with
$T_{\text{eff{,B}}}=2700$ K and $\log(g_{B})=5.0$. The adopted surface gravity
of the secondary is higher than the solution from VOSA to comply with the
mass-luminosity relations mentioned before.

###  3.3 Light curve modelling

The folded and binned light curve was modelled using phoebe2 (Prša et al.,
2016b). This code is the updated version of phoebe (Prša & Zwitter, 2005),
which includes several improvements, especially the inclusion of the library
of stellar atmospheric models, various observing passbands, and also valuable
tools for inverse problem solving (Horvat et al., 2018; Jones et al., 2020;
Conroy et al., 2020). To model the observable quantities as a function of time
or phase, the code creates a gravitationally affected surface mesh of model
stars. It populates the triangular mesh with quantities such as effective
temperature, intensity, and colour based on relevant physics. For a WDMS
binary, the white dwarf can be represented as a blackbody radiator, while the
atmosphere of the cooler star can be calculated using PHOENIX, covering
$T_{\text{eff{}}}$ down to $2300$ K (Husser et al., 2013). The TESS passband
covering $6000-10000$ Å was used to compute the expected variability more
accurately. Limb darkening of the primary component was represented by a
power-law model with appropriate coefficients from Claret et al. (2020). For
the secondary component, we used a quadratic limb darkening function with
appropriate coefficients from Claret (2018). Estimated $T_{\text{eff{}}}$ and
$\log(g)$ become the parameters for selecting the limb darkening coefficients.

The geometry of the folded light curve, especially the timing of ingress and
egress and the eclipse depth, provides constraints on some of the stellar
properties. Those include the radii of primary and secondary scaled by the
semimajor axis and the ratio of the effective temperatures of both components.
However, there is known degeneracy of the solutions due to the orbital
inclination. For example, a model with larger stellar radii and lower
inclination may produce a similar light curve profile as the model with
smaller radii and higher inclination. There are some ways to relieve this
degeneracy (Parsons et al., 2017), but those approaches require precise multi-
epoch spectroscopic data to characterise the kinematics of the components.
Consequently, light curve modelling must depend on prior knowledge established
by fitting synthetic spectra and SEDs. Nonetheless, light curve modelling can
provide additional consistency checks on parameters constrained by previous
methods.

A Markov Chain Monte Carlo (MCMC) sampling was performed to obtain posterior
probability distributions of parameters after the light curve modeling.
phoebe2 includes this capability by incorporating Affine-invariant MCMC
algorithms (Goodman & Weare, 2010) implemented in the emcee Python package
(Foreman-Mackey et al., 2013; Foreman-Mackey et al., 2019). In this approach,
the light curve models are governed by five free parameters sampled from the
initial parameter distributions. These are the orbital inclination $i$,
equivalent radii ($R$), and effective temperature ($T_{\text{eff{}}}$) of
primary and secondary components. The orbital eccentricity was set to zero
while the period was fixed. The masses of the components cannot be evaluated
directly using the light curve, but the masses constrain the semimajor axis,
which in turn scales the radii. We established the initial distribution of
those parameters based on the results of spectral and SED fitting. To achieve
reliable posteriors, we ran MCMC with 10 walkers through 5000 iterations.
During that run, we used binary models with 400 triangular mesh.

Table 2 summarises the way we generated the distribution of the priors. The
orbital period was fixed at $0.404807$ days, while the circular orbit was
assumed since there is no feature in the observed light curve to constrain the
eccentricity. If the size ratio between the primary and secondary is
$R_{B}/R_{A}\approx 10$ and the effective temperature ratio is
$T_{\text{eff{,B}}}/T_{\text{eff{,A}}}\approx 0.2$, then the expected
reduction in bolometric flux is less than two parts-per-thousand. With the
$T_{\text{mag}}\approx 17$, the photometric error of TESS is larger that the
secondary eclipse. Besides that, the out-of-eclipse light curve is relatively
flat, implying that the irradiation or reflection is insignificant. The
eclipse timing might not be exact, but initial sampling experimentation with
varied eclipse epoch $T_{0}$ revealed that the optimum solution deviates by a
negligible ${\lesssim}10^{-5}$ days. Consequently, $T_{0}$ was not included as
a free parameters during the final run.

