# Asteroseismology of SZ Lyn using Multi-Band High Time Resolution Photometry
from Ground and Space

J. Adassuriya1, S. Ganesh2, J. L. Gutiérrez3, G. Handler4, Santosh Joshi5 K.
P. S. C. Jayaratne6, K. S. Baliyan2
1Astronomy Division, Arthur C Clarke Institute, Katubedda, Moratuwa, Sri Lanka
2Astronomy and Astrophysics Division, Physical Research Laboratory, Ahmedabad,
India
3Department of Physics, Universitat Politècnica de Catalunya, Castelldefels,
Spain
4Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
5Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak-
Nainital, India
6Department of Physics, University of Colombo, Colombo, Sri Lanka
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report the analysis of high temporal resolution ground and space based
photometric observations of SZ Lyncis, a binary star one of whose components
is a high amplitude $\delta$ Scuti. UBVR photometric observations were
obtained from Mt. Abu Infrared Observatory and Fairborn Observatory; archival
observations from the WASP project were also included. Furthermore, the
continuous, high quality light curve from the TESS project was extensively
used for the analysis. The well resolved light curve from TESS reveals the
presence of 23 frequencies with four independent modes, 13 harmonics of the
main pulsation frequency of $8.296943\pm 0.000002$ d-1 and their combinations.
The frequency 8.296 d-1 is identified as the fundamental radial mode by
amplitude ratio method and using the estimated pulsation constant. The
frequencies 14.535 d-1, 32.620 d-1 and 4.584 d-1 are newly discovered for SZ
Lyn. Out of these three, 14.535 d-1 and 32.620 d-1 are identified as non-
radial lower order p-modes and 4.584 d-1 could be an indication of a g-mode in
a $\delta$ Scuti star. As a result of frequency determination and mode
identification, the physical parameters of SZ Lyn were revised by
optimizations of stellar pulsation models with the observed frequencies. The
theoretical models correspond to 7500 K $\leq$Teff $\leq$ 7800 K,
log(g)=3.81$\pm 0.06$. The mass of SZ Lyn was estimated to be close to 1.7–2.0
M⊙ using evolutionary sequences. The period-density relation estimates a mean
density $\rho$ of 0.1054$\pm 0.0016$ g cm-3.

###### keywords:

asteroseismology, techniques: photometry, stars: variables: $\delta$ Scuti,
stars: individual: SZ Lyn, stars: fundamental parameters

††pubyear: 2020††pagerange: Asteroseismology of SZ Lyn using Multi-Band High
Time Resolution Photometry from Ground and Space–Asteroseismology of SZ Lyn
using Multi-Band High Time Resolution Photometry from Ground and Space

##  1 Introduction

The study of multi-periodic stellar pulsations represents a unique methodology
for studying the interior of a star (Aerts et al., 2010; Aerts, 2019). Stellar
pulsations can be either radial and/or non-radial in nature and is driven
through either pressure waves (p-mode) or gravity waves (g-mode). In p-modes,
pressure is the primary restoring force for a star perturbed from equilibrium,
while in g-mode, buoyancy is the restoring force (Aerts et al., 2010). Diverse
pulsation group includes Beta Cephei ($\beta$ Cep), slowly pulsating B (SPB)
stars, gamma Doradus ($\gamma$ Dor), $\delta$ Scuti stars etc. Garrido et al.
(1990); Breger (2000b); Handler et al. (2006); Handler (2008) provide detailed
discussion on the ground based observations of these stars. The group of
$\delta$ Scuti stars is located in the region where the classical instability
strip intersects to the main sequence in the Hertzsprung-Russell (HR) diagram
(e.g., Murphy et al., 2019). Their spectral range is from A2 to F0
corresponding to temperatures of 7000 K < Teff < 9300 K (Uytterhoeven et al.,
2011). Typical periods for p-mode pulsations in $\delta$ Scuti stars range
from 15 min to 5 h (Uytterhoeven et al., 2011). $\delta$ Scuti stars which
simultaneously pulsate in p-modes and gravity g-modes excited by the $\kappa$
mechanism and the convective flux blocking mechanism, respectively, are
commonly referred to as hybrid stars (Bowman & Kurtz, 2018). These stars are
of intermediate mass ranging between 1.5 to 2.5 M⊙ (Aerts et al., 2010), hence
most of them are in the core hydrogen burning or shell hydrogen burning phase
and possess convective cores. (See Breger (2000b) for a detailed discussion on
these stars). Furthermore, $\delta$ Scuti type pulsations have been detected
in many pre- and post-main sequence (Antoci et al., 2019) stars while low
amplitude $\delta$ Scuti type pulsations have been detected in many metallic
A-type (Am) stars (e.g., Joshi et al., 2012, 2016, 2017; Smalley et al.,
2017). The complex interior structures of these stars can only be examined by
a study of the multi-periodic radial and non-radial pulsation modes, a
technique known as asteroseismology (Aerts et al., 2010; Joshi & Joshi, 2015).
Generally, the frequency spectrum of a $\delta$ Scuti star is consisting of
both g and p modes with many harmonics and combinations. Breger et al. (2011),
Murphy et al. (2013), Pápics et al. (2012) provide comprehensive analysis on
frequency combinations and harmonics of $\delta$ Scuti targets.

In this paper, we present a detailed analysis of SZ Lyncis, HD 67390 (RA=08h
09m 35.8s, DEC=+44∘ 28′ 17.6′′) a High Amplitude $\delta{}$ Scuti (HADS) type
binary star, mv= 9.1 mag, with reported pulsation and orbital period
0.12053793 days and 1173.5 days, respectively (Soliman et al., 1986). SZ Lyn
is the brighter component of a binary system where the fainter component is
not observed with spectroscopic techniques, and is hence characterized as a
single line spectroscopic binary (Gazeas et al., 2004). The fundamental
pulsation period of this star was first determined by Binnendijk (1968) and
later refined by Gazeas et al. (2004).

SZ Lyn has been studied on a number of occasions for the orbital and pulsation
parameters. Van Genderen (1967) reported that the linear ephemeris for the
time of pulsation maximum has not been accurately estimated. Barnes III &
Moffett (1975) suggested that the very long period orbital motion of SZ Lyn
affected the linear ephemeris due to the light-travel time across the orbit.
Using photometric and spectroscopic data, Moffett et al. (1988) computed non-
linear ephemeris for the times of light maxima and determined improved values
for the pulsation and orbital parameters. A similar analysis was made by
Paparó et al. (1988) and Li & Qian (2013). The observed minus calculated (O-C)
diagrams show periodic as well as secular variations, and Paparó et al. (1988)
concluded that the main pulsation period changes by $(2.25\pm 0.42)\times
10^{-12}$ day per cycle. This value was later refined to $(2.90\pm 0.22)\times
10^{-12}$ day per cycle by Gazeas et al. (2004).

Table 1: Physical parameters of SZ Lyn derived in literature. The parameters were used to generate pulsation and evolutionary models of SZ Lyn.  Parameter | Value | Reference
---|---|---
Teff (K) | 7540 | Langford (1976)
| 7235 | LAMOST
| 7799 | Gaia
| 6830$\pm$150 | Johnson (1966)
$\log(g)$ | 3.88 | Langford (1976)
| 3.94 | LAMOST
Mass (M⊙) | 1.57${}^{+0.17}_{-0.66}$ | Fernley et al. (1984)
Radius (R⊙) | 2.76${}^{+0.11}_{-0.46}$ | Fernley et al. (1984)
| 2.80 | Bardin & Imbert (1981)
Parallax (mas) | 2.49$\pm$0.07 | Gaia

The physical parameters of SZ Lyn determined by several authors are given in
Table 1. This star has nearly solar abundance (Alania, 1972; Langford, 1976).
Bardin & Imbert (1981) determined the radial velocity of SZ Lyn due to the
pulsation as 30 km s-1 using high time resolution radial velocity curves. Gaia
(Gaia Collaboration et al., 2016, 2018; Luri et al., 2018) determined a
parallax of $2.49\pm 0.07$ milli-arc seconds, which, according to Bailer-Jones
et al. (2018) places it at a distance of $397\pm 11$ pc. Most of the previous
studies of SZ Lyn reported the main pulsation period and a few attempts have
also been made to search for additional harmonics and secondary pulsation
periods. Apart from the main pulsation period, two harmonics of the
fundamental period were reported by Gazeas et al. (2004).

