11institutetext:  Department of Astrophysics, IMAPP, Radboud University
Nijmegen, P. O. Box 9010, 6500 GL Nijmegen, the Netherlands
11email<EMAIL_ADDRESS>22institutetext: Institute of Astronomy, KU
Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium 33institutetext: Department
of Physics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia

# Tidal perturbations and eclipse mapping in the pulsations in the
hierarchical triple system U Gru

C. Johnston 1122 A. Tkachenko 22 T. Van Reeth 22 D. M. Bowman 22 K. Pavlovski
33 H. Sana 22 S. Sekaran 22

(Received Date Month Year / Accepted Date Month Year)

###### Abstract

Context. Unambiguous examples of the influence of tides on self-excited, free
stellar pulsations have recently been observationally detected in space-based
photometric data.

Aims. We aim to investigate U Gru and contextualise it within the growing
class of tidally influenced pulsators. Initial analysis of U Gru revealed
frequencies spaced by the orbital frequency that are difficult to explain by
currently proposed tidal mechanisms.

Methods. We re-investigate the TESS photometry of U Gru alongside new uves
spectroscopy. We analyse the uves spectroscopy with least-squares
deconvolution and spectral disentangling techniques, and perform an
atmospheric analysis. We remove the binary signature from the light curve
using an effective model in order to investigate the pulsation signal in the
residuals. We track the amplitudes and phases of the residual pulsations as a
function of the orbital period to reveal their tidal influence.

Results. We establish that U Gru is likely a hierarchical triple system. We
identify a single p mode oscillation to exhibit amplitude and phase variation
over the binary orbit. We propose a toy model to demonstrate that the series
of frequencies separated by the orbital frequency can be reproduced by eclipse
mapping. We find no evidence of modulation to the other independent
oscillation modes.

Conclusions. We demonstrate that U Gru hosts at least one tidally perturbed
pulsation. Additionally we argue that eclipse mapping of the dominant, tidally
perturbed mode can produce the series of frequencies separated by the observed
orbital frequency. Our simulations show that the effects of eclipse mapping
are mode dependent, and are not expected to produce an observable signature
for all pulsation modes in an eclipse binary.

###### Key Words.:

asteroseismology – stars: binaries: eclipsing – stars: individual: U Gru –
stars: oscillations – stars: binaries: close

##  1 Introduction

Self-excited, free stellar pulsations give rise to surface flux variations
that are observable as brightness variations. The presence of a close binary
companion can cause modulations to these brightness variations as seen by the
observer. The modulations can be induced by: (1) directly modifying the
interior of the pulsating star giving rise to tidally perturbed pulsations
(e.g., Smeyers & Martens, 1983; Reyniers & Smeyers, 2003); (2) altering the
axis about which the pulsations occur (e.g., Handler et al., 2020; Fuller et
al., 2020); or (3) through preferentially altering the visibility of the
pulsating star’s photosphere along the observer’s line of sight (e.g. eclipse
mapping; Gamarova et al., 2003; Reed et al., 2005).

Recently, a new class of tidally tilted pulsators in binary systems has been
observationally established wherein the tidal forces are assumed to be strong
enough to re-align the pulsation axis with the line of apsides connecting the
centres of mass of each component (Fuller et al., 2020). Since the pulsation
axis is no longer aligned with the rotation axis in this scenario, a time-
dependent observed amplitude modulation is introduced as the pulsation nodal
lines periodically change visibility with orbital phase. This is comparable to
the configuration observed in rapidly oscillating Ap stars in which the
pulsation axis is inclined with respect to the rotation axis due to the strong
magnetic fields of these stars, commonly referred to as the oblique pulsator
model (Kurtz, 1982; Bigot & Dziembowski, 2002; Saio & Gautschy, 2004).
Furthermore, there is a subset of tidally tilted pulsators in which the tidal
deformation of the pulsating star effectively traps the pulsation to a single
side of the star due to mode coupling, i.e. tidally trapped pulsators (Fuller
et al., 2020). This creates an apparent orbital phase dependence of the
pulsation amplitude. When viewed in Fourier space, the amplitude modulation
results in pulsation multiplets which have components separated by the orbital
frequency.

The phenomenon of tidally tilted pulsations has, so far, been observed in a
handful of stars observed by NASA’s Transiting Exoplanet Survey Satellite
(TESS; Ricker et al., 2015). The cases of HD 74423, CO Cam, and HD 265435 show
pulsations that are preferentially trapped to one side of the star (Handler et
al., 2020; Kurtz et al., 2020; Jayaraman et al., 2022). The example of TIC
63328020, however, shows non-axisymmetric sectoral modes which have a tilted
pulsation axis and weak trapping (Fuller et al., 2020; Rappaport et al.,
2021), while the oscillating eclipsing Algol-type (oEA; Mkrtichian et al.,
2003) systems TZ Dra and HL Dra show evidence for a tidally tilted pulsation
axis (Shi et al., 2021; Kahraman Aliçavuş et al., 2022).

In addition to these clear examples of tidally tilted pulsators, there are a
handful of objects that demonstrate other forms of tidal perturbations to the
observed pulsations. Notably, the eclipsing binary RS Cha shows multiple
phenomena, including the tidal splitting of intrinsic pulsation modes induced
by the tidal bulge acting as the axis of symmetry, as well as observed
pulsation amplitude decrease due to the presence of eclipses (Balona, 2018;
Preece et al., 2019; Steindl et al., 2021). Furthermore, there are now several
(eclipsing) binaries pulsating in either pressure (p) or gravity (g) mode
oscillations that show tidal perturbations or other distortions that modify
the observed pulsation amplitude over the orbit (Jerzykiewicz et al., 2020;
Southworth et al., 2020, 2021; Lee, 2021; Van Reeth et al., 2022). While the
sample of tidally perturbed pulsators is increasing, there is no consensus on
a proposed underlying mechanism. There is no obvious underlying commonality in
orbital configuration, mass range, or previous evolutionary history amongst
the known sample of tidally influenced pulsating stars (Handler et al., 2022).

Given the recent increase in the identification of tidal signatures in a
diversity of pulsating stars, astronomers are starting to utilise the high
precision afforded by space-based telescopes to perform asteroseismology and
probe the influence of tides on the stellar interior. However, to be able to
exploit tidally influenced modes in asteroseismic analysis, we require the
unambiguous identification of the underlying tidal phenomena in question.
Therefore, complicated cases such as that of the oEA system U Gru, which shows
signatures of tidal effects as well as eclipse mapping (Reed et al., 2005;
Gamarova et al., 2003) in its pulsation modes (Bowman et al., 2019), need a
detailed investigation of combined time-series photometry and spectroscopy
before it can undergo a tidal asteroseismic analysis.

U Gru is a bright oEA system with a binary period of 1.8805 d, initially
discovered to be an eclipsing binary with an A5 spectral type by Brancewicz &
Dworak (1980). Recently, U Gru was observed by TESS in sectors 1 and 28,
yielding 54 days of high precision space based photometry. Using the initial
sector of TESS data, Bowman et al. (2019) report the discovery of several
independent p mode pulsations in addition to a series of significant
frequencies split by (multiples of) the orbital frequency. However, there is
no clear symmetry as to how the frequencies that are split by the orbital
frequency are distributed, making its interpretation as a tidal phenomenon
challenging given the currently proposed mechanisms (e.g., Balona, 2018;
Fuller et al., 2020). oEA systems are known to have recently undergone mass-
transfer, effectively rejuvenating the accretor and placing it closer to the
zero-age main-sequence (ZAMS). To further complicate matters, oEA systems are
commonly observed to have distant tertiary companions, potentially introducing
more complicated tidal forces in the system due to the ever changing tidal
potential (Fuller et al., 2013).

In this paper, we re-visit the TESS photometry of U Gru alongside new uves
spectroscopy in order to identify the mechanisms causing the tidal effects
observed in its pulsations. In Section 2 we present new spectroscopic
observations and atmospheric modelling. In Section 3 we discuss the analysis
of the existing TESS data. Finally, we discuss the presence and lack of tidal
signals and present a toy model to explain our observations in Section 4
before concluding in Section 5.