Table 2: Distribution of priors for MCMC random sampling. Parameter | Distribution | Unit
---|---|---
$T_{\text{eff{,A}}}$ | $\mathcal{N}(14050,370)$ | K
$T_{\text{eff{,B}}}$ | $\mathcal{N}(2700,150)$ | K
$R_{\text{equiv},1}$ | $\mathcal{N}(0.0131,0.0005)$ | R⊙
$R_{\text{equiv},2}$ | $\mathcal{N}(0.1330,0.0080)$ | R⊙
$i$ | $\mathcal{U}(85,90)$ | degree
$a$ | $2.0405$ | AU
$P$ | $0.404807$ | day
Table 3: Orbital parameters derived from the light curve of TIC 60040774. Period | $P$ |  $0.404807\pm 0.000149$ days
---|---|---
Time of eclipse | $T_{0}$ |  $2459201.762996$ BJD
Eclipse duration | $T_{14}$ |  $11.19$ min
Eclipse depth |  | $77.87\%$
Semimajor axis | $a$ |  $2.041\pm 0.039$ R⊙
Inclination | $i$ | $87.50\pm 0.16^{\circ}$
Table 4: Stellar properties derived from spectroscopic and photometric data
with remarks on the methods. Primary Component
---
Mass | $M_{A}$ | $0.598\pm 0.029$ | spectroscopy, cooling track
Radius | $R_{A}$ |  $0.0131\pm 0.0005$ R⊙ | spectroscopy, cooling track (LC prior)
|  |  $0.0137\pm 0.0005$ R⊙ | spectroscopy, light curve (LC posterior)
Roche-lobe filling factor |  | $1.3\%$ | photometry, light curve
Effective temperature | $T_{\text{eff},A}$ |  $14050\pm 360$ K | spectroscopy, light curve
Surface gravity | $\log(g_{A})$ | $7.98\pm 0.07$ | spectroscopy
Cooling age | $t_{\text{cool}}$ |  $0.243\pm 0.024$ Gyr | spectroscopy, cooling track
|  |  |
Secondary Component
Mass | $M_{B}$ | $0.107\pm 0.020$ | photometry, mass-luminosity relation
Radius | $R_{B}$ |  $0.1330\pm 0.0080$ R⊙ | photometry, radius-luminosity relation (LC prior)
|  |  $0.1385\pm 0.0040$ R⊙ | photometry, light curve (LC posterior)
Roche-lobe filling factor |  | $28.5\%$ | photometry, light curve
Effective temperature | $T_{\text{eff},B}$ |  $2700\pm 150$ K | photometry, SED fitting (LC prior)
|  |  $2759\pm 50$ K | photometry, light curve (LC posterior)
Surface gravity | $\log(g_{B})$ | $5.00\pm 0.50$ | photometry, SED fitting
Spectral class |  | M6.5 | photometry

Figure 5 displays the posterior distribution obtained from the MCMC run. From
that figure, some parameters correlate with each other. The highest
correlation is between the inclination and the radius of the secondary. The
solution of $R_{B}/a=0.0676$ and $i=87.5^{\circ}$ achieved through the MCMC
sampling is consistent with the solution from the SED fitting. The depth of
the light curve determines the radius of the secondary such that the posterior
distribution is more constrained than the prior (improvement from $6\%$ to
$2\%$ relative error). This also applies to the secondary’s effective
temperature even though weak positive correlations exist between
$T_{\text{eff{,B}}}$ and other parameters ($i$ and $R_{A}$). On the other
hand, the posterior distributions for $T_{\text{eff{,A}}}$ and $R_{A}$ are not
different to the prior distributions, implying that the single band light
curve provides insufficient information to constrain the parameters of the
primary component. Lastly, the light curve cannot provide additional
information to constrain the masses. Further spectroscopic observation that
produces a sufficient radial velocity profile is required to constrain the
masses of the components.

Figure 5: Corner plot showing the posterior distributions of some parameters
of TIC 60040774. Figure 6: The best fit eclipse model and its $1\sigma$
uncertainty for the primary eclipse of TIC 60040774 as observed by TESS.
Circles with error bars are 20-s binned data.