We have subjected SZ Lyn to asteroseismic analysis using high temporal
resolution, multi-band light curves obtained from the ground, as well as
highly-precise photometric data from the TESS space mission; we attempted to
find more frequencies in SZ Lyn and identify the pulsation modes, basically n
and l. The manuscript is organised as follows: The observational data and
reduction procedures are given in Sec. 2. The frequency analysis is performed
in Sec. 3. In Sec. 4, the mode identification procedures are discussed in
detail paying special attention to the theoretical model analysis. The
refinement of stellar parameters of SZ Lyn is presented in Sec. 5, followed by
the evolutionary and pulsation models. Finally, we have discussed the results
with a detailed consideration of the limitations and uncertainties of our
analysis in Sec. 6 and summarized the results under Sec. 7.

##  2 Photometric Observations

Photometry data from various sources (Mount Abu, APT, WASP and TESS) are
included in the current study. The Table 2 compiles the observing log along
with details of the filters used, the cadence, number of nights, and the total
number of data points. In the following subsections, the data are further
described. The observed ground based data are inconsistent with respect to the
comparison stars and magnitude scales. Due to these reasons, we refrained from
combining all three data sources into a single light curve and hence the
analyses were performed separately.

###  2.1 Mount Abu

A series of observations were carried out from Mt. Abu InfraRed Observatory
(MIRO). The observatory is located at the highest peak, Gurushikhar, of
Aravali range near Mount Abu in the western state of Rajasthan, India. The
observatory is located at 24.65∘ N, 72.78∘ E, at an altitude of 1680 meters
where one can obtain typical seeing of $\sim$1.2 arc seconds.

Observations were made with the Corrected Dall-Kirkham (CDK) 50 cm, f/6.8
equatorial mount telescope equipped with an Andor 1024$\times{}$1024 EMCCD,
thermo-electrically cooled to $-80^{\circ}{}$C. Details about the setup are
provided by Ganesh et al. (2013). The higher frame rates and negligible
readout noise ($<$1 electron with EM gain), characteristic of EMCCD, are ideal
for the observation of short period variable stars with high cadence. The
large EMCCD array provides a field of view of 13$\times{}$13 arc minutes. The
system was fully automated and the exposures were scripted using RTS2111RTS2 :
Remote Telescope System v2 is available at http://rts2.org/ which allowed to
obtain repeated sequences of B, V and R single frames with different exposure
times for each filter. The filter movement being automated, very little time
is lost in this process. After removing faulty frames, we have acquired a
total of 4328 frames in B, 4569 in V and 6836 in R during the seven nights of
observation between December 2013 and November 2016 (see Table 2). At least
one pulsation cycle of SZ Lyn was covered in B, V and R on each night.

Table 2: Photometric observation of SZ Lyn. The JD column shows exact date of observation at Mount Abu and the observation periods of the other three data sources. Observation | JD (2450000) | Effective nights | Band | Coverage (h) | Cadence (s) | Data points
---|---|---|---|---|---|---
Mount Abu | 6639.104 | 7 | B | 29.2 | 5 | 4328
(India) | 6640.153 |  | V | 28.9 | 2 | 4569
| 6664.134 |  | R | 55.3 | 2 | 6836
| 6684.118 |  |  |  |  |
| 6686.174 |  |  |  |  |
| 6693.097 |  |  |  |  |
| 7705.135 |  |  |  |  |
APT | 7480.634-7515.688 | 24 | U | 50.1 | 60 | 701
(USA) |  |  | B | 51.4 | 40 | 688
|  |  | V | 52.5 | 40 | 672
WASP | 4190.366-4575.408 | 39 | (400-750) nm | 216.0 | 60 | 2894
TESS | 8842.503-8868.827 | 25 (days) | (600-1000) nm | 631.8 | 120 | 16550

The basic reduction procedures of the Mount Abu data were performed using the
IRAF222IRAF is distributed by the National Optical Astronomy Observatory,
which is operated by the Association of Universities for Research in Astronomy
(AURA) under a cooperative agreement with the National Science Foundation.
software. The photometry was performed by running the phot package in IRAF.
The instrumental magnitudes were extracted for the BVR bands using an aperture
of size 3 to 4 times the FWHM of program star for different filters. An
annulus of 5 pixels width was used for all the bands to determine the
background. Two stars of magnitudes mV= 10.5 (TYC 2979-1329) and mV= 10.8 (TYC
2979-1343), in the same field of view were used as comparison stars for
differential photometry in each band.

The colour variations (B-V) and (V-R) were determined for the Mount Abu data.
The amplitude of the variations of SZ Lyn in the BVR bands were
$\Delta$B$\approx$0.7, $\Delta$V$\approx$0.5 and $\Delta$R$\approx$0.4
magnitudes. Both colour and magnitude variations reveal that magnitude change
in B is the highest, as expected for a $\delta$ Scuti star. Therefore, B band
was used as the reference filter in the UBVR system for the calculation of
observed and theoretical amplitude ratios presented in Sec. 4.

Figure 1: The high cadence light curves of ground and space observations for
0.25 day coverage of SZ Lyn. The phases are synchronized at the peaks and
hence the WASP and APT data do not start at 0. The power spectra in the
frequency range 2 - 50 d-1 is shown in the right panel corresponding to each
data source. V band light curves are shown for the Mount Abu and APT data. The
amplitude of the light curves are in magnitude scale where the ground based
data is differential and TESS fluxes are converted to magnitudes.

###  2.2 APT

Differential time-series photoelectric data were collected through the Johnson
UBV filters with the 0.75 m Automatic Photoelectric Telescope (APT) T6 at
Fairborn Observatory in Arizona (Strassmeier et al., 1997). Two comparison
stars, HD 67808 and HD 66113, were used. These stars are too distant from SZ
Lyn that they would have fitted onto the Mt. Abu CCD frames, but were chosen
to have colours and magnitudes similar to those of the target. Integration
times were 2 $\times$ 20 s in each filter, with the exception of the U filter
for which an integration time of 2 $\times$ 30 s was used to collect enough
photons.

The data were reduced following standard photoelectric photometry schemes.
First, the measurements were corrected for coincidence losses. Then, sky
background was subtracted within each target/local comparison star group.
Standard extinction coefficients were employed; small errors in their
assumption would be compensated when computing differential magnitudes.
Obviously, the same extinction correction was applied to each star. Some bad
measurements due to partly poor telescope tracking had to be eliminated.
Finally, differential magnitudes were computed by interpolation, and the
timings were converted to Heliocentric Julian Date. During the reductions it
turned out that HD 66113 could be a low-amplitude $\delta$ Scuti star
($f=12.573$ d-1, $A_{V}=2.1$ mmag), hence it was rejected from the computation
of the differential light curves of SZ Lyn. A total of 701/688/672 good
measurements for the U, B, and V filters, respectively, were obtained with an
estimated accuracy of 4.5/3.5/3.5 mmag per point. The time span of the
combined data set is 35.05 d, with data collected on 24 nights.

###  2.3 WASP

The Wide Angle Search for Planets (WASP) observatories consist of two
identical robotic telescopes, one located at La Palma and other at South
African Astronomical Observatory (Pollacco et al., 2006). The WASP program
also observed SZ Lyn during the period March 2007 to April 2008 in its own
wide pass-band (Table 2). During routine observations, two exposures of each
field with 30 seconds were obtained, and each field was sampled every 9–12
minutes; for SZ Lyn that resulted in a total of 2894 data points equivalent to
216 effective hours of observation. The flux were converted to magnitudes by
the WASP pipeline (Butters et al., 2010) so that the light curve in Fig. 1 is
in magnitude scale. The WASP data are neither equally distributed nor
differential, and the differential magnitudes from the APT data were computed
with respect to a much brighter and bluer star than Mount Abu data.

###  2.4 TESS

SZ Lyn was observed by TESS (Ricker et al., 2015) in sector 20, ID number TIC
192939152, from 24th December 2019 to 19th January 2020 in 120 seconds of
cadence. The light curves were generated using simple aperture photometry and
corrected by pre-search data conditioning (PDC). The PDC pipeline module uses
singular value decomposition to identify and correct for time-correlated
instrumental signatures such as, spacecraft pointing jitter, long-term
pointing drifts due to differential velocity aberration, and other stochastic
errors (Jenkins et al., 2016; Balona et al., 2019). The corrected flux from
TESS Asteroseismic Science Operation Center (TASOC) were converted to
magnitudes. A part of light curve of each data source for the duration of 0.25
days is shown in Fig. 1. The power spectra on the right side in the same
figure are a good illustration of how the nearly uninterrupted space data
result in a much cleaner spectral window function compared to the ground based
data.