##  2 UVES spectroscopy of U Gru

To characterise the components and the orbit of the system, we obtained high-
resolution spectroscopy with the Ultraviolet and Visual Échelle Spectrograph
(uves on UT2@VLT; Dekker et al., 2000) via a Director’s Discretionary Time
proposal (ID: 0104.D-0209; PI: C. Johnston). Our observations utilized the
standard DIC-2 blue arm setting, which has both a blue ($\lambda\sim 300-500$
nm; $\lambda_{c}=437$ nm; R$\sim$80 000) and red ($\lambda\sim 420-1100$ nm;
$\lambda_{c}=760$ nm; R$\sim$110 000) detector equipped on uves. All
observations were taken using a 1″slit with a 600 sec exposure time. We
obtained 12 epochs over a three month period, which are distributed to
maximize coverage across the orbital period of the U Gru binary system. The
observations are briefly summarized in Table 1.

The spectra were extracted, corrected, and calibrated with the standard
version of the eso uves pipeline (Sacco et al., 2014) and can be obtained
directly from the ESO
archive111http://archive.eso.org/wdb/wdb/adp/phase3_spectral/form?collection_name=UVES.
Further normalisation of the reduced spectra to the local continuum was
performed by fitting a 3rd order polynomial through a set of manually selected
continuum points. Barycentric velocity and time corrections were applied to
the spectra.

Table 1: Barycentric Julian Date, orbital phase, and S/N at 4500Åof the UVES observations. BJD | Phase | S/N
---|---|---
d |  |
2458707.80299 | 0.06 | 70
2458679.74695 | 0.14 | 114
2458698.73534 | 0.24 | 88
2458700.82038 | 0.35 | 123
2458672.73848 | 0.42 | 103
2458674.75670 | 0.49 | 103
2458661.73705 | 0.57 | 98
2458678.81785 | 0.65 | 130
2458633.88509 | 0.76 | 112
2458699.80944 | 0.81 | 112
2458720.68674 | 0.91 | 115
2458675.68874 | 0.99 | 37

###  2.1 Least-Squares Deconvolution

The first step of our spectroscopic analysis involves computing the least-
squares deconvolution (LSD; Donati et al., 1997) profiles with a customised
version of the LSD algorithm (Tkachenko et al., 2013). An LSD profile is an
average line profile that is characterised by a significantly enhanced signal-
to-noise ratio (S/N) compared to individual absorption lines in the stellar
spectrum. Additionally, the LSD profile captures any type of variability that
is common to all spectral lines involved in the calculation. In particular,
the LSD method provides an efficient way of detecting line profile variations
due to binarity and/or intrinsic variability of either of the binary
components in the form of rotational modulation and/or stellar pulsations
(e.g., Tkachenko et al., 2013; Johnston et al., 2021).

To compute the LSD profiles, we used multiple synthetic line lists for an
A-type star ($T_{\rm eff}=8\,000,\leavevmode\nobreak\ \log
g=4.0\leavevmode\nobreak\ {\rm dex},\leavevmode\nobreak\ {\rm[M/H]=0}$), a
G-type star ($T_{\rm eff}=6\,000,\leavevmode\nobreak\ \log
g=4.0\leavevmode\nobreak\ {\rm dex},\leavevmode\nobreak\ {\rm[M/H]=0}$), as
well as for a K-type star ($T_{\rm eff}=5\,000,\leavevmode\nobreak\ \log
g=4.0\leavevmode\nobreak\ {\rm dex},\leavevmode\nobreak\ {\rm[M/H]=0}$). The
line lists were constructed from the VALD database (Kupka et al., 2000),
covering a wavelength range of 4 000-7 000 Å, excluding Balmer lines. Although
we compute LSD profiles from the spectra obtained with both the blue and red
arms of the uves instrument, below we discuss the results obtained for the
blue arm only. We do this to avoid repeating the results given the strong
similarities obtained from the red part of the spectrum. Additionally, we do
not use the spectrum taken during primary eclipse as the S/N is low and we do
not see the signature of the primary. The orbital phase folded LSD profiles
for 11 spectra from the UVES blue arm are shown in Fig. 1. We note that the
systemic velocity has not been removed from the observations before we
calculate the LSD profiles. Two components are clearly visible in the LSD
profiles of the blue arm. One component is broadened and shows intrinsic line-
profile variability throughout the motion along the orbit. The other component
reveals a narrow-lined profile which appears stationary over the orbital
period. We identify the broad-lined profile as the rejuvenated A-type primary
(i.e. the $\delta$ Scuti pulsator), and identify the narrow-lined profile as a
potential slowly rotating tertiary component in the system due to its
negligible radial velocity (RV) variability. The cool faint secondary
component is not firmly detected in either of the two arms of the obtained
uves spectra.

Figure 1: The least-squares deconvolution profiles of U Gru computed from 11
blue arm uves spectra. The orbital phase resolved spectral contribution of the
broad-lined primary component is indicated with the dashed line.

###  2.2 Spectral disentangling

Spectroscopic observations of multiple star systems are comprised of the
composite light of all the $N$ components in the system. The analysis of such
a composite spectrum can be done by directly fitting $N$ components to the
spectrum with the appropriate RV shifts and light factors, or by separating
the components using techniques such as spectral disentangling. The method of
spectral disentangling allows one to simultaneously optimise for orbital
elements of the system and separate the individual spectra of the stars
forming the system (Simon & Sturm, 1994; Hadrava, 1995). For the purpose of
spectral disentangling, we employ the FDBinary software package (Ilijic et
al., 2004) using 11 out of the total 12 acquired blue arm spectra of U Gru. In
particular, we remove the spectrum taken at the time of the eclipse of the
more luminous primary component due to the strong variations in the spectral
line depths caused by the near total blockage of light from the primary
component. We perform the spectral disentangling in the wavelength region
between the $H_{\gamma}$ and $H_{\beta}$ Balmer profiles spanning 4 400-4 800
Å.

In the first step, we perform the spectral disentangling in a triple system
mode, primarily in an attempt to detect signatures of the faint secondary
component. However, similar to the results obtained with the LSD profiles, no
trace of the cool secondary component can be found with the spectral
disentangling technique. We therefore proceed with the spectral disentangling
in a ‘fake binary mode’ where the narrow-lined tertiary component is
constrained to have long periods such that it shows little to no RV
variability over the 3-month time span of our uves observations. The broad-
lined primary component is assumed to be on a circular orbit and is otherwise
unconstrained. This is supported from the initial TESS photometry and
preliminary binary modelling by Bowman et al. (2019). Both of these
assumptions are largely consistent with the scenario arising from the
properties of the LSD profiles outlined in Section 2.1. From this, we find the
RV semi-amplitude of the more luminous primary component to be $K_{\rm
1}=61.5\pm 0.7$ km s-1 and find no significant RV variability associated with
the tertiary component.

Using the orbital solution and the disentangled spectra of the primary and
tertiary components from the ‘fake binary mode’, we shift these disentangled
spectra by their individual RVs corresponding to the times of each of the uves
observations and then subtract each shifted disentangled spectrum from the
originally observed composite spectra. This way we obtain a time series of
eleven residual spectra in which spectral contributions of the primary and
tertiary components have been eliminated. In the next step, we explore these
residual spectra with the FDBinary algorithm in a spectroscopic single-lined
binary (SB1) mode in an attempt to search for extremely weak signatures of the
cool secondary component. This exercise reveals a tentative spectrum of a cool
star whose RV semi-amplitude $K_{\rm 2}$ is found to be between 195 and 200 km
s-1. Figure 2 shows the disentangled spectra of the primary (blue) and
tertiary (black) components of the U Gru system, as well as the tentative
disentangled spectrum of the secondary (red) alongside a synthetic spectrum
computed for $T_{\rm eff}=5500$ K and $\log\,g=3.5$ dex (orange) and
arbitrarily diluted to the level of 2% (in continuum units) for comparison. We
note that the disentangled secondary spectrum has been binned by a factor of
two to increase the signal to noise. Although we do not claim a firm detection
of the secondary component, we do identify some similarities between the
disentangled secondary spectrum and the synthetic spectrum in Fig. 2.

Figure 2: Disentangled spectra of the primary (blue) and tertiary (black)
components of the U Gru system. A representative synthetic spectrum
corresponding to $T_{\rm eff}=5500$ K and $\log\,g=3.5$ dex and diluted to the
level of 2% (in continuum units) is shown for comparison to the secondary
component (red). Note that a constant vertical shift have been applied to all
the spectra for visibility purposes.  Figure 3: Orbital solution for the
primary (blue), secondary (red), and tertiary (black) components of U Gru
obtained with the method of spectral disentangling. Note that the symbols do
not represent actual RV measurements. Instead, they indicate the distribution
of the observed spectra over the orbital cycle of the inner binary system,
while the orbital RV curve itself is computed from the best fit orbital
elements.