Figure 7: Mass-radius plot for the white dwarf primary (left) and secondary
(right) components of TIC 60040774 based on the solutions prior (triangle) and
after (square) light curve modeling. For the white dwarf, the plot is
complemented with white dwarf sample from Parsons et al. (2017) and
theoretical relation from (Tremblay et al., 2011). Darker circles are for
white dwarfs with $10,000\leq T_{\text{eff}}\leq 20,000$ K. For the secondary
M-dwarf, samples from (Parsons et al., 2018) are overplotted, together with
theoretical model from Baraffe et al. (2015) and semi-empirical relations from
Mann et al. (2015) and Mann et al. (2019)

.

The best-fit light curve model with uncertainty is displayed in Figure 6,
while the parameters are summarised in Table 3 and Table 4. For this model,
the goodness of fit is represented by the reduced $\chi^{2}$ of $1.05$. Based
on the adopted masses, we estimated that the Roche-lobe filling factor of the
primary and secondary are $1.3\%$ and $28.5\%$ respectively. This system is
fully detached with nearly spherical components. The posterior parameters are
also plotted in Figure 7. For the primary, it is seen that the MCMC run
provides a radius $5\%$ larger than the one derived from the cooling track.

As we can see in Figure 6, the ingress and egress profile are shallower than
the best fit model, implying that there is a possibility that the geometry of
either primary or secondary is not so simple. To explore this further we
experimented with different limb darkening profiles (quadratic and square-
root). However we find that different profiles cannot explain the shallower
ingress-egress without inflating the white dwarf radius. Shallower ingress-
egress may be produced by a bright accretion disc normally observed in
cataclysmic variables (e.g., Savoury et al., 2011). If this is the case, then
the light curve morphology implies that the disc extends to ${\gtrsim}5R_{A}$
and contributes ${\gtrsim}10\%$ to the observed luminosity. Nevertheless, this
explanation is not suitable for TIC 60040774 where the secondary is far from
filling its Roche-lobe. Additionally, the spectrum of this object does not
show any hydrogen emission.

Alternatively, a planetary debris disc may exist around the primary since the
chance of finding a young white dwarf (up to ${\sim}200$ Myr cooling age) with
planetary debris is ${\sim}27\%$ or higher (Koester et al., 2014). In most
cases, the identification and characterisation of planetary debris discs
around white dwarfs are based on infrared observations. The typical outer
radius of the disc is $0.3$ R⊙ (Farihi, 2016). Transiting planetary debris in
front of a white dwarf produces shallower and variable transit depths
(Vanderburg et al., 2015; Vanderbosch et al., 2021). The asymmetric profile of
ingress and egress may also lead to the notion that the secondary is
surrounded by debris, with a trailing part that is thicker than the leading
part. Alternatively, an extended atmosphere of the secondary may also produce
the shallower ingress-egress by gradually absorbing the radiation from the
compact primary (Day et al., 1988). Apart from these alternative explanations,
there is still a possibility that the observed profile comes from systematic
error. Consequently, follow-up observations with high cadence photometry
covering the complete eclipse stages are needed to clarify the true ingress-
egress profile of TIC 60040774.

###  3.4 Binary evolution of TIC 60040774

The white dwarf-main sequence binary TIC 60040774 is expected to have formed
as a system with higher total mass and wider orbit than those currently
observed. The system experienced binary evolution that includes a common-
envelope (CE) phase. This phase lasted for a very short time but significantly
shrank the orbit by orders of magnitude. At the end of the CE phase, the orbit
shrank further, mainly due to gravitational radiation. Inversely, we can trace
possible configuration of TIC 60040774 in the past from the observed
parameters.

First, the system’s orbital period just after the CE phase ($P_{CE}$) can be
estimated by considering the angular momentum loss due to gravitational
radiation. For the case of TIC 60040774 where the secondary is a fully-
convective star with mass $m_{2}<0.35$ M⊙, magnetic braking is inefficient in
shrinking the orbit. Following Schreiber & Gänsicke (2003),

$P_{CE}^{8/3}=\dfrac{256G^{5/3}(2\pi)^{8/3}}{5c^{5}}\dfrac{M_{A}M_{B}}{M_{A}+M_{B}}t_{\text{cool}}+P_{\text{orb}}^{8/3}$
(2)

where $G$ is the gravitational constant, $c$ is the speed of light in a
vacuum, $M_{A}$ and $M_{B}$ are the current masses of the primary and
secondary components, $P_{\text{orb}}$ is the observed orbital period, while
the $t_{\text{cool}}$ is the white dwarf cooling age. Using the parameters
presented in Table 3 and Table 4, we got $P_{CE}=0.40504$ days.

