# A precise asteroseismic age and metallicity for HD 139614: a pre-main-
sequence star with a protoplanetary disc in Upper-Centaurus Lupus

Simon J. Murphy,1,2 Meridith Joyce,3,4 Timothy R. Bedding1,2, Timothy R.
White1,2 E-mail<EMAIL_ADDRESS>(SJM<EMAIL_ADDRESS>(MJ) Mihkel Kama5,6
1 Sydney Institute for Astronomy (SIfA), School of Physics, University of
Sydney, NSW 2006, Australia
2 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, 8000 Aarhus C, Denmark
3 Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
4 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO
3D), Australia
5 Department of Physics and Astronomy, University College London, Gower
Street, London, WC1E 6BT, UK
6 Tartu Observatory, University of Tartu, Observatooriumi 1, 61602 Tõravere,
Tartumaa, Estonia

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

###### Abstract

HD 139614 is known to be a $\sim$14 Myr-old, possibly pre-main-sequence star
in the Sco-Cen OB association in the Upper Centaurus-Lupus subgroup, with a
slightly warped circumstellar disc containing ring structures hinting at one
or more planets. The star’s chemical abundance pattern is metal-deficient
except for volatile elements, which places it in the $\lambda$ Boo class and
suggests it has recently accreted gas-rich but dust-poor material. We identify
seven dipole and four radial pulsation modes among its $\delta$ Sct pulsations
using the TESS light curve and an échelle diagram. Precision modelling with
the MESA stellar evolution and GYRE stellar oscillation programs confirms it
is on the pre-main sequence. Asteroseismic, grid-based modelling suggests an
age of $10.75\pm 0.77$ Myr, a mass of $1.52\pm 0.02$ M⊙, and a global metal
abundance of $Z=0.0100\pm 0.0010$. This represents the first asteroseismic
determination of the bulk metallicity of a $\lambda$ Boo star. The precise age
and metallicity offer a benchmark for age estimates in Upper Centaurus–Lupus,
and for understanding disc retention and planet formation around intermediate-
mass stars.

###### keywords:

stars: chemically peculiar – variables: $\delta$ Scuti – stars: pre-main-
sequence – asteroseismology – protoplanetary discs – stars: fundamental
parameters

††pubyear: 2015††pagerange: A precise asteroseismic age and metallicity for HD
139614: a pre-main-sequence star with a protoplanetary disc in Upper-Centaurus
Lupus–A precise asteroseismic age and metallicity for HD 139614: a pre-main-
sequence star with a protoplanetary disc in Upper-Centaurus Lupus

##  1 Introduction

There are few initiatives in stellar astrophysics that would not benefit from
better ages. Better ages for open clusters would improve benchmarks for almost
every age determinant (Soderblom, 2010), with impacts reaching as far as
anchoring the initial-final mass relation of white dwarfs (Marigo et al.,
2020). Better ages from asteroseismology are reforming our understanding of
angular momentum transport in stars (Aerts et al., 2019; Eggenberger et al.,
2019; Ouazzani et al., 2019; den Hartogh et al., 2020), and better ages for
large populations of galactic field stars help to trace the chemical evolution
of the Galaxy (Bland-Hawthorn et al., 2019; Hayden et al., 2020; Lin et al.,
2020). The formation and evolution of protoplanetary discs is another area
where precise ages matter, with disc properties evolving on timescales of 3–5
Myr (Haisch et al., 2001; Kraus et al., 2012).

The evolution of young stars, their discs, and their planets are intimately
connected. For example, chemical peculiarities in $\lambda$ Boo stars probably
arise from a filtering of dust from gas in planet-forming discs (Kama et al.,
2015). For unclear reasons, some such discs appear to survive exceptionally
long, for $\gtrsim 5$ or even $10$ Myr (e.g., Vioque et al., 2018; Flaherty et
al., 2019). Accurate measurements of the stellar mass, age, and intrinsic
metallicity are crucial constraints for the theoretical work connecting
stellar mass build up, planet formation including migration, and disc
dissipation (GRAVITY Collaboration et al., 2019). Asteroseismology provides a
unique and independent way of measuring those parameters. Here, we use a light
curve from TESS (Ricker et al., 2015) in an asteroseismic analysis of the
young, disc-hosting, chemically peculiar star HD 139614