As the last exercise, we fix $K_{\rm 1}$ and set the initial guess for $K_{\rm
2}$ according to our previous findings, and once again perform spectral
disentangling in the SB3 triple system mode. The inner binary orbit is assumed
to be circular and we allow for small RV variability for the narrow-lined
tertiary component. We find the RV semi-amplitude of the secondary component
to be $K_{\rm 2}$=202$\pm$3 km s-1 while the disentangled spectra of all three
components do not show any significant differences with respect to the
solutions described above. We find that the tertiary RV semi-amplitude tends
towards 0 km s-1 in this solution. Together with $K_{\rm 1}$ the above value
of $K_{\rm 2}$ suggests a mass ratio of $q=0.304\pm 0.005$, and $\sim$2.7
$M_{\odot}$ and $\sim$0.8 $M_{\odot}$ for the mass of the primary and
secondary components, respectively, assuming an inclination of $i=90^{\circ}$.
All these values are consistent with the system configuration typical for
Algols and the mass of the primary is consistent with its late-A spectral
type. We note that given the lack of a robust detection of the secondary, we
do not attempt a full binary modelling in this work as any solution would be
degenerate without an independent constraint on the mass-ratio and semi-major
axis of the system.

Figure 3 shows the orbital RV curves for the primary and secondary components
of U Gru computed from the corresponding best fit orbital elements. Because
the method of spectral disentangling by-passes the step of RV determination
and instead solves for the orbital elements directly, the symbols shown in
Fig. 3 do not represent the actual RV measurements of the primary, secondary,
and tertiary components, respectively. Instead, the symbols represent the RVs
computed from the best fit orbital solution at the times of our uves
observations. We show this to demonstrate the orbital phase coverage achieved
with our spectroscopic observations.

###  2.3 Atmospheric parameters

We employ the Grid Search in Stellar Parameters (gssp; Tkachenko, 2015)
software package to infer atmospheric parameter values of the primary and
tertiary components from their disentangled spectra. We do not perform
spectrum analysis of the faint secondary component owing to the low S/N in its
disentangled spectrum. The gssp package is employed in its ‘unconstrained
fitting’ mode that allows us to analyse the disentangled spectra of stars as
if they were single. Even though gssp offers an option to fit the disentangled
spectra of binary (or triple system) components simultaneously and taking into
account wavelength dependence of the components’ light contributions, we do
not opt for that particular mode in our analysis. The main reason is because
we expect the secondary component to potentially still contaminate the
spectrum of the primary component. In turn, the uncertainty in the
disentangled spectrum of the primary component propagates into the results
obtained for the tertiary component when the two spectra are coupled
methodologically.

In our spectrum analysis, we explore different combinations of free and fixed
atmospheric parameters, of which we consider in total six for each of the
components: effective temperature $T_{\rm eff}$, surface gravity $\log\,g$,
microturbulent velocity $\xi$, projected rotational velocity $v\,\sin\,i$,
metallicity [M/H] (assuming an Asplund et al. (2005) solar composition), and
light dilution factor $\ell f_{\rm i}$. Immediately, we find that although the
light dilution factor is well constrained for the tertiary component, no
(realistic) optimal value can be obtained for the primary component.
Therefore, we start with the atmospheric analysis of the tertiary component
and then proceed with the analysis of the primary component where its light
dilution factor $\ell f_{\rm 1}$ is fixed according to the relation $\ell
f_{\rm 1}$ = 1 - $\ell f_{\rm 3}$. This relation assumes that the secondary
component is not present in the system. Instead, in the final stage of our
analysis, we re-determine atmospheric parameters of the primary and tertiary
components allowing for a few percent variations of the light factors $\ell
f_{\rm 1}$ and $\ell f_{\rm 3}$ from their optimal values. We do not consider
variations larger than a few percent because that is what we expect for the
light contribution from the faint secondary component. We account for
deviations of the atmospheric parameters of the primary and tertiary
components from their optimal values due to variable $\ell f_{\rm 1}$ and
$\ell f_{\rm 3}$ in the final reported parameter uncertainties.

Our spectrum analysis show that the most stable spectroscopic solution is
achieved when the $\xi$ and [M/H] parameters are fixed to 2 km s-1 and 0.0 dex
(solar value), respectively. We note that the metallicity [M/H] in the
atmospheric spectroscopic analysis refers to a joint contribution of all
elements heavier than helium. Otherwise, we find a statistically significant
difference in the metallicity of the two stars. This scenario is not expected
in a gravitationally bound binary (or triple) system where all components are
formed from the same natal cloud, despite the past history of binary
interaction. However, similar to the case with the light factors, we do
account for the effects of metallicity and microturbulent velocity in the
finally reported uncertainties for $T_{\rm eff}$, $\log\,g$, and $v\,\sin\,i$.
The final atmospheric parameters (and their uncertainties) of the primary
(A1V; Pecaut & Mamajek, 2013) and tertiary (F3V) component of U Gru are
reported in Table 2. Given the surface gravity and mass range of the primary
as derived above, the primary is rotating at less than 20% its critical rate.

Table 2: Atmospheric parameters of the primary and tertiary components of U Gru. See text for details. Parameter | Unit | Component
---|---|---
Primary | Tertiary
$T_{\rm eff}$ | K | 9000$\pm$550 | 6710$\pm$450
$\log\,g$ | dex | 4.40$\pm$0.35 | 4.35$\pm$0.5
$\xi$ | km s-1 | 2.0 (fixed)
$v\,\sin\,i$ | km s-1 | 59$\pm$5 | 6.8$\pm$1.9
${\rm[M/H]}$ | dex | 0.0 (fixed)
$f_{\rm i}$ | dex | 0.94 (fixed) | 0.06$\pm$ 0.02

##  3 TESS photometry

U Gru was observed by TESS in two sectors, separated by roughly two years.
During the first sector, U Gru was observed for 27.9 d in 2-min cadence mode,
whereas it was observed for 25.2 d at a 10-min cadence in the full-frame image
(FFI) data during sector 28. We find no indication of the tertiary object in
either sector of the TESS photometry. Using the 2-min cadence sector 1 data,
Bowman et al. (2019) reported 22 significant pulsation frequencies after
subtracting a multi-harmonic model to remove the eclipsing binary
contribution. However in this work we remove the strictly periodic eclipsing
signal by using an interpolated model constructed from the orbital phase-
folded light curve. In brief, we phase fold the original light curve on the
orbital period, and then bin the phased light curve. We then construct an
interpolated model from the phase-binned data and subtract that model from the
original light curve. The phase folded light curves from sectors 1 and 28 are
shown in black and grey in the top panel of Fig. 4. The interpolated models
are shown in red, and the residuals are shown in the bottom panel. The Lomb-
Scargle (LS) periodograms (Lomb, 1976; Scargle, 1982; Press & Rybicki, 1989;
Astropy Collaboration et al., 2018) of the interpolation model subtracted
residuals are shown in the black (sector 1) and grey (sector 28) in Fig. 5. As
can be seen, the series of frequencies separated by (multiples of) the orbital
frequency are still seen in the residuals from sector 28, suggesting that the
underlying phenomenon is stable across the two years separating the data sets.
Notably, the amplitudes observed in the periodogram of the sector 28 residuals
are generally lower than those observed in the sector 1 residuals. This is
likely caused by the change in photometric sampling rate (two min to 10 min)
between the sectors. There does appear to be an additional amplitude change
for the second highest amplitude frequency. Intrinsic pulsation mode amplitude
variability over multiple years is commonly observed in $\delta$ Scuti
pulsators and is attributed to mode coupling or changes to the internal
stellar structure (Bowman et al., 2016).

