11institutetext:  Department of Physics, University of Rome “La Sapienza”,
Piazzale Aldo Moro 2, 00185 Rome, Italy
11email<EMAIL_ADDRESS>22institutetext: INAF – Istituto di
Astrofisica e Planetologia Spaziali (INAF-IAPS), Via Fosso del Cavaliere 100,
I-00133, Rome, Italy 33institutetext: Department of Physics, University of
Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy
44institutetext: INAF – Turin Astrophysical Observatory, via Osservatorio 20,
10025, Pino Torinese, Italy 55institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, 69117 – Heidelberg, Germany 66institutetext:
Astrophysics Group, Keele University, Keele ST5 5BG, UK 77institutetext:
Instituto de Astronomia y Ciencias Planetarias de Atacama, Universidad de
Atacama, Copayapu 485, Copiapo, Chile 88institutetext: Department of Physics
and Astronomy, University of Florence, Largo Enrico Fermi 5, 50125 Firenze,
Italy 99institutetext: Centre for ExoLife Sciences, Niels Bohr Institute,
University of Copenhagen, Øster Voldgade 5, 1350 Copenhagen, Denmark
1010institutetext: Institute for Astronomy, University of Edinburgh, Royal
Observatory, Edinburgh EH9 3HJ, UK 1111institutetext: Department of Physics
“E.R. Caianiello”, University of Salerno, Via Giovanni Paolo II 132, 84084,
Fisciano, Italy 1212institutetext: Istituto Nazionale di Fisica Nucleare,
Sezione di Napoli, Napoli, Italy 1313institutetext: Universität Hamburg,
Department of Earth Sciences, Meteorological Institute, Bundesstrasse 55,
20146 Hamburg, Germany 1414institutetext: Centre for Exoplanet Science, SUPA,
School of Phys. & Astron., Univ. of St Andrews, North Haugh, St Andrews KY16
9SS, UK 1515institutetext: Millennium Institute of Astrophysics MAS, Nuncio
Monsenor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile
1616institutetext: Instituto de Astrofísica, Pontificia Universidad Católica
de Chile, Av. Vicuña Mackenna 4860, 7820436 Macul, Santiago, Chile
1717institutetext: University of Southern Denmark, Department of Physics,
Chemistry and Pharmacy, Campusvej 55, 5230 Odense M, Denmark
1818institutetext: Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg (ZAH), 69120 Heidelberg, Germany 1919institutetext:
Astronomy Research Center, Research Institute of Basic Sciences, Seoul
National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea
2020institutetext: Centro de Astronomía, Universidad de Antofagasta, Av.
Angamos 601, Antofagasta, Chile 2121institutetext: Departamento de Matemática
y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de la
Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile
2222institutetext: Department of Physics, Isfahan University of Technology,
Isfahan 84156-83111, Iran 2323institutetext: Centre for Electronic Imaging,
Department of Physical Sciences, The Open University, Milton Keynes, MK7 6AA,
UK 2424institutetext: ExoClock Project

# Star-spot activity, orbital obliquity, transmission spectrum, physical
properties, and TTVs of the HATS-2 planetary system

F. Biagiotti 1122 L. Mancini 334455 J. Southworth 66 J. Tregloan-Reed 77 L.
Naponiello 338844 U. G. Jørgensen 99 N. Bach-Møller 99 M. Basilicata 3344 M.
Bonavita 1010 V. Bozza 11111212 M. J. Burgdorf 1313 M. Dominik 1414 R. Figuera
Jaimes 151516161414 Th. Henning 55 T. C. Hinse 1717 M. Hundertmark 1818 E.
Khalouei 1919 P. Longa-Peña 2020 N. Peixinho 2121 M. Rabus 2121 S. Rahvar 1919
S. Sajadian 2222 J. Skottfelt 2323 C. Snodgrass 1010 Y. Jongen 2424 J.-P.
Vignes 2424

(Received ; accepted )

###### Abstract

Aims. Our aim in this paper is to refine the orbital and physical parameters
of the HATS-2 planetary system and study transit timing variations and
atmospheric composition thanks to transit observations that span more than ten
years and that were collected using different instruments and pass-band
filters. We also investigate the orbital alignment of the system by studying
the anomalies in the transit light curves induced by starspots on the
photosphere of the parent star.

Methods. We analysed new transit events from both ground-based telescopes and
NASA’s TESS mission. Anomalies were detected in most of the light curves and
modelled as starspots occulted by the planet during transit events. We fitted
the clean and symmetric light curves with the JKTEBOP code and those affected
by anomalies with the PRISM+GEMC codes to simultaneously model the photometric
parameters of the transits and the position, size, and contrast of each
starspot.

Results. We found consistency between the values we found for the physical and
orbital parameters and those from the discovery paper and ATLAS9 stellar
atmospherical models. We identified different sets of consecutive starspot-
crossing events that temporally occurred in less than five days. Under the
hypothesis that we are dealing with the same starspots, occulted twice by the
planet during two consecutive transits, we estimated the rotational period of
the parent star and, in turn the projected and the true orbital obliquity of
the planet. We find that the system is well aligned. We identified the
possible presence of transit timing variations in the system, which can be
caused by tidal orbital decay, and we derived a low-resolution transmission
spectrum.

###### Key Words.:

methods: data analysis – techniques: photometric – stars: fundamental
parameters – stars: individual: HATS-2 – planetary systems

##  1 Introduction

Due to their intrinsic size and proximity to their parent stars and despite
their rarity, transiting hot Jupiters are a class of exoplanets that have
received a great deal of attention from the scientific community since they
were discovered (see e.g. Raymond & Morbidelli, 2022, for a comprehensive
review). Such peculiar characteristics offer many advantages from an
observational point of view, and guarantee the possibility of characterising
most of the physical and orbital parameters of these planets with extreme
precision. They are also the best targets for atmospheric-characterisation
observations, as recently demonstrated by the JWST (Ahrer et al., 2023;
Rustamkulov et al., 2023; Alderson et al., 2023; Feinstein et al., 2023).

Even so, there are still some aspects of this population that are not yet well
understood and are under investigation, such as those regarding the physical
mechanisms that regulate the formation, accretion, and evolution processes or
those that cause their migration from the snowline ($\sim 3$ au) up to roughly
$0.01$ au from their host stars. For example, several mechanisms have been
advocated that are able to shrink the orbit of a giant planet, such as
dynamical interactions through planet-planet scattering (Rasio & Ford, 1996;
Davies et al., 2014), the Kozai mechanism (e.g. Wu & Murray, 2003), and the
disc-planet interaction (Lin et al., 1996; Ward, 1997). However, whatever the
responsible mechanism is, the planetary orbital eccentricity, $e$, and the
spin-orbit obliquity, $\psi$, should be affected. If planet-planet scattering
is the main mechanism that produces hot Jupiters, these scattering encounters
should randomise the orbital planes, whereas disc-planet interactions force
the planet’s orbit to be coplanar throughout its migration journey and we
should observe an excess of flat architectures (see e.g. Biazzo et al., 2022).

In this context, we have been carrying out a more-than-ten-year project
(Southworth et al., 2009) to better characterise this class of planets and
with the aim of measuring their physical and orbital parameters with an
accuracy of less than $5\%$. Our project is mainly based on high-quality
photometric observations of planetary-transit events and, in several cases,
was supported by high-resolution spectroscopic observations to characterise
the atmospheric properties and activity host star (see e.g. Mancini et al.,
2014a) or to measure the Rossitter-McLaughlin effect, which allows the
measurement of the project spin-orbit alignment, $\lambda$, (see e.g. Mancini
et al., 2018, 2022a).

In a few cases we faced the presence of anomalies on the transit light curves
in the form of small bumps, which are attributable to the presence of single
starspots or groups of starspots on the photosphere of the parent stars (see
e.g. Mancini et al., 2015, 2017). These bumps appear when the planet, during
its transits, temporarily hides the starspots, thus obscuring colder regions
of the photosphere (Silva, 2003; Winn, 2010). Appropriate modelling of the
light curve allows the parameters of the starspot, to be estimated, under the
simplistic hypothesis that this is a single circular spot.

In the case that the orbit of the transiting planet is well aligned with the
spin of its parent star and the planet has an orbital period much shorter than
the stellar rotation period, then the planet can occult the same starspot
multiple times. In these circumstances, the measurement of $\lambda$ becomes
feasible with photometric monitoring of two consecutive transits (Tregloan-
Reed et al., 2013). Instead, if there is a spin-orbit misalignment, the planet
will not occult the same starspot again at the subsequent transit, and we can
at most obtain a lower limit on the stellar obliquity (Dai & Winn, 2017).

In this work we report the results of our study of the planetary system HATS-2
that hosts a transiting hot Jupiter, HATS-2 b ($M_{\rm p}\approx 1.3\,M_{\rm
Jup}$; $R_{\rm p}\approx 1.2\,R_{\rm Jup}$), which orbits around a K V dwarf
star ($V=13.6$ mag) in roughly 1.35 days (Mohler-Fischer et al., 2013). The
authors of this discovery reported multi-band light curves of two transits in
which this planet occulted starspots and, for the first time ever as far as we
know, even a bright chromospheric region known as stellar plage. Using two
ground-based telescopes at the ESO Observatory of La Silla, we obtained new
high-quality light curves of the HATS-2 b transits and, thanks to TESS
photometric data, we recognised new starspots and characterised their
properties. We also reviewed the main physical and orbital parameters of this
planetary system and probed the chemical composition of the planetary
atmosphere. Possible transit time variations (TTVs) were, also investigated,
using these new and archival data.

The paper is organised as follows. In Sect. 2 we present the new photometric
follow-up observations used to characterise the system and the relative data-
reduction procedure. In Sect. 3 we present the analysis of the light curves,
while in Sect. 4 we revise the main physical properties of the HATS-2
planetary system, focusing on the analysis of the observed starspots events.
We analyse the transit times variations in Sect. 5, while in Sect. 6 we
investigate the variation in the planetary radius as a function of wavelength.
In Sect. 7 we analyse the TESS phase curves. Finally, we summarise our results
in Sect. 8.

##  2 Observations and data Reduction

Figure 1: Transits from the DK telescope. Left panel: Light curves of 11
transits (ten complete and one partial) of HATS-2 b observed with the Danish
1.54 m telescope, shown in chronological order. They are plotted against the
orbital phase and are compared to the best-fitting models. Labels indicate the
observation date and the filter (Bessell $I$ and $R$) that was used for each
data set. Starspot anomalies are visible in all the light curves. Right-hand
panel: The residuals of each fit. Figure 2: Light curves of four (three
complete and one partial) transits of HATS-2 b simultaneously observed in four
optical bands (Sloan $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, $z^{\prime}$)
with the GROND multi-band camera at the MPG 2.2 m telescope. They are shown in
date order. The light curves in the top panels are from Mohler-Fischer et al.
(2013), while those in the bottom panels are from this work. Starspot
anomalies are visible in the first three data sets. The light curves are
plotted against the orbital phase and are compared to the best-fitting models.
The residuals of the fits are shown at the base of each panel.

This work is mainly based on photometric follow-up observations of transit
events of the exoplanet HATS-2 b, which were performed using ground-based and
space telescopes. The ground-based data were collected starting in 2015, using
the DFOSC camera, installed at the Danish 1.54 m telescope and the GROND
multi-band camera, which is one of the instruments at the MPG 2.2 m telescope
and provides data in different passband filters simultaneously. The
observations were, therefore, carried out through different optical passbands
(covering the $400-1000$ nm wavelength range) to also investigate possible
variations in the transit depth with the wavelength and the dependence of the
starspot contrast with wavelength.

We observed 11 new transits with the Danish 1.54 m telescope and 2 new transit
events with the MPG 2-2 m telescope for a total of 19 new optical light
curves. All the observed transits were fully monitored except for the transit
on 2022 May 26 observed with the Danish telescope and the transit on 2015
January 29 observed with the MPG 2.2 m telescope. A summary of the
observations with their properties and observing conditions (airmass through
the night and moon illumination) is given in Table 1. We also reanalysed the
two transit light curves observed with GROND, which were reported in the
discovery paper (Mohler-Fischer et al., 2013).

Furthermore, HATS-2 was monitored by the _TESS_ space telescope (Ricker et
al., 2014) with the 2 min cadence during Sector 10 of its primary mission and
during Sectors 36 and 63 of its extended mission.

Many of the transits recorded by _TESS_ show anomalies that most likely are
connected to occultations of starspots by the planet. Similar anomalies are
also present in all the complete transit light curves recorded with the Danish
and MPG 2.2 m telescopes.

Table 1: Details of the ground-based transit observations presented in this
work.

Date of | Instrument | Start and | $N_{\rm obs}$ | $T_{\rm exp}$ | $T_{\rm obs}$ | Filter | Airmass | Moon | Aperture | $N_{\rm poly}$ | Scatter
---|---|---|---|---|---|---|---|---|---|---|---
first obs. |  | end times (UT) |  | (s) | (s) |  |  | illum. | sizes (px) |  | (mmag)
2015/01/26 | GROND | 03:52 $\rightarrow$ 08:59 | 117 | 90 | 157 | $g^{\prime}$ | 1.90$\rightarrow$1.00$\rightarrow$ 1.02 | 40% | 23,80,100 | 1 | 1.07
2015/01/26 | GROND | 03:52 $\rightarrow$ 08:59 | 117 | 90 | 157 | $r^{\prime}$ | 1.90$\rightarrow$1.00$\rightarrow$ 1.02 | 40% | 25,80,100 | 1 | 0.73
2015/01/26 | GROND | 03:52 $\rightarrow$ 08:59 | 117 | 90 | 157 | $i^{\prime}$ | 1.90$\rightarrow$1.00$\rightarrow$ 1.02 | 40% | 23,80,100 | 1 | 1.03
2015/01/26 | GROND | 03:52 $\rightarrow$ 08:59 | 117 | 90 | 157 | $z^{\prime}$ | 1.90$\rightarrow$1.00$\rightarrow$ 1.02 | 40% | 23,80,100 | 1 | 1.12
2015/01/30 | GROND | 03:54 $\rightarrow$ 06:42 | 76 | 85 | 106 | $g^{\prime}$ | 1.60$\rightarrow$1.05 | 80% | 20,80,100 | 1 | 0.88
2015/01/30 | GROND | 03:54 $\rightarrow$ 06:42 | 73 | 85 | 106 | $r^{\prime}$ | 1.60$\rightarrow$1.05 | 80% | 23,80,100 | 1 | 0.60
2015/01/30 | GROND | 03:54 $\rightarrow$ 06:42 | 72 | 85 | 106 | $i^{\prime}$ | 1.60$\rightarrow$1.05 | 80% | 23,80,100 | 1 | 1.01
2015/01/30 | GROND | 03:54 $\rightarrow$ 06:42 | 75 | 85 | 106 | $z^{\prime}$ | 1.60$\rightarrow$1.05 | 80% | 23,80,100 | 1 | 0.81
2016/04/16 | DFOSC | 23:33 $\rightarrow$ 03:59 | 149 | 100 | 112 | $R$ | 1.08$\rightarrow$1.05$\rightarrow$1.32 | 77$\%$ | 15,30,50 | 1 | 0.96
2017/05/30 | DFOSC | 22:49 $\rightarrow$ 03:03 | 137 | 100 | 113 | $R$ | 1.02$\rightarrow$1.02$\rightarrow$2.47 | 34$\%$ | 15,33,50 | 2 | 0.88
2017/07/11 | DFOSC | 22:54 $\rightarrow$ 02:28 | 103 | 100 | 125 | $R$ | 1.49$\rightarrow$1.63 | 92$\%$ | 17,35,55 | 2 | 1.21
2018/05/17 | DFOSC | 00:02 $\rightarrow$ 04:38 | 109 | 100 | 112 | $I$ | 1.02$\rightarrow$1.08$\rightarrow$1.63 | 9$\%$ |  9,17,40 | 2 | 1.13
2018/06/05 | DFOSC | 23:15 $\rightarrow$ 03:38 | 123 | 100 | 122 | $R$ | 1.07$\rightarrow$1.10$\rightarrow$4.94 | 57$\%$ | 15,33,53 | 2 | 1.38
2018/06/24 | DFOSC | 22:47 $\rightarrow$ 02:43 | 127 | 100 | 112 | $R$ | 1.17$\rightarrow$1.22 | 90$\%$ | 17,35,57 | 3 | 1.29
2019/05/19 | DFOSC | 23:16 $\rightarrow$ 03:59 | 122 | 100 | 111 | $I$ | 1.01$\rightarrow$1.02$\rightarrow$2.72 | 98$\%$ | 10,24,40 | 4 | 2.80
2019/05/23 | DFOSC | 01:18 $\rightarrow$ 05:19 | 112 | 100 | 115 | $I$ | 1.25$\rightarrow$1.32 | 74$\%$ | 14,32,45 | 2 | 1.23
2019/06/07 | DFOSC | 23:00 $\rightarrow$ 02:57 | 121 | 100 | 116 | $I$ | 1.06$\rightarrow$1.09$\rightarrow$3.10 | 25$\%$ | 13,33,55 | 3 | 1.12
2021/05/26 | DFOSC | 23:46 $\rightarrow$ 04:09 | 134 | 100 | 116 | $I$ | 1.06$\rightarrow$1.09$\rightarrow$4.35 | 99$\%$ | 13,34,52 | 3 | 1.03
2022/05/23 | DFOSC | 03:00 $\rightarrow$ 06:00 |  73 | 100 | 129 | $I$ | 1.01$\rightarrow$1.57$\rightarrow$2.00 | 94$\%$ | 18,40,60 | 2 | 1.42
Notes: $N_{\rm obs}$ is the number of observations, $T_{\rm exp}$ is the exposure time, $T_{\rm obs}$ is the observational cadence, and ‘Moon illum.’ is the geocentric fractional |
illumination of the Moon at midnight. The aperture sizes are the radii of the software apertures for the star, inner sky and outer sky, respectively. |
Scatter is the rms scatter of the data versus a fitted model. |

