# Asteroseismology of double-mode radial $\delta$ Scuti stars:
AE Ursae Majoris and RV Arietis

J. Daszyńska-Daszkiewicz, P. Walczak, W. Szewczuk, W. Niewiadomski
Instytut Astronomiczny, Uniwersytet Wrocławski, Kopernika 11, 51-622 Wrocław,
Poland
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We construct complex seismic models of two high-amplitude $\delta$ Sct stars,
AE UMa and RV Ari, each pulsating in two radial modes: fundamental and first
overtone. The models reproduce, besides the frequencies of two radial modes,
also the amplitude of bolometric flux variations (the non-adiabatic parameter
$f$) for the dominant mode. Applying the Monte Carlo–based Bayesian analysis,
we derive strong constraints, on the parameters of the model as well as on the
free parameters of the theory. A vast majority of seismic models of the two
stars are just at the beginning of hydrogen-shell burning and a small fraction
is at the very end of an overall contraction. The stars have a similar age of
about 1.6 Gyr for the hydrogen-shell burning phase. Both stars have unusual
low overshooting from the convective core; about 0.02 and 0.004 of the
pressure scale height for AE UMa and RV Ari, respectively. This result
presumably indicates that overshooting should vary with time and scale with a
decreasing convective core. The efficiency of convection in the envelope of
both stars is rather low and described by the mixing length parameter
$\alpha_{\rm MLT}$ of about 0.3$-$0.6. The third frequency of RV Ari,
confirmed by us in the TESS photometry, can only be associated with mixed
nonradial modes $\ell=1,~{}g_{4}-g_{8}$ or $\ell=2,~{}g_{10}-g_{12}$. We
include the dipole mode into our Bayesian modelling and demonstrate its huge
asteroseismic potential.

###### keywords:

stars: evolution – stars: oscillation – Physical Data and Processes: opacity,
convection– stars: individual: AE UMa, RV Ari

††pubyear: 2021††pagerange: Asteroseismology of double-mode radial $\delta$
Scuti stars:  AE Ursae Majoris and RV Arietis–13

##  1 Introduction

Asteroseismology of stars pulsating in more than one radial mode is of
particular importance because the period ratio of such modes takes values in a
very narrow range. The high-amplitude $\delta$ Scuti stars (HADS), a special
subclass of $\delta$ Sct variables, often pulsate in two radial modes, usually
in the fundamental and first overtone mode (e.g., Breger, 2000; McNamara,
2000; Furgoni, 2016; Yang et al., 2021). $\delta$ Scuti stars are classical
pulsating variables of AF spectral type and their instability is driven by the
opacity mechanism operating in the second helium ionization zone (Chevalier,
1971), with a small contribution from the hydrogen ionization region
(Pamyatnykh, 1999). The masses of $\delta$ Sct pulsators are in the range of
about 1.6 - 2.6 $\mathrm{M}_{\odot}$ and most of them are in the main-sequence
phase of evolution (e.g., Breger & Pamyatnykh, 1998; Bowman et al., 2016).
Radial and non-radial pulsations in pressure (p) and gravity (g) modes are be
excited.

HADS stars change their brightness in the $V$-passband in the range greater
than 0.3 mag. They are in an advanced phase of main-sequence evolution or,
usually, already in a post-main sequence phase (e.g., Breger, 2000). HADS
pulsators have typically low rotational velocities, below $V_{\rm rot}\sin
i=40~{}{\mathrm{km~{}s^{-1}}}$ (Breger, 2000), although there is at least one
exception, i.e., V2367 Cyg with the rotational velocity of about
$100~{}{\mathrm{km~{}s^{-1}}}$ (Balona et al., 2012).

From the fitting of frequencies of just two radial modes, one can already
obtain valuable constraints on global stellar parameters such as a mass,
effective temperature, luminosity (e.g., Petersen & Christensen-Dalsgaard,
1996; Daszyńska-Daszkiewicz et al., 2022; Netzel & Smolec, 2022). However, to
get more unambiguous solution and more information about a star, e.g., on
chemical composition, mixing processes or efficiency of convection, including
nonradial modes or other seismic tools is essential. In particular, the non-
adiabatic parameter $f$ is most suitable for obtaining reliable constraints on
convection in the outer layers of $\delta$ Sct stars (Daszyńska-Daszkiewicz et
al., 2003). The parameter $f$ gives the relative amplitude of the radiative
flux perturbation at the photosphereic level. Its diagnostic potential for
constraining the mixing length parameter $\alpha_{\rm MLT}$ has been already
demonstrated many times for the AF-type pulsators, e.g., $\beta$ Cas, AB Cas,
20 CVn (Daszyńska-Daszkiewicz et al., 2003; Daszyńska-Daszkiewicz, 2007), FG
Vir (Daszyńska-Daszkiewicz et al., 2005), SX Phe (Daszyńska-Daszkiewicz et
al., 2020; Daszyńska-Daszkiewicz et al., 2023), the prototype $\delta$ Sct
(Daszyńska-Daszkiewicz et al., 2021) and BP Peg (Daszyńska-Daszkiewicz et al.,
2022; Daszyńska-Daszkiewicz et al., 2023). The main results for AE UMa and RV
Ari were published by Daszyńska-Daszkiewicz et al. (2023), where we showed for
the four HADS stars that only the seismic models computed with the OPAL
opacities (Iglesias & Rogers, 1996) are caught within the observed error box
in the HR diagram. Seismic models computed with OP tables (Seaton, 2005) and
OPLIB tables (Colgan et al., 2016) were much cooler and less luminous.

Here, we present the details of complex seismic modelling of AE UMa and RV
Ari, which relies on the simultaneous fitting of the two radial modes and the
non-adiabatic parameter $f$ for the dominant mode. Besides, we present the
Fourier frequency analysis of the TESS space data, mode identification from
the phototometric observables and the asteroseismic potential of the nonradial
mode present in the star RV Ari.

Sect. 2 contains basic information about the stars and determination of main
observational parameters. In Sec. 3, we present the frequency analysis of the
TESS data of the two HADS pulsators. In the case of RV Ari, the ASAS
photometry is also analysed. In Sect. 4, we identify the degree $\ell$ of two
pulsational modes using the method based on the photometric amplitudes and
phases, to confirm, independently of the period ratio, their radial nature.
Sect. 5, presents the details of our complex seismic modelling of both HADS
stars based on the Bayesian analysis using Monte Carlo simulations. In Sect. 6
we include the nonradial mode into seismic modelling of RV Ari. The summary is
given in Sect. 7.

##  2 The two double-mode radial pulsators: AE UMa and RV Ari

Figure 1: The HR diagrams with the position of AE UMa (left panel) and RV Ari
(right panel). The evolutionary tracks were computed with Warsaw-New Jersey
code, assuming the OPAL opacity tables, the AGSS09 solar mixture and the
initial hydrogen abundance $X_{0}=0.70$. The metallicity is indicated in each
panel. The mixing length parameter was $\alpha_{\rm MLT}=0.5$ and zero-
overshooting from the convective core was adopted. The initial velocity of
rotation was $V_{\rm rot,0}=10~{}{\mathrm{km~{}s^{-1}}}$.

AE Ursae Majoris is an A9-spectral type star with the mean brightness in the V
passband of 11.35 mag (SIMBAD Astronomical Database). The variability of the
star was discovered by Greyer et al. (1955) and, firstly, is was classified as
a dwarf Cepheid by Tsesevich (1973) who determined the period of light
variations. The secondary period was found by Szeidl (1974) and Broglia &
Conconi (1975). Garcia et al. (1995) listed it as an SX Phe variable and this
classification is still in GCVS and on the SIMBAD website. However, already
Cox et al. (1979) postulated, on the basis of the period ratio, that AE UMa is
a Population I high-amplitude $\delta$ Sct star. Moreover, Rodriguez et al.
(1992) showed that the metallicty of AE UMa is [m/H]$=-0.3$, using the
photometric index $\delta m_{1}$. Hintz et al. (1997) determined [m/H] from
$-0.1$ to $-0.4$ using an approximate relationship between the metallicity and
the period ratio. Thus, there is no doubt that AE UMa belongs to Population I.
As a majority of HADS stars, AE UMa is a slow rotator with $V\sin i\approx
28~{}{\mathrm{km~{}s^{-1}}}$ (Jönsson et al., 2020). Pócs & Szeidl (2001)
analysing 25 years of photometric observations concluded that the period of
the fundamental radial mode is stable and the period of the first overtone is
decreasing with a rate ${\dot{P}}/P=-7.3\cdot 10^{-8}$ yr-1. According to the
authors amplitudes of both modes undergo only small changes.

Niu et al. (2017) constructed for the first time seismic models of AE UMa
based on the two radial modes and the period changes. From about 440 times of
maximum light, they determined the positive period change for the dominant
mode with a rate of ${\dot{P}}/P=+5.4(1.9)\cdot 10^{-9}$ yr-1. In their
seismic modelling of AE UMa, Niu et al. (2017) ignored all effects of rotation
and fixed the values of overshooting parameter from a convective core $f_{\rm
ov}=0.015$ and the mixing length parameter $\alpha_{\rm MLT}=1.89$. They
concluded that AE UMa is in the post-MS stage of evolution. Recently, Xue et
al. (2022) performed the frequency analysis of the TESS data of AE UMa made in
sector 21. They found two independent frequencies,
$11.6257(2)\,\mathrm{d}^{-1}$ and $15.0314(2)\,\mathrm{d}^{-1}$, as well as 63
harmonics and combinations of them. Using the times of maximum light from
about 46 years, they obtained ${\dot{P}}/P=+2.96(5)\cdot 10^{-9}$ yr-1 for the
dominant period. Xue et al. (2022) demonstrated also a prospect of using the
period changes in asteroseismic modelling and constructed such seismic models
for the fixed values of the mixing length parameter of $\alpha_{\rm
MLT}=1.89$. The authors ignored all effects of rotation and assumed zero-
overshooting from the convective core.

RV Arietis is the Population I star with an A spectral type and the mean
brightness in the V passband of 12.27 mag. The star was identified as variable
by Hoffmeister (1934). Broglia & Pestarino (1955) and Detre (1956) derived the
main period and detected the second mode from the modulation period. These two
periodic variations are explained by excitation of the fundamental and first
overtone radial modes (Cox et al., 1979). RV Ari is a slow rotator with the
projected rotational velocity of $V\sin i\approx 18~{}{\mathrm{km~{}s^{-1}}}$
(Rodriguez et al., 2000). Using the photometric index $\delta m_{1}$,
Rodriguez et al. (1992) determined for RV Ari the above-solar metallicity of
[m/H]$=+0.1$. Pócs et al. (2002) gathered an extensive BVRI photometry,
covering about 20 years, and obtained a decreasing period of the fundamental
mode with a rate ${\dot{P}}/P=-0.6\cdot 10^{-8}$ yr-1 and an increasing period
for the first overtone ${\dot{P}}/P=+0.9\cdot 10^{-8}$ yr-1. The opposite sign
of period changes for the two modes indicates some non-evolutionary effects.
The authors detected also the third signal in their photometry with the
frequency 13.6116 d-1 that can correspond only to nonradial mode.

Casas et al. (2006) presented the first seismic modelling of the star adopting
four discrete values of the mixing length parameter $\alpha_{\rm
MLT}=0.5,1.0,1.5,2.0$. They considered only main-sequence models and obtained
the constraints on effective temperature [7065, 7245] K and on age [1.19,
1.27] Gyr.

In our seismic analysis of both stars, we adopted the whole range of the
effective temperature found in the literature. To derive the luminosity, we
adopted distances determined on the basis of Starhorse2 model (Anders et al.,
2022), using the Gaia EDR3 observations (Gaia Collaboration et al., 2022). The
bolometric corrections were taken from Kurucz models for the microturbulent
velocity $\xi_{t}=2$ and $4\,{\mathrm{km~{}s^{-1}}}$. We considered the
metallicity [m/H]$=-0.5,~{}-0.3,~{}-0.2,~{}-0.1$ for AE UMa and
[m/H]$=0.0,~{}+0.1,~{}+0.3,~{}+0.5$ for RV Ari. The adopted parameters were as
follows:
$\bullet$ AE UMa: $\log T_{\rm eff}=3.88353(2922)$, $\log
L/$L${}_{\odot}=1.0907(896)$,
$\bullet$ RV Ari: $\log T_{\rm eff}=3.8787(367)$, $\log
L/$L${}_{\odot}=1.1029(263)$.

In Fig. 1, we show the position of both stars on the Hertzsprung-Russell
diagram. As one can see, the stars occupy a similar position but their
metallicity is quite different; AE UMa has [m/H] below the solar value and RV
Ari - above the solar value. For guidance, we show also a few evolutionary
tracks computed with Warsaw-New Jersey code described in Sect. 5. The tracks
were computed for the OPAL opacity table (Iglesias & Rogers, 1996) and the
solar chemical mixture ofAsplund et al. (2009), hereafter AGSS09.