HD 139614 is located in Upper Centaurus–Lupus (UCL), a subdivision of the
Sco–Cen association which, at 140 pc, is the nearest region to the Sun with
high-mass star formation (Blaauw, 1946; de Zeeuw et al., 1999). The relative
youth of UCL (median age = $16\pm 2$ Myr, Pecaut & Mamajek 2016), its observed
1$\sigma$ age spread of $\sim$7 Myr (Mamajek et al., 2002; Preibisch &
Mamajek, 2008; Pecaut & Mamajek, 2016), and its proximity make it an important
laboratory for high-angular-resolution studies of planet-forming discs of
various ages (e.g., Ansdell et al., 2016; Lieman-Sifry et al., 2016; Barenfeld
et al., 2017), each of which provide a snapshot of disc evolution. There is
mounting evidence that quite massive dust discs can survive for a wide range
of ages (e.g., Pfalzner et al., 2014), but determining those ages is
difficult. Ages of stars within UCL are known to vary with both location
within the association and with stellar mass (Hillenbrand et al., 2008; Pecaut
& Mamajek, 2016), which means there is a need for age calibrators throughout
the region.

HD 139614 is a well-studied member of UCL (membership probability 99.9%, Gagné
et al. 2018) whose protoplanetary disc is known to have dust gaps and a cavity
(Meeus et al., 1998; Matter et al., 2014, 2016; Carmona et al., 2017).
Scattered-light imagery shows a warped disc (Muro-Arena et al., 2020) that may
be evidence of embedded planets on inclined orbits (Nealon et al., 2018,
2019). Dust traps next to empty gaps or an inner cavity, which are also seen
in the HD 139614 system, may allow large quantities of dust to survive for
much longer than in many other disks (Pinilla et al., 2018, 2020). Embedded
planets are able to trap dust in the disc and prevent its accretion onto the
star (Jermyn & Kama, 2018), leaving fingerprints in the stellar spectrum of
dust-poor accretion (Kama et al., 2015). This is one hypothesized mechanism
for producing the chemical peculiarities that typify the $\lambda$ Boo class
(Venn & Lambert, 1990; Waters et al., 1992; King, 1994), which exhibit a
depletion of up to 2 dex in refractory elements alongside solar abundances of
volatiles (Baschek & Slettebak, 1988; Andrievsky et al., 2002; Kamp et al.,
2001). Indeed, VLTI and ALMA observations confirm the existence of planets
embedded in the discs of some $\lambda$ Boo stars (e.g. Matter et al., 2016;
Fedele et al., 2017; Cugno et al., 2019; Toci et al., 2020). Although no
planets are yet confirmed for HD 139614, it is known to be a $\lambda$ Boo
star based on its photospheric abundances (Folsom et al., 2012; Murphy et al.,
2015).

No conclusive assessment has yet been made as to whether $\lambda$ Boo stars
are globally metal-poor objects, though strong statistical arguments have been
made to the contrary (Paunzen et al., 2015; Murphy et al., 2020a). While
spectroscopy is able to determine the photospheric metallicity,
asteroseismology probes the stellar interior and can deliver precise bulk
metallicities (e.g. van Saders & Pinsonneault 2012), because the latter
affects the stellar structure to which the oscillation frequencies are
sensitive (e.g. Guzik et al. 1998; Aerts et al. 2010). The application of
asteroseismic techniques to $\lambda$ Boo stars is therefore highly sought
after (Paunzen et al., 1997). In addition, $\lambda$ Boo stars may make
particularly good asteroseismic targets since they are twice as likely as
normal stars of the same effective temperature to pulsate as $\delta$ Sct
stars (Murphy et al., 2020a), which are intermediate-mass stars pulsating
predominantly in pressure modes (Breger, 2000). While many studies have been
dedicated to the detection of pulsations in $\lambda$ Boo stars (e.g. Paunzen
et al., 1998; Koen et al., 2003; Breger et al., 2006; Paunzen et al., 2015),
somewhat fewer have succeeded in the first asteroseismic step of identifying
the pulsation modes (Paunzen et al., 2002; Mkrtichian et al., 2007; Bedding et
al., 2020; Murphy et al., 2020a), and only Casas et al. (2009) have made any
inferences on the stellar interior.