##  3 Frequency Analysis

For seismic analysis, the light curves observed from Mount Abu, APT WASP and
TESS data were searched for frequencies independently. We used the generalized
Lomb-Scargle (LS) (Zechmeister & Kürster, 2009) algorithm since ground based
observations were highly unevenly sampled and LS is specially designed to pick
out periodic variations in such kind of data. The LS periodogram tool is
available in the VARTOOLS software package (Hartman, 2012). The highest
frequency to search is the Nyquist frequency (fNq) and should be derived from
the data set. For TESS data this is approximately 360 d-1. fNq of the ground
based observations differ due to the uneven observation parameters. As we
could not detect frequencies above 120 d-1 up to the TESS upper limit of 360
d-1, we limit our search to the range 0 - 120 d-1 where frequency peaks
satisfying a SNR>4 significance criterion (Breger et al., 1993) occur in both
space and ground based data. In LS analysis, the frequency step size is
depending on 1/T, where T is the time span of the observation. The parameter
1/T in frequency analysis is termed as Rayleigh criterion (Aerts et al.,
2010). When T is higher the peaks of the power spectrum are narrower. Though
the ground based observation spreads over long range of time, the desired
narrowness could not be achieved due to the low duty cycle. This drastic
difference in ground and space based power spectra is clearly seen in Fig. 1.
Due to the different T in the four data sources, the sampling frequency of LS
analysis are different but we ensured that the sampling frequencies were kept
more than the highest frequency we expected to find. The light curves were
whitened at each peak and applied 5$\sigma$ clipping in the calculation of the
average and RMS of the power spectrum when computing the SNR value of a peak.
The power in LS periodogram is normalized to unity (Zechmeister & Kürster,
2009) so that the frequency peaks of the four data sources can be compared
graphically. Due to the unavailability of phase information of frequencies in
LS analysis, we performed the Discrete Fourier Transformation (DFT) using
Period04 (Lenz & Breger, 2004) keeping the same searching parameters as in LS
method and obtained amplitudes and phases of the frequencies. The power
spectra for the frequency range of 0 - 120 d-1 were obtained by both methods.
The frequencies determined by both methods are consistent. Period04 determined
the error of the frequencies, amplitudes and phases using 100 iterations of
Monte Carlo simulation. Due to the single-site observation, the power spectra
of the ground based observations were severely affected by $\pm 1$ d-1
aliasing and its multiples, $\pm 2$ d-1, $\pm 3$ d-1 etc. The power spectra in
the range of 0 - 100 d-1 for three data sources are shown in Fig. 2. For
clarity and save space we exclude the power spectra of WASP data in Fig. 2.

The identified frequencies of both space and ground based observations were
tabulated in Table 3. With the long time base observations of TESS, 23
frequencies are identified, while up to 10 of them are discovered in the Mount
Abu BVR bands, 8 in the APT UBV bands and 5 in WASP data. Though the frequency
extraction from the ground based observations is more complicated due to the
low observation span and gaps, we used pre-whitening and the killharm routine
in VARTOOLS to carefully remove the fundamental frequency of f1 8.296 d-1 and
its first three harmonics (corresponding to frequencies of 16.590, 24.890 and
33.187 d-1), as well as the $\pm 1$ d-1 day aliasing. The first four
frequencies were removed from the Mount Abu data using their Fourier
components and the residual light curve was obtained. The residuals were again
subjected to LS and DFT. The rest of the multiples (5$\times$f1 to
8$\times$f1) of ground based data were found using these residual light
curves. Eight common frequencies were clearly identified in Mount Abu and APT
power spectra while WASP detected only up to five frequencies. The main
pulsation frequency f1 and its seven harmonics were common in space and ground
observations. In addition, there are six more harmonics of f1 present in TESS
data. The harmonics above f3 are newly discovered for SZ Lyn in this work. The
f1 of 8.296 d-1, that we identify as the fundamental radial mode, had
previously been identified by Gazeas et al. (2004) and is further confirmed by
the asteroseismic technique of amplitude ratio method in the Sec. 4.

###  3.1 New frequencies of SZ Lyn

Most importantly, the frequencies f2 (14.535 d-1), f3 (32.620 d-1) and f4
(4.584 d-1), none of them harmonics of f1, are new for SZ Lyn. The truncated
frequency spectrum in Fig. 3 shows the linear combinations present due to the
frequencies f1, f2, f3 and f4. These linear combinations in one way support
that the newly discovered frequencies, f2, f3 and f4, are independent modes.
The possibility of these three frequencies being combinations of higher
frequency higher order l modes is also minimum due to lack of frequencies in
higher region of the frequency spectrum of TESS. Therefore, we further analyze
f2, f3 and f4 assuming they are base frequencies. Fig. 3 shows the
combinations made by f2, f3 with the well defined radial fundamental frequency
f1. In fact, radial overtones are common in $\delta$ Scuti stars, and so we
could in principle consider f2 as an overtone of fundamental. Nevertheless, we
estimated the first and second overtones for the well defined period ratios of
[0.772 - 0.776] range of Suárez et al. (2006) and [0.611 - 0.632] range of
Stellingwerf (1979), respectively. Given that the fundamental radial mode is
8.296 d-1, the calculated first and second overtones are 10.746 and 13.578 d-1
which are not found in the observed power spectra and f2 is too far from these
two overtones as well. Therefore, it is possible to conclude f2 is a nonradial
mode. To determine the spherical degree, l, of the nonradial mode, rotational
splitting is widely used (Kurtz et al., 2014; Breger et al., 2011). We noticed
that there are incomplete frequency multiplets around f2, one at 14.503 and
the other 14.435 d-1. Assuming this can be a sign of rotational splitting and
in order to confirm it, we fitted the 14.503 and 14.435 d-1 and pre-whitened.
The residuals still show several peaks which often occur when a signal changes
its amplitude and frequency slightly during the course of the observations.
Fourier decomposition of such a signal produces peaks that are real, but
resemble rotational splitting (Breger & Bischof, 2002; Bowman et al., 2016).
Another point supporting in this direction is that the combination of f2 with
the main pulsation, f1, at 22.830 d-1, also shows similar asymmetry. This is
another clue, although not unambiguous, that we see the beating of a time-
variable pulsation signal with the main radial mode.

The frequency f4 is located in the low frequency range and less than the
assumed fundamental radial mode of f1. Kurtz et al. (2014), Aerts et al.
(2010), Breger et al. (1999) strongly state that gravity modes (g-modes) are
located in the range 0 - 5 d-1. Therefore, the frequency f4 could be a medium
order gravity mode. Based on arguments given Kurtz et al. (2015), Kurtz et al.
(2014), f4 could be a combination of higher order p-modes or very low order
g-modes. Furthermore, Kurtz et al. (2014) show that the availability of
combinations $\nu_{1}\pm\nu_{g}$ with $\nu_{1}$ and $\nu_{g}$ being the
frequencies of the highest amplitude singlet p-mode and a g-mode respectively.
These combinations of frequencies produced by fundamental radial p-mode and a
g-mode can be naturally explained by non-linear effects that occur when the
p-mode and the g-modes are excited simultaneously. The combination frequencies
of the suspected g-mode with the dominant frequency are given by f${}_{1}+$f4
(12.880 d-1) and f${}_{1}-$f4 (3.703 d-1) which are symmetrically placed
around the fundamental p-mode frequency (f1) in Fig. 3. The availability of
these combinations in SZ Lyn supports the idea that f4 is a gravity mode and
indicates that both p-modes and g-modes are present in SZ Lyn.

Due to the lower precision of the ground-based data we could not clearly
detect the three frequencies, f2, f3 and f4 in these data. However, the power
spectra of Mount Abu were able to just resolve f2 peak despite the crowded
noisy field (See Fig. 2). But the amplitudes of f2 determined in UBVR colour
bands of ground based data are highly unreliable; therefore, we could not
include f2 in Sec. 4 to determine the spherical degree l using amplitude ratio
method. The amplitudes $A$ and the phases $\phi$ of the frequency f1
determined for UBVR colour bands using Period04 are shown in Table 4.