##  3 Frequency Analysis

Both stars, AE UMa and RV Ari, were observed in the framework of the TESS
mission (Ricker et al., 2015). Here, we used corrected 120 s cadence
observations delivered by TESS Science Processing Operations Center (SPOC,
Jenkins et al., 2016).

AE UMa was observed in the two 27 d sectors, S21 and S48, which are more than
2 years apart. Therefore, we decided to analyse each sector separately. The
Rayleigh resolution for each sector is about
$\Delta\nu_{R}=1/T=0.037\,\mathrm{d}^{-1}$.

RV Ari was observed in two sectors, 42 and 43, which span 51 days. Since these
sector are consecutive, we analyze them together. The Rayleigh resolution for
the combined sectors is $0.020\,\mathrm{d}^{-1}$. In addition to the space
TESS data, we analysed the ground-based ASAS-3 V-band photometry (Pojmanski,
2002) of RV Ari. ASAS data cover 2518 days what translates into the Rayleigh
resolution $\Delta\nu_{R}=0.0004\,\mathrm{d}^{-1}$).

In the first step, we normalized the TESS light curves by dividing them by a
linear fit. Only data points with quality flag 0 were used. The normalization
was done for each sector separately. In order to extract frequencies of the
light variability, we proceeded the standard pre-whitening procedure.
Amplitude periodograms (Deeming, 1975; Kurtz, 1985) were calculated up to the
Nyquist frequency for TESS 120 s cadence data, i.e., to 360 d-1. The fixed
frequency step in periodogram equal to $5\times 10^{-5}\,\mathrm{d}^{-1}$ was
used for both analyzed stars. In the case of TESS data, as a significance
criterion of a given frequency peak we chose the signal-to-noise ratio,
$S/N=5$. This threshold is higher than the standard value of 4 (Breger, 1993;
Kuschnig et al., 1997), but it corresponds to an estimate made by Baran & Koen
(2021) for TESS data. The noise $N$ was calculated as the mean value in a one
day window centred at the frequency before its extraction.

In the case of data with a high point-to-point precision, there is a risk of
artificially introducing spurious signals in the pre-whitening process.
According to Loumos & Deeming (1978), in their most conservative case,
frequencies that are separated less than 2.5 times the Rayleigh resolution
cannot be resolved properly and may be spurious. Therefore, we decided to skip
frequency with smaller amplitude in such close pairs.

Adopting the above criteria, in the case of AE UMa we found 59 significant
frequency peaks in the S21 data and 57 in the S48 data. In the case of each
sector, two peaks were rejected because of the adopted frequency resolution
$2.5\Delta\nu_{R}$. For RV Ari, we found 137 frequency peaks. Two frequencies
were rejected because of the adopted resolution.

In Fig. 2, we show the three amplitude periodograms calculated for the S48
data of AE UMa, i.e, 1) for the original data (top panel), 2) after
subtracting the frequency $\nu_{1}$ (middle panel) and 3) after subtracting
all significant frequencies (bottom panel). Periodograms for the S21 data are
visually indistinguishable. In Fig. 3, we show four amplitude periodograms for
the S42+S43 data of RV Ari. From top to bottom these are: 1) the periodogram
calculated for the original data, 2) after subtracting $\nu_{1}$, 3) after
subtracting $\nu_{1}$, $\nu_{2}$ and five combinations/harmonics, and 4) after
subtracting all significant signals.

Figure 2: The Fourier amplitude periodograms for AE UMa obtained from the TESS
light curve for the sector S48. The panels from top to bottom show: the
periodogram for the original data, after subtracting one term and after
subtracting 57 terms. A red line indicates the $5S/N$ level. Figure 3: The
Fourier amplitude periodograms for RV Ari obtained from the TESS light curve
from combined sectors S42 and S43. The panels from top to bottom show: the
periodogram for the original data, after subtracting one term, after
subtracting 7 terms and after subtracting 137 terms. A red line indicates the
$5S/N$ level. Figure 4: The Fourier amplitude periodograms for RV Ari obtained
from the ASAS photometry. The panels from top to bottom show: the periodogram
for the original data, after subtracting one term and after subtracting 5
terms. A red line indicates the $4S/N$ level.

Our final frequencies, amplitudes and phases were determined using the
nonlinear least-squares fit using the following formula:

$S(t)=\sum_{i=1}^{N}A_{i}\sin\left(2\pi\left(\nu_{i}t+\phi_{i}\right)\right)+c,$
(1)

where $N$ is the number of sinusoidal components, $A_{i}$, $\nu_{i}$,
$\phi_{i}$ are the amplitude, frequency and phase of the $i-$th component,
respectively, while the $c$ is an offset. Moreover, we applied the correction
to formal frequency errors as suggested by Schwarzenberg-Czerny (1991, the
post-mortem analysis). These corrections for both analyzed stars were of about
$1.5$.

Finally, the entire set of frequencies was searched for harmonics and
combinations. A given frequency was considered a combination if it satisfied
the equation

$\nu_{i}=m\nu_{j}+n\nu_{k}+o\nu_{l}$ (2)

within the Rayleigh resolution. In the case of two-parent combinations one of
the integers, $m$, $n$ or $o$, was set to zero, while in the case of harmonics
two integers were set to zero. Moreover, we assumed that $\nu_{j}$, $\nu_{k}$
and $\nu_{l}$ have higher amplitudes than $\nu_{i}$.

Out of all significant frequency peaks found in TESS data of AE UMa (see
Appendix A, Table 9), only two well known frequencies are independent. We give
their values with the amplitudes and S/N ratios in Table 1. Our results agree
with the recent analysis of the TESS S21 data by Xue et al. (2022). The
authors found two independent frequencies, $11.6257(2)\,\mathrm{d}^{-1}$ and
$15.0314(2)\,\mathrm{d}^{-1}$ as well as similar to ours harmonics and
combinations.

The time span between two sectors of AE UMa is about 740 d. The difference in
a dominant frequency $\nu_{1}$ obtained from S21 and S48 is
$\Delta\nu=\nu_{1}(S48)-\nu_{1}(S21)=-0.000164(7)$ d-1. The corresponding
difference in period amounts to $\Delta
P_{1}=P_{1}(48)-P_{1}(21)=+0.000001214(52)$ d. It gives the period change of
$\Delta P_{1}/\Delta t=+5.98(26)\cdot 10^{-7}$ [d/yr] and a rate of period
changes is ${\dot{P}_{1}}/P_{1}=+6.95(29)\cdot 10^{-6}$ [yr-1]. This rate is
about three orders of magnitude higher than the values obtained by Niu et al.
(2017) and Xue et al. (2022) who applied the O-C method to observations from
about 40 years. Such an abrupt change of period can have a local in time
character and results from nonlinear interactions of pulsational modes. The
rapid period changes over shorter time scales are observed for many $\delta$
Scuti stars (e.g., Breger & Pamyatnykh, 1998; Bowman et al., 2021). For the
first overtone mode we got also an increasing period with the rate of
${\dot{P}}_{2}/P_{2}=+8.54(99)\cdot 10^{-6}$ [yr-1].

Table 1: Independent frequencies found in the TESS observations of AE UMa from sectors 21 and 48. The following columns give the frequency ID, the frequency value, amplitude and signal-to-noise ratio. Formal errors are given in parentheses. ID | $\nu\,[\mathrm{d}^{-1}]$ | A [ppt] | $S/N$
---|---|---|---
S21 |
$\nu_{1}$ | 11.625687(3) | 131.88(2) | 9.5
$\nu_{2}$ | 15.03139(1) | 30.64(2) | 9.4
S48 |
$\nu_{1}$ | 11.625523(4) | 131.49(3) | 8.4
$\nu_{2}$ | 15.03113(2) | 30.77(3) | 8.4
Table 2: Independent frequencies found in the TESS observations of RV Ari from combined sectors 42 and 43. The following columns give the frequency ID, frequency value, amplitude and signal-to-noise ratio. Formal errors are given in parentheses.  ID | $\nu\,[\mathrm{d}^{-1}]$ | A [ppt] | $S/N$
---|---|---|---
$\nu_{1}$ | 10.737880(3) | 128.84(2) | 11.8
$\nu_{2}$ | 13.899137(9) | 39.63(2) | 10.5
$\nu_{3}$ | 13.61183(3) | 11.61(2) | 11.9
Table 3: The frequencies found in the ASAS data for RV Ari, with the amplitudes, signal-to-noise ratios and comments on possible combinations or harmonics. Errors are given in parenthesis. ID | $\nu\,[\mathrm{d}^{-1}]$ | A [mag] | $S/N$ | remarks
---|---|---|---|---
1 | 10.737895(6) | 0.207(5) | 10.8 |
2 | 13.89918(2) | 0.063(5) | 6.5 |
3 | 21.47576(2) | 0.058(6) | 5.2 | $2\nu_{1}$
4 | 3.16123(3) | 0.035(5) | 4.7 | $-1\nu_{1}+1\nu_{3}$
5 | 24.63706(3) | 0.035(5) | 4.7 | $1\nu_{1}+1\nu_{3}$

In the case of RV Ari, three frequencies appeared to be independent. Their
values, amplitudes and S/N ratios are given in Table 2. The rest of 132 peaks
can be explained by various combinations of these three independent
frequencies. The third frequency $\nu_{3}$ has been already suggested from
ground-based photometry by Pócs et al. (2002) and we confirm it in the space
data. Thus, RV Ari is one of a few HADS stars with an unquestionably existing
third independent frequency that can only be associated with a nonradial mode.

Next, we analyzed the ASAS-3 V-band photometry of RV Ari. These data consist
of the photometry made in five different apertures. We used the one with the
smallest mean error. Only points with quality flag A and B were retained. In
the case of ground-based photometry, we adopted $S/N=4$ as a threshold for
significant frequency peaks. The noise was calculated in a wider window of 10
d-1. The amplitude periodograms were calculated up to $150\,\mathrm{d}^{-1}$,
what covers frequency range found in TESS data. The step in periodograms was
set to $10^{-5}\,\mathrm{d}^{-1}$. We found five significant signals, two of
them are independent and the remaining three are combinations and harmonic.
These signals with the amplitudes and $S/N$ ratios are given in Table 3. In
Fig. 4, we show the amplitude periodograms calculated for original data (top
panel), after subtracting one frequency (middle panel) and after subtracting
all 5 significant frequencies (bottom panel).

##  4 Identification of the mode degree $\ell$

Figure 5: Identification of the mode degree $\ell$ for the frequencies
$\nu_{1}=11.6257$ d-1 (upper panels) and $\nu_{2}=15.0314$ d-1 (lower panels)
of AE UMa. The values of $\log T_{\rm eff}$ and $\log L/{\rm L}_{\sun}$
include the observed error box. Three combinations of the atmospheric
metallicity [m/H] and microturbulent velocity $\xi_{t}$ were considered.

Figure 6: A similar figure to Fig. 5 but for the frequencies $\nu_{1}=10.7377$
d-1 and $\nu_{1}=13.8985$ d-1 of RV Ari.

The period ratio corresponding to the frequencies $\nu_{1}$ and $\nu_{2}$
amounts to 0.77343 for AE UMa and 0.77256 for RV Ari. These values strongly
suggest that in each star $\nu_{1}$ corresponds to a radial fundamental mode
and $\nu_{2}$ to a radial first overtone. In this section, we independently
verify this hypothesis using the method of mode identification based on the
photometric amplitudes and phases. To this end, we use time-series photometry
in the Strömgren $uvby$ passbands made by Rodriguez et al. (1992). In Table 4,
we give the amplitudes and phases derived from these data.

Table 4: The photometric amplitudes and phases in the $uvby$ passbands for the two dominant frequencies of AE UMa and RV Ari determined from the data of Rodriguez et al. (1992). The last column give the number of observational data points $N$. star | frequency | $A_{\mathrm{u}}$ | $\phi_{\mathrm{u}}$ | $A_{\mathrm{v}}$ | $\phi_{\mathrm{v}}$ | $A_{\mathrm{b}}$ | $\phi_{\mathrm{b}}$ | $A_{\mathrm{y}}$ | $\phi_{\mathrm{y}}$ | $N$
---|---|---|---|---|---|---|---|---|---|---
| $[\mathrm{d}^{-1}]$ | [mag] | [rad] | [mag] | [rad] | [mag] | [rad] | [mag] | [rad] |
AE UMa | $\nu_{1}$ | 0.2312(17) | 4.301(7) | 0.2941(16) | 4.186(5) | 0.2569(14) | 4.191(5) | 0.2112(15) | 4.176(7) | 229
| $\nu_{2}$ | 0.0411(16) | 4.892(40) | 0.0508(16) | 4.828(32) | 0.0429(14) | 4.841(33) | 0.0348(14) | 4.861(42) |
RV Ari | $\nu_{1}$ | 0.2402(42) | 2.181(19) | 0.3083(39) | 2.103(14) | 0.2657(34) | 2.093(14) | 0.2213(33) | 2.072(16) | 140
| $\nu_{2}$ | 0.0730(43) | 3.115(61) | 0.0909(40) | 3.018(45) | 0.0809(35) | 3.012(44) | 0.0613(34) | 3.048(56) |

Here, we apply the method of Daszyńska-Daszkiewicz et al. (2003) based on a
simultaneous determination of the mode degree $\ell$, the intrinsic mode
amplitude $\varepsilon$ multiplied by $Y^{m}_{\ell}(i,0)$ and the non-
adiabatic parameter $f$ for a given observed frequency. The numbers $\ell$ and
$m$ are the spherical harmonic degree and the azimuthal order, respectively,
and $i$ is the inclination angle.