For a long time, the inability to successfully identify the radial order ($n$)
and the spherical degree ($\ell$) of the pulsations of $\delta$ Sct stars was
a major barrier to progress (e.g. Breger et al., 2004); many oscillation modes
were observed, but few had reliable mode identifications. The situation grew
worse with space photometry of $\delta$ Sct stars from CoRoT (García Hernández
et al., 2009; Poretti et al., 2010; Mantegazza et al., 2012) and Kepler (e.g.
Uytterhoeven et al., 2011), where unprecedented photometric precision revealed
many more oscillation frequencies. However, mode identifications are
traditionally obtained via time-series spectroscopy (Balona, 1986) or multi-
colour photometry (Watson, 1988) and techniques are limited with single pass-
band photometry alone (cf. Petersen & Christensen-Dalsgaard, 1996), but a
recent breakthrough made possible by the large number of stars observed at
rapid cadence by TESS (Ricker et al., 2015) has enabled mode identification
for some high-frequency $\delta$ Sct stars (Bedding et al., 2020).

At high radial orders, pressure modes (p modes) are asymptotically spaced in
frequency (Unno et al., 1989) by the so-called large spacing, $\Delta\nu$.
Most $\delta$ Sct stars oscillate at low frequencies, corresponding to low
radial orders and periods longer than $\sim$1 hr (Rodríguez et al., 2000;
Balona & Dziembowski, 2011), where spacings are not regular and modes are
therefore difficult to identify. Bedding et al. (2020) identified $\sim$60
TESS $\delta$ Sct stars oscillating at high frequencies with regular frequency
patterns. Gaia DR2 parallaxes (Gaia Collaboration et al., 2018) showed that
these stars were close to the ZAMS, and Bedding et al. (2020) found that
$\lambda$ Boo stars were overrepresented in this sample, consistent with the
idea that many $\lambda$ Boo stars are young objects.111Though not all are
young; Murphy & Paunzen (2017) showed with DR1 parallaxes that $\lambda$ Boo
stars are found at all main-sequence ages, and they may even exhibit infrared
excesses at a wide range of main-sequence ages (Gray et al., 2017; Murphy et
al., 2020b). A complete examination of southern $\lambda$ Boo stars observed
in the first year of TESS observations confirmed regular frequency spacings in
many of their Fourier spectra, including in HD 139614, and the fundamental
mode was identifiable in half the sample (Murphy et al., 2020a).

The recent progress on $\delta$ Sct stars offers the exciting prospect of
precise asteroseismic ages for intermediate mass stars and any discs or
planets they may host. Until now, such ages have generally been limited to
lower-mass stars via, e.g., empirical relations with stellar spindown (Barnes,
2007), activity decay (Baliunas et al., 1995), lithium depletion (Skumanich,
1972), or isochrone fitting (Valenti & Fischer 2005; see Soderblom 2010 for a
review), whereas for $\delta$ Sct stars, these methods are not generally
applicable (Russell, 1995; Pedersen et al., 2017; van Saders et al., 2019;
Zwintz, 2020). A related class of stars, the $\gamma$ Doradus stars that
pulsate in gravity modes (Guzik et al., 2000; Grigahcène et al., 2010), has
recently shown promise for asteroseismic ages because the gravity modes are
very sensitive to chemical discontinuities in the near-core region (Miglio et
al., 2008; Aerts, 2015; Van Reeth et al., 2015). Unfortunately, the $\gamma$
Dor instability strip is narrow (Dupret et al., 2005) and such detailed
asteroseismic information is only extractable from about 10% of $\gamma$ Dor
pulsators (Li et al., 2020). With the exception of certain $\delta$
Sct–$\gamma$ Dor hybrids (e.g. Kurtz et al., 2014; Saio et al., 2015), the
best $\delta$ Sct ages have often come from pre-main-sequence (pre-MS) members
of the class (Zwintz et al., 2014). While $\sim$50 pre-MS $\delta$ Sct stars
are known (Zwintz, 2017), a lack of reliable mode identifications has meant
that determining their evolutionary status has relied upon classical
parameters such as effective temperature ($T_{\rm eff}$), luminosity ($L$),
and metallicity ($Z$), and hindered the use of the pulsations to infer
metallicities or to improve the stellar models (Ripepi et al., 2011; Ripepi et
al., 2015; Chen & Li, 2018, 2019).