###  2.1 Danish 1.54 m telescope

Five transits of HATS-2 b were observed through a Bessell-$R$ filter, while
six transits (five complete and one incomplete) were observed through a
Bessell-$I$ filter with the DFOSC (Danish Faint Object Spectrograph and
Camera) imager mounted on the Danish 1.54 m Telescope at the ESO Observatory
in La Silla (Chile). The transit recorded on 2022 May 26 was not fully covered
due to technical problems at the beginning of the observations, whereas the
observations performed on 2019 May 19 were severely affected by weather
conditions.

The DFOSC is equipped with a focal reducer, which changes the original focal
length ($\approx$13 m) of the telescope and allows for a wider field of view
on the sky. The current camera of the DFOSC is an e2v CCD with $2048\times
4096$ pixels, a plate scale of $0.39^{\prime\prime}$ per pixel and 32-bit
encoding. In its current set-up, only half of the CCD is illuminated by the
incoming starlight, so the actual FOV is $13.7^{\prime}\times 13.7^{\prime}$.
The CCD has been further windowed in many of the observations to reduce the
readout time and, therefore, improve the sampling.

The data were analysed by using the IDL/ DEFOT pipeline (Southworth et al.,
2009, 2012b). Following a standard approach, which was already adopted in the
previous works of the series (e.g. see Southworth et al., 2012a, 2017), we
first calibrated the raw science images by master-bias and master-flat frames;
these frames were obtained by median combining a set of individual bias and
sky flat-field images, which were taken on the same night as each transit
observation. Then, we performed aperture photometry selecting an ensemble of
comparison stars and cross-correlating each image in a time series against a
reference image. In this step, the time stamps were converted from JDUTC to
BJDTDB, following the convention suggested by Eastman et al. (2010). Once the
aperture photometry analysis was completed, differential-magnitude light
curves were generated against each comparison star for each time series. Each
curve has a dimension equal to the number of observations $N_{\rm obs}$. For
the purpose of detrending the data, we fitted a polynomial with order between
$0<N_{\rm poly}<5$ to all the points before the ingress and after the egress,
for the set of the $n$ light curves. The parameters of the fit were the
$n\cdot(N_{\rm poly}+1)$ coefficients and the $n$ weights, while the number of
data points fitted are $n\cdot N_{\rm obs}$. The weights were simultaneously
optimised to minimise the scatter in the out-of-transit data points. Finally,
all the points were combined into one ensemble by a weighted flux summation,
obtaining the final light curve for each data set. All the light curves are
shown in Fig. 1.

The aperture radii, the order of the detrending polynomial, and the rms
scatter values of the data against a best-fitting model (see next section) are
listed in Table 1, showing the average good quality of these new ground-based
data. In all the final light curves, see Fig. 1, it is possible to observe the
presence of higher-order photometric effects (i.e. bumps connected to
starspots, as described in Sect. 3).

###  2.2 MPG 2.2 m telescope

Two transit events of HATS-2 b were recorded on 2015 January 25 and 2015
January 28 thanks to the GROND instrument, which is mounted on the Nasmyth
focus of the MPG 2.2 m Telescope at the ESO Observatory in La Silla (Chile).
GROND stands for Gamma-Ray Burst Optical/Near-Infrared Detector and is a
7-channel imager, which allows observations with different filters at the same
time (Greiner et al., 2008). Because of the connection of GRBs with high-
redshift galaxies, the available filters are the Sloan
$g^{\prime},\,r^{\prime},\,i^{\prime},\,z^{\prime}$ bands and the near-
infrared $J,\,H,\,K$ bandpass filters. The camera is essentially composed of a
system of dichroics and collimators, which split the incoming light towards
two arms (optical and NIR) and then in more beams so that all seven detectors
are illuminated at the same time. The first division is required because of
the differences between the NIR-detector arrays and optical CCDs, such as
temperature sensitivity. For our work, we only used the data from the optical
channel, which is composed of a system of four dichroics and four back-
illuminated E2V CCDs with $2048\times 2048$ pixel. For each CCD, the plate
scale is $0.158^{\prime\prime}$ per pixel and the FOV is $5.4\times 5.4$
arcmin2.

The first transit was completely monitored, whereas the second was only
partially monitored due to a crash of the TCS (telescope control system). The
observations were reduced in the same way as those from the Danish Telescope
(see Sect.2.1). Other two transit events obtained with GROND were reported in
the discovery paper (Mohler-Fischer et al., 2013) and reanalysed here. We have
not re-performed photometry for these observations. The 16 GROND light curves
are shown in Fig. 2.

###  2.3 TESS space telescope

The PDCSAP (Pre-search Data Conditioning Simple Aperture Photometry; Smith et
al., 2012; Stumpe et al., 2014) light curve of HATS-2 from both Sector 10 and
36 of the NASA _TESS_ mission was downloaded through the Python package
lightkurve (Lightkurve Collaboration et al., 2018), from the Mikulski Archive
for Space Telescopes (MATS). Using the same Python package, we downloaded also
raw PDCSAP data from Sector 63, for which the detrending was performed thanks
to the wotan Python package (Hippke et al., 2019). The detrended data from
each sector are shown in Fig. 3. Continuous observations of the target star
were obtained for two periods of roughly 25 days (from 2019 March 28 to 2019
April 21) and 24 days (from 2021 March 08 to 2021 March 31) for Sectors 10 and
36, respectively. For Sector 63, the target star was observed for a period of
25 days (from 2023 March 11 to 2023 April 05) collecting, however, more
transits. In fact, the data from Sector 10 and Sector 36 contain fifteen
transits of HATS-2 b, while Sector 63 contains eighteen transits. The sub-
optimal quality of TESS data is due to the relative faintness of the HATS-2
star ($V=13.6$ mag), which is at the limit of the working magnitude of TESS.
Nevertheless, several transits show clear anomalies, see Figs. 11, 12, 13.

Figure 3: Photometric monitoring of HATS-2 by the TESS space telescope. Top
panel: Data from Sector 10. Middle panel: Data from Sector 36. Bottom panel:
Data from Sector 63. Fifteen transit events of HATS-2 b were detected by TESS
in Sector 10 and 15 in Sector 36, while 18 transit events were detected in
Sector 63.

##  3 Light curves analysis

Since many of our obtained light curves contained anomalies that could be
connected to starspot complexes occulted by the exoplanet HATS-2 b during its
transits in front of its parent star, we had to model them with codes that are
optimised against the presence of starspots and plages. Previous works showed
that the presence of starspot anomalies not properly modelled could cause
measures that mimic TTVs (Oshagh et al., 2013), or an anomalous planetary
radius towards the bluest wavelengths (Oshagh et al., 2014).

The quality of our light curves is good enough to let us hypothesise that the
anomalies are caused by starspot-occultation events (this is also supported by
the spectral class of the host star) but not good enough to make assumptions
about the fine structures of the starspots (as discriminate between single
spots or groups of spots). So, it is normal practice to model the anomalies
assuming one or more circular spots. As in the previous works of our series
(e.g. Mancini et al., 2013, 2014b, 2015, 2017), we decided to use the PRISM
and GEMC codes111https://github.com/JTregloanReed/PRISM$\\_$GEMC (Tregloan-
Reed et al., 2013, 2015), which let us model the light curves fitting both the
transit and one or several spot-crossing events simultaneously thanks to a
bayesian approach and the use of genetic algorithms (Ter Braak, 2006).
PRISM+GEMC is set to use the quadratic Limb-Darkening (LD) law and, assuming a
circular orbit, can model a light curve fitting for the following parameters:
the sum of the fractional radii of the star and planet222$r_{\star}$ and
$r_{\rm p}$ are defined as the stellar radius $R_{\star}$ and the planetary
radius $R_{\mathrm{p}}$ scaled by the semi-major axis $a$, respectively.,
$r_{\star}+r_{\rm p}$, the ratio of the radii, $k=\frac{R_{\rm
p}}{R_{\star}}=\frac{r_{\rm p}}{r_{\star}}$, the orbital inclination, $i$, the
LD coefficients, $u_{1}$ and $u_{2}$, the time of mid-transit, $T_{0}$, the
longitude and latitude ($\phi\,,\theta$) of the centre of the starspot and,
finally, the radius, $r_{\rm s}$, and the contrast, $\rho$, of the starspot
(i.e. the ratio of the surface brightness of the starspot to that of the
surrounding photosphere).

We used PRISM+GEMC to analyse all the ground-based data sets presented in the
previous section, four transits from Sector 10 of the TESS mission, six
transits from Sector 36, and six transits from Sector 63. We also reanalysed
the two light curves that were recorded with the MPG 2.2 m telescope by
Mohler-Fischer et al. (2013). From now on, we will refer to the TESS transits
from Sectors 10 and 36 using a numerical identification between 1 and 15 for
both sectors, while those from Sector 63 between 1 and 18. The transit events
have been labelled following a chronological order, in other words, from the
left to the right referring to Fig. 3. All the transit events, with their
relative id number, are shown in Fig. 3.

By using PRISM+GEMC, we analysed the light curves $\\#$2, $\\#$4, $\\#$14, and
$\\#$15 from Sector 10 and those from $\\#$6 to $\\#$11 from Sector 36, as
they show clear transit anomalies. Regarding Sector 63, the curves that
presented anomalies were the ones labelled: $\\#3$, $\\#9$, $\\#11$, $\\#13$,
$\\#14$, and $\\#15$. We used 1000 generations and 256 chains during the burn-
in stage, and 128 chains during the burn-out stage (for details see Tregloan-
Reed et al., 2015). The uncertainties have been adjusted by multiplying each
data set by the square root of the reduced chi-squared value obtained during
the burn-in stage, that is $\beta=(\chi_{\rm red}^{2})^{1/2}$.

We always fitted for a single starspot on the stellar disc, except for the
cases of the 2012 June 1st $g^{\prime}$ transit from the MPG 2.2 m, the 2015
January 25 $g^{\prime}$ $r^{\prime}$ $i^{\prime}$ transits from the MPG 2.2 m,
and the $\\#9$ TESS transit from Sector 36. The results of the fit are listed
in Tables 2 and 9, while the best-fitting models and their residuals are shown
in Figs 1 and 2, and in the Appendix (see Figs. 11, 12, 13, 14, 15). Regarding
Sector 63, unbinned and 10 min cadence binned data were available, and we
decided to fit both for the curves that contained anomalies. In Fig.14 and 15
the reader can observe both the unbinned and 10 min cadence binned data. The
results were always consistent for the two different data sets and so we kept
the ones that gave us the smaller uncertainties regarding the starspot
properties. It is worth underlining that if we did not have the unbinned data
as a comparison, deciding to keep results from binned data would have been
dangerous as the binning process could introduce systematics and bias the
results (e.g. Kipping 2010).

The values in Tables 2 refer to the median values of the posterior-
probability-density (PDF) function of the parameters and not to the best-
fitting parameters. The large uncertainties in the starspot parameters are
caused by the fact that in some cases (as for the 2017 May 30 transit) the
anomalies amplitudes are too small ($<2$ mmag) and in other cases (TESS) the
scatter in the light curves is too high ($>2$ mmag). In both scenarios,
PRISM/GEMC can find reasonable starspot parameters but the associated error
bars are large because a wide range of solutions could produce such small or
unclear anomalies.

The other light curves of Sector 10 and 36, which do not show detectable
anomalies, have been analysed by using JKTEBOP (Southworth, 2008), which is
faster than PRISM+GEMC and can fit the same photometric parameters except for
the starspot parameters. The results are listed in Table 9 and the reader can
inspect the best-fitting models in Fig. 16 and Fig. 17. For Sector 63, all the
light curves (unbinned data) were analysed by using PRISM+GEMC. The results
are listed in Table 9 and shown in Fig. 18.

Due to the large scatter in TESS data, it is convenient to fit multiple
transits simultaneously, as pointed out by several authors (e.g. Mancini et
al., 2016; Southworth et al., 2022). This approach leads to more precise
measurements of the sum and ratio of the radii, the inclination and, above
all, the mid-transit times. In particular, we chose to perform a fit for each
of the four periods of continuous observations. We considered transits from
$\\#1$ to $\\#8$ and from $\\#9$ to $\\#15$ for the data in both Sectors 10
and 36, see Fig. 3. We applied the same reasoning to the Sector 63 TESS data.
In particular, we chose to perform four different fits considering in order:
($i$) transits from $\\#1$ to $\\#5$, ($ii$) from $\\#6$ to $\\#9$, ($iii$)
from $\\#10$ to $\\#14$, and ($iv$) from $\\#15$ to $\\#18$. Moreover, we
fixed the non-linear LD coefficient, while the linear one has been set as a
free parameter. Uncertainties were estimated using a Monte Carlo method. The
final values of the parameters were estimated by a weighted mean of all the
fits of all the light curves that are listed in Table (9).

Table 2: Starspot parameters derived from the PRISM+GEMC fits of the transit
light curves.