The intrinsic amplitude $\varepsilon$ of a mode is defined by the formula:

$\delta r(R,\theta,\varphi)=R{\rm Re}\\{\varepsilon
Y_{\ell}^{m}(\theta,\varphi){\rm e}^{-{\rm i}\omega t}\\},$ $None$

which gives the relative local radial displacement of the surface element
caused by a pulsational mode with the angular frequency $\omega$. Other
symbols have their usual meanings. The corresponding parameter $f$ is defined
by changes of the bolometric flux, ${\cal F}_{\rm bol}$ as

$\frac{\delta{\cal F}_{\rm bol}}{{\cal F}_{\rm bol}}={\rm Re}\\{\varepsilon
fY_{\ell}^{m}(\theta,\varphi){\rm e}^{-{\rm i}\omega t}\\}.$ $None$

Both, $\varepsilon$ and $f$ have to be regarded as complex numbers because
pulsations are non-adiabatic. The theoretical values of $f=(f_{R},~{}f_{I})$
are derived from linear non-adiabatic computations of stellar pulsations
whereas $\varepsilon$ is indeterminable under linear theory.

In the linear and zero-rotation approximation, the theoretical expression for
the complex amplitude of the relative total flux variation in a passband
$\lambda$, for a given pulsational mode, can be written in the form (e.g.
Daszyńska-Daszkiewicz et al., 2003; Daszyńska-Daszkiewicz et al., 2005):

${\cal A}_{\lambda}={\tilde{\varepsilon}}\left({\cal
D}_{\ell}^{\lambda}f+{\cal E}_{\ell}^{\lambda}\right),$ $None$

where

${\tilde{\varepsilon}}\equiv\varepsilon Y^{m}_{\ell}(i,0),$ $None$ ${\cal
D}_{\ell}^{\lambda}=b_{\ell}^{\lambda}\frac{1}{4}\frac{\partial\log({\cal
F}_{\lambda}|b_{\ell}^{\lambda}|)}{\partial\log T_{\rm{eff}}},$ $None$ ${\cal
E}_{\ell}^{\lambda}=b_{\ell}^{\lambda}\left[(2+\ell)(1-\ell)-\left(\frac{\omega^{2}R^{3}}{GM}+2\right)\frac{\partial\log({\cal
F}_{\lambda}|b_{\ell}^{\lambda}|)}{\partial\log g}\right],$ $None$

and

$b_{\ell}^{\lambda}=\int_{0}^{1}h_{\lambda}(\mu)\mu P_{\ell}(\mu)d\mu.$ $None$

The term ${\cal D}_{\ell}^{\lambda}$ describes the temperature effects and
${\cal E}_{\ell}^{\lambda}$ combines the geometrical and pressure effect.
$G,~{}M,~{}R$ have their usual meanings. In equation Eq. 5d,
$h_{\lambda}(\mu)$ stands for the limb darkening law and $P_{\lambda}(\mu)$ is
the Legendre polynomial. The partial derivatives of ${\cal F}_{\lambda}(T_{\rm
eff},\log g)$ in ${\cal D}_{\ell}^{\lambda}$ and ${\cal E}_{\ell}^{\lambda}$
as well as $b_{\ell}^{\lambda}(T_{\rm eff},\log g)$ and its derivatives have
to be calculated from model atmospheres. Their values are sensitive to the
metallicity [m/H] and microturbulent velocity $\xi_{t}$. Here, we used Vienna
model atmospheres (Heiter et al., 2002) that include turbulent convection
treatment from Canuto et al. (1996). For the limb darkening law,
$h_{\lambda}(\mu)$, we computed coefficients assuming the non-linear, four-
parametric formula of Claret (2000). The values of the photometric amplitudes
and phases themselves are given by $A_{\lambda}=|{\cal A}_{\lambda}|$ and
$\varphi_{\lambda}=arg({\cal A}_{\lambda})$, respectively.

Next, the system of equations (4) for the four passband $uvby$ was solved for
a given $\ell$ and $(T_{\rm eff},\log g)$ to determine $\tilde{\varepsilon}$
and $f$. We considered the degree $\ell$ and associated complex values of
$\tilde{\varepsilon}$ and $f$ as most probable if there is a clear minimum in
the difference between the calculated and observed photometric amplitudes and
phases. The goodness of the fit is measured by:

$\chi^{2}=\frac{1}{2N-N_{p}}\sum_{i=1}^{N}\frac{\left|{\cal
A}^{obs}_{\lambda_{i}}-{\cal
A}^{cal}_{\lambda_{i}}\right|^{2}}{|\sigma_{\lambda_{i}}|^{2}},$ $None$

where the superscripts $obs$ and $cal$ denote the observed and calculated
complex amplitude ${\cal A}_{\lambda}=A_{\lambda}{\rm e}^{{\rm
i}\varphi_{\lambda}}$, respectively. $N$ is the number of passbands $\lambda$
and $N_{p}$ is the number of parameters to be determined. $N_{p}=4$ because
there are two complex parameters, $\tilde{\varepsilon}$ and $f$. The
observational errors $\sigma_{\lambda}$ are computed as

$|\sigma_{\lambda}|^{2}=\sigma^{2}(A_{\lambda})+A_{\lambda}^{2}\sigma^{2}(\varphi_{\lambda}),$
$None$

where $\sigma(A_{\lambda})$ and $\sigma(\varphi)$ are the errors of the
observed amplitude and phase in a passband $\lambda$, respectively.

In Fig. 5, we show the values of the discriminant $\chi^{2}$ as a function of
$\ell$ for the two frequencies of AE UMa; top panels are for $\nu_{1}$ and
bottom panels for $\nu_{2}$. We considered several values of $T_{\rm eff}$ and
$\log L/$L☉ within the observed error box. We also checked the effect of the
atmospheric metallicity [m/H] and microturbulent velocity $\xi_{t}$. In case
of the dominant mode, for all values of ($\log T_{\rm eff},~{}\log L/{\rm
L}_{\sun}$) and all considered pairs of $\left({\rm[m/H]},~{}\xi_{t}\right)$
the clear minimum of $\chi^{2}$ is at $\ell=0$. Thus, there is no doubt that
$\nu_{1}$ is a radial mode. For the second frequency, the minimum at $\ell=0$
is not significantly smaller than at the other $\ell$’s. However, given that
the visibility of pulsational modes decreases very rapidly with increasing
$\ell$, it is reasonable to assume the $\ell=0$ identification for $\nu_{2}$.

The identification of $\ell$ of the two modes of RV Ari is presented in Fig.
6. Our photometric method clearly indicates that the dominant mode is radial.
For the second frequency the identification is not unambiguous. This is mostly
because of much larger errors in the photometric amplitudes and phases of RV
Ari (about three time larger comparing to AE UMa). It results from lower
number of observational data points and the fact that RV Ari is fainter than
AE UMa. However, from the plot $\chi^{2}$ vs. $\ell$ for $\nu_{2}$, one can
conclude that $\ell=0,1,2,3$ are equally possible. Combining this fact with
the period ratio and the largest visibility factor $b_{\ell}$ (cf. Eq. 5d) for
$\ell=0$, it is safe to assume for further analysis that $\nu_{2}$ is also a
radial mode.

##  5 Complex seismic modelling

The complex seismic modelling consists in the simultaneous matching of
pulsational frequencies and the corresponding values of the non-adiabatic
parameter $f$. The parameter $f$ gives the relative amplitude of the radiative
flux perturbation at the photosphereic level. Its theoretical value for a
given pulsational mode is obtained from non-adiabatic computations of stellar
pulsations and it is complex because there is a phase shift between the
radiative flux variation and radius variation. In the case of $\delta$ Sct
stellar models, the theoretical values of $f$ are very sensitive to the
efficiency of convection in the outer layers and to opacity data (e.g.,
Daszyńska-Daszkiewicz et al., 2003; Daszyńska-Daszkiewicz et al., 2023). By
comparing the theoretical and empirical values of $f$, one can get valuable
constraints on the physical conditions inside the star. Thus, the parameter
$f$ is a seismic tool that carries information about the stellar interior and
that is independent and complementary to the pulsation frequency.

The empirical values of $f$ and $\tilde{\varepsilon}$ were determined from the
amplitudes and phases in the $uvby$ passbands using the method outlined in
Sect. 4. In the case of radial modes $Y_{\ell}^{m}(i,0)=1$ and we have the
value of $\varepsilon$ itself (cf. Eq. 5a), i.e., we can say what are the
percentage changes in the radius caused by each pulsation mode (cf. Eq. 2). As
before, we adopted Vienna model atmospheres (Heiter et al., 2002). Models with
the microturbulent velocity $\xi_{t}=4~{}{\mathrm{km~{}s^{-1}}}$ gave the
smallest errors in the empirical values of $\varepsilon$ and $f$. Therefore,
we adopted $\xi_{t}=4~{}{\mathrm{km~{}s^{-1}}}$ whereas the atmospheric
metallicity [m/H] was changed consistently with the metallicity $Z$ in
evolutionary computations.

Table 5: The expected (E$X$) and median (Med) values of the parameters of the HSB seismic models of the stars AE UMa and RV Ari. The parameters were obtained from the Bayesian analysis based on Monte-Carlo simulations assuming the OPAL opacities and the Vienna model atmospheres with the microturbulent velocity of $\xi_{t}$=4 kms-1. star |  value | $M$ | $Z$ | $X_{0}$ | $\alpha_{\rm{mlt}}$ | age | $V_{\rm{0,rot}}$ | $V_{\rm{rot}}$ | $\log(T_{\rm eff}/{\rm K})$ | $\log{L/{\rm L}_{\sun}}$ | $\alpha_{\rm{ov}}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | [M☉] |  |  |  | [Gyr] | [km$\cdot$s-1] | [km$\cdot$s-1] |  |  |
AE UMa | E$X$ | 1.567(41) | 0.0130(10) | 0.698(17) | 0.43(17) | 1.587(73) | 18.4(11.5) | 18.4(11.5) | 3.8613(36) | 1.091(20) | 0.024(18)
| Med | $1.564^{+0.049}_{-0.040}$ | $0.0131^{+0.0010}_{-0.0009}$ |  $0.697^{+0.023}_{-0.018}$ |  $0.44^{+0.15}_{-0.18}$ |  $1.574^{+0.078}_{-0.057}$ | $17.9^{+12.9}_{-12.5}$ |  $17.8^{+12.9}_{-12.5}$ | $3.8617^{+0.0031}_{-0.0040}$ |  $1.092^{+0.020}_{-0.022}$ |  $0.021^{+0.020}_{-0.012}$
RV Ari | E$X$ | 1.629(43) | 0.0178(18) | 0.689(20) | 0.53(7) | 1.565(72) | 23.9(15.5) | 23.4(15.2) | 3.8484(50) | 1.093(27) | 0.004(2)
| Med | $1.631^{+0.037}_{-0.044}$ | $0.0179^{+0.0020}_{-0.0021}$ |  $0.685^{+0.024}_{-0.015}$ |  $0.53^{+0.07}_{-0.08}$ | $1.552^{+0.085}_{-0.059}$ | $22.1^{+20.8}_{-15.1}$ |  $21.5^{+20.6}_{-14.7}$ | $3.8501^{+0.0027}_{-0.0075}$ |  $1.100^{+0.017}_{-0.038}$ |  $0.004^{+0.002}_{-0.002}$

Table 6: The same as in Table 5 but for the OC seismic models of the stars AE UMa and RV Ari. star |  value | $M$ | $Z$ | $X_{0}$ | $\alpha_{\rm{mlt}}$ | age | $V_{\rm{0,rot}}$ | $V_{\rm{rot}}$ | $\log(T_{\rm eff}/{\rm K})$ | $\log{L/{\rm L}_{\sun}}$ | $\alpha_{\rm{ov}}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | [M☉] |  |  |  | [Gyr] | [km$\cdot$s-1] | [km$\cdot$s-1] |  |  |
AE UMa | E$X$ | 1.532(52) | 0.0120(11) | 0.701(20) | 0.33(17) | 1.706(114) | 17.7(13.3) | 17.5(13.2) | 3.8578(69) | 1.070(31) | 0.060(36)
| Med | $1.535^{+0.050}_{-0.058}$ | $0.0119^{+0.0014}_{-0.0011}$ |  $0.698^{+0.026}_{-0.017}$ |  $0.33^{+0.16}_{-0.17}$ |  $1.683^{+0.158}_{-0.094}$ | $14.6^{+18.0}_{-10.6}$ |  $14.3^{+18.0}_{-10.5}$ | $3.8587^{+0.0055}_{-0.0076}$ |  $1.072^{+0.027}_{-0.032}$ |  $0.055^{+0.049}_{-0.032}$
RV Ari | E$X$ | 1.657(26) | 0.0180(11) | 0.692(12) | 0.59(7) | 1.528(49) | 18.6(15.4) | 18.1(15.0) | 3.8525(49) | 1.115(24) | 0.004(3)
| Med | $1.657^{+0.027}_{-0.026}$ | $0.0180^{+0.0012}_{-0.0011}$ |  $0.693^{+0.012}_{-0.015}$ |  $0.59^{+0.07}_{-0.07}$ | $1.520^{+0.054}_{-0.041}$ | $16.5^{+18.7}_{-12.4}$ |  $15.9^{+18.3}_{-11.9}$ | $3.8532^{+0.0043}_{-0.0041}$ |  $1.118^{+0.021}_{-0.027}$ |  $0.004^{+0.003}_{-0.003}$