Figure 4: (Top): Phase folded light curve of U Gru for sectors 1 (black) and
28 (grey) with the interpolated binary model in red. Grey hatching denotes
phases of primary (forward hatching) and secondary (cross hatching) eclipse.
(Bottom): Residuals after subtraction of interpolation binary model for
sectors 1 (black) and 28 (grey). Figure 5: (Top): Lomb-Scargle periodogram of
binary subtracted residuals for sector 1. (Bottom): Lomb-Scargle periodogram
of binary subtracted residuals for sector 28. Labelled frequencies denote
frequencies that are tracked in Section 4.1 and shown in orange in Fig. 6

We conduct an iterative pre-whitening procedure on the residuals of the sector
1 data using the pythia code222https://github.com/colej/pythia in order to
remove the remaining periodic signals. We identify the signal to be removed at
each iteration as the peak with the highest S/N calculated via a 1 d-1 window
surrounding the frequency. The signal in question is modelled with a sinusoid,
where the optimised parameters are taken as the maximum a posteriori (MAP)
non-linear least-squares fit to the light curve. We perform the optimisation
using the pymc3 (Salvatier et al., 2016) and exoplanet (Foreman-Mackey et al.,
2021) python modules. We iterate until we extract all signals with S/N ¿ 4,
and then re-calculate the final model with all frequencies as free parameters.
We calculate the final S/N with respect to the residual light curve. We find
21 significant frequencies, which are reported in Table 3. The phases are
calculated with respect to $t_{0}=2458325.93900$, which is the reference epoch
of primary eclipse first reported by Bowman et al. 2019. We report
uncertainties on the frequencies, amplitudes, and phases according to
Montgomery & O’Donoghue (1999) accounting for the correlated nature of the
data according to Schwarzenberg-Czerny (2003). We recover 8 fewer frequencies
that are split by the orbital frequency than those originally reported in
Bowman et al. (2019). Five of these modes had a S/N ¡ 4 reported by Bowman et
al. (2019), whereas all eight were found to be formally insignificant in our
analysis. We find seven additional significant frequencies that Bowman et al.
(2019) did not recover in their analysis. The differences in extracted
frequencies is likely caused by the different methods used to subtract the
binary signal and the different methods for calculating the noise level around
a given frequency. We construct and show an Échelle diagram of our extracted
frequencies folded over the orbital frequency in Fig. 6, which shows a clear
series of frequencies split by (multiples of) the orbital frequency.

Table 3: Optimised frequencies, amplitudes, and phases of the significant frequencies identified via iterative pre-whitening. We also list the S/N of each extracted frequency, as well as its value modulo the orbital frequency. The zero-point in time was set to $t_{0}=2458325.93900$. | Frequency | F mod $f_{\rm orb}$ | Amplitude | Phase | S/N
---|---|---|---|---|---
| d-1 |  | ppt | rad |
$f_{01}$ | $22.945\pm 0.006$ | $0.079\pm 0.006$ | $0.16\pm 0.05$ | $1.06\pm 0.02$ | 5.5
$f_{02}$ | $23.468\pm 0.004$ | $0.070\pm 0.004$ | $0.22\pm 0.05$ | $1.72\pm 0.02$ | 8.2
$f_{03}$ | $24.002\pm 0.004$ | $0.072\pm 0.004$ | $0.21\pm 0.05$ | $1.67\pm 0.02$ | 9.7
$f_{04}$ | $24.535\pm 0.004$ | $0.073\pm 0.004$ | $0.26\pm 0.05$ | $1.53\pm 0.01$ | 10.7
$f_{05}$ | $25.063\pm 0.003$ | $0.070\pm 0.003$ | $0.28\pm 0.05$ | $1.76\pm 0.01$ | 10.2
$f_{06}$ | $25.596\pm 0.005$ | $0.071\pm 0.005$ | $0.19\pm 0.05$ | $1.50\pm 0.02$ | 7.2
$f_{07}$ | $26.130\pm 0.005$ | $0.073\pm 0.005$ | $0.17\pm 0.05$ | $1.67\pm 0.02$ | 7.5
$f_{08}$ | $26.670\pm 0.009$ | $0.080\pm 0.009$ | $0.10\pm 0.05$ | $1.23\pm 0.04$ | 4.6
$f_{09}$ | $28.789\pm 0.006$ | $0.073\pm 0.006$ | $0.15\pm 0.05$ | $-1.14\pm 0.02$ | 5.2
$f_{10}$ | $30.923\pm 0.005$ | $0.080\pm 0.005$ | $0.19\pm 0.05$ | $1.69\pm 0.02$ | 5.6
$f_{11}$ | $31.447\pm 0.001$ | $0.072\pm 0.001$ | $0.67\pm 0.05$ | $-1.70\pm 0.01$ | 17.4
$f_{12}$ | $31.984\pm 0.004$ | $0.078\pm 0.004$ | $0.22\pm 0.05$ | $1.97\pm 0.02$ | 5.9
$f_{13}$ | $32.947\pm 0.002$ | $0.509\pm 0.002$ | $0.47\pm 0.05$ | $1.274\pm 0.01$ | 6.6
$f_{14}$ | $33.046\pm 0.001$ | $0.076\pm 0.001$ | $1.49\pm 0.05$ | $-1.51\pm 0.01$ | 21.8
$f_{15}$ | $33.861\pm 0.001$ | $0.359\pm 0.001$ | $1.20\pm 0.05$ | $-1.30\pm 0.01$ | 26.9
$f_{16}$ | $34.017\pm 0.004$ | $0.515\pm 0.004$ | $0.24\pm 0.05$ | $-2.28\pm 0.02$ | 5.2
$f_{17}$ | $34.117\pm 0.004$ | $0.083\pm 0.004$ | $0.25\pm 0.05$ | $1.46\pm 0.02$ | 6.5
$f_{18}$ | $34.919\pm 0.006$ | $0.354\pm 0.006$ | $0.16\pm 0.05$ | $2.37\pm 0.02$ | 4.1
$f_{19}$ | $39.470\pm 0.004$ | $0.118\pm 0.004$ | $0.24\pm 0.05$ | $0.73\pm 0.02$ | 11.1
$f_{20}$ | $52.640\pm 0.009$ | $0.520\pm 0.009$ | $0.08\pm 0.05$ | $1.29\pm 0.05$ | 4.0
$f_{21}$ | $66.184\pm 0.004$ | $0.244\pm 0.004$ | $0.24\pm 0.05$ | $1.32\pm 0.02$ | 10.8
Figure 6: Échelle diagram of the significant frequencies extracted from the
residuals of sector 1 folded over the orbital period. The marker size scales
with amplitude. Orange markers denote frequencies labelled in Fig. 5 and
tracked in Section 4.1

##  4 Amplitude and phase tracking

###  4.1 Tidal effects

In all of the confirmed tidally tilted pulsating stars known to date, the
tidal splitting has resulted in a (generally) symmetric multiplet, split by
(multiples of) the orbital frequency. In the case of U Gru, however, Bowman et
al. 2019 identified a series of frequencies that did not appear to be centred
on a particular oscillation mode, posing a challenge for interpreting this
phenomenon as the result of a tidally tilted pulsation axis.

In this work we re-visit the pulsation analysis of U Gru, and track the
amplitude and phases of the several frequencies extracted from iterative pre-
whitening. We divide the Échelle diagram into distinct sections (e.g., 0-0.1
d-1, 0.1-0.2 d-1, etc.) as shown by the grey dashed lines in Fig. 6, and track
the highest amplitude mode for each section of the Échelle diagram, e.g.
$f_{13}$, $f_{14}$, $f_{15}$, $f_{19}$, and $f_{21}$. We also consider$f_{20}$
to be independent and track it. These modes are labelled in Fig. 5 and denoted
by orange markers in Fig. 6. To do this, we remove all other significant
frequencies from the other sections of the Échelle diagram before dividing the
orbital phase-folded light curve into 50 equally sized bins. We fit a sinusoid
with a fixed frequency to each bin and record the amplitude and phase
following Van Reeth et al. (2022). The tracked amplitudes and phases are shown
in Fig. 7, with orbital phase values of 0, 1 and 2 corresponding to primary
eclipse, and orbital phase values of 0.5 and 1.5 corresponding to secondary
eclipse. In order to compare the modulations for each frequency, we consider
the relative amplitude and phase modulations. As such, we divide the tracked
amplitude at each phase bin by the average amplitude $<A>$ across all orbital
phase-bins considered. Similarly, we subtract the average mode phase $<\phi>$
for each frequency. Only the highest amplitude pulsation at $f_{14}$=33.046
d-1 shows amplitude and phase modulation across the orbit, as shown in Fig. 7.
Additionally, we note that the remaining frequencies that we tracked
amplitudes and phases for show amplitude minima during primary eclipse.

Figure 7: (Top): Tracked mode amplitude for $f_{14}=33.046$ d-1 as a function
of orbital phase. (Bottom): Tracked relative mode phase for $f_{14}=33.046$
d-1 as a function of orbital phase. Phase values of 0, 1 and 2 correspond to
primary eclipse, and phase values of 0.5 and 1.5 correspond to secondary
eclipse.