In this paper, we perform mode identification and determine an asteroseismic
age and metallicity for HD 139614, which we conclude is a pre-MS star near to
the zero-age main sequence (ZAMS). In Sec. 2 we describe and summarise the
stellar parameters of HD 139614 available in the literature. We describe our
analysis of the TESS photometry, including mode identification, in Sec. 3. We
describe our modelling method in Sec. 4, and our results in Sec. 5. Detailed
discussions are presented in Sec. 6 before the conclusions in Sec. 7.

##  2 HD 139614 in the literature

The discovery of emission lines and a disc around HD 139614 has led to it
being very well studied (Henize, 1976; Meeus et al., 1998), with several
studies dedicated to observations and modelling of its disc (Matter et al.,
2014; Menu et al., 2015; Matter et al., 2016; Carmona et al., 2017; Seok & Li,
2017; Laws et al., 2020; Muro-Arena et al., 2020). In this section, we
describe the stellar atmospheric parameters available from the literature for
HD 139614, which we summarise in Table 2.

Although its abundances were determined by Dunkin et al. (1997) and again by
Acke & Waelkens (2004), the first suggestion that HD 139614 is a $\lambda$ Boo
star came from a spectroscopic study of some Herbig Ae stars by Folsom et al.
(2012). Its abundances were found to be typical of $\lambda$ Boo stars and it
was accepted as a member of the class shortly thereafter (Murphy et al.,
2015). With respect to the Asplund et al. (2009) solar photosphere, the Folsom
et al. (2012) Fe abundance for HD 139614 is [Fe/H] = $-0.57\pm 0.13$; other
abundances relevant to this classification are [C/H] = $-0.18$, [N/H] =
$0.00$, and [O/H] = $-0.02$. This metallicity ([Fe/H]) has generally been used
for HD 139614 in studies of its disc or for computing isochrones to infer the
stellar age, but it is likely that this metallicity applies neither to the
whole disc (Kama et al., 2015) nor the bulk of the star (Paunzen et al., 2015;
Murphy et al., 2020a), because it is the manifestation of a surface
peculiarity originating in a dust cavity in the disc. In this work, we did not
assume any particular metallicity for the star. Instead, we explored a range
of metallicities around the solar value $Z=0.014$, which appears to be
appropriate for stars in UCL (Lastennet & Valls-Gabaud, 2002; Preibisch &
Mamajek, 2008). Further details are given in Sec. 4.

Spectroscopic effective temperatures of 7600–8250 K have been determined for
HD 139614 by various groups (Dunkin et al., 1997; Acke & Waelkens, 2004;
Folsom et al., 2012; Fairlamb et al., 2015), with the most recent values being
$T_{\rm eff}$ $=7600\pm 300$ and $7750\pm 250$ K, respectively. A recent
$T_{\rm eff}$ from Strömgren photometry is in good agreement with this
($7626\pm 153$ K; Murphy et al., 2020a). We adopt $T_{\rm eff}=7650\pm 200$ K
for this study.