Telescope | Date or ID | Filter | starspot | $\phi\,(^{\circ})$ | $\theta\,(^{\circ})$ | $r_{\rm s}\,(^{\circ})$ | $\rho$ |  $T_{\rm spot}$ (K)
---|---|---|---|---|---|---|---|---
MPG 2.2 m | 2012/02/28 | $g^{\prime}$ | $\\#1$ | $7.11\pm 0.84$ | $5.24\pm 4.64$ | $4.94\pm 1.90$ | $0.11\pm 0.16$ | $3792\pm 739$
MPG 2.2 m | 2012/02/28 | $r^{\prime}$ | $\\#1$ | 5.28 $\pm$ 1.51 | 5.30 $\pm$ 5.72 | 6.65 $\pm$ 3.63 | 0.45 $\pm$ 0.23 | 4432 $\pm$ 504
MPG 2.2 m | 2012/02/28 | $i^{\prime}$ | $\\#1$ | 7.65 $\pm$ 1.80 | 3.12 $\pm$ 6.03 | 5.00 $\pm$ 3.37 | 0.43 $\pm$ 0.27 | 4247 $\pm$ 659
MPG 2.2 m | 2012/02/28 | $z^{\prime}$ | $\\#1$ | 4.59 $\pm$ 4.28 | 5.87 $\pm$ 10.80 | 5.79 $\pm$ 3.52 | 0.44 $\pm$ 0.34 | 4136 $\pm$ 883
MPG 2.2 m | 2012/06/01 | $g^{\prime}$ |  | $\\#1$
---
$\\#2$
| 36.15 $\pm$ 2.95
---
2.26 $\pm$ 4.33
| 4.59 $\pm$ 5.25
---
8.48 $\pm$ 6.17
| 17.14 $\pm$ 4.45
---
15.24 $\pm$ 6.74
| 0.64 $\pm$ 0.13
---
1.21 $\pm$ 0.10
| 4846 $\pm$ 237
---
5405 $\pm$ 182
MPG 2.2 m | 2012/06/01 | $r^{\prime}$ | $\\#1$ | 34.94 $\pm$ 1.43 | 7.28 $\pm$ 6.64 | 18.41 $\pm$ 2.72 | 0.79 $\pm$ 0.04 | 4964 $\pm$ 140
MPG 2.2 m | 2012/06/01 | $i^{\prime}$ | $\\#1$ | 33.95 $\pm$ 1.23 | 7.38 $\pm$ 5.77 | 17.57 $\pm$ 2.49 | 0.84 $\pm$ 0.07 | 4986 $\pm$ 194
MPG 2.2 m | 2012/06/01 | $z^{\prime}$ | $\\#1$ | 39.95 $\pm$ 2.11 | 32.36 $\pm$ 8.41 | 21.48 $\pm$ 5.35 | 0.81 $\pm$ 0.06 | 4870 $\pm$ 180
MPG 2.2 m | 2015/01/25 | $g^{\prime}$ |  | $\\#1$
---
$\\#2$
| -36.95 $\pm$ 4.52
---
9.27 $\pm$ 31.57
| 52.95 $\pm$ 13.98
---
9.60 $\pm$ 7.82
| 36.63 $\pm$ 12.44
---
2.92 $\pm$ 3.66
| 0.39 $\pm$ 0.23
---
0.43 $\pm$ 0.28
| 4492 $\pm$ 465
---
4559 $\pm$ 522
MPG 2.2 m | 2015/01/25 | $r^{\prime}$ |  | $\\#1$
---
$\\#2$
| -30.74 $\pm$ 2.77
---
18.16 $\pm$ 7.29
| 6.85 $\pm$ 5.40
---
31.29 $\pm$ 17.26
| 10.12 $\pm$ 5.39
---
25.37 $\pm$ 13.51
| 0.80 $\pm$ 0.09
---
0.90 $\pm$ 0.09
| 4977 $\pm$ 206
---
5106 $\pm$ 202
MPG 2.2 m | 2015/01/25 | $i^{\prime}$ |  | $\\#1$
---
$\\#2$
| -27.49 $\pm$ 9.23
---
22.13 $\pm$ 45.60
| 40.69 $\pm$ 15.86
---
47.33 $\pm$ 32.72
| 15.71 $\pm$ 11.84
---
21.64 $\pm$ 18.85
| 0.76 $\pm$ 0.18
---
0.83 $\pm$ 0.24
| 4862 $\pm$ 376
---
4974 $\pm$ 460
MPG 2.2 m | 2015/01/25 | $z^{\prime}$ | $\\#1$ | -27.29 $\pm$ 1.44 | 15.26 $\pm$ 7.68 | 5.19 $\pm$ 3.79 | 0.62 $\pm$ 0.16 | 4541 $\pm$ 397
DK 1.54 m | 2016/04/16 | $R$ | $\\#1$ | -0.01 $\pm$ 0.93 | 9.23 $\pm$ 6.84 | 9.19 $\pm$ 3.28 | 0.69 $\pm$ 0.17 | 4816 $\pm$ 328
DK 1.54 m | 2017/05/30 | $R$ | $\\#1$ | 2.48 $\pm$ 5.73 | 26.44 $\pm$ 16.79 | 14.21 $\pm$ 11.69 | 0.81 $\pm$ 0.18 | 4987 $\pm$ 324
DK 1.54 m | 2017/07/11 | $R$ | $\\#1$ | -37.28 $\pm$ 16.37 | 27.85 $\pm$ 20.33 | 17.92 $\pm$ 17.65 | 0.37 $\pm$ 0.26 | 4240 $\pm$ 622
DK 1.54 m | 2018/05/17 | $I$ | $\\#1$ | -13.91 $\pm$ 8.41 | 11.72 $\pm$ 11.32 | 16.22 $\pm$ 8.93 | 0.84 $\pm$ 0.03 | 4977 $\pm$ 141
DK 1.54 m | 2018/06/05 | $R$ | $\\#1$ | 9.80 $\pm$ 1.91 | 8.48 $\pm$ 7.60 | 9.47 $\pm$ 3.86 | 0.75 $\pm$ 0.20 | 4898 $\pm$ 368
DK 1.54 m | 2018/06/24 | $R$ | $\\#1$ | 8.01 $\pm$ 8.48 | 6.91 $\pm$ 8.38 | 4.00 $\pm$ 3.83 | 0.61 $\pm$ 0.27 | 4683 $\pm$ 504
DK 1.54 m | 2019/05/19 | $I$ | $\\#1$ | 63.36 $\pm$ 17.20 | 18.72 $\pm$ 24.66 | 42.58 $\pm$ 16.62 | 0.79 $\pm$ 0.18 | 4898 $\pm$ 382
DK 1.54 m | 2019/05/23 | $I$ | $\\#1$ | -48.08 $\pm$ 40.88 | 19.86 $\pm$ 26.27 | 9.79 $\pm$ 18.96 | 0.46 $\pm$ 0.25 | 4269 $\pm$ 620
DK 1.54 m | 2019/06/07 | $I$ | $\\#1$ | 24.89 $\pm$ 2.97 | 11.90 $\pm$ 13.57 | 5.50 $\pm$ 10.60 | 0.51 $\pm$ 0.23 | 4390 $\pm$ 546
DK 1.54 m | 2021/05/23 | $I$ | $\\#1$ | 0.70 $\pm$ 1.77 | 24.52 $\pm$ 12.49 | 16.91 $\pm$ 8.66 | 0.77 $\pm$ 0.15 | 4866 $\pm$ 336
DK 1.54 m | 2022/05/13 | $I$ | $\\#1$ | 2.38 $\pm$ 5.81 | 15.34 $\pm$ 13.18 | 9.68 $\pm$ 6.44 | 0.71 $\pm$ 0.22 | 4762 $\pm$ 464
TESS S10 |  $\\#$ 2 |  | $\\#1$ | 32.59 $\pm$ 43.65 | 20.28 $\pm$ 53.02 | 18.35 $\pm$ 34.91 | 0.60 $\pm$ 0.28 | 4575 $\pm$ 598
TESS S10 |  $\\#$ 4 |  | $\\#1$ | 31.79 $\pm$ 52.82 | 33.34 $\pm$ 54.25 | 38.50 $\pm$ 48.00 | 0.79 $\pm$ 0.28 | 4905 $\pm$ 539
TESS S10 |  $\\#$ 14 |  | $\\#1$ | -5.91 $\pm$ 5.15 | 18.74 $\pm$ 13.47 | 22.66 $\pm$ 6.89 | 0.72 $\pm$ 0.10 | 4789 $\pm$ 250
TESS S10 |  $\\#$ 15 |  | $\\#1$ | 16.88 $\pm$ 12.56 | 14.98 $\pm$ 27.90 | 7.15 $\pm$ 10.83 | 0.23 $\pm$ 0.27 | 3698 $\pm$ 920
TESS S36 |  $\\#$ 6 |  | $\\#1$ | -39.04 $\pm$ 51.64 | 35.41 $\pm$ 53.42 | 37.95 $\pm$ 44.08 | 0.73 $\pm$ 0.29 | 4806 $\pm$ 571
TESS S36 |  $\\#$ 7 |  | $\\#1$ | -14.43 $\pm$ 10.14 | 28.32 $\pm$ 28.73 | 16.76 $\pm$ 12.23 | 0.45 $\pm$ 0.24 | 4272 $\pm$ 589
TESS S36 |  $\\#$ 8 |  | $\\#1$ | 8.94 $\pm$ 9.21 | 32.43 $\pm$ 33.01 | 23.49 $\pm$ 16.39 | 0.64 $\pm$ 0.22 | 4648 $\pm$ 474
TESS S36 |  $\\#$ 9 |  |  | $\\#1$
---
$\\#2$
| -15.82 $\pm$ 25.76
---
-9.57 $\pm$ 48.03
| 24.39 $\pm$ 27.26
---
72.92 $\pm$ 53.11
| 41.89 $\pm$ 25.78
---
51.48 $\pm$ 33.69
| 0.76 $\pm$ 0.14
---
0.76 $\pm$ 0.27
| 4856 $\pm$ 314
---
4856 $\pm$ 529
TESS S36 |  $\\#$ 10 |  |  | $\\#1$
---
$\\#2$
| 9.47 $\pm$ 6.93
---
51.09 $\pm$ 52.14
| 21.76 $\pm$ 15.68
---
27.66 $\pm$ 52.45
| 13.18 $\pm$ 7.45
---
30.40 $\pm$ 23.19
| 0.48$\pm$ 0.22
---
0.51 $\pm$ 0.27
| 4337 $\pm$ 530
---
4399 $\pm$ 620
TESS S36 |  $\\#$ 11 |  | $\\#1$ | 32.11 $\pm$ 42.00 | 20.37 $\pm$ 48.72 | 23.13 $\pm$ 31.48 | 0.77 $\pm$ 0.23 | 4872 $\pm$ 462
TESS S63 |  $\\#$ 3 |  | $\\#1$ | 0.61 $\pm$ 44.00 | 16.92 $\pm$ 55.22 | 17.12 $\pm$ 46.76 | 0.70 $\pm$ 0.28 | 4755 $\pm$ 490
TESS S63 |  $\\#$ 9 |  | $\\#1$ | -7.79 $\pm$ 32.00 | 17.57 $\pm$ 52.35 | 12.70 $\pm$ 38.57 | 0.43 $\pm$ 0.28 | 4229 $\pm$ 631
TESS S63 |  $\\#$ 11 |  | $\\#1$ | 10.61 $\pm$ 50.86 | 24.02 $\pm$ 53.08 | 11.79 $\pm$ 41.32 | 0.62 $\pm$ 0.30 | 4613 $\pm$ 557
TESS S63 |  $\\#$ 13 |  | $\\#1$ | -15.12 $\pm$ 40.22 | 18.75 $\pm$ 54.57 | 11.25 $\pm$ 39.66 | 0.56 $\pm$ 0.29 | 4499 $\pm$ 567
TESS S63 |  $\\#$ 14 |  | $\\#1$ | 5.93 $\pm$ 54.84 | 22.41 $\pm$ 52.14 | 14.60 $\pm$ 41.39 | 0.60 $\pm$ 0.29 | 4575 $\pm$ 548
TESS S63 |  $\\#$ 15 |  | $\\#1$ | 33.04 $\pm$ 49.80 | 16.93 $\pm$ 56.49 | 31.61 $\pm$ 46.23 | 0.76 $\pm$ 0.28 | 4857 $\pm$ 471
Notes: The values in the last column were obtained using Eq. (1).

##  4 Physical and starspot properties of the HATS-2 exoplanetary system

###  4.1 Stellar and planetary parameters

To estimate the main physical parameters of the HATS-2 system, we used the
code jktabsdim 333https://www.astro.keele.ac.uk/jkt/codes/jktabsdim.html
(Southworth et al., 2009). The code makes use of the spectroscopic parameters,
the stellar radial-velocity (RV) semi-amplitude ($K_{\star}=268.9\pm 29.0$ m
s-1; Mohler-Fischer et al. 2013) and a set of theoretical stellar models. In
particular, jktabsdim uses tabulations from the Claret (Claret, 2004), Y2
(Demarque et al., 2004), and Padova (Girardi et al., 2000) models. jktabsdim
works in such a way that the velocity amplitude of the planet is iteratively
modified to maximise the agreement between the best additional constraint
derived from different theoretical models and the observed values of the
stellar density from transit, effective temperature and metallicity ($T_{\rm
eff}=5227\pm 95$ and ${\rm[Fe/H]}=0.15\pm 0.05$; Mohler-Fischer et al., 2013)
. A wide range of possible ages for the parent star is also considered during
this process. The code returns different estimates for each of the output
parameters, one for each set of theoretical models. The unweighted means are
considered the final values of the parameters. The systematic uncertainties,
which were caused by the use of theoretical models, were estimated, while the
statistical uncertainties were propagated from the uncertainties of the input
parameters. It is worth to underline that jktabsdim does not use Gaia
parallax, any apparent magnitudes, or interstellar extinction. As a
consistency check, we calculated the distance to the system using our
measurements of the stellar radius and effective temperature, apparent
magnitudes in the optical ($BV$) and in the infrared (2MASS $JHK$), and the
surface brightness relations from Kervella et al. (2004). An interstellar
reddening of $E(B-V)=0.08\pm 0.03$ mag was needed to align the distances in
the optical and infrared passbands, and the distance found in the $K$-band is
$332\pm 9$ pc. The Gaia DR3 distance of $337\pm 2$ pc is in good agreement
with this value.

The final values of the parameters are listed in Table 3 and are in good
agreement with those from Mohler-Fischer et al. (2013). The only parameter
that differs (although only at $\sim 1.5\,\sigma$) is the planetary radius,
which appears smaller in size. It is also possible to compare the final values
with the ones obtained considering a weighted mean of the results of only the
curves that do not present anomalies. The difference between these values can
be considered as a first-order approximation of the systematic error
introduced by the modelling of the starspots, which due to the low data
quality, is within the 1-sigma uncertainties. In this way we derived that the
systematic error is 0.85 $\%$ regarding the sum of the radii and 1.3 $\%$
regarding the ratio of the radii.