We find clear evidence that the amplitude and phase of the p mode at
$f_{14}=33.046$ d-1 is modulated over the binary orbit. We note that the
amplitude modulation displays an apparent minima during secondary eclipse,
with maxima occurring at $\frac{1}{3}$ phases of the orbit. The three observed
amplitude maxima are difficult to explain with the currently proposed tidal
tilting framework as a tilted mode with degree $\ell$ should produce 2$\ell$
amplitude maxima across the binary orbit (Fuller et al., 2020). Furthermore,
we observe phase modulations with a $\pi/2$ rad variation, similar to those
observed in distorted modes in roAp stars (e.g., Holdsworth et al., 2016) and
the g-mode pulsating binary V456 Cyg (Van Reeth et al., 2022). Finally, we
observe a complex behaviour of the amplitude and phase modulation at phases
surrounding primary eclipse. Following the oblique pulsator model assumed for
the case of tidal trapping, we should observe a symmetric multiplet split by
the orbital frequency from the intrinsic pulsation frequency as well as phase
variations with values of $0$, $\pi$, or $2\pi$. This model cannot explain
either the full series of frequencies split by (multiples of) the orbital
frequency nor the observed phased modulations of the dominant p mode in U Gru.

###  4.2 Eclipse mapping

Photometric observations of distant, point source stars are the total
integrated surface brightness of the projected stellar surface at a given
time. Stellar pulsations induce brightness variations in time by modifying the
surface brightness distribution according to the geometry of the pulsation
modes. In the case of an eclipsing binary, the projected stellar disk is
partially (or totally) obscured, resulting in a time-varying amount of the
stellar disk that is observed. This produces a time-dependent modification to
the partial cancellation of a varying mode over the stellar surface (Aerts et
al., 2010). U Gru exhibits total eclipses of the primary, wherein the
secondary component obscures nearly the entire visible disk of the pulsating
A-type primary. Therefore, there is an additional modulation of the observed
pulsation amplitude as the disk of the pulsating star is partially obscured
during ingress to primary eclipse, blocked totally during conjunction, and
then partially obscured again during egress from primary eclipse. This process
creates a complicated modulation of the observed pulsation amplitude in time
as part of the variable surface is preferentially obscured. This phenomenon is
sometimes referred to as spatial filtration (Gamarova et al., 2003; Rodríguez
et al., 2004; Gamarova et al., 2005) or eclipse mapping (Reed et al., 2001,
2005). To further complicate this scenario, the highest amplitude pulsation
mode observed in U Gru ($f_{14}$) shows evidence for being tidally perturbed,
adding an additional modulation to the observed pulsation amplitudes and
phases.

We construct a toy model to illustrate the impact that eclipses have on the
observed amplitudes and phases of non-tilted and tidally tilted pulsation
modes333https://github.com/colej/eb_mapping. The toy model is based on that
used by Van Reeth et al. (2022, in prep.), modified to our purposes. A full
explanation of the code is given in Appendix A. In brief, this toy model
assumes two spherical stars (with no deformation caused by the equilibrium
tide) in a Keplerian orbit. As we are simply trying to illustrate the effect
of time-dependent partial cancellation via eclipses and not attempting to
forward model the effect, the use of spherical, non tidally-distorted stars is
justified. The stars are inclined with respect to the observer according to
the orbit. The pulsating component has its surface perturbed according to
spherical harmonics of a given ($\ell$, $m$) combination. We then define a
series of points in time, and calculate the total flux as observed considering
the pulsation phase as well as the visibility of each star along the line of
sight. This simple model allows us to track the observed pulsation phase and
amplitude along the orbit as the pulsating star is eclipsed. Furthermore, we
allow for the pulsation axis to be inclined such than it can align with the
orbital plane to simulate the tidally tilted scenario, but we do not account
for mode trapping. This toy model does account for linear limb-darkening, but
does not account for the light travel time effect, which modulates the
observed phase of a periodic pulsation because of the physical change in
distance travelled by light from the pulsating star. In the case of U Gru,
this is a safe assumption as the peak-to-peak phase variation expected for
$f_{14}$ caused by the light travel time effect is $\sim 0.03$ rad, following
Murphy et al. (2014).

We define the orbit and components of our toy model to be qualitatively
similar to that of U Gru following an initial analysis with the PHOEBE code
(PHOEBE 1; Prša & Zwitter, 2005; Prša et al., 2011). This initial analysis
uses the TESS light curve and uves radial velocity solution (see Section 2)
and was adjusted by hand to match the light curve morphology and RVs of the
primary. As the model was not numerically optimized, we do not consider this a
full solution. We set the binary system to have $P_{\rm orb}=1.8805$ d, M1=2.6
M⊙, R1=2.8 R⊙, M2=0.8 M⊙, R2=2.6 R⊙, and $i$=85∘. We then simulate several
pulsation configurations, each with a different mode geometry ($\ell,m$) and
obliquity of the pulsation axis with respect to the rotation axis ($\beta$).
We investigate multiple pulsation geometries in order to demonstrate that the
differing pulsation geometries will produce different partial cancellation
effects, resulting in different observed pulsation amplitudes and phases. The
derived binary light curves, and pulsation amplitude and phase variations for
each ($\ell,m,\beta)$ configuration are shown in the top, middle, and bottom
panels of Fig. 8. The amplitude and phase modulations for the different
pulsation configurations clearly demonstrate that the presence of eclipses can
add an additional modulation to the pulsation amplitude and phase at orbital
phases when the pulsating component is eclipsed. Furthermore, the different
models in Fig. 8 demonstrate that the scale, complexity, and sign of the
modulation are dependent on the system inclination, the obliquity of the
pulsation axis, as well as the mode geometry (Reed et al., 2005).

Figure 8: (Top): Simulated light curves with different pulsation
configurations ($\ell,m,\beta$) phase folded over the binary period. The
points from the other simulated light curves are poorly visible due to the
strong overlap between the models and the size of the markers. $(Middle)$:
Mode amplitude variations for different pulsation configurations as seen by an
observer because of changing partial cancellation effects. $(Bottom)$: Mode
phase variations for different pulsation configurations as seen by an
observer.

Any additional periodic amplitude variability in a pulsation mode is captured
in the Fourier transform of the time series. To investigate the impact that
eclipse mapping has on the pulsation signal as seen in the periodogram, we
simulate the light curve for two of the configurations shown in Fig. 8 at the
time stamps of the sector 1 TESS data. These simulated light curves are
computed considering the full binary and pulsation signal, but have the
eclipsing signature removed for clarity. By computing the synthetic light
curves at the time stamps of the TESS data, we convolve the observational
window function into the resulting periodogram to make our simulations as
similar to the observations as possible. The periodograms of the subsequent
time series are shown in Fig. 9. As it is a mathematical decomposition of a
time series into a sum of sinusoids, the periodogram requires several
additional sinusoidal terms to represent any non-sinusoidal variations, such
as the amplitude modulations introduced by eclipse mapping. In this case, the
signal manifests as multiple additional peaks, separated from the intrinsic
pulsation frequency by integer multiples of the orbital frequency.

The total number of additional peaks, their amplitudes, and whether or not
they are symmetrically distributed about the intrinsic pulsation frequency
depends on the mode geometry and orbital configuration. We highlight this by
comparing the rows of Fig. 9. The top row considers an
($\ell=3,m=-2,\beta=0^{\circ}$) configuration where the two stars eclipse each
other. In the configuration for the middle row, the radius of the secondary
has been reduced so that the pulsating star is no longer eclipsed by the
secondary, and therefore does not display any eclipse mapping effects. In this
comparison, we note that the simulation without eclipses does not display the
same series of peaks split by the orbital frequency, whereas the simulation
with eclipses does. Furthermore, the bottom row displays a simulation for a
different pulsation configuration ($\ell=2,m=-1,\beta=90^{\circ}$), again with
the same orbital configuration as the top row that results in eclipses. This
simulation demonstrates that the resulting signature in the periodogram is
dependent on the pulsation geometry being considered, and that the resulting
signature may have too low amplitude to be observed in real data with multiple
noise components.