A precise parallax of $7.4243\pm 0.0533$ mas is available from Gaia DR2 (Gaia
Collaboration et al., 2018), for use in calculating a stellar luminosity.
Vioque et al. (2018) determined $\log L/L_{\odot}$ =
$0.773^{+0.032}_{-0.010}$, which equates to $L=5.93^{+0.45}_{-0.14}$ L⊙.
Conversely, Murphy et al. (2020a) determined $L=6.78\pm 0.29$ L⊙ ($\log L$
$=0.83$), which is consistent with the value of Fairlamb et al. (2015, $\log
L$ $=0.82$). The exact cause for the difference is unknown, but could arise
from different assumptions on metallicity or extinction in the luminosity
calculation. In this work we adopt $L$ = 6.7$\pm$0.5 L⊙ ($\log L/{\rm
L}_{\odot}=0.83\pm 0.03$); exploration of a broader luminosity range
($>1\sigma$) using seismic constraints shows that the lower luminosity of
Vioque et al. (2018) yields a poor fit (Sec. 5). It is also noteworthy that
Fairlamb et al. (2015) found a negligible accretion luminosity of
$-0.10^{+0.20}_{-0.31}$ L⊙.

The stellar age has been determined by many authors by reference to
isochrones, with varying degrees of precision (e.g. Folsom et al., 2012;
Fairlamb et al., 2015; Gagné et al., 2018; Vioque et al., 2018). The median
age, which also has the smallest uncertainty, is $14.5^{+1.4}_{-3.6}$ Myr
(Vioque et al., 2018). This is consistent with the age map of UCL by Pecaut &
Mamajek (2016, their figure 9), which suggests an age of $14\pm 2$ Myr for HD
139614. In our seismic modelling (Sec. 4), we applied an age prior of $<30$
Myr.

HD 139614 is an unusually slow rotator for a star of its mass ($1.49\pm 0.07$
M⊙; Fairlamb et al. 2015), with $v\sin i=24$–26 km s-1; (Dunkin et al., 1997;
Acke & Waelkens, 2004; Folsom et al., 2012; Alecian et al., 2013). The
rotational velocity distribution for A stars with mass 1.6 M⊙ (Zorec & Royer,
2012), has a mode at $v_{\rm eq}=150$ km s-1 (their figure 7), with the
1$\sigma$ and 2$\sigma$ ranges being $\sim$110–225, and $\sim$90–300 km s-1,
respectively. This suggests that HD 139614 is either an intrinsically slow
rotator (i.e. $v_{\rm eq}$ is small) or it has a low inclination, $i$, or
both. Disc modelling shows that the disc inclination is low, at $20\pm 2$ deg
(Matter et al., 2014), although there are misalignments within the disc of a
few degrees (Muro-Arena et al., 2020). The stellar inclination and the disc
inclination are not necessarily equal (Ansdell et al., 2020), particularly if
there is a companion orbiting within the disc (Matsakos & Königl, 2017). But
if the star and disc are aligned, then HD 139614 has $v_{\rm eq}\sim 70$ km
s-1, which is still slow for its mass. Since this is a small fraction of the
Keplerian break-up velocity ($\Omega=0.16\Omega_{\rm K}$), we expect the star
to have negligible oblateness.

Table 1: A summary of the stellar parameters for HD 139614 inferred from the literature. Some of these are used in our modelling (see Sec. 4 for details). Parameter | Value | Units | Section
---|---|---|---
$T_{\rm eff}$ | $7650\pm 200$ | K | Sec. 2
$L$ | $6.7\pm 0.5$ | ${\rm L}_{\odot}$ | Sec. 2
$Z$ |  $\sim$0.014 | — | Sec. 2
age |  $<$30 | Myr | Sec. 2
$v\sin i$ | 24–26 | km s-1 | Sec. 2
$v_{\rm eq}$ | $70\pm 10$ | km s-1 | Sec. 2
$\Delta\nu$ | $6.83\pm 0.02$ | d-1 | Sec. 3

##  3 Pulsation properties from TESS photometry

TESS observed HD 139614 for 25.2 d at 2-min cadence in Sector 12 (2019
May/Jun) during the first year of its nominal mission (it will observe it for
another $\sim$27 d in the extended mission in Sector 38 (2021 Apr/May). TESS
light curves are available in Simple Aperture Photometry (SAP) and Pre-Search
Data Conditioning (PDC) versions; the latter has undergone processing to
remove instrumental signals (Jenkins et al., 2016). We downloaded both
versions from the MAST portal222https://mast.stsci.edu/ using the lightkurve
package (Lightkurve Collaboration et al., 2018), and after visually checking
that the PDC algorithm had not removed astrophysical signals, we analysed this
version of the light curve. To minimise the uncertainties, we subtracted the
mid-time of the observations, 2 458 640.273 d.