Table 3: Median values and $68\%$ confidence intervals of the physical and orbital properties of the HATS-2 system, found using JKTABSDIM. Parameter | Symbol | Unit | This work | Mohler-Fischer et al. (2013)
---|---|---|---|---
Stellar mass | $M_{\star}$ | $M_{\sun}$ | $0.904\pm 0.051\pm 0.010$ | $0.882\pm 0.037$
Stellar radius | $R_{\star}$ | $R_{\sun}$ | 0.876 $\pm$ 0.020 $\pm$ 0.003 | 0.898 $\pm$ 0.019
Stellar logarithmic surface gravity | $\log{g_{\star}}$ | cgs | 4.510 $\pm$ 0.012 $\pm$ 0.002 | $4.48\pm 0.02$
Stellar density | $\rho_{\star}$ | $\rho_{\sun}$ | $1.346\pm 0.028$ | -
Stellar Age | $\tau_{\star}$ | Gyr | ${7.2}^{+3.7\,+2.9}_{-6.3\,-5.1}$ | $9.7\pm 2.9$
Stellar Rotational Period | $P_{\rm rot}$ | day | $22.46\pm 5.20$ | $30.32\pm 10.13$
Planetary mass | $M_{\rm p}$ | $M_{\rm Jup}$ | 1.37 $\pm$ 0.16 $\pm$ 0.01 | 1.345 $\pm$ 0.150
Planetary radius | $R_{\rm p}$ | $R_{\rm Jup}$ | 1.122 $\pm$ 0.024 $\pm$ 0.004 | 1.168 $\pm$ 0.030
Planetary surface gravity | $g_{\rm p}$ | m s-2 | 27.0 $\pm$ 2.9 | 24.6 $\pm$ 2.8
Planetary density | $\rho_{\rm p}$ | $\rho_{\rm Jup}$ | 0.91 $\pm$ 0.10 | 0.79 $\pm$ 0.11
Planetary equilibrium temperature | $T_{\rm eq}$ | K | 1550 $\pm$ 29 | 1577 $\pm$ 31
Orbital inclination | $i$ | ∘ | 87.83 $\pm$ 0.18 | 87.2 $\pm$ 0.7
Semi-major axis | $a$ | au | 0.02317 $\pm$ 0.00043 $\pm$ 0.00009 | 0.0230 $\pm$ 0.0003
Inclination angle of the stellar spin | $\sin i_{\star}$ |  | 0.76 $\pm$ 0.31 | -
Sky-projected spin-orbit angle | $\left|\lambda\right|$ | ∘ | 2.72 $\pm$ 17.84 | 8 $\pm$ 8
True spin-orbit angle | $\psi$ | ∘ | $38.49\pm 27.13$ | -
Notes: Results from Mohler-Fischer et al. (2013) are also reported for
comparison. Where two error bars are
given, the first refers to the statistical uncertainties, while the second to
the systematic errors. Other derived
parameters, i.e. the stellar rotational period, the sky-projected obliquity,
the inclination angle of the stellar spin,
and the true obliquity are also listed.

###  4.2 Starspot analysis

Figure 4: Variation in the starspot contrast with wavelength. The vertical
bars represent the errors in the measurements and the horizontal bars show the
FWHM transmission of the passbands used. In the upper panels, the colour of
the data points is in agreement with Fig. 2. In the top left panel, the data
taken on 2012 February 28 with the GROND multi-band camera are represented
together with the ATLAS9 (Kurucz, 1979) stellar atmospherical models computed
considering a stellar surface at 5227 K and a starspot at 4200 and 4400 K. In
the top right panel, the data taken on 2012 June 1st with the GROND multi-band
camera are represented together with the ATLAS9 stellar atmospherical models
computed considering a stellar surface at 5227 K and a starspot at 4600, 4850,
and 4980 K. In the bottom panels, the points represent the contrast values
obtained using the Danish 1.54 m transits (as explained in the plot legend),
while the curves are the same ATLAS9 models used in the upper panels.

Having modelled the HATS-2 transit light curves containing starspot anomalies
with the PRISM-GEMC codes, we were able to derive the best-fitting values for
the position of the starspots on the stellar disc, beyond their angular size
and contrast. These estimations are reported in Table 2, in whose last column
the reader can also find the starspot-temperature values, which were derived
considering the stellar photosphere and starspots as black bodies and using
equation 1 from Silva (2003), that is

$\rho=\frac{B_{\nu}(T_{\rm spot})}{B_{\nu}(T_{\rm eff})}=\frac{{\rm
exp}\left(h\nu/k_{\rm B}T_{\rm eff}-1\right)}{{\rm exp}\left(h\nu/k_{\rm
B}T_{\rm spot}-1\right)}\,,$ (1)

where $\nu$ is the central frequency of each filter
band444http://ulisse.pd.astro.it/Astro/ADPS/, $h$ is the Planck constant,
while $k_{\rm B}$ is the Boltzmann constant. All the values are consistent
with starspots cooler than the stellar photosphere and lie between 4000 and
5000 K, within the uncertainties.

The only exception is related to the first anomaly on the $g^{\prime}$ light
curve of the transit observed on 2012 June 1st, which was identified by
Mohler-Fischer et al. (2013) as a plage (see the top right-hand panel in Fig.
2). In the case of the Sun, plages are usually visible through H-$\alpha$ or
calcium (Ca) K line wavelengths by using appropriate filters. In K-type stars,
such as HATS-2, the Ca II lines are much stronger than the H-$\alpha$ ones and
fall in the wavelength transmission range defined by the $g^{\prime}$ filter,
$\lambda=478.8\pm 137.9$ nm. We, therefore, agree with the interpretation of
Mohler-Fischer et al. (2013) that the anomaly could be caused by a plage
connected with the following starspot.

In general, we noted that the angular sizes of the starspots vary from a
minimum of $2.9^{\circ}$ to a maximum of $40^{\circ}-50^{\circ}$. Considering
that an angular radius of $90^{\circ}$ covers half of the stellar hemisphere,
the starspots we have found cover from a minimum of $1.6\%$ to a maximum of
$20\%-30\%$ of the stellar disc. As in the case of, e.g. TrEs-1 and WASP-19,
our measurements are similar to those found for other G-type and K-type stars
(Rabus et al., 2009; Mancini et al., 2013) and are in good agreement with the
sizes of very large sunspots (Sun et al., 2015). However, we stress that some
of the starspots, which we have detected and modelled with circular shapes,
might actually be a group of starspots, according to the Zurich and McIntosh
classifications (McIntosh, 1990).

Since starspots are darker in the ultraviolet than in the infrared and we
observed transits through different filters, we can check if the starspot
contrast varies as expected as a function of the wavelength. In Fig. 4 we
compare the starspot contrasts estimated by PRISM+GEMC and reported in Table 2
with theoretical expectations. Using ATLAS9 atmospheric models (Kurucz, 1979),
we modelled a stellar photosphere of $5227$ K and starspots with five
different temperature $4200$ K, $4400$ K, $4600$ K, $4850$ K and $4980$ K.
Although the error bars of the starspot contrasts are quite large, the trend
in each panel of Fig. 4 is that for which the starspots are brighter in the
redder passbands than in the bluer one, for both all three complete
simultaneous GROND multi-band observations and the two-colours observations
with the Danish telescope.

###  4.3 Spin-orbit alignment

We investigated the possibility that some of the observed starspot-crossing
events have been caused by the same starspot because, then, as explained in
Sect. 1, the spin-orbit alignment of the planetary system can be derived.

Mohler-Fischer et al. (2013) already argued that the anomalies observed during
the 2012 February 28 and 2012 June 1st transits could have been caused by the
same starspot. Studying the values reported in Table 2, we note that, for
these two transits, the differences in the starspot latitudes in all the GROND
bands fall within $1\sigma$ but their angular sizes are not similar. Hence, it
is possible that we are dealing with a starspot complex that evolved following
the Zurich classification. However, the two transits are separated by 94 days
and, using empirical laws, Mohler-Fischer et al. (2013) estimated that a group
of starspots with this size could have a lifetime of $\simeq 130$ days. With
these caveats, the authors estimated a sky-projected obliquity of
$8^{\circ}\pm 8^{\circ}$ and a rotational period of $31\pm 10$ days.

Using the new data presented in this work, we identified eleven consecutive
starspot-crossing events, separated by less than 5 days, and for which the
relative latitudes and angular sizes differences fall within $1\sigma$,
referring to the values listed in Table 2. 10 out of 11 events have been
identified in the TESS photometric data. Of particular interest are the
consecutive transits that we labelled with $\\#7$ and $\\#8$ from TESS Sector
36 (see Fig. 5). Investigating the light curves (see Fig. 12), we note that
the anomalies that appear in the transits $\\#7$ and $\\#8$ have similar
amplitudes and duration, whereas the time of the maximum of the anomalies
changes with the increase of the orbital phase. These facts suggest that the
anomalies are due to the same starspot rotating around the surface of the
star.

Figure 5: Representation of the stellar disc, starspot position, and transit
chord for the two consecutive transit events, $\\#7$ (left panel) and $\\#8$
(right panel) with starspot crossings, obtained using PRISM/GEMC. The grey
scale of each starspot is related to its contrast. The starspot properties
refer to the best-fitting models in Fig. 12 (see Appendix).

This hypothesis is also supported by the modelling results from PRISM/GEMC.
Indeed, the best-fitting circular starspots for the transit events $\\#7$ and
$\\#8$ have similar sizes and contrasts (see Table 2). The transit designed as
number $\\#6$ also supports this hypothesis, although its small size does not
allow us to constrain the size of the starspot with the required accuracy.

The other TESS consecutive transit events were identified as $\\#$14 and
$\\#$15 from Sector 10 (Fig. 11), as $\\#$9, $\\#$10, and $\\#$11 from Sector
36 (Fig. 13), and as $\\#$13, $\\#$14, and $\\#$15 from Sector 63 (Fig. 14).

The two consecutive starspot-crossing events, which were identified using our
data from ground-based telescopes, are those from the Danish 1.54 m telescope
observed on 2019 May 19 and 2019 May 23 (see Fig. 1).

Using the values listed in Table 2, it is possible to calculate the values of
the sky-projected obliquity, $\lambda$, and the stellar rotational period
$P_{\rm rot}$ for each of the identified events by using the following
equations:

$\left|\lambda\right|=\arctan{\left(\left|\frac{\sin\theta_{2}-\sin\theta_{1}}{\cos\theta_{2}\sin\phi_{2}-\cos\theta_{1}\sin\phi_{1}}\right|\right)}\,,$
(2) $P_{\rm rot}=\frac{2\pi\Delta t}{\sqrt{D}}\,,$ (3)
$D=2\left(1-\sin\theta_{1}\sin\theta_{2}-\cos\theta_{1}\cos\theta_{2}\sin\phi_{1}\sin\phi_{2}\right)\\\
+\cos^{2}\theta_{2}\left[\sin^{2}\phi_{2}-1\right]+\cos^{2}\theta_{1}\left[\sin^{2}\phi_{1}-1\right]\,,$
(4)

where $\theta_{1,2}$ and $\phi_{1,2}$ are the latitudes and longitudes of the
two consecutive starspots and $\Delta t$ is the time interval between the two
starspot-crossing events. The quantity $\sqrt{D}$ represents the arc length on
the stellar photosphere between the two positions of the starspot. The
mathematical derivation of these equations is given in the Appendix. The
values of $\lambda$ and $P_{\rm rot}$ are reported in Table 4 for each event.
The large uncertainties associated with the $\left|\lambda\right|$ and $P_{\rm
rot}$ values derived from some of the TESS light curves must be once again
intended, as explained in Section 3, as a consequence of the poor quality
(high scatter) of the data. Our final estimations were computed by performing
a weighted mean, obtaining $\left|\lambda\right|=2^{\circ}.72\pm
17^{\circ}.84$ and $P_{\rm rot}=22.46\pm 5.20$ days. This value is consistent
with the value of $P_{\rm rot,\,TESS}=21.44$ days, obtained with a
FAP$<0.1\,\%$ (False Alarm Probability) performing a GLS (Generalised Lomb-
Scargle Periodgram, VanderPlas 2018) analysis of the SAP flux from all the
three sectors of the TESS Mission, see Fig. 20. The estimation of the sky-
projected obliquity resulted consistent with zero degrees and, therefore, with
a well-aligned system. It is represented in Fig. 6 as a function of the
parent-star effective temperature, together with a sample of values taken from
the TEPCat catalogue555The TEPCat catalogue is available at
http://www.astro.keele.ac.uk/jkt/tepcat (Southworth, 2011)..

Table 4: All the spot-crossing events identified over two transits in this
work.

Telescope | Consecutive Events |  $\left|\lambda\right|$ (∘) |  $P_{\rm rot}$ (days)
---|---|---|---
Danish | May 19-23, 2019 | $0.70\pm\leavevmode\nobreak\ \leavevmode\nobreak\ 22.0$ | $22.01\pm\leavevmode\nobreak\ \leavevmode\nobreak\ 7.06\leavevmode\nobreak\ $
TESS s10 |  $\\#$14 and $\\#$15 | $9.40\pm 76.20$ | $22.21\pm 14.32$
TESS s36 |  $\\#$6 and $\\#$7 | $20\pm 176\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $27.3\pm 49.5$
TESS s36 |  $\\#$6 and $\\#$8 | $3.83\pm 81.47$ | $26.34\pm 26.8$
TESS s36 |  $\\#$7 and $\\#$8 | $10.01\pm 104.00\leavevmode\nobreak\ \leavevmode\nobreak\ $ | $23.90\pm 16.33$
TESS s36 |  $\\#$9 and $\\#$10 | $6.01\pm 71.80$ | $21.10\pm 21.41$
TESS s36 |  $\\#$9 and $\\#$11 | $4.97\pm 68.53$ | $22.71\pm 22.30$
TESS s36 |  $\\#$10 and $\\#$11 | $3.75\pm 136.63$ | $24.57\pm 44.57$
TESS s63 |  $\\#$13 and $\\#$14 | 9.90 $\pm$ 201.64 | 24.47 $\pm$ 77.67
TESS s63 |  $\\#$13 and $\\#$15 | 2.25 $\pm$ 97.00 | 22.12 $\pm$ 27.82
TESS s63 |  $\\#$14 and $\\#$15 | 11.93 $\pm$ 161.79 | 19.53 $\pm$ 52.54
|  | 2.72 $\pm$ 17.84 | 22.46 $\pm$ 5.20
Notes: In the table are listed the sky-projected and stellar rotational period
values obtained using Eq. (2) and (3). The final values were
calculated by a weighted mean.

Considering that HATS-2 has $T_{\rm eff}=5227\pm 95$ K, our result is in
agreement with the trend for which the orbits of the hot Jupiters hosted
around cool dwarf stars tend to present a low obliquity (Winn et al., 2010).
Moreover, this can be an indication that HATS-2 b has reached its actual
position through a Type II migration, or that tidal interactions with its
parent star damped a non-zero obliquity caused by Lidov-Kozai oscillations
induced by a distant companion (Raymond & Morbidelli, 2022). The second
scenario is preferable as HATS-2 is a quite old ($\sim 7$ Gyr) K-type star, so
both the age and the convective envelope enhance the tidal dissipation
efficiency. The analysis of the TTVs, which is discussed in the next section,
will provide other insights to discriminate between the two formation
scenarios.

Knowing the rotational period of the parent star and using
$(v\sin{i_{\star}})$ from Mohler-Fischer et al. (2013), we determined the sine
of inclination of the stellar-rotation axis by means of

$\sin{i_{\star}}=P_{\rm rot}\frac{(v\sin{i_{\star}})}{2\pi R_{\star}}=0.76\pm
0.31\,.$ (5)

Then, using the following formula (Winn et al., 2007)

$\cos{\psi}=\cos{i}\cos{{i_{\star}}}+\sin{i}\sin{{i_{\star}}}\cos{\lambda},$
(6)

we derived the true orbital obliquity of HATS-2 b, which resulted to be
$\psi=38^{\circ}.49\pm 27^{\circ}.13$. This measurement excludes that HATS-2
is in a pole-on configuration.