###  4.3 Combined tidal and eclipse mapping effects

Our toy model produces two quantifiable outputs: 1) complex behaviour in the
tracked mode amplitude and phase that occurs specifically during eclipse of
the pulsating component, and 2) a series of peaks in the periodogram that are
separated by the orbital frequency and potentially asymmetrically distributed
around the true pulsation frequency. The observed chaotic mode amplitude and
phase behaviour during phases of primary eclipse (e.g. Fig. 7) is
qualitatively similar to the behaviour predicted by our eclipse mapping model.
Given this, we propose that the asymmetrically distributed series of peaks
separated by the orbital frequency observed in the residual periodogram of U
Gru is also caused by eclipse mapping, similar to the results in Fig. 9.
Following this interpretation, we can use the mathematical framework presented
by Jayaraman et al. (2022) to investigate the observed amplitude and phase of
this mode across the orbit and during primary eclipse. This framework assumes
that any frequencies separated from the intrinsic mode by a multiple of the
orbital frequency can be used to derive information about the phase and
amplitude modulation of the intrinsic pulsation mode. While this methodology
was developed to interpret the case of tidally tilted pulsators, it can be
generalised for other pulsations that show orbital phase dependence. We
simultaneously fit for the amplitude, and phase of the mode at
$f_{14}=33.046\leavevmode\nobreak\ $d-1, as well as the amplitudes and phases
of those frequencies located at integer multiples of the orbital frequency
such $f_{j}=f_{1}\pm n_{j}$ with $n$ being a real integer. We fit for the 14
frequencies with $f{\rm mod}f_{\rm orb}<0.1$ d-1 simultaneously, all of which
are taken from Table 3. In addition to the frequency at
$f_{14}=33.046\leavevmode\nobreak\ $d-1, we fit 12 sinusoids that are at
frequencies below the main frequency, and only one at higher frequencies.
While there may be other components to this series of frequencies, they do not
have high enough amplitudes to be considered significant in our frequency
analysis, and are therefore ignored. Although Handler et al. 2020; Kurtz et
al. 2020; Rappaport et al. 2021; Jayaraman et al. 2022 intentionally set the
reference epoch for the pulsations to occur when the $f_{j}=n\pm$1 frequencies
have maximum amplitude, we cannot follow the same procedure as the frequencies
we consider are asymmetrically distributed in the case of U Gru. Instead, we
retain the same reference epoch as that of superior conjunction reported in
Bowman et al. (2019).

The resulting amplitude and phase modulation over the orbit are shown in the
middle and bottom panels of Fig. 10. The blue trend represent the fits using
sector 1 data. The general phase and amplitude behaviour recovered by this
method agrees with the behaviour recovered by fitting a sinusoid with
$f=33.0456\leavevmode\nobreak\ $d-1 to different phase bins carried out in
Section 4.1 (shown in black in Fig. 10). We also find complex amplitude and
phase modulations around primary eclipse in Fig. 10. We further construct a
model considering only those frequencies separated from the intrinsic mode by
$-3\leq N\leq 3$ (if significant). This model is shown in orange in the middle
and bottom panels of Fig. 10, respectively. When only considering frequencies
separated from the intrinsic mode by $\pm 3$ times the orbital frequency, we
only recover the three amplitude maxima and phase sign changes, and not the
complex behaviour near primary eclipse. This suggests that the remaining
asymmetrically distributed frequencies separated by larger integer multiples
of the orbital frequency are the result of the complex behaviour at primary
eclipse. We argue that this further supports the eclipse mapping origin of the
asymmetrically distributed series of frequencies separated by the orbital
frequency. We find the same results when using the sector 28 data.

Figure 9: Left column display simulated light curves of the pulsation mode
only. Right column displays periodogram of the accompanying light curve. These
light curves do not consider the light variations caused by the binary itself.
Grey hatching denotes phases of primary (forward hatching) and secondary
(cross hatching) eclipse. Figure 10: (Top): Phase folded light curve of U Gru
for sector 1. (Middle): Amplitude modulation of $f_{14}=33.046$ d-1 as
reconstructed according to the fitting method developed by Jayaraman et al.
(2022). The blue considers all 14 significant frequencies split from $f_{14}$
listed in Table 3. The orange lines only consider those frequencies with
$f=f_{14}\pm Nf_{\rm orb}$, where $-3\leq N\leq 3$, if they were found to be
significant in the iterative pre-whitening procedure. Tracked mode amplitudes
from Section 4 shown as black x’s. (Bottom): Same as the middle panel but for
the phase modulation. Grey hatching denotes phases of primary (forward
hatching) and secondary (cross hatching) eclipse.

##  5 Discussion and conclusions

To date there are multiple examples of pulsating stars in binaries that
exhibit behaviour consistent with the tidally tilted or trapped pulsator
model. In addition to this, there are an increasing number of systems that
exhibit pulsational behaviour that cannot be explained by the existing tidally
tilted or trapped models. In this work we increase the diversity of observed
behaviour with the case of the bright triple system U Gru.

We used uves spectroscopy to confirm the Algol-type status of U Gru and
further discover that the system consists of an apparently rejuvenated A-type
star and cool companion in a tight 1.8805 d orbit with a wide F3V tertiary
companion. The atmospheric solution for the primary is consistent with a
moderately rotating, early A-type star that has been rejuvenated by recent
mass transfer. Unfortunately, due to its low light contribution, we could not
robustly detect the signature of the cool secondary component. The lack of RV
shifts and eclipse timing variations in the TESS data suggests that the orbit
of the tertiary is much longer than the two year time-base covered by the TESS
data. To this end, we consider the separation sufficiently large as to not
introduce additional tidal forcing from the distant third body.

The first frequency analysis of the TESS light curve by Bowman et al. (2019)
revealed a series of significant frequencies separated by the orbital
frequency, or integer multiples thereof. In this work, we conducted a follow-
up analysis of the TESS light curves for both sectors 1 and 28. We removed the
dominant binary signal via an interpolated model and ran an iterative pre-
whitening frequency analysis on the residual light curves, recovering the
independent pulsations as well as the series of frequencies split by integer
multiples of the orbital frequency. This demonstrated that the mechanism which
produced the series of frequencies separated by multiples of the orbital
frequencies is stable over the 2-yr TESS time base.

The multiperiodic pulsations of U Gru present a challenge for interpretation
from only a single tidal mechanism. Only one pulsation mode
($f_{14}=33.046\leavevmode\nobreak\ $d-1) shows evidence for amplitude and
phase modulation across the binary orbit similar to what is expected in the
case of a tidally tilted pulsation axis. However, the odd number of amplitude
minima and maxima and phase zero-crossings, combined with the $\pi/2$ rad
phase variation are not consistent with predictions from the tidally tilted or
trapped pulsator models (Fuller et al., 2020). Though, it is possible that
future developments of this model that include the Coriolis force could
produce predictions that more closely match our observations.

Currently, the $\pi/2$ rad phase variations we find are most similar to that
of V456 Cyg, where small phase and normal amplitude variations were observed
in a subset of g modes in a circular, synchronised binary (Van Reeth et al.,
2022). Because of this, the authors argue that the modes are tidally perturbed
and have contributions from non-axisymmetric components. Despite the
similarities, it is not clear whether or not the same mechanism that modifies
the g modes in V456 Cyg can modify the p modes in U Gru. Alternatively, due to
the moderate rotation of the pulsating A-star, we could be observing the
effects of rotation modifying and confining p modes to ‘island modes’ (e.g.,
Ouazzani et al., 2012; Reese et al., 2013, 2017). Moderate to rapid stellar
rotation can preferentially trap pulsations to specific latitudes and
longitudes, depending on the mode geometry and rotation rate (Aerts et al.,
2010). As such, moderate rotation that is synchronised with the binary orbit
could, in theory, produce an observed amplitude modulation that is
commensurate with the orbital period. However, there is no prediction for the
observed phase behaviour of such rotationally modified modes. It is also worth
noting that while U Gru hosts multiple independent pulsation modes, most do
not show any amplitude or phase variation with the orbit. This suggests that
the mechanism behind the observed amplitude and phase variation in the
dominant pulsation mode is not ubiquitous for all modes. Interestingly, other
systems such as RS Cha and HD 265435 also host some pulsations that display
amplitude and phase modulation and some others that do not (Steindl et al.,
2021; Jayaraman et al., 2022).