We performed forward modelling using the Binary Star Evolution (BSE) code
(Hurley et al., 2002) which includes single star evolution recipe (Hurley et
al., 2000) and various physical mechanisms involved in the interacting binary
star evolution. Synthetic binaries with a fixed secondary mass of
$m_{2}=0.107$ M⊙ and various progenitor masses of $1.80\leq m_{1}\leq 3.80$ M⊙
were generated and evolved until they reached the common-envelope phase. Those
binaries have circular orbits with initial periods of $400\leq
P_{\text{init}}\leq 1000$ days or separations between $1.33$ to $3.07$ AU. The
resolution of input parameter grids is $0.01$ M⊙ in mass and $5$ days in
period. We adopted solar metallicity and set the remaining stellar evolution
parameters to default values. We used the common-envelope efficiency
$\alpha_{CE}=1.0$ and the binding energy factor $\lambda=0.5$. These two
parameters have a significant effect on the outcome of the common-envelope
evolution. According to Davis et al. (2012), the $\alpha_{CE}$ correlates with
the observed white dwarf mass and secondary component. The adopted values for
the BSE models are consistent with the prescribed relation assuming that the
binding energy of the envelope is purely gravitational. The adopted $\lambda$
is based on the calculation by Dewi & Tauris (2000) for the star of mass 3 M⊙.

The simulation produced some important parameters at the end of the CE phase,
especially the orbital period ($P_{CE}$) and the mass of the white dwarf to be
compared to the ones derived from the observation. The best estimate of the
progenitor mass is $m_{1}=2.59\pm 0.33$ M⊙ while the initial period and
separation are $829\pm 115$ days and $2.40\pm 0.24$ AU. When the primary
started to reach the giant branch, the wind from the primary induced drag and
shrank the orbit to the point of Roche-lobe overflow. This process was
followed by the CE phase.

Beyond the common-envelope stage, the evolution of TIC 60040774 is
straightforward without any dynamical complexity. The white dwarf is getting
cooler while the low-mass secondary evolves slowly, and it will not fill its
Roche-lobe in the next couple of billion years. Gravitational-wave radiation
with a time scale of 400 Gyr becomes the only mechanism reducing the orbital
separation unless the envelope ejection is not 100% efficient ($\alpha_{CE}\ll
1$) and the existence of a debris disc accelerates the orbital shrinkage
(Kashi & Soker, 2011).

###  3.5 TIC 60040774 among other eclipsing PCEBs

Having a primary mass of $0.598$ M⊙, a secondary mass of $0.107$ M⊙, and
orbital period of $9.71$ hours, TIC 60040774 can be regarded as a typical WDMS
binary as it is located close to the central distribution of WDMS in terms of
white dwarf mass, temperature of the secondary component, and also the orbital
period (Rebassa-Mansergas et al., 2016, 2021; Kruckow et al., 2021). However,
TIC 60040774 gains its uniqueness if we place it in the context of eclipsing
PCEB.

Figure 8 shows the distribution of mass and period of PCEBs listed in Parsons
et al. (2015) and Parsons et al. (2017). The distribution of the primary mass
shows a single peak at around $0.4-0.45$ M⊙, while the distribution of
secondary mass is double-peaked with a clear valley around $0.35$ M⊙. This
valley is associated with the upper limit of fully-convective stars. Systems
with such low-mass secondaries remain detached for an extended period since
the orbital shrinkage is only caused by gravitational wave radiation
(Zorotovic et al., 2014). Consequently, the probability of finding such
systems is higher. TIC 60040774 is among the systems with a low-mass
secondary.

If we look at the period distribution, eclipsing PCEBs tend to have a shorter
orbital period while TIC 60040774 is at the right-end of the distribution. We
can say that TIC 60040774 is a relatively rare case because the probability of
finding eclipsing system is inversely proportional to the orbital separation
(and period). Besides this trend, a multimodal distribution is seen in Figure
8. However, this distribution is likely to be caused by selection bias as the
Monte Carlo simulation of the PCEBs (Zorotovic et al., 2014) did not produce
the multimodality.