Figure 6: Sky-projected obliquity as a function of the host stars’ effective
temperature for an ensemble of exoplanetary systems. The star is the value of
the sky-projected obliquity of the HATS-2 system found in this work, while the
dots are the values from the TEPCAT catalogue.

##  5 Transit Timing Variations

Table 5:  Times of mid-transit for HATS-2 b and relative O-C residuals.

Telescope | Epoch | O (BJDTDB) | O-C (min)
---|---|---|---
MPG | -1924.0 | 55985.731172 $\pm$ 0.000090 | -0.501 $\pm$ 0.130
MPG | -1854.0 | 56080.520649 $\pm$ 0.000109 | -0.340 $\pm$ 0.157
MPG | -1139.0 | 57048.727133 $\pm$ 0.000117 | 0.847 $\pm$ 0.168
DK | -809.0 | 57495.591314 $\pm$ 0.000161 | 0.891 $\pm$ 0.232
DK | -507.0 | 57904.539218 $\pm$ 0.000168 | 0.170 $\pm$ 0.241
DK | -476.0 | 57946.517242 $\pm$ 0.000175 | -0.007 $\pm$ 0.251
DK | -247.0 | 58256.613740 $\pm$ 0.000180 | -0.209 $\pm$ 0.259
DK | -233.0 | 58275.571993 $\pm$ 0.000189 | 0.338 $\pm$ 0.273
DK | -219.0 | 58294.529570 $\pm$ 0.000198 | -0.088 $\pm$ 0.285
TESS | -10.0 | 58577.543615 $\pm$ 0.000203 | 0.031 $\pm$ 0.292
TESS | 0.0 | 58591.084807 $\pm$ 0.000227 | -0.178 $\pm$ 0.327
DK | 24.0 | 58623.584665 $\pm$ 0.000282 | 0.752 $\pm$ 0.406
DK | 27.0 | 58627.646453 $\pm$ 0.000282 | -0.130 $\pm$ 0.406
DK | 38.0 | 58642.542160 $\pm$ 0.000301 | 0.204 $\pm$ 0.434
DSC/ExoClock | 468.0 | 59224.819071 $\pm$ 0.000317 | -0.682 $\pm$ 0.457
SSO/ExoClock | 500.0 | 59268.151393 $\pm$ 0.000332 | -0.623 $\pm$ 0.478
TESS | 514.0 | 59287.109786 $\pm$ 0.000379 | 0.124 $\pm$ 0.546
TESS | 526.0 | 59303.359703 $\pm$ 0.000383 | 0.574 $\pm$ 0.551
DSC/ExoClock | 535.0 | 59315.546858 $\pm$ 0.000383 | 0.503 $\pm$ 0.552
DSC/ExoClock | 538.0 | 59319.608897 $\pm$ 0.000403 | -0.018 $\pm$ 0.581
DK | 569.0 | 59361.587384 $\pm$ 0.000424 | 0.469 $\pm$ 0.610
DSC/ExoClock | 736.0 | 59587.726556 $\pm$ 0.000451 | -1.217 $\pm$ 0.650
DSC/ExoClock | 1010.0 | 59958.758793 $\pm$ 0.000461 | -1.823 $\pm$ 0.664
TESS | 1054.0 | 60018.341413 $\pm$ 0.000484 | -0.767 $\pm$ 0.696
TESS | 1058.0 | 60023.758037 $\pm$ 0.000507 | -0.639 $\pm$ 0.730
TESS | 1063.0 | 60030.529038 $\pm$ 0.000584 | -0.162 $\pm$ 0.841
TESS | 1068.0 | 60037.298643 $\pm$ 0.001085 | -1.693 $\pm$ 1.562
Notes: The values refer to Case ($b$), and are calculated using equation (8).

Thanks to the analysis of the light curves presented in the previous sections,
we have a set of mid-transit times, which are listed in Table 5. The values
derived from the GROND light curves refer to a weighted mean of the $T_{0}$
values obtained in each of the four optical filters, while the values derived
from the TESS light curves refer to the result of the eight simultaneous fits
with JKTEBOP.

We also used a series of light curves from the ExoClock database (Kokori et
al., 2022) 666https://www.exoclock.space/database/planets/HATS-2b/.. In
particular, we selected the most well-sampled (i.e. with well-sampled ingress,
plateau and egress) and high-quality (i.e. low scatter; see the last column of
Table 1) light curves from those presented by Edwards et al. (2021) and Kokori
et al. (2023). In detail, we selected three light curves collected at the Deep
Sky Chile Observatory on 2021 January 10, 2021 April 11, and 2021 April 15
(Kokori et al., 2023), and one at the Sliding Spring Observatory on 2021
February 22 (Edwards et al., 2021).

We also considered two unpublished light curves once more collected by amateur
astronomers within the ExoClock collaboration; one was again observed with the
Deep Sky Chile 40 cm telescope, while the other with the Hakos Astro Farm 50
cm telescope777https://www.exoclock.space/database/observations_by_observer..

All these extra light curves were fitted with the same procedures presented in
the previous sections thanks to the JKTEBOP and PRISM+GEMC codes. The best-
fitting models are presented in Fig. 21 together with the detrended light
curves. The results are listed in Table 8. In the end, our final ensemble of
mid-transit times covers a time baseline extending over more than 10 years. It
is, therefore, possible to refine the ephemerides by considering various
models for fitting the timings.

We set the middle transit of the second half of the Sector 10 TESS data to be
the zeroth epoch because most of its light curve does not show the presence of
anomalies connected to starspots within the experimental uncertainties and has
been estimated by a simultaneously fit with the ‘neighbour’ transits.

We considered two different data sets: ($a$) fitting only GROND, DFOSC and
TESS data without ExoClock times, and ($b$) fitting all the data listed in
Table 5. The first model that we tried is a simple straight line, $T_{0}^{\rm
lin}=\kappa_{0}+\kappa_{1}\times E$, which represents a linear ephemeris. In
case ($a$), we obtained

$T_{0}^{\rm lin}=\,{\rm BJD_{TDB}}\,58591.08501(11)+1.35413385(9)\times E\,,$
(7)

while in case ($b$)

$T_{0}^{\rm lin}=\,{\rm BJD_{TDB}}\,58591.084931(96)+1.35413379(8)\times E\,.$
(8)

Here, $E$ represents the epoch of the transit and the bracketed quantities
indicate the uncertainties in the preceding digits. Variations of the order of
minutes are evident from the O-C residuals of both of the two fits shown in
Fig. 7. This means that the linear ephemeris is a good approximation but seems
to not exactly reproduce the dynamic state of the system.

Since physical phenomena may hide in these variations, we also tried to fit
the data with

  * $\bullet$

a quadratic polynomial that represents a quadratic ephemeris and could be
connected to a tidal-decay scenario

$T_{0}^{\rm quad}=\kappa_{0}+\kappa_{1}\times E+\kappa_{2}\times E^{2}\,;$ (9)

  * $\bullet$

a sinusoidal function that could be seen as a linear ephemeris periodically
perturbed with an amplitude $A_{\rm TTV}$, a period $P_{\rm
TTV}={2\pi}/{\Omega}$, and a phase $\varphi$. This ephemeris could be
connected to an apsidal precession scenario or the presence of a planetary or
sub-stellar companion (e.g. a brown dwarf)

$T_{0}^{\rm sin}=\kappa_{0}+\kappa_{1}\times
E+A\sin\left(\Omega\,E+\varphi\right).$ (10)

All the fits were performed using the lmfit (Newville et al., 2014) Python
package. The results are listed in Table 6 and their O-C residuals versus the
best-fitting linear model are represented in Fig. 7, for both scenarios. The
sinusoidal and quadratic ephemeris seems to fit better the data with respect
to the linear one as confirmed by statistical indicators, such as the Reduced
Chi-Squared $\chi_{\rm red}$, the Bayesian information criterion (BIC) and the
Akaike information criterion (AIC), which are, respectively, defined as

$\chi_{\rm red}^{2}=\frac{\chi^{2}}{N_{\rm dof}}=\frac{\chi^{2}}{N-j}\;,$ (11)
${\rm BIC}=N\ln\left(\frac{\chi^{2}}{N}\right)+N_{\Theta}\ln N\;,$ (12) ${\rm
AIC}=N\ln\left(\frac{\chi^{2}}{N}\right)+2N_{\Theta}\;,$ (13)

where $N$ is the number of data points and $N_{\Theta}$ is the number of
parameters of the best-fitting model. $N_{\rm dof}$ is the number of degrees
of freedom and usually is set as the number of data points $N$ minus the
number of free parameters $N_{\Theta}$. The $\chi_{\rm red}^{2}$, BIC and AIC
values for each of the considered ephemeris models and groups of data are
listed in Table 7.

Table 6: Best-fit parameters of the ephemeris models for the HATS-2 b mid-
transit time residuals.

Model | $\kappa_{0}$ | $\kappa_{1}$ | $\kappa_{2}$ | $A$ | $\Omega$ | $\varphi$
---|---|---|---|---|---|---
| [days] | [days/epoch] | [days/epoch2] | [days] | [rad/epoch] | [rad]
|  |  | Data from professional telescopes – Case ($a$) |  |  |
Linear | $58591.08501\pm 0.00011$ | $1.35413385\pm 0.00000009$ |  |  |  |
Quadratic | $58591.08510\pm 0.00009$ | $1.35413340\pm 0.00000010$ | $(-3.01\pm 0.85)\times 10^{-10}$ |  |  |
Sinusoidal | $58591.08510\pm 0.00010$ | $1.35413424\pm 0.00000016$ |  | $(7.98\pm 1.04)\times 10^{-4}$ | $0.00144\pm 0.00018$ | $3.33\pm 0.19$
|  |  | Data from professional telescopes + ExoClock – Case ($b$) |  |  |
Linear | $58591.084931\pm 0.000096$ | $1.35413379\pm 0.00000008$ |  |  |  |
Quadratic | $58591.085000\pm 0.000070$ | $1.35413330\pm 0.00000010$ | $(-3.37\pm 0.75)\times 10^{-10}$ |  |  |
Sinusoidal | $58591.085000\pm 0.000200$ | $1.35413420\pm 0.00000096$ |  | $(8.6\pm 7.0)\times 10^{-4}$ | $0.00137\pm 0.00066$ | $3.25\pm 0.23$

Table 7: Goodness of fit of the three ephemeris models.

Data set | Model | $N_{\Theta}$ | $\chi^{2}$ | $\chi_{\rm red}^{2}$ | BIC | AIC
---|---|---|---|---|---|---
Case ($a$) | Linear Model | 2 | 72.54 | 3.82 | 32.12 | 30.03
| Quadratic Model | 3 | 39.66 | 2.20 | 22.49 | 19.35
| Sinusoidal Model | 5 | 30.91 | 1.93 | 23.34 | 18.12
Case ($b$) | Linear Model | 2 | 85.83 | 3.30 | 38.03 | 35.36
| Quadratic Model | 3 | 44.48 | 1.78 | 22.95 | 18.96
| Sinusoidal Model | 5 | 34.96 | 1.52 | 22.87 | 16.22

Figure 7: Top panel: O-C residuals for HATS-2 b in the case ($a$),
superimposed with the best-fitting linear ephemeris model (black line), the
best-fitting quadratic ephemeris model (red line) and the best-fitting
cosinusoidal ephemeris model (cyan line). The purple points represent the
times derived from the Danish 1.54 m telescope observations, the yellow points
the times derived from the MPG/ESO 2.2m telescope data, and the blue points
the times derived from TESS data. Bottom panel: Same as the top panel but for
case ($b$), i.e. also considering the data from the ExoClock database, which
are represented as green points.

All the $\chi_{\rm red}^{2}$ values listed in Table 7 are larger than unity.
This is not a sign of a bad fit but is in line with our previous TTV analysis,
(see e.g. Mancini et al. 2022b and references within). As pointed out in
various works (e.g. Patra et al. 2017; Mancini et al. 2022b), the difference
between the BIC and AIC values of two models, $\eta=\Delta{\rm BIC}$ and
$\xi=\Delta{\rm AIC}$, are good statistical indicators. In particular,
$\eta>10$ and a $\xi>10$ indicate that the evidence favouring the lower BIC
model against the other is very strong.

First, we focus on the comparison between the linear and the quadratic
ephemeris models. Without the ExoClock data, the quadratic model is favoured
by a $\eta=9.6$ and a $\xi=10.7$, so we are at the limit of the statistical
significance. Adding the ExoClock data, the quadratic model is favoured by a
$\eta=15.1$ and a $\xi=16.4$. So, the tidal-decay scenario seems to be highly
preferred with respect to a simple linear-ephemeris scenario and this makes
HATS-2 a good candidate in the context of the search for secular timing
variations in hot Jupiters (Hagey et al. 2022). Focusing now on the second
scenario results, by using the values listed in Table 6 and the following
formula (Patra et al., 2017):

$\frac{1}{2}\frac{{\rm d}P}{{\rm
d}E}E^{2}=-\frac{27\pi}{4\,Q_{\star}^{\prime}}\frac{M_{\rm
p}}{M_{\star}}\left(\frac{R_{\star}}{a}\right)^{5}PE^{2}\,,$ (14)

we derived a $\frac{dP}{dE}=-(5.48\pm 2.48)\times 10^{-10}$ days per orbital
cycle, and therefore the period derivative is
$\dot{P}=\frac{1}{P}\frac{dP}{dE}=-15.7\,\pm\,3.5$ ms yr-1, consistent with an
orbital period that shrink to zero in a time within
$\frac{P}{dP/dt}=\frac{P^{2}}{dP/dE}=7.5\,\pm\,1.7$ Myr. This is reasonable
for a planet that orbits around its host star with a semi-major axis $\simeq
2$ times the Roche radius and so close to tidal disruption. Using Eq. (14), we
also derived a limit to the modified stellar tidal dissipation quality factor
of $Q_{\star}^{\prime}>(1.99\pm 0.26)\times 10^{4}$; this result is based on
the 95 per cent confidence lower limits on $\dot{P}$, while the uncertainties
come from propagating the errors in $M_{\rm p}/M_{\star}$ and $R_{\star}/a$.
This result is consistent with theoretical works such that by Ahuir et al.
(2021). In Fig. 8, we plot the obtained limit as a function of stellar age and
planetary equilibrium temperature, together with the values obtained for other
systems hosting hot Jupiters that can be considered good candidates to study
orbital decay. These are: KELT-16 (Mancini et al., 2022b), HATS-18 (Southworth
et al., 2022), WASP 18 and WASP 19 (Rosário et al., 2022), WASP-12 (Wong et
al., 2022), WASP-4 (Turner et al., 2022), WASP-103 (Barros et al., 2022), and
TrES-1, TrES-5 and HAT-P-19 (Hagey et al., 2022). The stellar ages were taken
from Bonomo et al. (2017). In Hagey et al. (2022) only the orbital period
derivates were presented, so we had to compute the $Q_{\star}^{\prime}$ limits
via Eq. (14); the planetary and stellar parameters have been taken from the
literature (i.e. Stassun et al. 2017; Maciejewski et al. 2016; Johnson et al.
2009; Hartman et al. 2011). The results show a correlation between the derived
tidal quality factor and the planetary equilibrium temperature, in agreement
with Eq. (14). In this context, the value derived for HATS-2 occupies the
right place in the diagram of the tidal quality factor versus stellar age and
equilibrium temperature. The relation between the stellar tidal quality factor
and the stellar age is not clear and further observations may give new
insights.