The light curve shows peak-to-peak variability of $\pm$5 mmag (Fig. 1a), with
periods in the range 0.4–1.2 h typical of $\delta$ Sct stars near the ZAMS.
There is no evidence of transient dimming by circumstellar material, as seen
in UXOR variables (Herbst et al., 1994) and their less extreme counterparts,
the ‘young dippers’ (Cody et al., 2014; Ansdell et al., 2016; Stauffer et al.,
2017). This supports the view that the disc is seen nearly face-on (Meeus et
al., 1998).

Figure 1: (a): TESS Sector 12 light curve of HD 139614, showing $\delta$ Sct
pulsations with 0.4–1.2 h periods. (b): The Fourier transform of the TESS
light curve. The frequencies show a characteristic spacing, especially at high
frequency, which we identify as the large separation of $\Delta\nu$ $=6.83$
d-1. Mode IDs are also shown for the modes we could identify.

Bedding et al. (2020) showed how the frequencies of some $\delta$ Sct stars
can lie on ridges in an échelle diagram when the correct large spacing
($\Delta\nu$) is used, according to the angular degree of the modes. Murphy et
al. (2020a) applied this technique to HD 139614, showing two clear ridges at
$\Delta\nu$ $=6.83$ d-1. The échelle diagram is shown in Fig. 2. Using only
these observations, we confirm the large spacing of $\Delta\nu=6.83\pm 0.02$
d-1 ($79.05\pm 0.23$ $\upmu$Hz), which is determined as the value that makes
the dipole mode ridge vertical over orders $n=3$–8 (Bedding et al., 2020),
with convention that $n=1$ is the fundamental mode. Since $n=8$ is below the
canonical “asymptotic regime” for p modes, our observed $\Delta\nu$ can be
expected to differ from the typical output of stellar evolution codes such as
MESA (discussed in detail in Sec. 4.2). Bedding et al. (2020) found that the
radial mode ridge curves at low frequency, while the dipole ridge remains
vertical, providing a way to identify these ridges. In this way, we identified
the dipole modes as the peaks along the left side of Fig. 2, and the radial
modes as the ones down the centre and curving to the right. Some later
refinement to this mode identification was made by comparing with preliminary
models (Sec. 4); our final mode IDs are shown in Figs 1b and 2.

Figure 2: Échelle diagram of HD 139614, with large separation,
$\Delta\nu=6.83$ d-1. A repeated overlap region is shown on the right for
clarity. The greyscale indicates the Fourier amplitude at that frequency, and
some smoothing has been applied. Circles show radial modes, triangles show
dipole modes, and observed peaks without corresponding mode IDs are shown as
crosses.

The $\ell=1$ modes appear to be singlets, rather than rotationally split
doublets or triplets. As mentioned in Sec. 2, the inclination of the star is
expected to be low, so singlets corresponding to $m=0$ are expected for the
dipole modes (Gizon & Solanki, 2003). This offers a loose constraint of
$i<30^{\circ}$, independent of the $i_{\rm disc}=20$ deg measurement, and is
consistent with star–disc alignment.

For precise frequency modelling, we determined the pulsation frequencies and
amplitudes via simultaneous least-squares fitting using PERIOD04 (Lenz &
Breger, 2004). We extracted peaks above 5 d-1 down to an amplitude limit of
0.03 mmag, corresponding to signal-to-noise ratios (SNR) of 3–4 for the
weakest peaks. At frequencies below 5 d-1, there are few significant peaks
(Fig. 1) and no means of identifying the modes or distinguishing them from
instrumental signals. The extracted frequencies are shown in Table 2 along
with their identifications. The table also includes a weak peak at 61.97 d-1,
which could be identified because of its location on the dipole mode ridge in
the échelle diagram.