Figure 8: Plot of the orbital periods and masses of 40 eclipsing PCEBs listed
in (Parsons et al., 2015) where the primary and secondary masses are in black
and red respectively. The associated histograms are displayed where the gray
histogram is for the complete sample.

With the orbital separation of ${\sim}2$ AU, TIC 60040774 is also a simple
detached system where the irradiation from the primary star does not affect
the secondary. The effective temperature of the white dwarf is around five
times that of the secondary, while the scaled radius of this compact body is
just $0.00671$. This condition implies that the irradiation from the white
dwarf is just $1.5\%$ that of the intrinsic flux of the M-dwarf secondary.
Consequently, the SED of the secondary is close to the theoretical model.

Eclipsing PCEBs with low-mass secondary become a good target for finding
circumbinary planets since the sharpness of the light curve implies a great
precision of measuring the eclipse timing. Systems like NN Ser, QS Vir, RR
Cae, GK Vir, and DE CVn are known to have circumbinary planet(s) with orbital
period of more than ten years (Beuermann et al., 2010; Qian et al., 2012; Han
et al., 2018; Almeida et al., 2020). The existence of a Jupiter-like planets
modulates the orbital period of the binary with typical amplitudes of a few
tens of seconds by inducing light time travel effects (e.g., Marsh et al.,
2014). Either the first generation formation scenario (Bear & Soker, 2014) or
the post-common-envelope planetary formation (Schleicher & Dreizler, 2014) may
explain the existence of planets orbiting PCEBs. Belongs to the same class,
TIC 60040774 may also host a planet or planetary system. However, Bear & Soker
(2014) speculated that the absence of detection of a planet orbiting PCEB with
${\sim}0.6$ M⊙ primary may be a consequence of evolutionary process. At this
moment, we have not identified any extra transit in the TESS light curve,
although this only covers one-month of observations. Even though there are
more than 60 eclipses observed for TIC 60040774, the unbinned light curve from
a single eclipse is noisy such that identifying short period modulation from
this data is difficult. Similar to the case of GK Vir (Almeida et al., 2020),
follow-up observations using a meter-class telescope for more than a decade is
required to reveal the existence of a planet orbiting TIC 60040774.

##  4 Conclusion

We derived orbital and physical parameters of TIC 60040774, one of the
eclipsing post common-envelope binary with low-mass secondary located $134$ pc
from the Sun. Based on spectroscopic data, we estimated that the primary
component of this system is a white dwarf with a mass of $0.598\pm 0.029$ M⊙
and an effective temperature of $14050\pm 360$ K. The secondary component is
an M6.5 dwarf with a mass of $0.107\pm 0.020$ M⊙ and a temperature of $2759\pm
50$ K. Physical parameters for the secondary were derived through SED fitting
combined with the MCMC sampling to fit the light curve from TESS. With the
period of 9.71 hours, TIC 60040774 becomes one of the eclipsing PCEBs with
relatively long orbital period. Additionally, we estimated that the system
started from binary with a late B primary star of mass $2.5$ M⊙ and an initial
separation of $2.4$ AU. The total age of the system is $1.024\pm 0.300$ Gyr
with the common-envelope stage occuring $0.243$ Gyr ago.

Our estimate of the parameters of the TIC 60040774 is not independent of some
theoretical and semi-empirical models. However, the orbital separation is
relatively large such that the irradiation from the white dwarf does not
affect the secondary. Consequently, the expected properties are consistent to
the models. However, we found that the ingress-egress profile acquired from
the TESS photometry seems to be shallower than the one expected from the best
model consisting of components with typical geometrical structure. This
profiles may indicate the existence of debris disc in TIC 60040774 though this
hypothesis needs to be evaluated using additional data.

## Acknowledgment

This paper includes data collected by the _TESS_ mission. Funding for the
_TESS_ mission is provided by the NASA’s Science Mission Directorate. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of VOSA, developed under the Spanish Virtual Observatory project supported by
the Spanish MINECO through grant AyA2017-84089. VOSA has been partially
updated by using funding from the European Union’s Horizon 2020 Research and
Innovation Programme, under Grant Agreement no 776403 (EXOPLANETS-A). The SALT
observations were obtained under the SALT transient followup programme
2018-2-LSP-001 (PI: DAHB). DAHB and JB acknowledge reseach support from the
National Research Foundation.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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