Investigating Table 7, it is possible to observe that also a sinusoidal model
can explain the data well. In fact, in both scenarios, the relative $\eta$ and
$\xi$ are always smaller than 5. The sinusoidal model can be connected to the
presence of apsidal precession induced by the tidal interactions between the
planet and the host star (Ragozzine & Wolf 2009; Gimenez & Garcia-Pelayo
1983):

${\rm O-C}\simeq-\frac{eP_{\rm a}}{\pi}\,\cos{\omega}=-\frac{eP_{\rm
a}}{\pi}\,\cos{\left(\omega_{0}+\frac{d\omega}{dE}E\right)}\,,$ (15)
$\frac{d\omega}{dE}\simeq 15\pi\,k_{2,p}\frac{M_{\star}}{M_{\rm
p}}\left(\frac{R_{\rm p}}{a}\right)^{5}\,,$ (16)

where $P_{\rm a}$ indicates the sidereal period, $\omega$ the argument of the
periastron, $k_{2,p}$ is the planet’s Love number and $d\omega/dE$ the
precession rate. Plugging the values listed in Table 6 into Eq. (16), we
obtained a value of $k_{2,p}$ greater than 1.5 that has no physical meaning.
The other physical explanation is the presence of a third body in the system
(Agol et al. 2005). However, modelling TTVs induced by a third body is a
highly degenerate problem and the quality of our data and the temporal
coverage are not good enough to face it. In any case, future and systematic
recordings of new mid-transit times will certainly give useful insights to
entangle the problem. Fig. 7 tells us that it should be possible to
discriminate between the quadratic and the sinusoidal model by exploiting new
transit measures in the next two years.

Figure 8: Stellar tidal quality factor as a function of stellar age for HATS-2
and an ensemble of host stars. These stars were selected considering systems
where tidal decay has been observed or is a possible explanation for the
observed TTVs (Hagey et al. 2022; Rosário et al. 2022; Wong et al. 2022;
Mancini et al. 2022b; Barros et al. 2022).

##  6 Variation in the planetary radius with wavelength

Due to their proximity to their parent star, hot Jupiters are strongly
irradiated and their spectra are expected to show several absorption features
at optical wavelengths. Some of the features that are usually detected in the
optical bands include sodium ($\sim 590$ nm), potassium ($\sim 770$ nm) and
water vapour ($\sim 950$ nm). In several cases, Rayleigh scattering at bluer
wavelengths has been found as well as the presence of strong absorbers, such
as gaseous titanium oxide (TiO) between 450 and 800 nm.

The transit depth is a parameter that depends on the wavelength and,
therefore, on the possible absorption of the chemical species present in the
exoplanetary atmosphere. Multi-colour photometry can be a useful tool for
investigating how $k$ changes with wavelength, probing in this way the
chemical composition at the terminator of planetary atmospheres during transit
events. _Transmission photometry_ has, in general, a very low spectral
resolution but is suitable for ground-based telescopes with smaller apertures
and exoplanets orbiting faint stars. This type of study can be, therefore,
useful to get information about the absorption behaviour of the exoplanetary
atmosphere and guide more precise investigation via _transmission
spectroscopy_ , being the latter a most effective technique.

Using the light curves of the transits of HATS-2 b, which we obtained through
observations at different passbands, we attempted to get a low-resolution
transmission spectrum of HATS-2 b. Following a general approach, we run
PRISM/GEMC for each of the ground-based light curves again to calculate the
ratio of the radii in each passband, being the other photometric and starspots
parameters fixed to the best-fitting values listed in Tables 9 and 2. This
yielded a set of $k$ values which are directly comparable and whose error bars
exclude common sources of uncertainty. In particular, for fixing the values of
the LD coefficients, we calculated the weighted means of the values, based on
their colour. As a last step, we averaged the ratio of the radii values taken
with the same filters. The weighted mean was performed using the scatter in
the light curves to set the size of the errorbar. In this way, we can be sure
that the uncertainties will not be underestimated. We excluded the TESS light
curves because the long-pass filter of TESS is just too wide ($>500$ nm) for
our purposes. The results are:

  * $k_{g^{\prime}}=0.13192\pm 0.00094$;

  * $k_{r^{\prime}}=0.13310\pm 0.00063$;

  * $k_{R}=0.13258\pm 0.00057$;

  * $k_{i^{\prime}}=0.13419\pm 0.00067$;

  * $k_{I}=0.13405\pm 0.00064$;

  * $k_{z^{\prime}}=0.13252\pm 0.00083$.

These values are shown in Fig. 9 and are compared with three 1D model
atmospheres, which were obtained by Fortney et al. (2010).

In particular, the red line represents an equilibrium-chemistry spectrum,
which was calculated for Jupiter’s gravity ($25$ m s-2) with a base radius of
$1.25\,R_{\rm Jup}$ at the 10 bar level and at 1500 K. This model is dominated
by H2/He Rayleigh scattering in the blue and pressure-broadened neutral atomic
lines of sodium and potassium at 770 nm, and the opacity of TiO and VO
molecules were included. The blue line represents a model similar to the
previous one but the opacity TiO and VO were excluded. Another spectrum was
computed from the latter by changing the intensity of Rayleigh scattering,
which physically corresponds to considering different compositions and
thicknesses of the external atmosphere layers. In particular, the dashed dark-
green line was obtained by increasing the Rayleigh scattering by a factor of
10.

A variation in the planetary radius was found between the $g^{\prime}$ and
$i^{\prime}$ bands. The variation is roughly 6.5 pressure scale heights888The
pressure scale height of a planetary atmosphere is defined as $H=k_{\rm
B}T_{\rm eq}/\mu_{\rm m}g_{\rm p}$, where $k_{\rm B}$ is the Boltzmann
constant and $\mu_{\rm m}$ is the mean molecular weight, for which the value
of 2.3 amu is usually adopted for giant planets (Lecavelier Des Etangs et al.,
2008). but its significance is below the $2\,\sigma$ level, which preserve us
to claim any absorption detection.

Figure 9: Top panel: Variation in the planetary radius, in terms of the
planet-to-star radius ratio, with wavelength. The points are from the ground-
based transit observations presented in this work (black from the MPG 2.2 m
telescope and grey from the Danish 1.54 m telescope). The vertical bars
represent the errors in the measurements and the horizontal bars show the FWHM
transmission of the passbands used. The observational points are compared with
three synthetic spectra from Fortney et al. (2010) (Red: equilibrium-
chemistry; Blue: no TiO, VO opacity; Green: enhanced Rayleigh scattering). The
same offset is applied to all four models to provide the best fit to the
measurements. The size of 10 atmospheric pressure scale heights ($10\times H$)
is shown on the right of the plot. Bottom panel: Transmission curves for the
Bessell $I$ and $R$ filters and the total efficiencies of the GROND filters.

##  7 Flux ratio of HATS-2 b from TESS phase curve

Figure 10: HATS-2b phase curve obtained by phase-binning the data from Sectors
10, 36 and 63 of NASA’s TESS mission plotted against theoretical curves
computed thanks to the batman Python package.

We investigated the possibility to measure the occultation depth, in other
words the planet-to-star flux ratio $F_{\rm p}/F_{\star}$, in the continuous
time series data of TESS (see Fig. 3). For this purpose, we phased the data
from every sector considering the best-fitting linear ephemeris found in the
previous sections, Eq. (8), and binned them using a weighted sigma-clipping
mean, which means that the data are iteratively kept or rejected in terms of
standard deviation. The resulting phase curve is presented in Fig. 10,
together with different theoretical phase curves generated thanks to the
batman Python package (Kreidberg, 2015). To simulate the occultation curves,
batman needs two free parameters, the occultation depth and the occultation
mid-time, and a series of fixed parameters as the mid-transit time, the ratio
of the radii, the scaled stellar radius, the period, the eccentricity, and the
inclination of the orbit. In turn, the occultation depth at visible
wavelengths (such as the one at which the TESS bandpass is centred, $\simeq$
786 nm) can be connected to the planetary day-side temperature $T_{\rm p}$ and
geometric albedo $A_{\rm g}$ (Kovacs et al., 2022):

$\delta_{\rm occ}\simeq\delta_{\rm refl}+\delta_{\rm th}\simeq A_{\rm
g}\left(\frac{R_{\rm p}}{a}\right)^{2}+k^{2}\frac{B_{\nu}(T_{\rm
p})}{B_{\nu}(T_{\rm eff})},$ (17)

where $B_{\nu}$ is the Planck function. If we use Eq. (17) together with the
values listed in Table 3 we can generate different theoretical light curves
considering different scenarios and corresponding values for $A_{\rm g}$ and
$T_{\rm p}$. In particular, we considered three scenarios: ($i$) $A_{\rm
g}=0.16$ and $T_{\rm p}=1600$ K, ($ii$) $A_{\rm g}=0.5$ and $T_{\rm p}=1600$
K, and ($iii$) $A_{\rm g}=0.05$ and $T_{\rm p}=2500$ K. The first scenario
should be that closer to reality if we consider typical values of $A_{\rm g}$
found in the literature for hot Jupiters and a conservative atmospherical
circulation. The second and third scenarios, instead, represent two opposite
extremes. The second represents the situation where most of the light that we
receive from the planet in the TESS bandpass is just reflected light; the
third, instead, represents the case where the day-side temperature greatly
exceeds the equilibrium temperature, for example, due to the presence of
absorbers or tidal locking. In any case, none of the generated theoretical
curves seems to be close to the phased TESS light curve for which a non-
detection of the occultation is the most compatible scenario. Moreover, the
uncertainty on the mean flux (unbinned) is close to 1 ppt (1000 ppm), which
makes any tentative of measuring the occultation depth extremely difficult. In
the end, as expected due to the faintness of the parent star, we did not
report any detection of a secondary eclipse in the HATS-2 TESS photometry.
This work and TESS photometry place an upper limit on the occultation depth of
HATS-2b, $\delta_{\rm occ}< 400$ ppm.

##  8 Summary and conclusions

In this work, we studied the physical and orbital properties of the hot
Jupiter HATS-2 b, a giant transiting exoplanet with $P_{\rm orb}\approx 1.35$
days, and therefore, an interesting target to investigate in relation to a
possible orbital-decay process. We reported the photometric monitoring of 13
new transit events of HATS-2 b, which were observed with two medium-class
telescopes through six different optical passbands (see Figs. 1 and 2). The
transits were observed using the defocusing technique, achieving a photometric
precision of 0.6 mmag per observation in the best case. Two transits were
simultaneously observed in four optical bands thanks to the GROND multi-band
camera. In total, we collected 19 new light curves. We also considered the
data from two other transits observed with GROND (Mohler-Fischer et al., 2013)
and four published and two new light curves (see Fig. 21) that were recorded
by amateur astronomers within the ExoClock project. We also considered the
photometric data collected by TESS and analysed 48 complete transits recorded
by this space telescope (see Fig. 3). Most of the light curves presented
features that break the classical symmetry of transit events. Such features
are attributable to starspot complexes that are revealed by the planet HATS-2
b when it transits in front of its parent star, and thus making a temperature
scan of a chord of the stellar photosphere. The main results that we obtained
are as follows.

  * •

Thanks to the TESS light curves and the new ground-based light curves, we
reviewed the main physical and orbital parameters of the HATS-2 planetary
system. Our results are shown in Table 3 and are in very good agreement with
those obtained by Mohler-Fischer et al. (2013).

  * •

Almost all of the light curves were analysed with PRISM+GEMC, a code able to
simultaneously fit a transit light curve containing one or more starspot
features. This modelling of the light curves allowed us to determine the
position of the starspots on the stellar disc as well as their size and
contrast (see Table 2). Having monitored several transits in different
filters, we verified that the variations in the starspot contrast with
wavelength were in agreement with the expectations from theoretical models
(see Fig. 4). Based on theoretical assumptions, we also estimated the
temperature of each starspot (see Table 2).

  * •

We detected starspot features on the light curves of consecutive transits of
HATS-2 b both from the ground and from space. In particular, we identified ten
and one consecutive starspot-crossing events from the space and ground
observations, respectively. All these transits are separated by less than five
days, and the latitudes and angular size differences of the starspots fall
within $1\sigma$. They also have similar amplitudes, contrast and duration. We
concluded that for each of these 11 consecutive transits, the same starspot
was occulted by the planet after having rotated a bit on the surface of the
parent star. Based on this sensible hypothesis, we easily determined the
stellar rotational period, $P_{\rm rot}=22.46\pm 5.20$ days, and the sly-
projected spin-orbit obliquity, $\lambda=2^{\circ}.72\pm 17^{\circ}.84$. We
also placed a weak constraint on the true orbital obliquity
$\psi=38^{\circ}.49\pm 27^{\circ}.13$.

  * •

We estimated the mid-transit time for each of the transit events of HATS-2 b
that we presented plus others from the ExoClock archive, obtaining a list of
27 epochs that covers a time baseline extending over more than ten years (see
Table 5). We used these timings to update the ephemeris of the orbital period
and to search for possible variations in the mid-transit times. We found that
models that predict both orbital decay and apsidal precession fit the data
better than the linear-ephemeris scenario. The observations of many other
transits of HATS-2 b and new radial-velocity measurements are needed to firmly
confirm our indication that its orbit is decaying or is affected by a third
body in the system.

  * •

Thanks to the multi-band photometric observations of HATS-2 b transits, we
were able to obtain a low-resolution optical transmission spectrum of the
planet. We found a variation in the planetary radius between the $g^{\prime}$
and $i^{\prime}$ bands. This variation is roughly 6.5 pressure scale heights
but its significance is below the $2\,\sigma$ level. Therefore, we cannot
claim the presence of strong optical absorbers (either Na and K, or TiO) in
the atmosphere of HATS-2 b. More accurate data are needed for this purpose.

  * •

We analysed the continuous HATS-2 time series data of TESS to measure the
occultation depth, that is the planet-to-star flux ratio. We obtained a phase
curve of HATS-2 b by phase-binning the data from three sectors, and then we
compared it against different theoretical models testing various values for
the planetary day-side temperature and geometric albedo. Our analysis did not
lead to any conclusive results since the occultation is not seen in the TESS
data and none of the theoretical curves is able to explain the phased TESS
light curve.

###### Acknowledgements.

This work is based on data collected with the MPG 2.2 m telescope and within
the MiNDSTEp program with the Danish 1.54 m telescope. Both telescopes are
located at the ESO Observatory in La Silla (Chile). The GROND camera was built
by the high-energy group of MPE in collaboration with the LSW Tautenburg and
ESO, and is operated as a PI instrument at the MPG 2.2 m telescope. This work
includes data collected with the TESS mission, obtained from the MAST data
archive at the Space Telescope Science Institute (STScI). Funding for the TESS
mission is provided by the NASA Explorer Program. STScI is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5–26555. We acknowledge the use of public TESS data from
pipelines at the TESS Science Office and at the TESS Science Processing
Operations Center. Resources supporting this work were provided by the NASA
High-End Computing (HEC) Program through the NASA Advanced Supercomputing
(NAS) Division at Ames Research Center for the production of the SPOC data
products. We acknowledge funding from the Novo Nordisk Foundation
Interdisciplinary Synergy Program grant no. NNF19OC0057374 and from the
European Union H2020-MSCA-ITN-2019 under grant No. 860470 (CHAMELEON). Support
for this project is provided by ANID’s Millennium Science Initiative through
grant ICN12_009, awarded to the Millennium Institute of Astrophysics (MAS),
and by ANID’s Basal project FB210003. L. M. acknowledges support from the
MIUR-PRIN project No. 2022J4H55R. T.C.H. acknowledges funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No. 871149. The Horizon 2020 programme is supported by the
Europlanet 2024 RI provides free access to the world’s largest collection of
planetary simulation and analysis facilities, data services and tools, a
ground-based observational network and programme of community support
activities. The following internet-based resources were used in research for
this paper: the ESO Digitized Sky Survey; the NASA Astrophysics Data System;
the SIMBAD database operated at CDS, Strasbourg, France; and the ar$\chi$iv
scientific paper preprint service operated by the Cornell University.