Figure 2: The comparison of LS power spectra of space and ground base observation. The low amplitude frequencies were recovered by whitening the previously determined frequencies f1, 2f1, 3f1 etc. The horizontal red lines in the last row indicates the $4\sigma$ threshold for three data sources. Table 3: Frequencies and amplitudes of SZ Lyn. V band data only are shown for Mount Abu and APT. The amplitude is in milli-magnitude. The errors of the amplitudes were obtained by Monte Carlo simulation using Period04. The 4$\sigma$ threshold limit for TESS is 0.08 which indicates by short horizontal line.  | TESS |  | Mount Abu |  | APT |  | WASP |
---|---|---|---|---|---|---|---|---
ID | Frequency | Amplitude | Frequency | Amplitude | Frequency | Amplitude | Frequency | Amplitude
| (c/d) | (mmg) | (c/d) | (mmg) | (c/d) | (mmg) | (c/d) | (mmg)
$f_{1}$ | 8.296 | 145.850$\pm$0.02 | 8.296 | 229.0$\pm$3.0 | 8.296 | 232.0$\pm$3.0 | 8.296 | 224.0$\pm$3.0
$2f_{1}$ | 16.592 | 53.320$\pm$0.01 | 16.590 | 76.0$\pm$3.0 | 16.592 | 82.0$\pm$3.0 | 16.593 | 76.0$\pm$3.0
$3f_{1}$ | 24.889 | 20.020$\pm$0.01 | 24.890 | 32.0$\pm$1.0 | 24.889 | 31.0$\pm$1.0 | 24.890 | 24.0$\pm$1.0
$4f_{1}$ | 33.186 | 8.220$\pm$0.01 | 33.186 | 11.0$\pm$1.0 | 33.185 | 13.0$\pm$1.0 | 33.185 | 15.0$\pm$1.0
$5f_{1}$ | 41.482 | 4.080$\pm$0.01 | 41.483 | 9.0$\pm$1.0 | 41.481 | 6.0$\pm$1.0 | 41.567 | 7.0$\pm$1.0
$6f_{1}$ | 49.779 | 2.420$\pm$0.01 | 49.849 | 3.0$\pm$1.0 | 49.777 | 4.0$\pm$1.0 |  |
$7f_{1}$ | 58.075 | 1.460$\pm$0.01 | 58.065 | 7.0$\pm$1.0 | 58.157 | 3.0$\pm$1.0 |  |
$8f_{1}$ | 66.372 | 0.800$\pm$0.01 | 66.365 | 4.0$\pm$1.0 | 66.339 | 3.0$\pm$1.0 |  |
$f_{2}$ | 14.535 | 0.800$\pm$0.01 | 14.221 | 6.0$\pm$1.0 |  |  |  |
$9f_{1}$ | 74.668 | 0.470$\pm$0.01 | 73.120 | 3.0$\pm$1.0 |  |  |  |
$f_{2}+f_{1}$ | 22.830 | 0.309$\pm$0.008 |  |  |  |  |  |
$f_{3}$ | 32.620 | 0.293$\pm$0.008 |  |  |  |  |  |
$10f_{1}$ | 82.965 | 0.277$\pm$0.008 |  |  |  |  |  |
$f_{2}-f_{1}$ | 6.237 | 0.223$\pm$0.010 |  |  |  |  |  |
$f_{4}$ | 4.584 | 0.164$\pm$0.007 |  |  |  |  |  |
$f_{3}+f_{1}$ | 40.888 | 0.157$\pm$0.010 |  |  |  |  |  |
$11f_{1}$ | 91.261 | 0.154$\pm$0.009 |  |  |  |  |  |
$f_{3}-f_{1}$ | 24.308 | 0.144$\pm$0.007 |  |  |  |  |  |
$f_{1}-f_{4}$ | 3.703 | 0.143$\pm$0.009 |  |  |  |  |  |
$f_{1}+f_{4}$ | 12.880 | 0.120$\pm$0.006 |  |  |  |  |  |
$12f_{1}$ | 99.564 | 0.105$\pm$0.008 |  |  |  |  |  |
$13f_{1}$ | 107.824 | 0.068 |  |  |  |  |  |
$14f_{1}$ | 116.104 | 0.042 |  |  |  |  |  |
Table 4: Observed amplitudes ($A$ in magnitude) and phases ($\phi$ in radians) for the main frequency f1=8.296 c/d obtained from PERIOD04. Only f1 is included in the table as this is the only independent frequency available in multi-band photometry of ground based data. The observed amplitude ratios in the middle row were used in Sec. 4. The last row is phase differences in degrees. The phase zero is at 2457480.63391813 HJD. | U | B | V | R
---|---|---|---|---
A | 0.275$\pm$0.003 | 0.290$\pm$0.004 | 0.229$\pm$0.003 | 0.171$\pm$0.002
$\phi$ | 0.834$\pm$0.003 | 0.805$\pm$0.003 | 0.803$\pm$0.003 |
| $A_{U}/A_{B}$ | $A_{B}/A_{B}$ | $A_{V}/A_{B}$ | $A_{R}/A_{B}$
| 0.95$\pm$0.02 | 1.00 | 0.79$\pm 0.02$ | 0.59$\pm 0.02$
| $\phi_{U}-\phi_{B}$ | $\phi_{U}-\phi_{V}$ |  |
| 1.7$\pm 0.2$ | 1.8$\pm 0.2$ |  |
Figure 3: New set of frequencies and their combinations with the fundamental
frequency f1 detected in SZ Lyn using TESS data. The blue lines indicate the
location of the whitened frequencies, f1 and its first four harmonics. The
discontinuity at the frequency f1 is due to the removal of strong side-lobes
created by f1. The horizontal red line is the $4\sigma$ threshold line for cut
off.

##  4 Amplitude Ratios

It is possible to determine the spherical degree l using the amplitude ratios
and phase differences in different wavelength bands (e.g., Balona & Evers,
1999; Aerts et al., 2010). The process is model-dependent because the
amplitude and phase variations not only depend on the spherical harmonic
degree l (Balona & Evers, 1999), but also –to a lesser extent– on the
effective temperatures and gravities of the models as shown in Dupret et al.
(2003). In general, flux changes in pulsating stars originate from the
temperature and gravity variations as a function of radius (Garrido et al.,
1990). The mode identification needs to be done through a sequence of
theoretical modeling of different combinations of parameters of the star.
Theoretical aspects of flux change of the model star were discussed in detail
with the consideration of non-adiabatic parameters and limb darkening effect
by Watson (1988) and Heynderickx (1994).

The spherical degree can be identified by matching the observed amplitude
ratios with the theoretically predicted amplitude ratios for the different
stellar model atmospheres. The magnitude variation of a pulsating star,
$\Delta m_{\lambda}$, at wavelength $\lambda$ with a spherical harmonic degree
l at a pulsating frequency $\omega$ can be expressed as in Aerts et al.
(2010):

$\Delta m_{\lambda}=A_{0}P_{lm}(\cos\
i)b_{l\lambda}(T_{1}+T_{2}+T_{3})e^{i\omega t}$ (1)

$T_{1}=(1-l)(l+2)$
$T_{2}=f_{T}(\alpha_{T\lambda}+\beta_{T\lambda})e^{-i\psi_{T}}$
$T_{3}=-f_{g}(\alpha_{g\lambda}+\beta_{g\lambda})$

where $P_{lm}$ is the associated Legendre function of degree $l$ and azimuthal
number $m$, $A_{0}$ is related to the amplitude of oscillations of the
photosphere, and $i$ is the inclination angle between the stellar axis and the
direction towards the observer. For the derivation and the details of the
components in Eq. 1, refer to Watson (1988), Heynderickx (1994), Balona &
Evers (1999) and Dupret et al. (2003). The component $T_{1}$ is the
contribution of the magnitude variation due to the different pulsation modes.
$T_{2}$ is the temperature dependent component of the magnitude which consists
of $f_{T}$, the amplitude of temperature variation function relative to the
normalized radial displacement at the photosphere and $\psi_{T}$, the phase
difference between maximum temperature and maximum radial displacement.
$T_{3}$ represents gravity variation where $f_{g}$ is the amplitude of gravity
variation function to the normalized radial displacement at the photosphere
(Dupret et al., 2003). The determination of $f_{T}$ is highly dependent on the
non-adiabatic parameter R (Garrido, 2000a). The other two parameters,
$\psi_{T}$ and $f_{g}$ are also unknown for any stellar model. The method of
mode identification therefore depends on the correct combination of these
parameters which were done earlier by different techniques, approximating the
observations to the theoretical models for reasonable ranges of $f_{T}$ and
$\psi_{T}$ (Balona & Evers, 1999). The value of R was estimated by Heynderickx
(1994) by adjusting it for best fit of the observed light curves, while
Garrido et al. (1990) redefined it as a range of interest keeping R and
$\psi_{T}$ as free parameters in the range [0.25 $-$ 1] and [
$90^{\circ}-135^{\circ}$] respectively for $\delta$ Scuti stars. Dupret et al.
(2003) introduced Time Dependent Convection (TDC) non-adiabatic treatment of
the model atmosphere and proposed a method of determination of amplitudes
$f_{T}$, $f_{g}$ of the eigenfunctions and the phase $\psi_{T}$ for different
degree l for a range of effective temperatures Teff, $\log(g)$ and mass of a
star. In this work, we used the outputs of TDC non-adiabatic computations
provided by M. A. Dupret (private communication) for SZ Lyn. The input
parameters Teff=7540 K, $\log(g)$=3.88 (Langford, 1976) and M=1.57 M⊙ (Fernley
et al., 1984) with the observational uncertainties (see Table 1) were passed
to the non-adiabatic code to generate the pulsation models which give
frequencies close to the observed main frequency of 8.296 d-1. Details of this
iterative process can be found in Dupret et al. (2005). Two models which found
very close to our observation results are given in Table 5. The output of the
non-adiabatic code also provides the amplitudes of $f_{T}$, $f_{g}$ and phase
angle $\psi_{T}$ for frequencies for different degree l. Fig. 4 represents the
distribution of $f_{T}$, $f_{g}$ and $\psi_{T}$ with the model frequencies
generated by non-adiabatic code. The interpolations of the best fit
polynomials determine the $f_{T}$, $f_{g}$ and $\psi_{T}$ for the observed
frequency f1. We have shown only the $l=0$ mode for the present purpose in
Fig. 4 .