We performed an extensive complex seismic modelling of AE UMA and RV Ari by
fitting the two radial mode frequencies and the non-adiabatic parameter $f$
for the dominant modes. In the case of both stars, the second modes had too
low amplitudes to determine the empirical values of $f$ with enough accuracy.
To find seismic models, we used the Bayesian analysis based on Monte Carlo
simulations. Our approach is shortly described in Appendix B and the details
can be found in Daszyńska-Daszkiewicz et al. (2022); Daszyńska-Daszkiewicz et
al. (2023). Here, we just recall the adjustable parameters: mass $M$, initial
hydrogen abundance $X_{0}$, metallicity $Z$, initial rotational velocity
$V_{\rm rot,0}$, convective overshooting parameter $\alpha_{\rm ov}$ and the
mixing length parameter $\alpha_{\rm MLT}$. Because only computations with the
OPAL data give consistent results with the observational values of $(T_{\rm
eff},~{}L/{\rm L}_{\sun})$ (Daszyńska-Daszkiewicz et al., 2023), we adopted
these opacities in all computations. At lower temperature range, i.e., for
$\log T<3.95$, opacity data from Ferguson et al. (2005) were used.

Evolutionary computations were performed using the Warsaw-New Jersey code,
(e.g., Pamyatnykh, 1999). The code takes into account the mean effect of the
centrifugal force, assuming solid-body rotation and constant global angular
momentum during evolution. Because both stars are slow rotators, neglecting
differential rotation is justified. Convection in stellar envelope is treated
in the framework of standard mixing-length theory (MLT) and its efficiency is
measured by the mixing length parameter $\alpha_{\rm MLT}$. The solar chemical
mixture was adopted from Asplund et al. (2009) and the OPAL2005 equation of
state was used (Rogers et al., 1996; Rogers & Nayfonov, 2002). Overshooting
from a convective core in the code is implemented according to Dziembowski &
Pamyatnykh (2008). Their prescription takes into account both the distance of
overshooting $d_{\rm ov}=\alpha_{\rm ov}H_{p}$, where $H_{p}$ is the pressure
scale height and $\alpha_{\rm ov}$ is a free parameter, as well as a hydrogen
profile $X(m)$ in the overshoot layer.

Non-adiabatic stellar pulsations were computed using a linear code of
Dziembowski (1977). The code assumes that the convective flux does not change
during pulsations which is justified if convection is not very efficient in
the envelope. The effects of rotation on pulsational frequencies are taken
into account up to the second order in the framework of perturbation theory.

We calculated evolutionary and pulsational models for each set of randomly
selected parameters $\left(M,~{}X_{0},~{}Z,~{}V_{\rm rot,0},~{}\alpha_{\rm
MLT},~{}\alpha_{\rm ov}\right)$. The number of simulations was about 360 000
for each star. In the case of initial hydrogen abundance $X_{0}$, we assumed a
beta function $B(2,2)$ as a prior probability to limit its value to the
reasonable range, i.e., from 0.65 to 0.75 with $X_{0}=0.7$ as the most
probable. For other parameters we used uninformative priors, i.e., uniform
distributions. The vast majority of our seismic models of the two stars, that
have the values of ($T_{\rm eff},~{}L/{\rm L}_{\sun}$) consistent with the
observational determinations, are already at the beginning of hydrogen-shell
burning (HSB). Only a small fraction of seismic models with proper values of
($T_{\rm eff},~{}L/{\rm L}_{\sun}$) is an overall contraction (OC) phase, at
its very end. In all seismic modes both radial modes, fundamental and first
overtone, are unstable in both stars. In Table 5, we give the expected and
median values of determined parameters of the seismic models in the HSB phase
for the two HADS stars. The errors in parentheses at the expected values are
standard deviations. The median errors were estimated from the 0.84 and 16
quantiles, which correspond to the one standard deviation from the mean value
in the case of a normal distribution. Table 6 contains the same statistics for
the seismic models in the OC phase. The corresponding corner plots and
histograms for the HSB seismic models are presented in Appendix B, in Figs.
B1-B4. The histograms for the OC seismic models look qualitatively similar.

Figure 7: The structure of two seismic models of AE UMa. Their positions on
the HR diagram is indicated in top panels. The models reproduce two radial
mode frequencies and the non-adiabatic parameter $f$ for the dominant mode.
Left panel presents the model in the HSB phase and right panel the model in
the overall contraction. Middle panels show the run of main three gradients
and the mean Rosseland opacity inside the models whereas bottom panels show
the local radiative and convective luminosity.

Two stars have a very similar position in the HR diagram but RV Ari is more
massive and has higher metallicity than AE UMa. The age of both HADS stars is
quite similar and amounts to about 1.6 Gyr, if the stars are in the HSB phase
of evolution. Seismic models in the OC phase are about 100 Myr older in the
case of AE UMa and about 30 Myr younger in the case of RV Ari.

We obtained, that convection in the outer layers of the two stars is not very
efficient. For the HSB seismic models, the mixing length parameter amounts to
about 0.4 for AE UMa and about 0.5 for RV Ari. The OC seismic models of AE UMa
and RV Ari, have $\alpha_{\rm MLT}$ of about 0.3 and 0.6, respectively.

The most striking result is a very small overshooting from the convective
core. This result may indicate that the overshooting parameter should depend
on time (evolution) and, presumably, should scale with a mass and size of the
convective core.

In Fig. 7, we show the structure of two representative complex seismic models
of AE UMa. Seismic models of RV Ari have qualitatively the same structure.
Left panels show the model in the HSB phase and right panels the model in the
OC phase. Both models have the following common parameters:
$X_{0}=0.70,~{}\alpha_{\rm ov}=0.0$ and $\alpha_{\rm MLT}=0.5$. The other
parameters are: $M=1.546M_{\sun},~{}Z=0.0127,~{}\log T_{\rm
eff}=3.8566,~{}\log L/{\rm L}_{\sun}=1.067$ for the HSB model and
$M=1.576M_{\sun},~{}Z=0.0135,~{}\log T_{\rm eff}=3.8575,~{}\log L/{\rm
L}_{\sun}=1.076$ for the OC model. In the top panels, we show the positions of
these models on the HR diagram with the corresponding evolutionary tracks.
Middle panels show the run of main gradients (actual $\nabla$, radiative
$\nabla_{\rm rad}$, adiabatic $\nabla_{\rm ad}$) and the mean Rosseland
opacity $\kappa$ inside each model. The local radiative and convective
luminosity is presented in the bottom panels.

In the case of HSB model, the small helium core has the size of 3% of the
stellar radius and the interior up to $\log(T/{\rm K})\approx 4.8$ is
radiative. The whole energy comes from hydrogen-burning in the shell
proceeding via the CNO cycle. An overproduction of energy in the shell
$L_{r}/L>1$ is used for expansion of the envelope. The radiative gradient
become very large around $\log(T/{\rm K})=4.0$ where the actual gradient split
into the two maxima corresponding to the hydrogen ionization and first
ionization of helium. In this narrow layer the local convective luminosity
becomes important. In the zone of second helium ionization $\log(T/{\rm
K})\approx 4.7$, where pulsational driving occurs, $\nabla$ is only slightly
larger than $\nabla_{\rm ad}$ and the local convective luminosity is zero. In
the case of the OC models, the hydrogen is burned in a small convective core
with the radius of about $3\%R$. The structure of the OC model above the core
is very similar to that of the HSB model.

Table 7: The modal values of the empirical intrinsic mode amplitude $|\varepsilon|$ (a fraction of the radius changes) and the resulting radial velocity amplitude due to pulsations for the two radial modes of AE UMa and RV Ari. | $\nu_{1}$ | $\nu_{2}$
---|---|---
| $\ell=0,~{}p_{1}$ | $\ell=0,~{}p_{2}$
star | $|\varepsilon|$ | $A(V_{\rm puls})$ | $|\varepsilon|$ | $A(V_{\rm puls})$
|  | [km$\cdot$s-1] |  | [km$\cdot$s-1]
AE UMa | 0.0189(24) | 17.7(2.2) | 0.0017(6) | 2.0(7)
RV Ari | 0.0153(4) | 13.9(4) | 0.0029(20) | 3.4(2.3)

As we mentioned at the beginning of this section, from our analysis, we
obtained also the empirical values of the intrinsic mode amplitude
$\varepsilon$ for both radial modes. These values cannot be compared with
theoretical predictions, because we use the linear theory, but it provides us
with an estimate of the relative radius changes and the expected amplitude of
radial velocity variations $A(V_{\rm puls})$. In Table 7, we give the values
of $\varepsilon$ and the corresponding radial velocity amplitude for the two
radial modes of both stars. These are the modal values, i.e., is the most
frequently occurring values in computed seismic models. As one can see, the
radial fundamental modes of AE UMa and RV Ari cause the radius changes of
about 1.9% and 1.5%, respectively. In turn, these radius changes cause the
radial velocity variations with an amplitude of about 18 and 14
${\mathrm{km~{}s^{-1}}}$, respectively. The first overtone modes have much
smaller radius variations of about 0.2% and 0.3% for AE UMa and RV Ari,
respectively. The amplitude of radial velocity variations caused by the first
overtone modes are about 2.0 and 3.4 ${\mathrm{km~{}s^{-1}}}$ for AE UMa and
RV Ari, respectively.

##  6 Including the nonradial mode of RV Ari into seismic modelling

Table 8: Examples of seismic models of RV Ari with the characteristic of nonradial modes $\ell=1,2$ having frequencies closest to the observed value $\nu_{3}=13.6118$ d-1. All seismic models have $\alpha_{\rm ov}=0.0$ and were computed with the OPAL opacities. The 8th column indicates the evolutionary phases. The last three columns contain a ratio of the energy in gravity propagation zone to the total kinetic energy, the instability parameter $\eta$ and the Ledoux constant. The models indicated with asterisks will be discussed later in the text.  $M$ | $Z$ | $X_{0}$ | $\alpha_{\rm MLT}$ | $\log(T_{\rm eff}/{\rm K})$ | $\log{L/{\rm L}_{\sun}}$ | age | phase | $\nu_{\rm rot}$ | $\nu_{\rm puls}$ | mode | $E_{\rm k,g}/E_{\rm k}$ | $\eta$ | $C_{\rm n\ell}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$[{\rm M}_{\sun}]$ |  |  |  |  |  | [Gyr] |  | [d-1] | [d-1] |  |  |  |
1.590 | 0.015 | 0.70 | 0.55 | 3.8469 | 1.081 | 1.6435 | HSB | 0.206 | 13.69773 | $\ell=1,g_{7}$ | 0.82 | 0.090 | 0.426
|  |  |  |  |  |  |  |  | 13.87880 | $\ell=2,g_{12}$ | 0.77 | 0.089 | 0.130
1.624* | 0.016 | 0.70 | 0.55 | 3.8485 | 1.093 | 1.5973 | HSB | 0.203 | 13.70427 | $\ell=1,g_{6}$ | 0.83 | 0.089 | 0.432
|  |  |  |  |  |  |  |  | 13.44373 | $\ell=2,g_{12}$ | 0.76 | 0.090 | 0.147
1.652 | 0.017 | 0.70 | 0.55 | 3.8491 | 1.100 | 1.5684 | HSB | 0.202 | 14.16947 | $\ell=1,g_{5}$ | 0.43 | 0.088 | 0.216
|  |  |  |  |  |  |  |  | 13.35265 | $\ell=2,g_{11}$ | 0.76 | 0.090 | 0.151
1.640 | 0.019 | 0.68 | 0.64 | 3.8493 | 1.098 | 1.5205 | HSB | 0.210 | 13.37437 | $\ell=1,g_{5}$ | 0.90 | 0.089 | 0.468
|  |  |  |  |  |  |  |  | 13.34091 | $\ell=2,g_{10}$ | 0.75 | 0.089 | 0.152
1.557* | 0.018 | 0.68 | 0.50 | 3.8274 | 0.995 | 1.7090 | OC | 0.399 | 14.3346 | $\ell=1,g_{1}$ | 0.07 | 0.066 | 0.030
|  |  |  |  |  |  |  |  | 13.2824 | $\ell=2,g_{4}$ | 0.61 | 0.076 | 0.144

Figure 8: Top panels: the propagation diagram for the HSB (left) and OC
(right) seismic models of RV Ari with the parameters given in the 2nd and 5th
line of Table 8. The horizontal line indicates the value of the observed
frequency $\nu_{3}=13.61183$ d-1. Lower panels: the distribution of the
kinetic energy density for nonradial modes listed in Table 8. The vertical
lines mark the edge of a core.