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##  Appendix A PRISM/GEMC and JKTEBOP fits of TESS data

Figure 11: Left panel: Light curves of HATS-2 obtained from the Sector 10 of
NASA’s TESS mission data and used in the analysis of the physical and starspot
parameters of the system. They are plotted against the orbital phase and are
compared to PRISM/GEMC best-fitting models. The labels indicate the
observational ID (see Fig. 3). Right panel: Residuals of the fits represented
with the same notation used in the left panel. Figure 12:  Left panel: Light
curves of HATS-2 obtained from the Sector 36 of NASA’s TESS mission data, and
used in the analysis of the physical and starspot parameters of the system. In
particular, the image refers only to the transits labelled $\\#6$, $\\#7$, and
$\\#8$ (see Fig. 3 and Fig. 5). They are plotted against the orbital phase and
are compared to PRISM/GEMC best-fitting models. Right panel: Residuals of the
fits represented with the same notation used in the left panel.  Figure 13:
Left panel: Remaining transit light curves containing spot-crossing events
from Sector 36 of NASA’s TESS mission data that were not represented in Fig.
12. They are plotted against the orbital phase and are compared to new
PRISM/GEMC best-fitting models. The labels indicate the observational ID.
Right panel: Residuals of the fits represented with the same notation used in
the left panel. Figure 14:  Left panel: The light curves of HATS-2 obtained
from Sector 63 of NASA’s TESS mission data, and used in the analysis of the
physical and starspot parameters of the system. In blue are represented the
unbinned data, while in red the 10 min cadence binned data. In particular, the
image refers only to the transits labelled $\\#13$, $\\#14$, and $\\#15$ (see
Fig. 3). They are plotted against the orbital phase and are compared to
PRISM/GEMC best-fitting models. Right panel: Residuals of the fits represented
with the same notation used in the left panel.  Figure 15: Left panel:
Remaining transit light curves containing spot-crossing events from Sector 63
of NASA’s TESS mission data that were not represented in Fig. 14. In blue are
represented the unbinned data, while in red the 10 min cadence binned data.
They are plotted against the orbital phase and are compared to new PRISM/GEMC
best-fitting models. The labels indicate the observational ID. Right panel:
Residuals of the fits represented with the same notation used in the left
panel. Figure 16: Left panel: Light curves of eight transits of HATS-2b
observed with NASA’s TESS space telescope in Sector 10 of its primary mission.
The labels indicate the observation ID (see Fig. 3). They are plotted against
the orbital phase and are compared to the best-fitting JKTEBOP models. Right
panel: Residuals of each fit. Figure 17: Left panel: Light curves of eight
transits of HATS-2b observed with NASA’s TESS space telescope in Sector 36 of
its primary mission. The labels indicate the observation ID (see Fig. 3). They
are plotted against the orbital phase and are compared to the best-fitting
JKTEBOP models. Right panel: Residuals of each fit. Figure 18: Left panel:
Light curves of 12 transits of HATS-2b observed with NASA’s TESS space
telescope in Sector 63 of its primary mission. The labels indicate the
observation ID (see Fig. 3). They are plotted against the orbital phase and
are compared to the best-fitting PRISM+GEMC models. Right panel: Residuals of
each fit.

##  Appendix B Mathematical derivation of Section 4.3 equations

Using spherical trigonometry, the relation that connects the longitude $\phi$
and latitude $\theta$ to Cartesian coordinates for a point on a sphere is:

$\begin{cases}x=R_{\star}\cos\theta\cos\phi\\\
y=R_{\star}\cos\theta\sin\phi\\\ z=R_{\star}\sin\theta\end{cases}\,.$ (18)

Referring to Fig. 19, the sky-projected obliquity is given by:

$\lambda=\arctan{\frac{\rm{BC}}{\rm{AC}}}=\arctan{\frac{z_{2}-z_{1}}{y_{2}-y_{1}}}.$
(19)

To obtain the rotational period a proportion must be considered: $P_{\rm
rot}\,:\,2\pi R_{\star}=\Delta t\,:\,\rm{AB}$. Using Pythagorean theorem is
possible to derive the Equation in Section 4.3.

Figure 19: Geometry of the problem. The rotation axis differs from the normal
to the sky-projected plane, so two consecutive spots are observed at different
latitudes and longitudes.

##  Appendix C GLS fits of TESS photometry

Figure 20: TESS total light curve (Sectors 10, 36, and 63) in phase with its
combined GLS peak of 21.4 days (FAP¡0.1$\%$ evaluated with the bootstrap
method), from the simple aperture photometry (SAP) which was not corrected
from long-term trends. The fit of the phased light curve is represented by a
blue curve, while the transits of HATS-2 b have been removed.

##  Appendix D PRISM/GEMC and JKTEBOP fits of ExoClock data

Figure 21: Left: Light curves of the transits of HATS-2b observed by the
astronomers involved in the ExoClock collaboration. They are plotted against
the orbital phase and are compared to the best-fitting JKTEBOP and PRISM+GEMC
models. The labels indicate in order: observation date, observatory, and
observer. DSC stands for Deep Sky Chile, while SSO stands for Sliding Spring
Observatory. Right: Residuals of each fit. Table 8:  Parameters of the
PRISM/GEMC and JKTEBOP best fits of the HATS-2 light curves from the ExoClock
database.

Telescope | Date or ID | Code | $r_{\star}+r_{\rm p}$ | $r_{\rm p}/r_{\star}$ | $i\,(^{\circ})$ |  $\phi$ (∘) |  $\theta$ (∘) |  $r_{s}$ (∘) | $\rho$
---|---|---|---|---|---|---|---|---|---
DSC | 2021/01/10 | PRISM+GEMC | 0.204 $\pm$ 0.010 | 0.1303 $\pm$ 0.0040 | 88.4 $\pm$ 1.7 | 23.7 $\pm$ 15.0 | 13.0 $\pm$ 20.0 | 5.9 $\pm$ 10.0 | 0.29 $\pm$ 0.15
SSO | 2021/02/22 | PRISM+GEMS | 0.216 $\pm$ 0.019 | 0.1268 $\pm$ 0.0059 | 86.4 $\pm$ 2.3 | -60.5 $\pm$ 45.0 | 35.8 $\pm$ 50.0 | 31.2 $\pm$ 20.0 | 0.75 $\pm$ 0.21
DSC | 2021/04/11 | PRISM+GEMC | 0.202 $\pm$ 0.013 | 0.1281 $\pm$ 0.0040 | 88.2 $\pm$ 1.9 | -16.8 $\pm$ 2.0 | 11.2 $\pm$ 10.0 | 7.75 $\pm$ 3.57 | 0.46 $\pm$ 0.22
DSC | 2021/04/15 | PRISM+GEMC | 0.206 $\pm$ 0.012 | 0.1331 $\pm$ 0.0059 | 87.0 $\pm$ 1.8 | 45.3 $\pm$ 30.0 | 15.8 $\pm$ 30.0 | 8.6 $\pm$ 15.0 | 0.46 $\pm$ 0.29
DSC | 2022/01/08 | JKTEBOP | 0.212 $\pm$ 0.016 | 0.1200 $\pm$ 0.0063 | 85.8 $\pm$ 3.4 | – | – | – | –
DSC | 2023/01/14 | JKTEBOP | 0.257 $\pm$ 0.028 | 0.1330 $\pm$ 0.0100 | 81.5 $\pm$ 2.6 | – | – | – | –

##  Appendix E Extra Table

Table 9: Parameters of the PRISM/GEMC and JKTEBOP best fits for the HATS-2
light curves used in this work. The final row gives, in bold, the weighted
means of the results for the individual data sets.