Figure 4: $f_{T}$, $f_{g}$ and $\psi_{T}$ as a function of the pulsation frequency (d-1) for the mode of degree l = 0 for Model 1 and Model 2. The dots are the values generated by non-adiabatic code and continuous lines are the best fit polynomials. The dashed line indicates the observed frequency f1. The frequencies, f2, f3 and f4 are not marked since they were not identified in multi-band observation and hence could not include in Sec. 4. Table 5: Best-fitting DTC non-adiabatic pulsation models for the observed frequencies of SZ Lyn. | Model 1 |
---|---|---
$M/M_{\odot}$=2.00 |  $T_{\rm eff}$=7522 K |  $\log(L/L_{\odot}$)=1.425
$\log(g)$=3.77 |  $R/R_{\odot}$ = 3.00 | age(yr)=$1.0\times 10^{9}$
X=0.72 | Z=0.014 |  $\alpha_{\rm MLT}$ = 1.7
| Model 2 |
$M/M_{\odot}$=1.90 |  $T_{\rm eff}$=7557 K |  $\log(L/L_{\odot}$)=1.322
$\log(g)$=3.86 |  $R/R_{\odot}$ = 2.68 | age(yr)=$1.1\times 10^{9}$
X=0.72 | Z=0.014 |  $\alpha_{\rm MLT}$ = 1.7

For the partial derivatives of the monochromatic flux, $\alpha_{T\lambda}$
(rate of change in flux due to temperature) and $\alpha_{g\lambda}$ (rate of
change in flux due to gravity) (Eq. 1), we used ATLAS9 model atmospheres and
fluxes by Kurucz (1993), Castelli (2003). Due to the complexity of lookup
tables in all UBVR bands of Kurucz grid for the computation of
$\alpha_{T\lambda}$ and $\alpha_{g\lambda}$, we produced a code, AlphaTg, to
readout the model fluxes from the grid for the desired range of effective
temperature and $\log(g)$ with a step size of 250 K and 0.5 respectively.
Since the non-adiabatic parameters of $f_{T}$, $f_{g}$ and $\psi_{T}$ were
computed for the solar metallicity of $Z=0.014$, we used the solar metal
abundance of $[M/H]=0.0$ model from the Kurucz grid for the partial
derivatives. Furthermore, Garrido (2000a) has shown that the flux derivatives
have no significant change with the metallicity except in blue band. The
variation of flux derivatives in Fig. 5 also provides similar results as the
relative variation is much higher for shorter wavelengths. Balona & Evers
(1999) pointed out that amplitude discrimination is more effective in shorter
wavelength bands because of this variation in flux derivatives. All the flux
models were computed with the turbulence velocity fixed at 2 km/s and a mixing
length parameter $(\alpha_{\rm MLT})$ of 1.25. Breger et al. (1998) showed the
mixing length parameter, $\alpha_{\rm MLT}$, is in between 1 and 2 for
$\delta$ Scuti stars and less effective on hot stars while later according to
Bowman et al. (2018) the $\alpha_{\rm MLT}$ was revised as close to 2 for
$\delta$ Scuti stars. Hence the flux derivatives were calculated for
$\alpha_{\rm MLT}$ = 1.25 which is the highest of two options, 1.25 and 0.5
given in ATLAS9 models. The partial derivatives, $\alpha_{T\lambda}$ and
$\alpha_{g\lambda}$ computed within the observational error box of SZ Lyn
using AlphaTg code are shown in Fig. 5. It is clear that the temperature
contribution ($\alpha_{T\lambda}$) to the magnitude variation and hence to the
theoretical amplitudes is much more effective than the gravity component
($\alpha_{g\lambda}$), as seen in Fig. 5.

Figure 5: Flux derivative as a function of temperature (upper panel) and as a
function of gravity (lower panel) generated by AlphaTg code. The two dashed
line indicates the two models proposed for SZ Lyn in Table 5. The derivatives
taken at these two models were used for the calculation of theoretical
amplitudes.

Limb darkening effect was added to the theoretical amplitude by computing limb
darkening integral using the simple linear limb darkening law given by Claret
& Hauschildt (2003)

$b_{l\lambda}=\int_{0}^{1}\mu I(\mu)P_{l}d\mu$ (2)

$\frac{I(\mu)}{I(1)}=1-u(1-\mu)$

where $u$ is the limb darkening coefficient.

The limb darkening integral $\textit{b}_{l\lambda}$ depends on the mode of the
oscillation l. From the grids of limb darkening coefficients provided by
Claret & Hauschildt (2003), the limb darkening integrals were computed using
AlphaTg code for UBVR bands and hence obtained derivatives, $\beta_{T\lambda}$
and $\beta_{g\lambda}$, for three spherical degrees of $l$ ($l$=0,1,2) in the
range of effective temperatures and gravity of SZ Lyn. The results are shown
in Fig. 6. The limb darkening derivatives of temperature in upper panel of
Fig. 6 are more sensitive to l = 0 degree as well as for lower wavelengths
while derivatives are less effective for l = 2. Besides, the gravity
counterpart of limb darkening derivative (lower panel, Fig. 6) compared to the
temperature is very much less particularly in the range of $\log(g)$ of SZ
Lyn. The effect of gravity change is less effective for mode discrimination
than that of temperature, suggesting that the effect of gravity on limb
darkening is negligible compared to the effect of temperature. This effect is
previously shown by Garrido (2000a). However, we consider limb darkening
derivatives due to gravity, even though negligible, in our calculations of
theoretical amplitudes because of the model parameters of SZ Lyn in Table 5
are very close to each other. Based on the computations, The theoretical
amplitude ratios of $l$ = 0, 1 and 2 for Model 1 and Model 2 given in Table 5
were calculated and subsequently compared with the observed amplitude ratios
given in the Table 4 for the frequency f1. The graphical representation of the
comparison of amplitude ratios for Model 1 and Model 2 are shown in Fig. 7 and
Fig. 8, respectively. Since observations are scattered and the two model
parameters are very close to each other, in order to better assess the
appropriateness of the different values for $l$ as well as to converge for the
best model, we computed the $\chi^{2}$ function in Eq. 3 (Daszyńska-
Daszkiewicz & Walczak, 2009) for different degree l for the two models:

$\chi^{2}(l)=\sum_{j=1}^{\rm filters}\Big{(}\frac{A_{j,\rm th}/A_{\rm
ref,th}-A_{j,\rm obs}/A_{\rm ref,obs}}{\sigma_{j,\rm obs}}\Big{)}^{2}$ (3)

From the $\chi^{2}$ minimization shown in Fig. 9 we can conclude that the
frequency f1 is better fitted to $l=0$ degree of Model 2 and therefore Model 2
is more appropriate to represent the physical properties of SZ Lyn.