We determined uncertainties using a 100-process Monte Carlo simulation in
PERIOD04. The Fourier noise (0.0068 mmag) was taken to be the median Fourier
amplitude in the residuals of our multi-frequency fit at frequencies > 5 d-1,
and was used to calculate the SNR of each peak. The observational frequency
uncertainties are as low as $10^{-4}$ d-1 in some cases, with strong
dependence on pulsation amplitude, as expected (Montgomery & Odonoghue, 1999;
Kjeldsen & Bedding, 2012).

Table 2: Significant frequencies in the TESS light-curve of HD 139614. The first column is a list of labels used in the text. Modes lying on the radial or dipole ridge have been identified with their radial order $n$ and degree $\ell$. The fundamental radial mode, with $n=1$ and $\ell=0$, is not observed. Entries are ordered by mode ID, then by those mentioned in the text (F12–F14), then by frequency. Label | Freq. |  $\sigma$Freq. | Amp. |  $\sigma$Amp. | SNR | $n$ | $\ell$
---|---|---|---|---|---|---|---
| d-1 | d-1 | mmag | mmag |  |  |
F1 | 21.36513 | 0.00013 | 1.152 | 0.006 | 169.4 | 1 | 1
F2 | 27.95328 | 0.00041 | 0.256 | 0.005 | 37.6 | 2 | 1
F3 | 34.77609 | 0.00006 | 1.557 | 0.005 | 228.9 | 3 | 1
F4 | 41.53747 | 0.00274 | 0.040 | 0.007 | 5.9 | 4 | 1
F5 | 48.37933 | 0.00161 | 0.085 | 0.006 | 12.5 | 5 | 1
F6 | 55.25743 | 0.00132 | 0.104 | 0.006 | 15.3 | 6 | 1
F7 | 61.97086 | 0.15366 | 0.021 | 0.007 | 3.1 | 7 | 1
F8 | 26.73244 | 0.00020 | 0.705 | 0.027 | 103.6 | 2 | 0
F9 | 32.55347 | 0.06198 | 0.414 | 0.136 | 60.9 | 3 | 0
F10 | 38.40601 | 0.00100 | 0.119 | 0.006 | 17.5 | 4 | 0
F11 | 58.46088 | 0.00321 | 0.059 | 0.006 | 8.6 | 7 | 0
F12 | 28.24243 | 0.00017 | 0.885 | 0.006 | 130.1 |  |
F13 | 27.01307 | 0.00027 | 0.458 | 0.007 | 67.3 |  |
F14 | 20.59936 | 0.00036 | 0.409 | 0.006 | 60.2 |  |
F15 | 18.54655 | 0.00272 | 0.047 | 0.006 | 6.9 |  |
F16 | 18.69868 | 0.05470 | 0.034 | 0.012 | 4.9 |  |
F17 | 19.17648 | 0.00204 | 0.046 | 0.007 | 6.8 |  |
F18 | 19.50079 | 0.00092 | 0.107 | 0.006 | 15.7 |  |
F19 | 19.80056 | 0.00440 | 0.031 | 0.005 | 4.6 |  |
F20 | 20.72404 | 0.00280 | 0.043 | 0.006 | 6.3 |  |
F21 | 22.15081 | 0.00148 | 0.080 | 0.007 | 11.8 |  |
F22 | 22.25558 | 0.00080 | 0.167 | 0.007 | 24.6 |  |
F23 | 22.79990 | 0.13974 | 0.030 | 0.008 | 4.5 |  |
F24 | 23.13894 | 0.00173 | 0.066 | 0.006 | 9.7 |  |
F25 | 23.30843 | 0.03089 | 0.046 | 0.012 | 6.8 |  |
F26 | 24.46210 | 0.31248 | 0.052 | 0.019 | 7.7 |  |
F27 | 24.72077 | 0.96335 | 0.062 | 0.038 | 9.1 |  |
F28 | 26.67124 | 0.57532 | 0.064 | 0.034 | 9.4 |  |
F29 | 27.57177 | 0.00334 | 0.046 | 0.006 | 6.8 |  |
F30 | 29.12031 | 0.00194 | 0.058 | 0.006 | 8.5 |  |
F31 | 29.28361 | 0.00120 | 0.108 | 0.006 | 15.9 |  |
F32 | 29.80605 | 0.01082 | 0.024 | 0.007 | 3.5 |  |
F33 | 30.27912 | 0.00082 | 0.152 | 0.006 | 22.4 |  |
F34 | 32.31897 | 1.26274 | 0.049 | 0.020 | 7.2 |  |
F35 | 58.50036 | 0.00638 | 0.031 | 0.005 | 4.5 |  |