Our results of the Fourier analysis of the space TESS data confirmed that the
third frequency of RV Ari, $\nu_{3}=13.61183$ d-1, proposed by Pócs et al.
(2002), is a real and independent signal. This frequency can only be
associated with a nonradial mode because of its proximity to the second
frequency $\nu_{2}=13.899137$ d-1 which corresponds to the first overtone
radial mode. The $\nu_{3}$ does not appear in the Rodriguez et al. (1992)
data, so we cannot even try to identify its pulsational mode from the
photometric amplitudes and phases. On the other hand, taking into account the
very small amplitude of this frequency (more than 3 times smaller than the
amplitude of $\nu_{2}$), we doubt that such an attempt would be successful.
For this purpose, new time-series multi-colour photometry or spectroscopy is
required. However, guided by the fact that the visibility of modes in
photometry decreases rapidly with the degree $\ell$, $\nu_{3}$ is quite likely
a dipole or quadrupole mode. Therefore, we will make such working hypothesis
in this Section.

In Table 8, we list the parameters of the four HSB and one OC seismic models
of RV Ari with the main characteristic of dipole and quadrupole modes having
frequencies closest to the observed value of $\nu_{3}=13.61183$d-1. We provide
the ratio of the kinetic energy in the gravity propagation zone to the total
kinetic energy $E_{\rm k,g}/E_{\rm k}$, the normalized instability parameter
$\eta$ and the Ledoux constant $C_{\rm n\ell}$. As one can see, despite of
quite high frequencies, in the case of the HSB seismic models, the modes have
a very strong gravity character with $E_{\rm k,g}$ greater than 70% of the
total $E_{\rm k}$ in all cases but one. Only in the case of HSB model with
$M=1.652$ M☉, the mode $\ell=1,~{}g_{5}$ has this ratio slightly below 0.5. In
the case of the OC seismic model, $\ell=1,~{}g_{1}$ is almost a pure pressure
mode but its frequency is quite far from $\nu_{3}$. The mode $\ell=2,~{}g_{4}$
is mixed with $E_{\rm k,g}$ of about 60%. All modes are pulsational unstable.

In Fig. 8, we show the propagation diagram (top panels) and the distribution
of kinetic energy density of the dipole and quadruple mode (lower panels) for
the seismic models indicated with asterisk in Table 8 (the 2nd and 5th model)
. All quantities are plotted as a function of the fractional radius in a
logarithmic scale. The 2nd model is in the phase of hydrogen-shell burning
whereas the 5th model is in the overall contraction phases. The Lamb frequency
$L_{\ell}$ was depicted for $\ell=1$ and 2. The horizontal line in the top
panels corresponds to the observed frequency $\nu_{3}=13.61183$ d-1. In the
case of the HSB model, the maximum of the Brunt-Väisälä frequency $N^{2}$
occurs at the edge of a small helium core. The small convective core of the OC
model is precisely defined by $N^{2}<0$. The core edges are marked as a
vertical line. The middle and bottom panels show the kinetic energy density
for the modes $\ell=1$ and $\ell=2$, respectively. As one can see, the kinetic
energy density of both modes of the HSB models is large and strongly
concentrated within the helium core of a size of $0.03R$. Modes with such
property have a very strong potential for probing near-core conditions and
chemical composition. In the case of the OC model, the mode $\ell=1,g_{1}$ is
almost pure pressure and its kinetic energy concentrates in the outer layers.
The mixed mode $\ell=2,g_{4}$ has the kinetic energy concentrated within the
chemical gradient zone.

In the next step, we constructed seismic models that fit simultaneously the
two radial mode frequencies, the complex parameter $f$ of the dominant mode
and $\nu_{3}$ as a dipole axisymmetric mode. Again, the Bayesian analysis
based on Monte Carlo simulations was applied. We obtained very narrow ranges
of determined parameters $M$, $X_{0}$, $Z$ with errors about 2 to 5 times
smaller that in Sect. 5. whereas the values of overshooting and mixing length
parameters are again around 0.0 and 0.5, respectively, with a similar
uncertainty. Interestingly, in the case of HSB seismic models our simulations
converge to the two solutions for $(M,~{}X_{0},~{}Z)$ with the following
expected values:

I$.~{}~{}~{}~{}M=1.644(7)\,{\rm M}_{\sun},~{}X_{0}=0.689(10),~{}Z=0.0186(4)$

II$.~{}~{}~{}~{}M=1.626(8)\,{\rm M}_{\sun},~{}X_{0}=0.696(7),~{}Z=0.0164(3)$

This dichotomy is most evident for metallicity $Z$. In the Appendix B, in Fig.
B5, we plot the metallicity $Z$ as a function of the model number. As one can
see, independently of the starting value, the simulations converge only to the
two values of $Z$ given above. In the first solution a dipole mode is always
$g_{7}$ and in the second solution it is always $g_{8}$.

For OC seismic models we got one solution with a higher $Z$:

I$.~{}~{}~{}~{}M=1.640(10)\,{\rm M}_{\sun},~{}X_{0}=0.693(7),~{}Z=0.0181(5)$,

and a dipole mode is $g_{4}$. As in the case of fitting the two radial modes,
the OC seismic models that reproduce also the third frequency $\nu_{3}$ as a
dipole mode are a definite minority.

The interesting result is that the obtained range of the rotational velocity
differs significantly between the HSB and OC seismic models. For the HSB
models, the expected value of the current rotation velocity is in the range
$V_{\rm rot}\in(5,~{}46)\,{\mathrm{km~{}s^{-1}}}$ whereas for the OC models we
obtained the range $V_{\rm rot}\in(32,~{}47)\,{\mathrm{km~{}s^{-1}}}$. Thus,
if we had independent information on the rotation rate, e.g., from the
rotational splitting of dipole modes, then perhaps a choice between HSB and OC
siesmic models would be possible.

##  7 Summary

We presented the analysis of TESS space data and detailed complex seismic
modelling of the two high-amplitude $\delta$ Scuti stars AE UMa and RV Ari.
The Fourier analysis of the TESS light curves revealed the two well-known
frequencies for each HADS. The important result is a confirmation of the third
frequency of RV Ari that has been detected from the ground-based photometry.

The ratio of two dominant frequencies of AE UMa and RV Ari strongly indicates
that two radial modes, fundamental and first overtone, are excited in both
stars. We verified this hypothesis using the method of mode identification
based on the multi-colour photometric amplitudes and phases.

Our seismic modelling of the two HADS stars consisted in simultaneous fitting
of the two radial mode frequencies as well as the complex amplitude of
relative bolometric flux variations of the dominant mode, the so-called
parameter $f$. To this end, the Bayesian analysis based on Monte-Carlo
simulations was used. Our extensive seismic modelling allowed to constrain the
global parameters as well as free parameters. The mixing length parameter
$\alpha_{\rm MLT}$, that describes the efficiency of envelope convection,
amounts to about 0.3$-$0.6. Determination of the narrow range of $\alpha_{\rm
MLT}$ was possible only due to the inclusion of the parameter $f$ into seismic
modelling. All models are in the post-main sequence phase of evolution,
however the question whether it is the HSB or OC phase cannot be unequivocally
resolved. On the other hand, the HSB seismic models account for the vast
majority, so it can be assumed that this phase is much more likely.

An interesting result is a very small value of the overshooting parameter
$\alpha_{\rm ov}$, which describes the amount of mixing at the edge of
convective core during the phase of main-sequence and overall contraction.
This result may be due to the fact that $\alpha_{\rm ov}$ is assumed to be
constant during evolution. Presumably, it should depend on time and scale with
a decreasing convective core.

The third frequency of RV Ari can only be associated with non-radial mode
because of its proximity to the second frequency, which is the first overtone
radial mode. It is very likely a dipole or quadrupole mode because of the disk
averaging effect in photometric amplitudes. We made a working hypothesis that
$\nu_{3}$ is a dipole axisymmetric mode and repeated seismic modelling. Thus,
we fitted the three frequencies and the parameter $f$ for the dominant
frequency. Including the nonradial mode constrained enormously, in particular,
the global stellar parameters. As before, the number of seismic models in the
HSB phase is much larger. Moreover, the HSB and OC seismic models have
different ranges of the rotational velocity. Thus, perhaps independent
information on the rotation would finally decide between these two phases of
evolution.

## Acknowledgements

The work was financially supported by the Polish NCN grant
2018/29/B/ST9/02803. Calculations have been partly carried out using resources
provided by Wroclaw Centre for Networking and Supercomputing
(http://www.wcss.pl), grant No. 265. This paper includes data collected by the
TESS mission. Funding for the TESS mission is provided by the NASA’s Science
Mission Directorate. This work has made use of data from the European Space
Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

## Data Availability

The TESS data are available from the NASA MAST portal
https://archive.stsci.edu/. The ASAS observations are available at the website
of http://www.astrouw.edu.pl/asas. Theoretical computations will be shared on
reasonable request to the corresponding author.

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##  Appendix A The frequencies from the Fourier analysis of the TESS light
curves

List of extracted frequencies for AE UMa and RV Ari in the order they were
found in the TESS data. In Table A1, we give the frequencies of AE UMa
obtained from the data separately from the sector S21 and S48. Table A2
contains the frequencies of RV Ari obtained from the two combined sectors, S42
and S43.