Figure 6: Limb darkening derivatives. The upper panel shows the variation of
limb darkening derivative with temperature for all three modes of l and for
UBVR colour bands for $\log(g)=4$. The colours, Violet, Blue, Green and Red
represent the UBVR bands respectively. Lower panel shows limb darkening
derivatives with gravity for $l$ and UBVR for Teff = 7500 K. Figure 7: The
comparison of the observed amplitude ratios of the frequency f1 with the
computed amplitude ratios for the Model 1 for the frequency $f_{1}$. Dots with
error bars are the observed amplitude ratios and the lines are the theoretical
predictions for spherical degree, l. Figure 8: The comparison of the observed
amplitude ratios of the frequency f1 with the computed amplitude ratios for
the Model 2 for the frequency $f_{1}$. Dots with error bars are the observed
amplitude ratios and the lines are the theoretical predictions for spherical
degree, l. Figure 9: $\chi^{2}$ minimization of the theoretical and
observational amplitudes for the frequency f1 against the spherical degree l
for Model 1 and Model 2.

##  5 Physical Parameters of SZ Lyn

We have tried to refine the physical parameters of SZ Lyn in this
asteroseismic investigation. Although the fundamental radial mode, f1, is
clearly defined in early literature, we used asteroseismic approach with
state-of-art TDC non adiabatic models to determine the spherical degree of the
frequency f1 through the mode identification method in Sec. 4. The advantage
of this approach is the ability to investigate stellar parameters through
different models of different pulsation codes as well as the possibility of
perturbation of parameters over the acceptable ranges within the models; hence
we can obtain the best set of physical parameters.

###  5.1 HELAS model oscillations

Furthermore, we considered
HELAS333http://helas.astro.uni.wroc.pl/deliverables.php?active=opalmodel&lang=en
pulsation models with the observed frequencies of SZ Lyn. We searched the
HELAS pulsation models produced by the linear non-adiabatic code (Pamyatnykh,
1999a; Dziembowski, 1977) using OP opacities Seaton (2005) and AGSS09 chemical
composition Asplund et al. (2009) for Z=0.015 and Z=0.02, from ZAMS to early
post-main sequence stages, and for radial and non-radial oscillations.

The HELAS pulsation models of mass range from M=$1.8$ M⊙ to M=$2.0$ M⊙ with
the step size of M=$0.1$ M⊙ are consistent with the observed radial mode f1,
8.296 d-1 (see Table 6). In addition, the other two frequencies, f2, f3 are
also overlapping with the model frequencies of l=2 mode of the model of mass
M=$2.0$ M⊙. Therefore, our identification of f2 and f3 as non-radial modes in
both our frequency analysis and our pulsation constant calculations is
strengthened by these model calculations, which have non-radial modes at
frequencies close to the observed values for f2 and f3. Although this is not
sufficient for a solid conclusion of the spherical degree of f2 and f3 as l=2,
we can still conclude that they likely are non-radial modes. From the models
with predicted fundamental radial node n=1, l=0 given in Table 6, the model
M=1.9 M⊙ is in line with the parameters of the TDC non-adiabatic Model 2
approximated by amplitude ratio method.

Therefore, HELAS models also provide evidence favouring a star of mass
M$\approx$1.9 M⊙ which oscillates in radial fundamental mode close to 8.296
d-1 as well as having some non-radial modes close to the observed frequencies.
Hence, considering all the results of this section, we infer that the mass of
SZ Lyn should be within 1.8 M⊙ $\leq$ M $\leq$ 2.0 M⊙.

Table 6: HELAS fundamental radial pulsation models (n=1,l=0) close to SZ Lyn. Model | Mass | Teff | log (L/L⊙) | $\log(g)$ | Z | Freq.
---|---|---|---|---|---|---
| (M⊙) | (K) |  |  |  | (d-1)
H1 | 1.8 | 7246 | 1.324 | 3.76 | 0.015 | 8.230
H2 | 1.9 | 7574 | 1.414 | 3.77 | 0.015 | 8.269
H3 | 2.0 | 7892 | 1.501 | 3.78 | 0.015 | 8.274
H4 | 1.9 | 7205 | 1.326 | 3.77 | 0.02 | 8.250
H5 | 2.0 | 7518 | 1.413 | 3.78 | 0.02 | 8.299

###  5.2 Evolutionary status of SZ Lyn

We have computed evolutionary sequences using version 11701 of the MESA
stellar evolution code (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018,
2019) for stars with masses ranging from $1.70$ M⊙ to $2.0$ M⊙ in steps of
$0.1$ M⊙. The evolutionary models were computed for the standard solar
composition of (Asplund et al. (2009), hereinafter AGSS09) scaled to $Z=0.01$
and $Z=0.02$ to bracket the reasonable values of metallicity for SZ Lyn. All
the sequences use OPAL opacities (Rogers & Iglesias, 1996) and a single fixed
value for the mixing length parameter of $\alpha_{\rm MLT}=2.0$. The
differences between OPAL and the OP opacities used in HELAS models are rather
small in the range of effective temperature of SZ Lyn (Seaton & Badnell,
2004), thus allowing a comparison between the two sets. We have adopted the
model of exponential decay (Herwig, 2000) to account for core overshooting. In
this model, raising convective elements exponentially decay after crossing the
boundary marked by Schwarzschild’s convective criterion:

$D_{\rm ov}=D_{{\rm conv},0}\exp{\left(-\frac{2z}{f_{\rm ov}H_{{\rm
P},0}}\right)}$ (4)

where $D_{{\rm conv},0}$ is the diffusion coefficient near to the convection
boundary, $H_{{\rm P},0}$ the pressure scale height (both determined according
to the MLT). In our sequences, we have used three values of the parameter
$f_{\rm ov}$: 0.00, 0.007, and 0.014 to model several extents of overshooting,
from none ($f_{\rm ov}=0.00$) to a significant overshooting ($f_{\rm
ov}=0.014$).

The nuclear network is pp_and_cno_extras.net, that includes 25 isotopes
ranging from 1H to 24Mg. We have neither considered mass-loss nor rotation for
the evolutionary sequences.

The full set of evolutionary sequences is shown in Figure 10. The tracks for
low metallicity are of higher luminosity, and the ones for higher metallicity
are of lower luminosity. Increasing core overshooting ($f_{\rm
ov}=0.00,0.007,0.014$) causes an increase in the duration of the main
sequence, as well as an evolution at a slightly higher luminosity. Dashed
lines show the limits of the instability strip (Breger & Pamyatnykh, 1998). An
analysis of this figure shows that, given the ample range of parameters
studied, there is not a single match for the TDC non-adiabatic pulsation
models (Model 1 is marked as a triangle, Model 2 as a circle), and that
reasonable values of the mass can range from 1.7 to 2.0 solar masses (1.7 to
1.9 M⊙ if $Z=0.01$, and 1.8 to 2.0 M⊙ if $Z=0.02$). During the main sequence
evolution of stars with M$\geq$1.2 M⊙ a convective core appears, but its
extension has not been still accurately determined (see, for example, the
results on some of the stars analysed in Deheuvels et al. (2016)). As
expected, the effect of increasing overshooting is to extend the duration of
the main sequence, as a larger amount of H becomes available. For our purposes
this effect obscures the current evolutionary state of SZ Lyn, which could be
at the last phases of the main sequence, or at the first stages of the post-
main sequence. Nevertheless, the evolutionary sequences indicate that the
likely mass of SZ Lyn might be in the range 1.7–2.0 M⊙, significantly larger
than the old value of 1.57 M⊙ found by Fernley et al. (1984). Further
observational refinements in the metallicity of the star would allow
improvements in the mass inference.

###  5.3 Pulsation constant

The pulsation constant ($Q$) can be determined for the frequency $f_{1}$ using
the Eq. 5 given by Breger (1990) for the two models mentioned in Table 5.

$\log(Q)=\log(P)+\frac{1}{2}\ \log(g)+\frac{1}{10}\ M_{\rm bol}+\log(T_{\rm
eff})-6.454$ (5)

The bolometric magnitude Mbol of SZ Lyn is estimated for two models using the
values found in Table 5. We take for the solar bolometric absolute magnitude
the standard value of M⊙bol=4.74 (Bessell et al., 1998; Torres, 2010) and
solar luminosity L${}_{\odot}=3.85\times 10^{33}$ erg s-1. The pulsation
constant $Q$ is then found to be 0.032 for Model 1 and 0.038 for Model 2. The
stellar parallax of SZ Lyn obtained by Gaia mission provides an absolute
magnitude MV of +1.14. and the bolometric correction (BC) is -0.094 as given
by Cox (2000), thus resulting in a bolometric absolute magnitude for SZ Lyn of
Mbol +1.05. This results in an alternative set of $Q$ values of 0.031 and
0.035 for Model 1 and Model 2 respectively. The pulsation constant for
fundamental radial p-modes in $\delta$ Scuti stars is in the range of
$0.022\leq Q\leq 0.033$ (Breger, 1975) and an even narrower range of
$0.0327\leq Q\leq 0.0332$ for the fundamental radial mode pulsation (Fitch,
1981).