The observed series of frequencies that are separated by (multiples of) the
orbital frequency cannot be explained by any currently proposed tidal
mechanism. Instead, we present a toy model to demonstrate that the observed
signals can be explained by eclipse mapping. The results of the toy model
demonstrate that eclipse mapping can produce a series of frequencies that are
separated by the orbital frequency and that are asymmetrically distributed
around the intrinsic pulsation frequency, as is observed in the case of U Gru.
This model also demonstrates that the specific resulting signature in the
periodogram depends on the mode geometry, such that some mode geometries might
not experience any observable effect from eclipse mapping for a given orbital
configuration. The mode geometry dependence can explain why we observe some
pulsation modes in U Gru to experience eclipse mapping effects, while others
to not although they originate from the same system. This logic can be further
extended to other pulsating stars in eclipsing binaries such as RS Cha, TIC
63328020, TZ Dra, HL Dra, V1031 Ori, and VV Ori, where some show complex
series of frequencies separated by the orbital frequency and others do not
(Steindl et al., 2021; Rappaport et al., 2021; Lee, 2021; Southworth et al.,
2021; Shi et al., 2021; Kahraman Aliçavuş et al., 2022).

While it has been hypothesised that eclipse mapping can serve as a means of
mode identification for pulsations in well characterised eclipsing binary
systems, there are numerous observational considerations to account for
(Rodríguez et al., 2004; Reed et al., 2005; Gamarova et al., 2005). Primarily,
the amplitude of the eclipse mapping effect depends on the mode geometry and
amplitude, as well as the specifics of the binary orbit. To this end,
unambiguous mode identification through eclipse mapping would require the
observer to be able to identify all of the frequencies distributed about a
given pulsation frequency. Given the typical instrumental and astrophysical
noise components in astronomical time series, this task is challenging.
Furthermore, if any additional tidal phenomena are active which produce
frequencies separated from the pulsation frequency by the orbital frequency, a
comparison to theoretical predictions would need to account for their
contributions as well. Although, compared to the model dependent methods of
mode identification via amplitude ratios or phase differences from multi-
colour photometry (Garrido, 2000; Handler, 2008), eclipse mapping provides an
alternative that does not depend on the underlying detailed stellar model.

U Gru’s status as an oEA star places it in a growing subset of stars with some
form of tidally influenced pulsations that have a recent history of mass
transfer. Other oEA stars in this category include TZ Dra, HL Dra, V1031 Ori,
and V456 Cyg. Future studies that focus on pulsations in oscillating Algol
type systems would benefit from investigating for the signature of tidally
tilted pulsation axes, or otherwise tidally influenced pulsations in a
systematic manner. As we are observing the whole sky with high precision
space-based data, we expect to find more systems with an increasing diversity
of tidally related phenomena. Increasing the sample of such pulsators will
enable us to investigate the underlying physical mechanisms, as well as
investigate their dependence on stellar properties and potentially their
evolutionary history. Future missions such as PLATO (Rauer et al., 2014) will
be of great benefit in this realm as it will deliver multi-colour photometry.
With high-cadence multi-colour photometry, future work will be able to
independently constrain mode identification to further scrutinise the
underlying mechanisms behind the increasing diversity of observed tidally
related phenomena. Indeed, future work should investigate the wavelength
dependence to the observed effects of eclipse mapping, and the extent of the
combined power of mode identification through the combination of eclipse
mapping and mulit-filter amplitude and phase ratios, which often suffer from
limitations when individually applied (Daszyńska-Daszkiewicz et al., 2002;
Dupret et al., 2003).

## Acknowledgements

We thank the anonymous referee for their comments which improved this
manuscript. CJ acknowledges support from the Netherlands Research School of
Astronomy (NOVA). The research leading to these results has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement N∘670519: MAMSIE; CJ,
DMB, AT), from the Research Foundation Flanders (FWO) under grant agreement
G0A2917N (BlackGEM; CJ), from the KU Leuven Research Council (grant
C16/18/005: PARADISE; AT), from the Research Foundation Flanders (FWO) under
grant agreements G0H5416N (ERC Runner Up Project; AT), as well as from the
BELgian federal Science Policy Office (BELSPO) through PRODEX grant PLATO
(AT). TVR and DMB gratefully acknowledge financial support from FWO under
grant agreement numbers 12ZB620N and 1286521N, respectively.

This work is based on observations collected at the European Southern
Observatory under ESO programme 0104.D-0209. This paper includes data
collected by the TESS mission. Funding for the TESS mission is provided by the
NASA’s Science Mission Directorate. This paper includes data collected by the
TESS mission, which are publicly available from the Mikulski Archive for Space
Telescopes (MAST). This work has made use of the VALD database, operated at
Uppsala University, the Institute of Astronomy RAS in Moscow, and the
University of Vienna.

This work has made use of the numpy (Harris et al., 2020), matplotlib (Hunter,
2007), pymc3 (Salvatier et al., 2016), and exoplanet (Foreman-Mackey et al.,
2021) python packages.