Table 9: The whole set of frequencies found in the TESS observations of AE UMa from sector 21 and from sector 48. The following columns give the frequency ID, frequency value, amplitude and signal-to-noise ratio. Formal errors are given in parentheses. S21 - 57 significant peaks |
---|---
ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$ | ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$
$\nu_{1}$ | 11.625687(3) | 131.88(2) | 9.5 | $6\nu_{1}+2\nu_{2}$ | 99.8153(6) | 0.69(2) | 8.8
$2\nu_{1}$ | 23.251413(9) | 49.07(2) | 9.4 | $3\nu_{1}-2\nu_{2}$ | 4.8158(7) | 0.64(2) | 8.3
$\nu_{2}$ | 15.03139(1) | 30.64(2) | 9.4 | $5\nu_{1}-\nu_{2}$ | 43.0954(7) | 0.61(2) | 8.6
$\nu_{1}+\nu_{2}$ | 26.65705(2) | 21.36(2) | 9.4 | $7\nu_{1}+2\nu_{2}$ | 111.4422(7) | 0.59(2) | 8.6
$-\nu_{1}+\nu_{2}$ | 3.40569(2) | 20.06(2) | 9.4 | $6\nu_{1}-\nu_{2}$ | 54.7204(8) | 0.56(2) | 7.9
$3\nu_{1}$ | 34.87682(2) | 19.42(2) | 9.4 | $\nu_{1}+3\nu_{2}$ | 56.7190(8) | 0.52(2) | 8.2
$2\nu_{1}+\nu_{2}$ | 38.28254(4) | 12.16(2) | 9.4 | $4\nu_{1}-2\nu_{2}$ | 16.4426(9) | 0.50(2) | 7.9
$4\nu_{1}$ | 46.50241(5) | 9.04(2) | 9.4 | $9\nu_{1}+\nu_{2}$ | 119.6604(9) | 0.49(2) | 7.5
$3\nu_{1}+\nu_{2}$ | 49.90797(6) | 6.87(2) | 9.4 | $7\nu_{1}-\nu_{2}$ | 66.3453(9) | 0.48(2) | 7.9
$2\nu_{1}-\nu_{2}$ | 8.22030(7) | 6.64(2) | 9.4 | $3\nu_{2}$ | 45.0946(9) | 0.46(2) | 7.8
$5\nu_{1}$ | 58.1278(1) | 4.26(2) | 9.4 | $9\nu_{1}$ | 104.629(1) | 0.41(2) | 7.7
$4\nu_{1}+\nu_{2}$ | 61.5335(1) | 3.86(2) | 9.4 | $8\nu_{1}+2\nu_{2}$ | 123.068(1) | 0.39(2) | 7.5
$2\nu_{2}$ | 30.0628(1) | 3.34(2) | 9.4 | $8\nu_{1}-\nu_{2}$ | 77.973(1) | 0.38(2) | 7.3
$\nu_{1}+2\nu_{2}$ | 41.6885(1) | 3.30(2) | 9.3 | $5\nu_{1}-2\nu_{2}$ | 28.064(1) | 0.36(2) | 7.3
$3\nu_{1}-\nu_{2}$ | 19.8458(2) | 2.68(2) | 9.3 | $10\nu_{1}+\nu_{2}$ | 131.286(1) | 0.33(2) | 7.2
$5\nu_{1}+\nu_{2}$ | 73.1590(2) | 2.59(2) | 9.2 | $9\nu_{1}+2\nu_{2}$ | 134.694(1) | 0.32(2) | 6.9
$6\nu_{1}$ | 69.7533(2) | 2.30(2) | 9.4 | $2\nu_{1}+3\nu_{2}$ | 68.345(1) | 0.32(2) | 7.4
$2\nu_{1}+2\nu_{2}$ | 53.3137(2) | 1.99(2) | 9.2 | $3\nu_{1}+3\nu_{2}$ | 79.969(2) | 0.28(2) | 5.9
$-\nu_{1}+2\nu_{2}$ | 18.4369(2) | 1.88(2) | 9.4 | $4\nu_{1}+3\nu_{2}$ | 91.595(2) | 0.26(2) | 6.1
$6\nu_{1}+\nu_{2}$ | 84.7848(2) | 1.75(2) | 9.2 | $10\nu_{1}$ | 116.254(2) | 0.25(2) | 6.0
$3\nu_{1}+2\nu_{2}$ | 64.9390(3) | 1.40(2) | 9.0 | $5\nu_{1}+3\nu_{2}$ | 103.221(2) | 0.25(2) | 6.1
$7\nu_{1}$ | 81.3789(3) | 1.35(2) | 9.2 | $9\nu_{1}-\nu_{2}$ | 89.598(2) | 0.25(2) | 5.8
$4\nu_{1}-\nu_{2}$ | 31.4720(4) | 1.22(2) | 9.1 | $-\nu_{1}+3\nu_{2}$ | 33.470(2) | 0.22(2) | 5.9
$7\nu_{1}+\nu_{2}$ | 96.4101(4) | 1.15(2) | 9.0 | $7\nu_{1}+3\nu_{2}$ | 126.474(2) | 0.21(2) | 6.1
$4\nu_{1}+2\nu_{2}$ | 76.5645(4) | 1.09(2) | 8.9 | $10\nu_{1}+2\nu_{2}$ | 146.317(2) | 0.20(2) | 6.0
$5\nu_{1}+2\nu_{2}$ | 88.1901(5) | 0.92(2) | 8.9 | $11\nu_{1}+\nu_{2}$ | 142.911(2) | 0.21(2) | 5.5
$8\nu_{1}$ | 93.0043(5) | 0.79(2) | 8.5 | $6\nu_{1}-2\nu_{2}$ | 39.694(2) | 0.21(2) | 5.1
$-2\nu_{1}+2\nu_{2}$ | 6.8111(6) | 0.79(2) | 8.8 | $6\nu_{1}+3\nu_{2}$ | 114.844(2) | 0.20(2) | 5.7
$8\nu_{1}+\nu_{2}$ | 108.0344(6) | 0.74(2) | 8.7 |  |  |  |
S48 - 55 significant peaks |
ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$ | ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$
$\nu_{1}$ | 11.625523(4) | 131.49(3) | 8.4 | $6\nu_{1}+2\nu_{2}$ | 99.8167(6) | 0.78(3) | 8.0
$2\nu_{1}$ | 23.25104(1) | 48.68(3) | 8.4 | $8\nu_{1}$ | 93.0057(6) | 0.76(3) | 8.1
$\nu_{2}$ | 15.03113(2) | 30.77(3) | 8.4 | $3\nu_{1}-2\nu_{2}$ | 4.8139(7) | 0.67(3) | 7.5
$\nu_{1}+\nu_{2}$ | 26.65662(2) | 21.42(3) | 8.4 | $5\nu_{1}-\nu_{2}$ | 43.0980(8) | 0.63(3) | 7.9
$-\nu_{1}+\nu_{2}$ | 3.40558(2) | 19.90(3) | 8.4 | $7\nu_{1}+2\nu_{2}$ | 111.4416(9) | 0.57(3) | 7.4
$3\nu_{1}$ | 34.87683(3) | 19.39(3) | 8.4 | $1\nu_{1}+3\nu_{2}$ | 56.7190(9) | 0.55(3) | 7.5
$2\nu_{1}+\nu_{2}$ | 38.28240(4) | 12.26(3) | 8.4 | $9\nu_{1}+\nu_{2}$ | 119.6611(9) | 0.51(3) | 7.9
$4\nu_{1}$ | 46.50250(5) | 9.07(3) | 8.3 | $6\nu_{1}-\nu_{2}$ | 54.7249(9) | 0.51(3) | 7.8
$3\nu_{1}+\nu_{2}$ | 49.90817(7) | 6.93(3) | 8.4 | $7\nu_{1}-\nu_{2}$ | 66.3499(10) | 0.49(3) | 7.2
$2\nu_{1}-\nu_{2}$ | 8.21952(7) | 6.57(3) | 8.3 | $3\nu_{2}$ | 45.094(1) | 0.47(3) | 7.1
$5\nu_{1}$ | 58.1284(1) | 4.29(3) | 8.4 | $4\nu_{1}-2\nu_{2}$ | 16.438(1) | 0.44(3) | 7.1
$4\nu_{1}+\nu_{2}$ | 61.5339(1) | 3.89(3) | 8.3 | $9\nu_{1}$ | 104.631(1) | 0.43(3) | 7.2
$2\nu_{2}$ | 30.0622(1) | 3.42(3) | 8.4 | $8\nu_{1}+2\nu_{2}$ | 123.069(1) | 0.41(3) | 7.4
$\nu_{1}+2\nu_{2}$ | 41.6877(1) | 3.36(3) | 8.4 | $2\nu_{1}+3\nu_{2}$ | 68.345(1) | 0.41(3) | 6.6
$3\nu_{1}-\nu_{2}$ | 19.8452(2) | 2.70(3) | 8.3 | $8\nu_{1}-\nu_{2}$ | 77.974(1) | 0.36(3) | 6.9
$5\nu_{1}+\nu_{2}$ | 73.1594(2) | 2.59(3) | 8.4 | $9\nu_{1}+2\nu_{2}$ | 134.691(1) | 0.34(3) | 6.5
$6\nu_{1}$ | 69.7543(2) | 2.32(3) | 8.2 | $5\nu_{1}-2\nu_{2}$ | 28.064(1) | 0.34(3) | 6.5
$2\nu_{1}+2\nu_{2}$ | 53.3138(2) | 2.03(3) | 8.3 | $10\nu_{1}+\nu_{2}$ | 131.290(2) | 0.31(3) | 6.2
$-\nu_{1}+2\nu_{2}$ | 18.4370(3) | 1.91(3) | 8.3 | $-\nu_{1}+3\nu_{2}$ | 33.466(2) | 0.29(3) | 5.9
$6\nu_{1}+\nu_{2}$ | 84.7857(3) | 1.75(3) | 8.2 | $10\nu_{1}$ | 116.258(2) | 0.26(3) | 6.1
$3\nu_{1}+2\nu_{2}$ | 64.9394(3) | 1.45(3) | 8.1 | $3\nu_{1}+3\nu_{2}$ | 79.969(2) | 0.26(3) | 5.8
$7\nu_{1}$ | 81.3800(4) | 1.36(3) | 8.2 | $9\nu_{1}-\nu_{2}$ | 89.599(2) | 0.24(3) | 5.2
$4\nu_{1}-\nu_{2}$ | 31.4702(4) | 1.24(3) | 8.0 | $5\nu_{1}+3\nu_{2}$ | 103.221(2) | 0.24(3) | 5.5
$7\nu_{1}+\nu_{2}$ | 96.4111(4) | 1.17(3) | 8.2 | $4\nu_{1}+3\nu_{2}$ | 91.595(2) | 0.24(3) | 5.8
$4\nu_{1}+2\nu_{2}$ | 76.5648(4) | 1.11(3) | 8.1 | $10\nu_{1}+2\nu_{2}$ | 146.318(2) | 0.22(3) | 5.4
$5\nu_{1}+2\nu_{2}$ | 88.1910(5) | 0.89(3) | 7.7 | $6\nu_{1}+3\nu_{2}$ | 114.849(2) | 0.22(3) | 5.2
$8\nu_{1}+\nu_{2}$ | 108.0364(6) | 0.79(3) | 7.6 | $7\nu_{1}+3\nu_{2}$ | 126.469(3) | 0.19(3) | 5.2
$-2\nu_{1}+2\nu_{2}$ | 6.8116(6) | 0.77(3) | 8.1 |  |  |  |
Table 10: The whole set of 135 frequencies found in the TESS observations of RV Ari from combined sectors 42 and 43. The following columns give the frequency ID, frequency value, amplitude and signal-to-noise ratio. Formal errors are given in parentheses. ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$ | ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$
---|---|---|---|---|---|---|---
$\nu_{1}$ | 10.737880(3) | 128.84(2) | 11.8 | $3\nu_{1}+\nu_{2}-\nu_{3}$ | 32.5010(5) | 0.68(2) | 9.6
$2\nu_{1}$ | 21.475767(9) | 42.22(2) | 11.8 | $8\nu_{1}+2\nu_{2}$ | 113.7017(5) | 0.68(2) | 8.9
$\nu_{2}$ | 13.899137(9) | 39.63(2) | 10.5 | $7\nu_{1}-\nu_{2}$ | 61.2661(6) | 0.62(2) | 9.7
$\nu_{1}+\nu_{2}$ | 24.63702(1) | 25.92(2) | 11.3 | $\nu_{1}+2\nu_{2}+\nu_{3}$ | 52.1476(6) | 0.63(2) | 9.0
$-\nu_{1}+\nu_{2}$ | 3.16127(1) | 25.82(2) | 11.1 | $2\nu_{2}+\nu_{3}$ | 41.4103(6) | 0.62(2) | 9.2
$3\nu_{1}$ | 32.21365(2) | 16.26(2) | 11.7 | $2\nu_{1}+2\nu_{2}+\nu_{3}$ | 62.8861(6) | 0.59(2) | 9.1
$2\nu_{1}+\nu_{2}$ | 35.37489(3) | 13.77(2) | 10.8 | $5\nu_{1}+3\nu_{2}$ | 95.3866(6) | 0.59(2) | 8.4
$\nu_{3}$ | 13.61183(3) | 11.61(2) | 11.9 | $4\nu_{1}+\nu_{2}-\nu_{3}$ | 43.2389(6) | 0.58(2) | 9.5
$2\nu_{1}-\nu_{2}$ | 7.57660(4) | 9.91(2) | 11.0 | $4\nu_{1}+3\nu_{2}$ | 84.6488(6) | 0.57(2) | 7.6
$4\nu_{1}$ | 42.95153(5) | 7.52(2) | 11.4 | $-\nu_{1}+3\nu_{2}$ | 30.9597(7) | 0.56(2) | 7.2
$3\nu_{1}+\nu_{2}$ | 46.11283(5) | 6.97(2) | 10.6 | $3\nu_{1}-\nu_{2}+\nu_{3}$ | 31.9264(7) | 0.56(2) | 9.6
$\nu_{1}+2\nu_{2}$ | 38.53621(6) | 5.76(2) | 10.3 | $3\nu_{1}+3\nu_{2}$ | 73.9107(7) | 0.54(2) | 7.0
$2\nu_{2}$ | 27.79831(6) | 5.67(2) | 9.7 | $6\nu_{1}+3\nu_{2}$ | 106.1244(7) | 0.54(2) | 8.2
$3\nu_{1}-\nu_{2}$ | 18.31451(6) | 5.61(2) | 10.9 | $8\nu_{1}$ | 85.9029(7) | 0.53(2) | 8.4
$-\nu_{1}+\nu_{3}$ | 2.87400(7) | 5.23(2) | 11.8 | $9\nu_{1}+\nu_{2}$ | 110.5396(7) | 0.51(2) | 8.6
$\nu_{1}+\nu_{3}$ | 24.34969(8) | 4.43(2) | 11.6 | $2\nu_{3}$ | 27.2230(7) | 0.53(2) | 9.5
$4\nu_{1}+\nu_{2}$ | 56.85068(9) | 3.97(2) | 10.8 | $5\nu_{1}-\nu_{3}$ | 40.0776(7) | 0.50(2) | 8.7
$5\nu_{1}$ | 53.6895(1) | 3.49(2) | 10.9 | $7\nu_{1}+3\nu_{2}$ | 116.8627(7) | 0.49(2) | 7.8
$4\nu_{1}-\nu_{2}$ | 29.0525(1) | 3.38(2) | 10.6 | $\nu_{1}+2\nu_{3}$ | 37.9620(8) | 0.48(2) | 9.1
$2\nu_{1}+\nu_{3}$ | 35.0876(1) | 3.37(2) | 11.8 | $9\nu_{1}+2\nu_{2}$ | 124.4400(8) | 0.47(2) | 8.3
$-\nu_{1}+2\nu_{2}$ | 17.0603(1) | 3.34(2) | 9.7 | $4\nu_{1}-\nu_{2}+\nu_{3}$ | 42.6646(8) | 0.47(2) | 8.9
$2\nu_{1}+2\nu_{2}$ | 49.2740(1) | 2.85(2) | 9.1 | $3\nu_{1}+2\nu_{2}+\nu_{3}$ | 73.6236(8) | 0.45(2) | 8.6
$5\nu_{1}+\nu_{2}$ | 67.5886(1) | 2.74(2) | 11.2 | $-2\nu_{1}+\nu_{2}+\nu_{3}$ | 6.0349(9) | 0.43(2) | 8.6
$\nu_{2}+\nu_{3}$ | 27.5109(1) | 2.52(2) | 10.8 | $5\nu_{1}+\nu_{2}+\nu_{3}$ | 81.1997(8) | 0.44(2) | 9.6
$2\nu_{1}-\nu_{3}$ | 7.8641(2) | 2.17(2) | 11.5 | $5\nu_{1}-\nu_{2}+\nu_{3}$ | 53.4028(9) | 0.43(2) | 8.4
$5\nu_{1}-\nu_{2}$ | 39.7904(2) | 2.02(2) | 10.6 | $-\nu_{1}+2\nu_{2}+\nu_{3}$ | 30.6718(9) | 0.40(2) | 7.8
$3\nu_{1}+2\nu_{2}$ | 60.0120(2) | 1.97(2) | 8.9 | $5\nu_{1}+\nu_{2}-\nu_{3}$ | 53.9774(9) | 0.40(2) | 8.1
$3\nu_{1}+\nu_{3}$ | 45.8255(2) | 1.95(2) | 11.8 | $5\nu_{1}-2\nu_{2}$ | 25.8919(9) | 0.40(2) | 7.2
$\nu_{1}+\nu_{2}+\nu_{3}$ | 38.2489(2) | 1.87(2) | 10.5 | $4\nu_{1}-2\nu_{2}$ | 15.1540(9) | 0.39(2) | 7.2
$6\nu_{1}+\nu_{2}$ | 78.3264(2) | 1.82(2) | 11.1 | $8\nu_{1}+3\nu_{2}$ | 127.6006(10) | 0.38(2) | 7.7
$6\nu_{1}$ | 64.4274(2) | 1.77(2) | 10.2 | $5\nu_{1}+\nu_{3}$ | 67.3016(10) | 0.38(2) | 8.5
$4\nu_{1}+2\nu_{2}$ | 70.7499(2) | 1.59(2) | 9.7 | $4\nu_{1}+2\nu_{2}+\nu_{3}$ | 84.361(1) | 0.36(2) | 8.2
$2\nu_{1}+\nu_{2}+\nu_{3}$ | 48.9868(2) | 1.60(2) | 10.6 | $8\nu_{1}-\nu_{2}$ | 72.004(1) | 0.35(2) | 7.7
$-\nu_{1}+\nu_{2}+\nu_{3}$ | 16.7731(2) | 1.52(2) | 11.1 | $6\nu_{1}+\nu_{2}+\nu_{3}$ | 91.938(1) | 0.35(2) | 8.1
$-2\nu_{1}+2\nu_{2}$ | 6.3228(3) | 1.32(2) | 10.1 | $6\nu_{1}-2\nu_{2}$ | 36.629(1) | 0.34(2) | 6.9
$5\nu_{1}+2\nu_{2}$ | 81.4876(3) | 1.37(2) | 10.0 | $9\nu_{1}$ | 96.642(1) | 0.32(2) | 7.0
$3\nu_{1}-\nu_{3}$ | 18.6017(3) | 1.31(2) | 11.1 | $6\nu_{1}-\nu_{2}+\nu_{3}$ | 64.140(1) | 0.32(2) | 7.2
$\nu_{spurious}$ | 0.0633(3) | 1.37(2) | 5.0 | $-2\nu_{1}+3\nu_{2}$ | 20.222(1) | 0.31(2) | 6.6
$7\nu_{1}+\nu_{2}$ | 89.0646(3) | 1.20(2) | 10.5 | $6\nu_{1}-\nu_{3}$ | 50.816(1) | 0.32(2) | 6.8
$6\nu_{1}+2\nu_{2}$ | 92.2254(3) | 1.19(2) | 10.1 | $2\nu_{1}+2\nu_{2}-\nu_{3}$ | 35.664(1) | 0.32(2) | 7.5
$6\nu_{1}-\nu_{2}$ | 50.5285(3) | 1.18(2) | 9.9 | $2\nu_{1}+2\nu_{3}$ | 48.698(1) | 0.31(2) | 7.0
$\nu_{1}+3\nu_{2}$ | 52.4349(3) | 1.22(2) | 8.9 | $3\nu_{1}+2\nu_{2}-\nu_{3}$ | 46.400(1) | 0.31(2) | 6.7
$\nu_{1}-\nu_{2}+\nu_{3}$ | 10.4502(3) | 1.16(2) | 8.8 | $10\nu_{1}+\nu_{2}$ | 121.279(1) | 0.30(2) | 7.6
$3\nu_{1}+\nu_{2}+\nu_{3}$ | 59.7243(3) | 1.14(2) | 10.4 | $3\nu_{1}-\nu_{2}-\nu_{3}$ | 4.702(1) | 0.29(2) | 7.4
$7\nu_{1}$ | 75.1653(4) | 0.92(2) | 10.0 | $7\nu_{1}-2\nu_{2}$ | 47.366(1) | 0.28(2) | 6.5
$7\nu_{1}+2\nu_{2}$ | 102.9636(4) | 0.92(2) | 9.8 | $6\nu_{1}+\nu_{2}-\nu_{3}$ | 64.714(1) | 0.28(2) | 6.8
$2\nu_{1}-\nu_{2}+\nu_{3}$ | 21.1885(4) | 0.88(2) | 9.6 | $9\nu_{1}+3\nu_{2}$ | 138.337(1) | 0.29(2) | 6.5
$4\nu_{1}+\nu_{3}$ | 56.5627(4) | 0.86(2) | 10.2 | $10\nu_{1}+2\nu_{2}$ | 135.176(1) | 0.28(2) | 7.0
$4\nu_{1}-\nu_{3}$ | 29.3397(4) | 0.84(2) | 11.1 | $4\nu_{1}+2\nu_{2}-\nu_{3}$ | 57.139(1) | 0.28(2) | 7.0
$3\nu_{2}$ | 41.6977(4) | 0.84(2) | 7.7 | $5\nu_{1}+2\nu_{2}+\nu_{3}$ | 95.100(1) | 0.26(2) | 7.1
$2\nu_{1}+3\nu_{2}$ | 63.1732(5) | 0.81(2) | 7.6 | $7\nu_{1}+\nu_{2}+\nu_{3}$ | 102.676(1) | 0.26(2) | 6.3
$8\nu_{1}+\nu_{2}$ | 99.8021(5) | 0.77(2) | 9.9 | $\nu_{1}+2\nu_{2}-\nu_{3}$ | 24.924(1) | 0.25(2) | 5.9
$\nu_{1}+\nu_{2}-\nu_{3}$ | 11.0251(5) | 0.76(2) | 7.3 | $6\nu_{1}+4\nu_{2}$ | 120.024(1) | 0.26(2) | 5.6
$2\nu_{1}+\nu_{2}-\nu_{3}$ | 21.7632(5) | 0.76(2) | 9.9 | $\nu_{1}+3\nu_{2}+\nu_{3}$ | 66.047(1) | 0.25(2) | 5.2
$3\nu_{1}-2\nu_{2}$ | 4.4152(5) | 0.71(2) | 8.7 | $\nu_{1}+4\nu_{2}$ | 66.335(1) | 0.26(2) | 7.0
$4\nu_{1}+\nu_{2}+\nu_{3}$ | 70.4626(5) | 0.69(2) | 10.2 | $7\nu_{1}+2\nu_{2}+\nu_{3}$ | 116.576(1) | 0.25(2) | 6.3
Table 11: continued ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$ | ID | $\nu\,(\mathrm{d}^{-1})$ | A (ppt) | $S/N$
---|---|---|---|---|---|---|---
$6\nu_{1}+2\nu_{2}+\nu_{3}$ | 105.839(2) | 0.24(2) | 6.6 | $-\nu_{1}+2\nu_{3}$ | 16.488(2) | 0.19(2) | 5.5
$2\nu_{2}-\nu_{3}$ | 14.185(2) | 0.23(2) | 5.7 | $7\nu_{1}+4\nu_{2}$ | 130.763(2) | 0.19(2) | 5.4
$4\nu_{2}$ | 55.594(2) | 0.22(2) | 5.3 | $-2\nu_{1}+2\nu_{2}+\nu_{3}$ | 19.937(2) | 0.18(2) | 5.7
$2\nu_{1}+4\nu_{2}$ | 77.072(2) | 0.22(2) | 5.2 | $\nu_{1}+\nu_{2}+2\nu_{3}$ | 51.859(2) | 0.19(2) | 6.2
$5\nu_{1}+4\nu_{2}$ | 109.288(2) | 0.22(2) | 5.2 | $11\nu_{1}+\nu_{2}$ | 132.016(2) | 0.18(2) | 5.8
$6\nu_{1}+\nu_{3}$ | 78.037(2) | 0.21(2) | 5.9 | $8\nu_{1}+4\nu_{2}$ | 141.501(2) | 0.18(2) | 5.4
$8\nu_{1}+\nu_{2}+\nu_{3}$ | 113.416(2) | 0.21(2) | 6.4 | $5\nu_{1}+2\nu_{2}-\nu_{3}$ | 67.875(2) | 0.18(2) | 6.0
$5\nu_{1}-\nu_{2}-\nu_{3}$ | 26.176(2) | 0.21(2) | 5.5 | $2\nu_{1}+\nu_{2}+2\nu_{3}$ | 62.598(2) | 0.17(2) | 5.2
$4\nu_{1}+4\nu_{2}$ | 98.547(2) | 0.20(2) | 5.3 | $9\nu_{1}-\nu_{2}$ | 82.743(2) | 0.16(2) | 5.4
$10\nu_{1}+3\nu_{2}$ | 149.077(2) | 0.20(2) | 5.9 | $7\nu_{1}-\nu_{3}$ | 61.554(2) | 0.15(2) | 5.3
$\nu_{2}+2\nu_{3}$ | 41.123(2) | 0.20(2) | 5.9 | $7\nu_{1}+\nu_{3}$ | 88.773(3) | 0.14(2) | 5.0
$3\nu_{1}+2\nu_{3}$ | 59.436(2) | 0.19(2) | 5.3 |  |  |  |