The calculations show that the $Q$ value for Model 1 is within the allowed
range given by Breger (1975) and close to the lower limit of the range given
by Fitch (1981), and this gives reinforces our confidence that the frequency
f1 is the radial fundamental mode of stellar pulsation. On the other hand, the
$Q$ value of radial fundamental mode of Model 2 parameters is slightly
deviated from the standard intervals.

###  5.4 Mean density

As we have only determined a single radial mode without any strongly confirmed
overtones, we cannot use the period ratios to determine the average density of
the star. Nevertheless, we could estimate the average density of SZ Lyn using
the $Q$ values, calculated in Sec. 5.3, using the relation:

$P=Q\;(\rho/\rho_{\odot})^{-1/2}$ (6)

where $P$ is the period of the fundamental oscillation in days and $Q$ is the
pulsation constant. The average density of the Sun, $\rho_{\odot}$= 1.4103 g
cm-3, and predicted $Q$ values of 0.031 and 0.035 in Sec. 5.3 results in an
average density of $\rho$=0.0933 gcm-3 and $\rho$=0.1189 gcm-3 for the Model 1
and Model 2 respectively. Though the fundamental period P is been determined
with a high accuracy, the uncertainty of Q is obtained from a propagation of
errors of Eq. 5 which could not be determined accurately due to the non-
availability of model uncertainties of Teff and $\log(g)$. Therefore, the
determination of average density using Q values calculated in Sec. 5.3 is
unreliable. Instead, with the accurate fundamental period of SZ Lyn,
$P=0.120526$ d and for more narrow range of $Q$ values $0.0327\leq Q\leq
0.0332$ (Fitch, 1981), we find the average density and error as
$\rho=0.1054\pm 0.0016$ gcm-3 from the upper and lower limits of the above
range. Suárez et al. (2014) computed the mean densities of $\delta$ Scuti
stars using their virtual observatory tool, TOUCAN, and found that the
fundamental frequency range of 95 - 113 $\mu$Hz (0.121832 - 0.102425 d) has
relative densities ($\rho/\rho_{\odot}$) less than 0.11. Hence we can conclude
that the density of $\rho=0.1054\pm 0.0016$ gcm-3 is an appropriate value for
SZ Lyn.

Figure 10: Main and post-main sequence evolutionary tracks of stars in the
range 1.7-2.0 M⊙ for the scaled AGSS09 chemical composition $Z=0.01$ (upper
set of tracks in each panel) and $Z=0.02$ (lower set of tracks in each panel)
for $\alpha_{\rm MLT}=2$. Curves of green and black colours correspond to
$f_{\rm ov}=0.00$, dark blue and orange to $f_{\rm ov}=0.007$, and light blue
and red to $f_{\rm ov}=0.014$. The black triangle and dot represent TDC Model
1 and Model 2, respectively (Table 5). The purple and green squares represent
Models H2 and H5 from Table 6, respectively. The rectangle shows observation
error box. Instability strip is shown in dashed lines. Figure 11: Perturbation
of input parameters Teff, chemical composition (Z) and mixing length
($\alpha_{\rm MLT}$) on l=0 of Model 2. Left; Deviation of theoretical
amplitude ratios at l=0 with observational error of $\pm$150 K at constant
Z=0.014 and $\alpha_{\rm MLT}$=1.7. Middle; Deviations of amplitudes with Z at
Teff=7557 K and and $\alpha_{\rm MLT}$=1.7. Right; Deviation of amplitudes
with $\alpha_{\rm MLT}$ at Teff=7557 K and Z=0.014. The $\chi^{2}$
minimization with the observation for each deviation is included.

##  6 Discussion

SZ Lyn has been extensively studied by several authors who have analysed their
results considering both its binary nature as well as its pulsating nature.
Though its binary orbital characteristics have been studied in detail, a
comprehensive asteroseismic study of SZ Lyn was lacking so far. We reconsider
SZ Lyn giving special attention to recovering more frequencies apart from the
main frequency, and determining the degree of the oscillation. With this
information, we have tried to refine its stellar parameters.

We confirmed 23 frequencies, 14 of which are multiples of the fundamental
frequency f1. Gazeas et al. (2004) also confirmed the frequency f1 and its
first two harmonics, and we discovered 11 new harmonics in SZ Lyn. Therefore
to confirm the radial mode, we adopted the amplitude ratio method as it
provides a consistent result with the observed amplitudes. In addition to the
amplitudes, $\delta$ Scuti stars show wavelength dependence on the phase. A
method proposed by Garrido et al. (1990) showed that mode can be determined by
plotting amplitude ratio versus phase difference for two colors. This method
was performed for Strömgren photometric system taking two bands each time and
plotting the so-called area of interest for a broad range of parameters of
$\psi_{T}$ and non-adiabaticity R. Although the phase difference method is
more appropriate for $\delta$ Scuti stars to determine the degree of the
modes, we did not represent it graphically because of the availability of one
frequency (f1) in multi-band photometry and there are no other frequencies to
compare. But the phase difference of the frequency f1 shown in Table 4,
$\phi_{\rm U}-\phi_{\rm B}$ and $\phi_{\rm U}-\phi_{\rm V}$ both are greater
than zero (Breger, 2000a; Garrido, 2000b), thus further confirming that f1
corresponds to a radial mode of l=0. Furthermore, the TESS observations
clearly revealed the presence of independent frequencies f2=14.535 d-1,
f3=32.620 d-1 and f4=4.584 d-1. Unfortunately these frequencies could not be
recovered in the multi-band ground based data. Therefore, we could not perform
the amplitude ratio method to determine the spherical degree l. In some cases,
mode identification can be done without multi-band photometry as more often
the space data has shown rotational splitting and period spacing patterns in
the frequency spectra (Aerts, 2019; Aerts et al., 2010).

Provided that the frequency f2 is nonradial, in principle, the effect of
rotational splitting should place side lobes centered at f2. We estimated the
first-order effect of rotational splitting using the measured upper limit of
$v\sin i<40$ km s-1 (McNamara, 1985). Given that this is also a general limit
to the projected rotational velocity of high-amplitude $\delta$ Scuti stars
(Breger, 2000b), we can assume this as an intrinsic limit to $v_{\rm rot}$.
Using R∗=2.86 R⊙ from our Model 2, the upper limit of first order rotational
splitting for nonradial modes was estimated with a negligible Ledoux constant,
Cn.l (Ledoux, 1951). This upper limit amounts to 0.276 d-1. We were not able
to find signatures of a recurring frequency splitting within this limit around
f2 in SZ Lyn. The absence of rotational splitting and period spacings in the
frequency spectra of SZ Lyn shows the importance of continuous multi-band
ground based observations to determine l using amplitude ratios and phase
differences. However. we were able to conclude that f2 and f3 are non-radial
modes through the period ratio method and pulsation constant calculations. In
addition, the HELAS pulsation models give some evidences that f2 and f3 are
close to l=2 mode but we refrain from confirming the mode of f2 and f3 as
further investigations are needed. The discovery of frequency f4 is remarkable
because it is identified as medium-order g-mode pulsation which is rare in
$\delta$ Scuti stars.

As a consequence of the determination of independent frequencies and their
modes, this investigation is used to refine the stellar parameters of SZ Lyn.
In the process of determining the degree of the mode l of the frequency f1
using amplitude ratio method, it is possible to narrow down the stellar
parameters of SZ Lyn. We produced pulsation models using TDC non adiabatic
models (Dupret et al., 2003) as well as linear non adiabatic models
(Pamyatnykh, 1999b; Dziembowski, 1977). The introduction of state-of-art TDC
non adiabatic models was an important step forward in the modelling of
$\delta$ Scuti stars (Dupret et al., 2003, 2005). Murphy et al. (2012) gave a
successful first application of these models to an individual $\delta$ Scuti
star observed by Kepler. Finally, evolutionary sequences were computed using
MESA and overlapped the models to provide a further insight on the stellar
parameters. Although the determination of stellar parameters through the
combination of different models is more accurate, the inconsistencies with
models clearly stand out. Therefore, the behaviour of the models was
considered by perturbing the input parameters.