## References

  * Aerts et al. (2010) Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology
  * Asplund et al. (2005) Asplund, M., Grevesse, N., & Sauval, A. J. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 336, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ed. I. Barnes, Thomas G. & F. N. Bash, 25
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Balona (2018) Balona, L. A. 2018, MNRAS, 476, 4840
  * Bigot & Dziembowski (2002) Bigot, L. & Dziembowski, W. A. 2002, A&A, 391, 235
  * Bowman et al. (2019) Bowman, D. M., Johnston, C., Tkachenko, A., et al. 2019, ApJ, 883, L26
  * Bowman et al. (2016) Bowman, D. M., Kurtz, D. W., Breger, M., Murphy, S. J., & Holdsworth, D. L. 2016, MNRAS, 460, 1970
  * Brancewicz & Dworak (1980) Brancewicz, H. K. & Dworak, T. Z. 1980, Acta Astron., 30, 501
  * Daszyńska-Daszkiewicz et al. (2002) Daszyńska-Daszkiewicz, J., Dziembowski, W. A., Pamyatnykh, A. A., & Goupil, M. J. 2002, A&A, 392, 151
  * Dekker et al. (2000) Dekker, H., D’Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4008, Optical and IR Telescope Instrumentation and Detectors, ed. M. Iye & A. F. Moorwood, 534–545
  * Donati et al. (1997) Donati, J. F., Semel, M., Carter, B. D., Rees, D. E., & Collier Cameron, A. 1997, MNRAS, 291, 658
  * Dupret et al. (2003) Dupret, M. A., De Ridder, J., De Cat, P., et al. 2003, A&A, 398, 677
  * Foreman-Mackey et al. (2021) Foreman-Mackey, D., Luger, R., Agol, E., et al. 2021, exoplanet: Gradient-based probabilistic inference for exoplanet data & other astronomical time series, if you use this software, please cite it as below.
  * Fuller et al. (2013) Fuller, J., Derekas, A., Borkovits, T., et al. 2013, MNRAS, 429, 2425
  * Fuller et al. (2020) Fuller, J., Kurtz, D. W., Handler, G., & Rappaport, S. 2020, MNRAS, 498, 5730
  * Gamarova et al. (2005) Gamarova, A. Y., Mkrtichian, D. E., & Rodríguez, E. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 333, Tidal Evolution and Oscillations in Binary Stars, ed. A. Claret, A. Giménez, & J. P. Zahn, 258
  * Gamarova et al. (2003) Gamarova, A. Y., Mkrtichian, D. E., Rodriguez, E., Costa, V., & Lopez-Gonzalez, M. J. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 292, Interplay of Periodic, Cyclic and Stochastic Variability in Selected Areas of the H-R Diagram, ed. C. Sterken, 369
  * Garrido (2000) Garrido, R. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 210, Delta Scuti and Related Stars, ed. M. Breger & M. Montgomery, 67
  * Hadrava (1995) Hadrava, P. 1995, A&AS, 114, 393
  * Handler (2008) Handler, G. 2008, Communications in Asteroseismology, 157, 106
  * Handler et al. (2022) Handler, G., Jayaraman, R., Kurtz, D. W., Fuller, J., & Rappaport, S. A. 2022, arXiv e-prints, arXiv:2201.01722
  * Handler et al. (2020) Handler, G., Kurtz, D. W., Rappaport, S. A., et al. 2020, Nature Astronomy, 4, 684
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Holdsworth et al. (2016) Holdsworth, D. L., Kurtz, D. W., Smalley, B., et al. 2016, MNRAS, 462, 876
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90
  * Ilijic et al. (2004) Ilijic, S., Hensberge, H., Pavlovski, K., & Freyhammer, L. M. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 318, Spectroscopically and Spatially Resolving the Components of the Close Binary Stars, ed. R. W. Hilditch, H. Hensberge, & K. Pavlovski, 111–113
  * Jayaraman et al. (2022) Jayaraman, R., Handler, G., Rappaport, S. A., et al. 2022, ApJ, 928, L14
  * Jerzykiewicz et al. (2020) Jerzykiewicz, M., Pigulski, A., Handler, G., et al. 2020, MNRAS, 496, 2391
  * Johnston et al. (2021) Johnston, C., Aimar, N., Abdul-Masih, M., et al. 2021, MNRAS, 503, 1124
  * Kahraman Aliçavuş et al. (2022) Kahraman Aliçavuş, F., Handler, G., Aliçavuş, F., et al. 2022, MNRAS, 510, 1413
  * Kupka et al. (2000) Kupka, F. G., Ryabchikova, T. A., Piskunov, N. E., Stempels, H. C., & Weiss, W. W. 2000, Baltic Astronomy, 9, 590
  * Kurtz (1982) Kurtz, D. W. 1982, MNRAS, 200, 807
  * Kurtz et al. (2020) Kurtz, D. W., Handler, G., Rappaport, S. A., et al. 2020, MNRAS, 494, 5118
  * Lee (2021) Lee, J. W. 2021, PASJ, 73, 809
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447
  * Mkrtichian et al. (2003) Mkrtichian, D. E., Nazarenko, V., Gamarova, A. Y., et al. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 292, Interplay of Periodic, Cyclic and Stochastic Variability in Selected Areas of the H-R Diagram, ed. C. Sterken, 113
  * Montgomery & O’Donoghue (1999) Montgomery, M. H. & O’Donoghue, D. 1999, Delta Scuti Star Newsletter, 13, 28
  * Murphy et al. (2014) Murphy, S. J., Bedding, T. R., Shibahashi, H., Kurtz, D. W., & Kjeldsen, H. 2014, MNRAS, 441, 2515
  * Ouazzani et al. (2012) Ouazzani, R. M., Dupret, M. A., & Reese, D. R. 2012, A&A, 547, A75
  * Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9
  * Preece et al. (2019) Preece, H. P., Jeffery, C. S., & Tout, C. A. 2019, MNRAS, 489, 3066
  * Press & Rybicki (1989) Press, W. H. & Rybicki, G. B. 1989, ApJ, 338, 277
  * Prša et al. (2011) Prša, A., Matijevic, G., Latkovic, O., Vilardell, F., & Wils, P. 2011, PHOEBE: PHysics Of Eclipsing BinariEs, Astrophysics Source Code Library, record ascl:1106.002
  * Prša & Zwitter (2005) Prša, A. & Zwitter, T. 2005, ApJ, 628, 426
  * Rappaport et al. (2021) Rappaport, S. A., Kurtz, D. W., Handler, G., et al. 2021, MNRAS, 503, 254
  * Rauer et al. (2014) Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249
  * Reed et al. (2001) Reed, M., Kawaler, S. D., & Kleinman, S. J. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 226, 12th European Workshop on White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald, & S. Goodchild, 181
  * Reed et al. (2005) Reed, M. D., Brondel, B. J., & Kawaler, S. D. 2005, ApJ, 634, 602
  * Reese et al. (2017) Reese, D. R., Lignières, F., Ballot, J., et al. 2017, A&A, 601, A130
  * Reese et al. (2013) Reese, D. R., Prat, V., Barban, C., van ’t Veer-Menneret, C., & MacGregor, K. B. 2013, A&A, 550, A77
  * Reyniers & Smeyers (2003) Reyniers, K. & Smeyers, P. 2003, A&A, 404, 1051
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Rodríguez et al. (2004) Rodríguez, E., García, J. M., Gamarova, A. Y., et al. 2004, MNRAS, 353, 310
  * Sacco et al. (2014) Sacco, G. G., Morbidelli, L., Franciosini, E., et al. 2014, A&A, 565, A113
  * Saio & Gautschy (2004) Saio, H. & Gautschy, A. 2004, MNRAS, 350, 485
  * Salvatier et al. (2016) Salvatier, J., Wiecki, T. V., & Fonnesbeck, C. 2016, PeerJ Computer Science, 2, e55
  * Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835
  * Schwarzenberg-Czerny (2003) Schwarzenberg-Czerny, A. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 292, Interplay of Periodic, Cyclic and Stochastic Variability in Selected Areas of the H-R Diagram, ed. C. Sterken, 383
  * Shi et al. (2021) Shi, X.-d., Qian, S.-b., Li, L.-j., & Liao, W.-p. 2021, MNRAS, 505, 6166
  * Simon & Sturm (1994) Simon, K. P. & Sturm, E. 1994, A&A, 281, 286
  * Smeyers & Martens (1983) Smeyers, P. & Martens, L. 1983, A&A, 125, 193
  * Southworth et al. (2021) Southworth, J., Bowman, D. M., & Pavlovski, K. 2021, MNRAS, 501, L65
  * Southworth et al. (2020) Southworth, J., Bowman, D. M., Tkachenko, A., & Pavlovski, K. 2020, MNRAS, 497, L19
  * Steindl et al. (2021) Steindl, T., Zwintz, K., & Bowman, D. M. 2021, A&A, 645, A119
  * Tkachenko (2015) Tkachenko, A. 2015, A&A, 581, A129
  * Tkachenko et al. (2013) Tkachenko, A., Van Reeth, T., Tsymbal, V., et al. 2013, A&A, 560, A37
  * Van Reeth et al. (2022) Van Reeth, T., Southworth, J., Van Beeck, J., & Bowman, D. M. 2022, A&A, 659, A177

##  Appendix A Pulsating binary model

We construct a binary system consisting of two spherical, non-tidally
distorted stars in a Keplerian orbit. The orbit is defined by the masses of
the two stars (M1,2), their orbital period (Porb), reference epoch of superior
conjunction ($t_{0,sc})$ , and separation ($a$), and the inclination of their
orbit with respect to the observers line of sight ($i$). In addition to their
masses, the stars are characterised by their radii (R1,2) and what percent of
the total light seen by the observer they contribute ($\ell_{1,2}$).
Furthermore, we include a third light contribution that has ambiguous origin.

The surface of each star is discretized on a spherical grid with radius r=R⋆,
latitude $-\pi\leq\theta\leq\pi$, and longitude $0\leq\phi\leq 2\pi$. At any
point in time, only a particular range of latitudes and longitudes on the
stellar surface are visible, denoted as ($\theta_{0},\phi_{0}$)t. The user is
allowed to decide if either or both stars are pulsating, and assign a mode
identification ($\ell,m$) and frequency ($f_{\rm pulse})$ to the pulsation
mode. For this model, only one pulsation per star is allowed. The system, and
therefore the surface of each star is inclined according to the binary
inclination, changing the visible range of latitudes and longitudes on the
stellar surface. The inclined surface ranges are denoted as
($\theta_{i},\phi_{i}$)t. Finally, we allow for the surface to be inclined
again by some angle $\beta$ to simulate the case of a tidally tilted
pulsation. The final visible surface is then denoted by
($\theta_{\beta},\phi_{\beta}$)t. We then calculate the perturbation to the
surface brightness following the given mode geometry according to:

$\begin{gathered}L_{r}=a_{r}Y_{\ell}^{m}\left(\theta,\phi\right)\exp\left(-i2\pi
ft\right)\\\ L_{\theta}=b_{r}\frac{\partial
Y_{\ell}^{m}\left(\theta,\phi\right)}{\partial\theta}\exp\left(-i2\pi
ft\right)\\\ L_{\theta}=\frac{b_{r}}{\sin\theta}\frac{\partial
Y_{\ell}^{m}\left(\theta,\phi\right)}{\partial\phi}\exp\left(-i2\pi
ft\right),\\\ \end{gathered}$ (1)

where $Y_{\ell}^{m}\left(\theta,\phi\right)$ are spherical harmonics, and
$a_{r}$ and $b_{r}$ are the radial and horizontal amplitudes at the stellar
surface, related through the pulsation frequency in the corotating frame
$f_{co}$ by:

$\frac{a_{r}}{b_{r}}=\frac{GM_{\star}}{f_{co}^{2}R_{\star}^{3}}.$ (2)

For simplicity, we set $a_{r}=1$ and calculate $b_{r}$ through Eq. 2.

We initialise a grid of time points, and progress the positions of the stars
according to the orbit. At each time point, we check if the surface of either
star is being eclipsed, and if so apply a simple boolean mask to the points on
the stellar surface that are in eclipse. We calculate the summed fluxes of the
stars according to their visible (and potentially pulsating) surfaces and
scale them according to their light contributions considering any potential
third light. We provide the total summed binary and pulsation flux, as well as
the individual fluxes of each component.