##  Appendix B Seismic models from the Bayesian analysis based on Monte Carlo
simulations

To perform seismic modelling with the most accurate sampling of various
parameters, we used Bayesian analysis based on Monte Carlo simulations. This
analysis was based on the Gaussian likelihood function defined as (e.g.,
Jørgensen & Lindegren, 2005; da Silva et al., 2006; Rodrigues et al., 2017;
Jiang & Gizon, 2021)

${\cal
L}(E|{\mathbf{H}})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\cdot{\rm
exp}\left(-\frac{({\cal O}_{i}-{\cal M}_{i})^{2}}{2\sigma_{i}^{2}}\right),$
(3)

where ${\mathbf{H}}$ is the hypothesis that represents adjustable parameters,
i.e., mass $M$, initial hydrogen abundance $X_{0}$, metallicity $Z$, initial
rotational velocity $V_{\rm rot,0}$, the mixing length parameter $\alpha_{\rm
MLT}$ and convective overshooting parameter $\alpha_{\rm ov}$. The evidence
$E$ represents the calculated observables ${\cal M}_{i}$, i.e., the effective
temperature $T_{\rm eff}$, luminosity $L/L_{\odot}$, pulsational frequencies,
$\nu_{1}$ and $\nu_{2}$, and the non-adiabatic parameter $f$ for the dominant
mode. that can be directly compared with the observed parameters ${\cal
O}_{i}$ determined with the errors $\sigma_{i}$.

All seismic models of both stars reproduce the exact value of the dominant
frequency as the radial fundamental mode. The second frequency, corresponding
the first radial overtone, is reproduced with an accuracy of at least 0.0005
d-1, which is approximately equivalent to numerical accuracy. The empirical
values of $f$, both its real and imaginary part, for the dominant mode is
reproduced at least within the $3\sigma$ errors. With these criteria, the vast
majority of seismic models, that have the values of $T_{\rm eff},L/{\rm
L}_{\sun}$ consistent with the observational values, are in the phase of
hydrogen-shell burning (HSB). In Fig. B1, we show the corner plots for the
parameters of the HSB seismic models of AE UMa. The corresponding histograms
are presented in Fig. B2. The histograms were normalised to 1.0, so the
numbers on the Y-axis times 100 are the percentage of models with a given
parameter range. Figs. B3 and B4 show the same plots for the parameters of the
HSB seismic models of RV Ari.

As we mentioned in the main text, for RV Ari we made the working hypothesis
that its third frequency $\nu_{3}=13.6118$ d-1 is a dipole axisymmetric mode.
We constructed seismic models that reproduce the $\nu_{3}$ with an accuracy of
0.0005 d-1. In the case of HSB models our simulations converged to the two
well-constrained solutions in ($M,X_{0},Z$). This dichotomy is best seen for
metallicity $Z$ what is shown in Fig. B5, where the values of $Z$ are plotted
as a function of the model number. Bluish colours correspond to the models
that converge to a higher $Z$ solution, while greenish colours to the models
that converge to a lower $Z$ solution,

Figure 9: The corner plots for parameters obtained from complex seismic
modeling of the star AE UMa. All models are in the hydrogen-shell burning
phases of evolution and were computed with the OPAL opacities.