We searched the best model for SZ Lyn centering the error box at Teff=7500 K
and $\log(g)=4.0$ and stepping in Teff ($\pm$250 K) and in $\log(g)$
($\pm$0.5) in ATLAS9. The best fit polynomials (minimum RMSE) of AlphaTg code
determine the flux and limb darkening derivatives for any Teff and $\log(g)$
so that the error contribution from flux and limb darkening derivatives to the
theoretical amplitudes is minimized. However, the perturbation of Teff on
eigenvalues $f_{T}$ and $\psi_{T}$ is significant and hence change the
theoretical amplitudes considerably (Dupret et al., 2005). The temperature
component (T2) has a major contribution in theoretical amplitudes. Thus the
best model was perturbed by the observational uncertainty of $\pm$150 K. This
uncertainty propagates to the non adiabatic parameters $f_{T}$ and $\psi_{T}$
and thus deviated the amplitudes of l=0 as shown in the left panel of Fig. 11.
Dupret et al. (2005) investigated the dependency of non adiabatic parameters
$f_{T}$ and $\psi_{T}$ for effective temperature range of $\delta$ Scuti stars
and showed that in the lower effective temperature region ($\approx$ 7250 -
6500 K) both $f_{T}$ and $\psi_{T}$ drop significantly. We performed the
extrapolation to the lower temperature region and observed the amplitude
ratios of all pass bands were further reduced and hence deviated more from the
observations. The temperature perturbation of Model 2 shown in Fig. 11
revealed that the $\chi^{2}$ value is minimized around 7550 K and deviated
from the minimization at higher and lower Teff. This temperature approximation
is consistent with linear non adiabatic models, H2 and H5, in Table 6.

By means of linear regression Bowman & Kurtz (2018) obtained new statistics
for $\delta$ Scuti stars using short cadence Kepler observations. In this
analysis the stars are classified in three categories with respect to
$\log(g)$ and 3.5$\leq\log(g)\leq$4.0 is defined as Middle Age Main Sequence
(MAMS). By means of the linear regression coefficients, the Teff was estimated
for the observed frequency f1 = 8.296 d-1 as 7620 K for revised KIC values
from Huber et al. (2014) and as 7469 K for Brown et al. (2011) KIC values.
Though the specific Teff is not possible to be given with model uncertainties,
it is rather enough evidences supporting that the temperature of SZ Lyn
converge to 7500 K$\leq$Teff $\leq$7800 K resulting in the conclusion that SZ
Lyn is more close to blue edge than thought before.

Similarly we investigated the model dependency on Z and $\alpha_{\rm MLT}$ in
middle and right panels of Fig. 11 respectively. The higher the metallicity
(Z), the more efficient the $\kappa$ mechanism in lowering the luminosity in
the driving region. Hence the amplitude of the luminosity variation and the
local effective temperature variation $f_{T}$ at photosphere are small. In
addition, The phase difference between the local effective temperature and
radial displacement $\psi_{T}$ is close to 180∘ (Dupret et al., 2003). These
individual variations were accounted in the computation of amplitudes in pass
bands and overall effect on the amplitudes can be seen in the Fig. 11. The
left panel of Fig. 11 is the dependency of Model 2 on the mixing length
parameter ($\alpha_{\rm MLT}$). The deviations of $f_{T}$ and $\psi_{T}$ for
$\alpha_{\rm MLT}$ = 1.0 were taken from Dupret et al. (2005). $\alpha_{\rm
MLT}$ = 1.0 is somewhat low for $\delta$ Scuti stars; a value of 1.8 to 2.0
provides a reasonably good agreement between theoretical models and
observations of $\delta$ Scuti stars (Bowman & Kurtz, 2018). This is close to
our choice of $\alpha_{\rm MLT}=1.7$ in Fig. 11. However, it is important to
have more individual observations of $\delta$ Scuti stars, with a similar
analysis, for a better understanding of the behaviour of these parameters.

In this process the pulsation models of SZ Lyn were evaluated with an
appropriate set of evolutionary sequences with different stellar parameters.
We computed evolutionary sequences with MESA to obtain an independent value
for the mass of SZ Lyn. Placing Models 1 and 2 in the HR diagram allowed us to
compare with the evolutionary tracks of stars between 1.7–2.0 M⊙,
$Z=0.01-0.02$, and different values for overshooting. We believe that these
ranges of mass, metallicity and overshooting encompass the likely values for
SZ Lyn. The results show that both models are compatible with stars in these
ranges at the last phases of the central H burning, or at the beginning of
shell burning. Nevertheless, without a better determination of metallicity
–and of the still harder to determine overshooting– we cannot refine the mass
determination beyond this 1.7–2.0 solar masses range. This range is in
reasonable agreement with HELAS’s models which converged to 2 M⊙ model with
the observation.

Table 7: Best model of SZ Lyn. The $\log(g)$ is the average of all the pulsation models. Physical Property | Value
---|---
$T_{\rm eff}$ | 7500 K$\leq$Teff$\leq$ 7800 K
$\log(g)$ | 3.81$\pm$0.06
Mass (M) | 1.7-2.0 M⊙
Radius (R) | 2.68 R⊙
Metallicity (Z) | 0.014 - 0.020
Mean density ($\rho$) | 0.1054$\pm$0.0016 g cm-3

##  7 Conclusions

A comprehensive analysis of ground and space based photometry of the
$\delta$Scuti star SZ Lyn is presented and discussed with inputs from various
theoretical models.

The well resolved TESS data allow the identification of 23 frequencies in SZ
Lyn. Apart from the fundamental radial mode of 8.296 d-1 and its first two
harmonics, we found 3 independent modes and 10 harmonics of the fundamental
frequency. Though the ground based data are not up to the level of quality of
TESS, most of the harmonics and one independent mode are present in the Mount
Abu light curve also. The presence of the dominant p-mode of the radial
fundamental at 8.296 d-1, non-radial p-modes at 14.535 d-1, 32.620 d-1 and a
g-mode at 4.584 d-1 indicate that all possible oscillations can be
simultaneously present in the same star. Additionally, the HELAS pulsation
models give indication that 14.535 d-1 and 32.620 d-1 are close to spherical
degree l = 2 mode but this conclusion needs to be confirmed with more
observational evidence.

We attempted to refine the physical parameters of SZ Lyn through a rigorous
analysis of stellar pulsation and evolutionary models. Due to the large number
of input parameters it is obvious that inconsistencies between different
models make this approach difficult. Nevertheless, a suitable exploration of
parameters can be a powerful means to determine approximate values for the
effective temperature and mass of SZ Lyn. Our analysis points to a T${}_{\rm
eff}\approx$ 7500 K – 7800 K and a mass in the range 1.7 M⊙–2.0 M⊙.

We also note that even with the availability of high-quality space-based data,
UBVR ground-based observations are essential to constrain the different
theoretical models.

## Acknowledgements

We acknowledge the anonymous referee for very pertinent suggestions. We thank
M. A. Dupret for providing non-adiabatic computation results. JA acknowledges
the support of the Centre for International Cooperation in Science, Govt. of
India for his work and stay at PRL when some of the work presented here was
completed. JA’s earlier work at PRL was under a UN sponsored CSSTEAP (Centre
for Space Science Technology and Education in Asia Pacific) program. We thank
the staff at the Mount Abu IR Observatory for their technical support during
the course of this work. We are also very grateful to the staff of Arthur C
Clarke Institute for their invaluable administrative support. We thank Prof U
C Joshi and Prof H O Vats, for their support and useful discussion at various
times through this work. SJ acknowledges the discussion with Dr. Dogus Ozuyar
on evolutionary aspects. We are grateful to Dominic Bowman and Kosmas Gazeas
for comments on the manuscript.

This work is supported by the Department of Space, Govt. of India. GH
gratefully acknowledges funding through NCN grant 2015/18/A/ST9/00578.

"This paper makes use of data from the first public release of the WASP data
(Butters et al. 2010) as provided by the WASP consortium and services at the
NASA Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program."

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program. Funding for the TESS
Asteroseismic Science Operations Centre is provided by the Danish National
Research Foundation (Grant agreement no.: DNRF106), ESA PRODEX (PEA
4000119301) and Stellar Astrophysics Centre (SAC) at Aarhus University. We
thank the TESS team and staff and TASC/TASOC for their support of the present
work.

JG’s research has been partly supported by the Spanish project PID
2019-109363GB-100

## Data Availability

The TESS and WASP observations of SZ Lyn are publicly available from the
respective data archives. The model data received from Dr Dupret and the other
results computed by the authors are already included in the article.
Intermediate results and the underlying observational data, from Mount Abu and
APT, will be shared on reasonable request to the corresponding author.

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