Telescope | Date or ID | Filter | Code | $r_{\star}+r_{\rm p}$ | $r_{\rm p}/r_{\star}$ | $i\,(^{\circ})$ | $u_{1}$ | $u_{2}$
---|---|---|---|---|---|---|---|---
MPG 2.2m | 2012/02/28 | $g^{\prime}$ | PRISM/GEMC | 0.2053 $\pm$ 0.0049 | 0.1370 $\pm$ 0.0019 | 87.89 $\pm$ 0.93 | 0.82 $\pm$ 0.09 | 0.10 $\pm$ 0.09
MPG 2.2m | 2012/02/28 | $r^{\prime}$ | PRISM/GEMC | 0.2059 $\pm$ 0.0060 | 0.1359 $\pm$ 0.0023 | 88.07 $\pm$ 1.11 | 0.69 $\pm$ 0.13 | 0.10 $\pm$ 0.15
MPG 2.2m | 2012/02/28 | $i^{\prime}$ | PRISM/GEMC | 0.1990 $\pm$ 0.0044 | 0.1342 $\pm$ 0.0017 | 88.48 $\pm$ 1.10 | 0.39 $\pm$ 0.13 | 0.28 $\pm$ 0.23
MPG 2.2m | 2012/02/28 | $z^{\prime}$ | PRISM/GEMC | 0.2026 $\pm$ 0.0043 | 0.1370 $\pm$ 0.0023 | 88.46 $\pm$ 1.07 | 0.33 $\pm$ 0.14 | 0.38 $\pm$ 0.25
MPG 2.2m | 2012/06/01 | $g^{\prime}$ | PRISM/GEMC | 0.1895 $\pm$ 0.0053 | 0.1340 $\pm$ 0.0025 | 86.19 $\pm$ 0.82 | 0.41 $\pm$ 0.19 | 0.93 $\pm$ 0.23
MPG 2.2m | 2012/06/01 | $r^{\prime}$ | PRISM/GEMC | 0.1927 $\pm$ 0.0038 | 0.1329 $\pm$ 0.0016 | 87.41 $\pm$ 0.73 | 0.47 $\pm$ 0.10 | 0.34 $\pm$ 0.18
MPG 2.2m | 2012/06/01 | $i^{\prime}$ | PRISM/GEMC | 0.1872 $\pm$ 0.0034 | 0.1321 $\pm$ 0.0014 | 88.37 $\pm$ 0.82 | 0.51 $\pm$ 0.09 | 0.07 $\pm$ 0.14
MPG 2.2m | 2012/06/01 | $z^{\prime}$ | PRISM/GEMC | 0.1891 $\pm$ 0.0042 | 0.1308 $\pm$ 0.0016 | 87.98 $\pm$ 0.89 | 0.32 $\pm$ 0.11 | 0.09 $\pm$ 0.17
MPG 2.2m | 2015/01/25 | $g^{\prime}$ | PRISM/GEMC | 0.2009 $\pm$ 0.0049 | 0.1294 $\pm$ 0.0045 | 87.53 $\pm$ 0.97 | 0.64 $\pm$ 0.10 | 0.06 $\pm$ 0.13
MPG 2.2m | 2015/01/25 | $r^{\prime}$ | PRISM/GEMC | 0.2080 $\pm$ 0.0052 | 0.1300 $\pm$ 0.0026 | 87.42 $\pm$ 0.96 | 0.59 $\pm$ 0.14 | 0.26 $\pm$ 0.20
MPG 2.2m | 2015/01/25 | $i^{\prime}$ | PRISM/GEMC | 0.2204 $\pm$ 0.0083 | 0.1327 $\pm$ 0.0034 | 85.35 $\pm$ 1.17 | 0.33 $\pm$ 0.14 | 0.41 $\pm$ 0.25
MPG 2.2m | 2015/01/25 | $z^{\prime}$ | PRISM/GEMC | 0.2105 $\pm$ 0.0076 | 0.1295 $\pm$ 0.0024 | 87.18 $\pm$ 1.35 | 0.31 $\pm$ 0.13 | 0.57 $\pm$ 0.23
MPG 2.2m | 2015/01/29 | Combined | PRISM/GEMC | 0.1973 $\pm$ 0.0117 | 0.1350 $\pm$ 0.0015 | 87.74 $\pm$ 0.99 | 0.50 $\pm$ 0.08 | 0.07 $\pm$ 0.13
Danish 1.54m | 2016/04/16 | $R$ | PRISM/GEMC | 0.2081 $\pm$ 0.0068 | 0.1299 $\pm$ 0.0022 | 86.80 $\pm$ 1.10 | 0.44 $\pm$ 0.14 | 0.45 $\pm$ 0.21
Danish 1.54m | 2017/05/30 | $R$ | PRISM/GEMC | 0.2033 $\pm$ 0.0036 | 0.1359 $\pm$ 0.0022 | 87.92 $\pm$ 0.96 | 0.64 $\pm$ 0.09 | 0.10 $\pm$ 0.15
Danish 1.54m | 2017/07/11 | $R$ | PRISM/GEMC | 0.1958 $\pm$ 0.0031 | 0.1268 $\pm$ 0.0047 | 89.18 $\pm$ 0.87 | 0.57 $\pm$ 0.08 | 0.06 $\pm$ 0.14
Danish 1.54m | 2018/05/17 | $I$ | PRISM/GEMC | 0.2004 $\pm$ 0.0054 | 0.1312 $\pm$ 0.0021 | 88.57 $\pm$ 1.22 | 0.53 $\pm$ 0.17 | 0.10 $\pm$ 0.24
Danish 1.54m | 2018/06/05 | $R$ | PRISM/GEMC | 0.2000 $\pm$ 0.0059 | 0.1297 $\pm$ 0.0021 | 88.14 $\pm$ 1.22 | 0.47 $\pm$ 0.14 | 0.24 $\pm$ 0.22
Danish 1.54m | 2018/05/24 | $R$ | PRISM/GEMC | 0.2024 $\pm$ 0.0052 | 0.1324 $\pm$ 0.0020 | 88.63 $\pm$ 1.14 | 0.75 $\pm$ 0.09 | 0.05 $\pm$ 0.11
Danish 1.54m | 2019/05/19 | $I$ | PRISM/GEMC | 0.2066 $\pm$ 0.0100 | 0.1257 $\pm$ 0.0038 | 87.00 $\pm$ 1.72 | 0.22 $\pm$ 0.18 | 0.40 $\pm$ 0.24
Danish 1.54m | 2019/05/23 | $I$ | PRISM/GEMC | 0.1968 $\pm$ 0.0031 | 0.1321 $\pm$ 0.0024 | 88.66 $\pm$ 0.81 | 0.45 $\pm$ 0.08 | 0.07 $\pm$ 0.11
Danish 1.54m | 2019/06/07 | $I$ | PRISM/GEMC | 0.1996 $\pm$ 0.0036 | 0.1311 $\pm$ 0.0021 | 88.91 $\pm$ 0.96 | 0.44 $\pm$ 0.11 | 0.17 $\pm$ 0.20
Danish 1.54m | 2021/05/26 | $I$ | PRISM/GEMC | 0.1994 $\pm$ 0.0047 | 0.1336 $\pm$ 0.0026 | 88.36 $\pm$ 1.14 | 0.31 $\pm$ 0.15 | 0.32 $\pm$ 0.25
Danish 1.54m | 2022/05/13 | $I$ | PRISM/GEMC | 0.1889 $\pm$ 0.0208 | 0.1301 $\pm$ 0.0028 | 88.25 $\pm$ 1.37 | 0.16 $\pm$ 0.13 | 0.56 $\pm$ 0.24
TESS s10 |  $\\#$1 |  | JKTEBOP | 0.2207 $\pm$ 0.0210 | 0.1132 $\pm$ 0.0130 | 89.09 $\pm$ 3.34 | 0.78 $\pm$ 0.52 | 0.38 $\pm$ 0.95
TESS s10 |  $\\#$2 |  | PRISM/GEMC | 0.2097 $\pm$ 0.0289 | 0.1322 $\pm$ 0.0066 | 87.51 $\pm$ 2.58 | 0.21 $\pm$ 0.22 | 0.37 $\pm$ 0.23
TESS s10 |  $\\#$4 |  | PRISM/GEMC | 0.2024 $\pm$ 0.0259 | 0.1251 $\pm$ 0.0070 | 87.21 $\pm$ 2.84 | 0.38 $\pm$ 0.22 | 0.27 $\pm$ 0.23
TESS s10 |  $\\#$5 |  | JKTEBOP | 0.2378 $\pm$ 0.0492 | 0.1320 $\pm$ 0.0156 | 83.07 $\pm$ 5.91 | 0.61 $\pm$ 0.73 | fixed
TESS s10 |  $\\#$6 |  | JKTEBOP | 0.2039 $\pm$ 0.0302 | 0.1126 $\pm$ 0.0135 | 89.49 $\pm$ 1.67 | 0.89 $\pm$ 0.64 | 0.13 $\pm$ 1.25
TESS s10 |  $\\#$7 |  | JKTEBOP | 0.2034 $\pm$ 0.0204 | 0.1216 $\pm$ 0.0126 | 88.89 $\pm$ 3.27 | 0.39 $\pm$ 0.85 | 0.44 $\pm$ 1.50
TESS s10 |  $\\#$8 |  | JKTEBOP | 0.2159 $\pm$ 0.0315 | 0.1186 $\pm$ 0.0051 | 85.38 $\pm$ 4.40 | fixed | fixed
TESS s10 |  $\\#$9 |  | JKTEBOP | 0.1894 $\pm$ 0.0163 | 0.1273 $\pm$ 0.0048 | 89.92 $\pm$ 3.11 | fixed | fixed
TESS s10 |  $\\#$10 |  | JKTEBOP | 0.2494 $\pm$ 0.0394 | 0.1361 $\pm$ 0.0078 | 82.08 $\pm$ 3.80 | 0.04 $\pm$ 0.45 | fixed
TESS s10 |  $\\#$11 |  | JKTEBOP | 0.2418 $\pm$ 0.0494 | 0.1398 $\pm$ 0.0117 | 82.03 $\pm$ 5.73 | 0.24 $\pm$ 0.53 | fixed
TESS s10 |  $\\#$12 |  | JKTEBOP | 0.1831 $\pm$ 0.0148 | 0.1295 $\pm$ 0.0063 | 89.95 $\pm$ 2.54 | 0.29 $\pm$ 0.30 | fixed
TESS s10 |  $\\#$13 |  | JKTEBOP | 0.1912 $\pm$ 0.0201 | 0.1260 $\pm$ 0.0076 | 89.59 $\pm$ 3.26 | 0.44 $\pm$ 0.40 | fixed
TESS s10 |  $\\#$14 |  | PRISM/GEMC | 0.2044 $\pm$ 0.0131 | 0.1307 $\pm$ 0.0052 | 86.78 $\pm$ 1.80 | 0.49 $\pm$ 0.22 | 0.25 $\pm$ 0.20
TESS s10 |  $\\#$15 |  | PRISM/GEMC | 0.2100 $\pm$ 0.0247 | 0.1290 $\pm$ 0.0066 | 87.57 $\pm$ 2.78 | 0.71 $\pm$ 0.25 | 0.13 $\pm$ 0.23
TESS s36 |  $\\#$1 |  | JKTEBOP | 0.1939 $\pm$ 0.0185 | 0.1304 $\pm$ 0.0063 | 88.11 $\pm$ 2.91 | 0.19 $\pm$ 0.40 | fixed
TESS s36 |  $\\#$2 |  | JKTEBOP | 0.2324 $\pm$ 0.0431 | 0.1322 $\pm$ 0.0138 | 83.55 $\pm$ 5.88 | 0.56 $\pm$ 0.87 | fixed
TESS s36 |  $\\#$3 |  | JKTEBOP | 0.2604 $\pm$ 0.0447 | 0.1373 $\pm$ 0.0071 | 81.27 $\pm$ 3.85 | fixed | fixed
TESS s36 |  $\\#$4 |  | JKTEBOP | 0.1917 $\pm$ 0.0175 | 0.1357 $\pm$ 0.0123 | 89.43 $\pm$ 2.50 | 0.37 $\pm$ 0.90 | -0.11 $\pm$ 2.45
TESS s36 |  $\\#$5 |  | JKTEBOP | 0.1992 $\pm$ 0.0218 | 0.1240 $\pm$ 0.0111 | 89.42 $\pm$ 2.68 | 0.45 $\pm$ 0.49 | fixed
TESS s36 |  $\\#$6 |  | PRISM/GEMC | 0.2057 $\pm$ 0.0170 | 0.1289 $\pm$ 0.0067 | 87.83 $\pm$ 2.26 | 0.20 $\pm$ 0.21 | 0.28 $\pm$ 0.24
TESS s36 |  $\\#$7 |  | PRISM/GEMC | 0.2355 $\pm$ 0.0293 | 0.1345 $\pm$ 0.0083 | 84.69 $\pm$ 2.97 | 0.36 $\pm$ 0.23 | 0.33 $\pm$ 0.24
TESS s36 |  $\\#$8 |  | PRISM/GEMC | 0.2148 $\pm$ 0.0181 | 0.1318 $\pm$ 0.0062 | 86.40 $\pm$ 2.32 | 0.22 $\pm$ 0.20 | 0.36 $\pm$ 0.23
TESS s36 |  $\\#$9 |  | PRISM/GEMC | 0.1948 $\pm$ 0.0139 | 0.1291 $\pm$ 0.0063 | 88.43 $\pm$ 1.92 | 0.24 $\pm$ 0.21 | 0.25 $\pm$ 0.23
TESS s36 |  $\\#$10 |  | PRISM/GEMC | 0.2089 $\pm$ 0.0268 | 0.1263 $\pm$ 0.0083 | 86.99 $\pm$ 2.62 | 0.25 $\pm$ 0.23 | 0.59 $\pm$ 0.24
TESS s36 |  $\\#$11 |  | PRISM/GEMC | 0.2011 $\pm$ 0.0198 | 0.1283 $\pm$ 0.0059 | 87.38 $\pm$ 2.44 | 0.32 $\pm$ 0.21 | 0.37 $\pm$ 0.23
TESS s36 |  $\\#$12 |  | JKTEBOP | 0.1958 $\pm$ 0.0181 | 0.1192 $\pm$ 0.0100 | 89.94 $\pm$ 2.50 | 0.56 $\pm$ 0.43 | fixed
TESS s36 |  $\\#$13 |  | JKTEBOP | 0.1938 $\pm$ 0.0149 | 0.1350 $\pm$ 0.0030 | 89.17 $\pm$ 2.86 | -0.09 $\pm$ 0.27 | fixed
TESS s36 |  $\\#$14 |  | JKTEBOP | 0.1942 $\pm$ 0.0230 | 0.1243 $\pm$ 0.0133 | 89.82 $\pm$ 2.93 | 0.41 $\pm$ 0.83 | 0.43 $\pm$ 1.80
TESS s36 |  $\\#$15 |  | JKTEBOP | 0.1968 $\pm$ 0.0170 | 0.1249 $\pm$ 0.0082 | 89.87 $\pm$ 2.38 | 0.46 $\pm$ 0.37 | fixed
TESS s63 |  $\\#$1 |  | PRISM/GEMC | 0.2017 $\pm$ 0.0172 | 0.1301 $\pm$ 0.0044 | 88.47 $\pm$ 2.24 | 0.32 $\pm$ 0.20 | 0.21 $\pm$ 0.23
TESS s63 |  $\\#$2 |  | PRISM/GEMC | 0.2082 $\pm$ 0.0234 | 0.1320 $\pm$ 0.0047 | 86.21 $\pm$ 2.64 | 0.19 $\pm$ 0.20 | 0.19 $\pm$ 0.23
TESS s63 |  $\\#$3 |  | PRISM/GEMC | 0.1967 $\pm$ 0.0167 | 0.1376 $\pm$ 0.0067 | 87.65 $\pm$ 2.14 | 0.16 $\pm$ 0.14 | 0.27 $\pm$ 0.23
TESS s63 |  $\\#$4 |  | PRISM/GEMC | 0.2214 $\pm$ 0.0512 | 0.1190 $\pm$ 0.0143 | 85.88 $\pm$ 4.63 | 0.76 $\pm$ 0.26 | 0.11 $\pm$ 0.22
TESS s63 |  $\\#$5 |  | PRISM/GEMC | 0.2220 $\pm$ 0.0459 | 0.1195 $\pm$ 0.0139 | 85.74 $\pm$ 4.30 | 0.74 $\pm$ 0.25 | 0.13 $\pm$ 0.23
TESS s63 |  $\\#$6 |  | PRISM/GEMC | 0.2230 $\pm$ 0.0264 | 0.1297 $\pm$ 0.0053 | 84.53 $\pm$ 2.95 | 0.23 $\pm$ 0.21 | f 0.34 $\pm$ 0.24
TESS s63 |  $\\#$7 |  | PRISM/GEMC | 0.1913 $\pm$ 0.0096 | 0.1381 $\pm$ 0.0042 | 88.80 $\pm$ 1.60 | 0.14 $\pm$ 0.16 | 0.14 $\pm$ 0.22
TESS s63 |  $\\#$8 |  | PRISM/GEMC | 0.2538 $\pm$ 0.0302 | 0.1393 $\pm$ 0.0061 | 81.29 $\pm$ 3.21 | 0.10 $\pm$ 0.20 | 0.18 $\pm$ 0.24
TESS s63 |  $\\#$9 |  | PRISM/GEMC | 0.2035 $\pm$ 0.0157 | 0.1319 $\pm$ 0.0064 | 88.12 $\pm$ 2.17 | 0.22 $\pm$ 0.20 | 0.38 $\pm$ 0.24
TESS s63 |  $\\#$10 |  | PRISM/GEMC | 0.2302 $\pm$ 0.0266 | 0.1338 $\pm$ 0.0051 | 84.21 $\pm$ 2.91 | 0.09 $\pm$ 0.18 | 0.20 $\pm$ 0.24
TESS s63 |  $\\#$11 |  | PRISM/GEMC | 0.2022 $\pm$ 0.0271 | 0.1243 $\pm$ 0.0066 | 87.32 $\pm$ 2.91 | 0.41 $\pm$ 0.21 | 0.31 $\pm$ 0.24
TESS s63 |  $\\#$12 |  | PRISM/GEMC | 0.2016 $\pm$ 0.0244 | 0.1297 $\pm$ 0.0050 | 88.26 $\pm$ 2.75 | 0.31 $\pm$ 0.21 | 0.31 $\pm$ 0.24
TESS s63 |  $\\#$13 |  | PRISM/GEMC | 0.2065 $\pm$ 0.0222 | 0.1351 $\pm$ 0.0061 | 86.87 $\pm$ 2.45 | 0.25 $\pm$ 0.18 | 0.25 $\pm$ 0.24
TESS s63 |  $\\#$14 |  | PRISM/GEMC | 0.2128 $\pm$ 0.0220 | 0.1330 $\pm$ 0.0057 | 86.00 $\pm$ 2.54 | 0.21 $\pm$ 0.20 | 0.23 $\pm$ 0.23
TESS s63 |  $\\#$15 |  | PRISM/GEMC | 0.2076 $\pm$ 0.0282 | 0.1283 $\pm$ 0.0050 | 88.37 $\pm$ 2.98 | 0.27 $\pm$ 0.21 | 0.17 $\pm$ 0.23
TESS s63 |  $\\#$16 |  | PRISM/GEMC | 0.1937 $\pm$ 0.0270 | 0.1314 $\pm$ 0.0057 | 87.59 $\pm$ 2.82 | 0.49 $\pm$ 0.23 | 0.21 $\pm$ 0.24
TESS s63 |  $\\#$17 |  | PRISM/GEMC | 0.1986 $\pm$ 0.0293 | 0.1302 $\pm$ 0.0059 | 87.60 $\pm$ 3.09 | 0.39 $\pm$ 0.22 | 0.35 $\pm$ 0.23
TESS s63 |  $\\#$18 |  | PRISM/GEMC | 0.2054 $\pm$ 0.0380 | 0.1311 $\pm$ 0.0081 | 88.55 $\pm$ 3.74 | 0.59 $\pm$ 0.24 | 0.16 $\pm$ 0.23
TESS s10 | combined $\\#1-\\#8$ |  | JKTEBOP | 0.2020 $\pm$ 0.0121 | 0.1272 $\pm$ 0.0031 | 87.60 $\pm$ 2.61 | 0.37 $\pm$ 0.13 | fixed
TESS s10 |  combined $\\#9-\\#15$ |  | JKTEBOP | 0.2115 $\pm$ 0.0144 | 0.1346 $\pm$ 0.0029 | 85.47 $\pm$ 2.81 | 0.24 $\pm$ 0.13 | fixed
TESS s36 | combined $\\#1-\\#8$ |  | JKTEBOP | 0.2021 $\pm$ 0.0089 | 0.1318 $\pm$ 0.0029 | 87.73 $\pm$ 2.21 | 0.26 $\pm$ 0.14 | fixed
TESS s36 |  combined $\\#9-\\#15$ |  | JKTEBOP | 0.1959 $\pm$ 0.0091 | 0.1296 $\pm$ 0.0024 | 89.24 $\pm$ 2.15 | 0.28 $\pm$ 0.12 | fixed
TESS s63 | combined $\\#$1-$\\#$5 |  | JKTEBOP | 0.2050 $\pm$ 0.0170 | 0.1246 $\pm$ 0.0030 | 86.6 $\pm$ 3.2 | 0.28 $\pm$ 0.18 | fixed
TESS s63 | combined $\\#$6-$\\#$9 |  | JKTEBOP | 0.1990 $\pm$ 0.0190 | 0.1271 $\pm$ 0.0027 | 87.0 $\pm$ 3.2 | 0.14 $\pm$ 0.20 | fixed
TESS s63 | combined $\\#$10-$\\#$14 |  | JKTEBOP | 0.2040 $\pm$ 0.0160 | 0.1260 $\pm$ 0.0025 | 86.7 $\pm$ 3.0 | 0.18 $\pm$ 0.20 | fixed
TESS s63 | combined $\\#$15-$\\#$18 |  | JKTEBOP | 0.1970 $\pm$ 0.0110 | 0.1210 $\pm$ 0.0037 | 89.8 $\pm$ 1.8 | 0.54 $\pm$ 0.14 | fixed
Final Results |  |  |  | 0.19901 $\pm$ 0.00090 | 0.13164 $\pm$ 0.00039 | 87.83 $\pm$ 0.18 |  |