Figure 10: The corresponding histograms for determined parameters of AE UMa.
Figure 11: The corner plots for parameters obtained from complex seismic
modeling of the star RV Ari. All models are in the hydrogen-shell burning
phases of evolution and were computed with the OPAL opacities.

Figure 12: The corresponding histograms for determined parameters of RV Ari.
Figure 13: The metallicity $Z$ as a function of the model number for HSB
models of RV Ari obtained from the Bayesian analysis, which aims to also fit
the third frequency $\nu_{3}$ as a dipole axisymmetric mode. Bluish and
greenish colours correspond to simulations converging to the higher and lower
$Z$, respectively.


# The rUNSWift SPL Field Segmentation Dataset

Wentao Lu School of Computer Science and Engineering
The University of New South Wales
<EMAIL_ADDRESS>
###### Abstract

In RoboCup SPL, soccer field segmentation has been widely recognised as one of
the most critical robot vision problems. Key challenges include dynamic light
condition, different calibration status for individual robot, various camera
prospective and more. In this paper, we propose a dataset that contains 20
videos recorded with Nao V5/V6 humanroid
robots111https://www.softbankrobotics.com/emea/en/nao by team rUNSWift under
different circumstances. Each of the videos contains several consecutive high
resolution frames and the corresponding labels for field. We propose this
dataset to provide training data for the league to overcome field segmentation
problem. The dataset will be available online for download. Details of
annotation and example of usage will be explained in later sections.

##  1 Introduction

Since the beginning of RoboCup SPL competition, field segmentation/recognition
remains a key problem of robot vision system. In early years, field line
detection with RANSAC[3] based field recognition is widely used. This method
can roughly find the field area as per the rule, field lines are 50
centimetres away from the field boundary. This method works but one critical
disadvantage is in some cases, the robot is not able to see the field lines.
In late years, many teams have proposed colour detection based field
recognition[5] which appears to be more robust and flexible to use. To detect
a certain colour, the method must overcome several challenges. For example,
since 2016, our league has introduced outdoor competitions where the light
condition varies and could affect the colour detection[2]. Recently, thanks to
the introduction of Nao V6 robot, the on-board computation power enriches
which allows us to run more complicatied learning algorithms to recognise
ball, robots, and also the soccer field. Learning algorithms require large
amount of data. While many teams have kindly shared the ball and robot image
dataset, datasets for field segmentation remains few. To make contribution to
this particular problem, we propose this dataset and corresponding labels.
The complete dataset can be downloaded from
https://drive.google.com/file/d/17EDkVGvqW46Cn8a7_9uGuXDlqvweOvE4/view?usp=sharing.

##  2 Dataset Specifications

In this section, we will explain how the data look like and how we collect the
data. In Section 2.1, we will show the format and other specifications of the
dataset. Data collection details will be discussed in Section 2.2.

###  2.1 Dataset format

Table 1 shows the data format and resolution for each video in the dataset.
Our dataset contains 7 short videos and 13 longer ones. Each video is
decomposed into consecutive image frames where the image format is listed in
Table 1 as well. Due to the various situation when we collect the data, only
short videos have bottom frames attached while the longer ones we only keep
top frames as field segmentation is required mainly on top camera. Resolution
for top camera remains 1280 * 960 while the corresponding label is downsampled
by 5 times to 256 * 192. Resolution for bottom images of shorter videos is
halfed to 640 * 480. Raw image format from the robot is YUV and for most
videos in our dataset, the image frames have been converted to BMP format.

Index | Resolution(Top) | Label(Top) | Resolution(Bottom) | Image Format | #Frames
---|---|---|---|---|---
1 | 1280 * 960 | 256 * 192 | 640 * 480 | YUV | 19
2 | 1280 * 960 | 256 * 192 | 640 * 480 | YUV | 19
3 | 1280 * 960 | 256 * 192 | 640 * 480 | YUV | 19
4 | 1280 * 960 | 256 * 192 | 640 * 480 | BMP | 5
5 | 1280 * 960 | 256 * 192 | 640 * 480 | BMP/YUV | 6
7 | 1280 * 960 | 256 * 192 | 640 * 480 | BMP/YUV | 19
8 | 1280 * 960 | 256 * 192 | 640 * 480 | BMP/YUV | 7
9 | 1280 * 960 | 256 * 192 | N/A | BMP | 259
10 | 1280 * 960 | 256 * 192 | N/A | BMP | 199
11 | 1280 * 960 | 256 * 192 | N/A | BMP | 121
12 | 1280 * 960 | 256 * 192 | N/A | BMP | 271
13 | 1280 * 960 | 256 * 192 | N/A | BMP | 113
14 | 1280 * 960 | 256 * 192 | N/A | BMP | 68
15 | 1280 * 960 | 256 * 192 | N/A | BMP | 119
16 | 1280 * 960 | 256 * 192 | N/A | BMP | 298
17 | 1280 * 960 | 256 * 192 | N/A | BMP | 251
18 | 1280 * 960 | 256 * 192 | N/A | BMP | 210
19 | 1280 * 960 | 256 * 192 | N/A | BMP | 181
20 | 1280 * 960 | 256 * 192 | N/A | BMP | 213
21 | 1280 * 960 | 256 * 192 | N/A | BMP | 188
Table 1: Dataset specs

###  2.2 Data collection

In this section, we will discuss how we collect those data in detail. We use
Nao V5/V6 robots to run different behaviour in different conditions to collect
data. When the robot running, the raw images, usually a video, will be
recorded with other data. The combined data will be logged into a dump file
which will later be downloaded to local machine. Team rUNSWift has two tools
to record and decode the dump file, which are Offnao and Vatnao[4]. More
information about these tools can be founded in rUNSWift’s team report. Vatnao
loads saved dump file and decodes the video into consecutive frames which can
be saved in selected image format.
Table 2 shows location, light condition and behaviour setting for each video.
7 shorter videos were recorded in Germany when team rUNSWift participated in
RoboCup German Open 2019. All those videos are recorded in indoor field with
different light condition and robot behaviour. Longer videos were recorded in
Sydney, Australia with 3 different locations which are UNSW robot laboratory,
Internation Conference Centre’s indoor field and outdoor field. Those data
recorded in outdoor field also cover different time where we aim to include
more natural and various light condition.
Table 2 also includes body motion and recording type. Body motion explains
whether the robot that is recording the dump file moves. Recording type
demonstrates what condition is while we record the data. Single-testing means
only one robot is operating on the field while recording data. Multi-testing
means multiple robots are running where those robots can occupy parts of the
field which is a harder case for field segmentation.

index | #frame | location | light condition | body motion | recording type
---|---|---|---|---|---
1 | 19 | Indoor(Germany) | Constant(low) | Static | Multi-testing
2 | 19 | Indoor(Germany) | Constant(low) | Static | Multi-testing
3 | 19 | Indoor(Germany) | Constant(low) | Static | Multi-testing
4 | 5 | Indoor(Germany) | Constant(high) | Static | Single-testing
5 | 5 | Indoor(Germany) | Constant(high) | Static | Single-testing
7 | 19 | Indoor(Germany) | Constant(low) | Dynamic | Single-testing
8 | 7 | Indoor(Germany) | Constant(low) | Dynamic | Game
9 | 259 | Indoor(UNSW) | Constant(medium) | Dynamic | Single-testing
10 | 199 | Outdoor(Sydney) | Night | Static | Single-testing
11 | 121 | Indoor(UNSW) | Uneven(medium) | Dynamic | Single-testing
12 | 271 | Indoor(UNSW) | Uneven(medium) | Dynamic | Multi-testing
13 | 113 | Indoor(UNSW) | Uneven(medium) | Dynamic | Multi-testing
14 | 68 | Outdoor(Sydney) | Night | Static | Single-testing
15 | 119 | Outdoor(Sydney) | Night | Static | Single-testing
16 | 298 | Indoor(Sydney) | Constant(low) | Dynamic | Multi-testing
17 | 251 | Outdoor(Sydney) | Morning(uneven) | Static | Single-testing
18 | 210 | Indoor(UNSW) | Extremely uneven | Dynamic | Single-testing
19 | 181 | Indoor(UNSW) | Extremely uneven | Dynamic | Single-testing
20 | 213 | Indoor(UNSW) | Extremely uneven | Dynamic | Multi-testing
21 | 188 | Indoor(UNSW) | Extremely uneven | Dynamic | Multi-testing
Table 2: Dataset Information

##  3 Data Annotation

In this section, we will discuss how we label the field and how to utilise the
provided image label. In Section 3.2, a brief statistics for each video will
be demonstrated. A simple tool will also be introduced in Section 3.3, which
allows user to label their own image data.

###  3.1 Labelling system

Figure 1 and 2 show two examples regarding how we label the field. The image
label is a 256 * 192 monochrome picture which is downsampled by 5 times from
full resolution(1280 * 960). Within the image label, the soccer field is
marked as white pixels while all other stuff marked as black. Generally, lines
within the soccer field are also marked with white pixels due to workload
restrictions.

(a) Raw Image

(b) Label

Figure 1: Data pair collected in Germany

(a) Raw Image

(b) Label

Figure 2: Data pair collected in UNSW

###  3.2 Field Statistics

In this section, we will run through a list of statistics to demonstrate how
the data look like from a mathematical prospective.

Figure 3: Statistics for soccer field coverage

Figure 3 shows a statistics for coverage of soccer field of the whole dataset.
As the statistics shown, most images have a field coverage around 50% and 90%
where the average value is 66%. 155 images of total 2390 are field-less(no
field covered) with pure black labels. Individual statistics for each video
can be found in Appendix A.

###  3.3 Tools

A labelling tool is provided along with the dataset. This tool is written in
Python 3 with OpenCV2. When given the image folder path, the function will
load the image and start a mouse click tracking procedure. By clicking left
mouse button, user can draw key points on the image. If all key points marked,
user can click middle mouse button to show the connected component created by
those key points which is the labelled field. If user satisfies the marked
result, click ”S” button on keyboard to ask the tool to save the label file.
If not, click ”Q” button will let the tool quit the labelling function.

(a) Raw Image

(b) Mask

(c) ROI

(d) Points Marked

Figure 4: A sample labelling process

##  4 Usage Example

This dataset can be utilised in different vision applications. Mainly, this
dataset is designed to help solving the field segmentation problem under
various conditions. Machine learning algorithms as well as handcrafted vision
algorithms can both utilise these data for development or validation.
Videos are recorded under a variety of light conditions. This feature allows
user to work on a colour recognition algorithm to find particular colour, such
as Green for the field or White for robots, balls and lines.
Field features are also included in some images, such as corners, center
circles and lines. Using the provided tool, user can generate their own Field
Feature dataset. Since 2019, a statistical learning named Class Conditional
Gaussian Mixture Model[1] has been used by rUNSWift for detecting the field
features where parts of the data used are generated by this dataset and tool.

##  5 Conclusion

In this paper, we propose a dataset collected under different environment
condition. This dataset can be used for developing SPL robot vision algorithms
especially soccer field segmentation.

##  Appendix A Individual Statistics

(a) Statistics for Video index 1

(b) Statistics for Video index 2

(a) Statistics for Video index 3

(b) Statistics for Video index 4

(a) Statistics for Video index 5

(b) Statistics for Video index 7

(a) Statistics for Video index 8

(b) Statistics for Video index 9

(a) Statistics for Video index 10

(b) Statistics for Video index 11

Figure 9:

(a) Statistics for Video index 12

(b) Statistics for Video index 13

Figure 10:

(a) Statistics for Video index 14

(b) Statistics for Video index 15

Figure 11:

(a) Statistics for Video index 16

(b) Statistics for Video index 17

Figure 12:

(a) Statistics for Video index 18

(b) Statistics for Video index 19

Figure 13:

(a) Statistics for Video index 20

(b) Statistics for Video index 21

Figure 14:

## References

  * [1] Ashar, J., Brameld, K., Jones, E. R., Kaur, T., Li, L., Lu, W., Pagnucco, M., Sammut, C., Sheh, Q., Schmidt, P., and et al. runswift team report 2019, 2019.
  * [2] Committee, R. T. Robocup standard platform league (nao) rule book.  https://spl.robocup.org/wp-content/uploads/downloads/Rules2016.pdf.
  * [3] Fischler, M. A., and Bolles, R. C. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography.  Commun. ACM 24, 6 (June 1981), 381–395.
  * [4] rUNSWift.  Tools - robocup spl 2020 documentation.  https://runswift.readthedocs.io/en/latest/tools/index.html.
  * [5] TJArk.  Tjark team description paper & research report 2016.  https://github.com/TJArk-Robotics/coderelease_2016/blob/master/TJArk%20Team%20Research%20Report%202016.pdf.