##  4 Modelling Method

We performed grid-based modelling using stellar tracks computed with the
Modules for Experiments in Stellar Astrophysics (MESA) software version 11701
and theoretical frequency spectra computed with the GYRE stellar oscillation
program version 5.2 (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018,
2019; Townsend & Teitler, 2013). We computed the stellar tracks and
oscillation spectra separately, using structural models from particular
evolutionary timesteps as initial conditions for higher-precision oscillation
calculations. The details of these two types of calculation are described
below.

###  4.1 Evolutionary Tracks

Our grid considered variations over a number of parameters that dictate the
star’s evolutionary trajectory. Mass, global metal abundance ($Z$), and the
convective mixing length ($\alpha_{\text{MLT}}$), were varied at initial
resolutions of $0.1M_{\odot}$, $0.001$, and 0.1 $H_{\rm p}$,333pressure scale
heights respectively. Our best models are reported with final precisions of
$0.01M_{\odot}$ and $0.0005$ in $Z$. We aimed first to isolate the parameter
ranges and combinations that produced stellar tracks consistent with the
luminosity and effective temperature of HD 139614 (i.e. “classical”
constraints) independently of assumptions about the star’s age or seismic
features. Following the approach of Joyce & Chaboyer (2018b), grid resolution
was increased adaptively in regions of the parameter space where good
agreement was found with HD 139614’s classical observational constraints.

Our models adopted a simple photosphere (evaluated at an optical depth of
$\tau=2/3$) for atmospheric surface boundary conditions, and we tested the
effects of different model atmosphere choices using the best fitting model
(Sec. 4.3). We used the scaled solar abundances and opacities of Asplund et
al. (2009). We used the MESA equation of state (EOS), a basic nuclear reaction
network with additional rates for 56Fe and 58Ni,444basic_plus_fe56_ni58.net,
in MESA and the Henyey implementation of the mixing length formalism (Henyey
et al., 1965; Paxton et al., 2011). All of our stellar tracks assumed the
solar value for the initial helium abundance, $Y=0.29$. Although there is a
known degeneracy between mass, composition, and mixing length when attempting
to reproduce the surface features of stars (cf. Joyce & Chaboyer 2018a’s
analysis of HD 140283), our models were found to be insufficiently sensitive
to metallicity to suggest that variations in $Y$ would have a noticeable
impact on our results. Hence, we did not consider variations in $Y$. Our
models were likewise found to be insensitive to the choice of mixing length at
the 20% level, making such considerations less important than in the cases
studied by Joyce & Chaboyer (2018a) and Joyce & Chaboyer (2018b).

The $1\sigma$ ranges of temperature and luminosity measurements (Table 1) are
shown on the HR diagram in Fig. 3, along with five stellar tracks of different
mass. All models shown have $\alpha_{\text{MLT}}=1.7$ and $Z=0.010$. All were
terminated at an age of $30$ Myr, with the exception of the track with mass
1.50 $M_{\odot}$, which was continued until an age of 1 Gyr. As the 1.50-M⊙
track demonstrates, models in this mass and metallicity regime are only
consistent with the observational constraints during the pre-main sequence.
However, it is straightforward to match the classical observables with models
of different but plausible mass and metallicity at each of the three
subsequent evolutionary phases: main sequence, convective turn off, and base
of the subgiant branch. Without imposing age restrictions or taking seismic
quantities into account, the range of possible masses, metallicities, and ages
is quite large. When seismology (in the form of MESA’s $\Delta\nu$
approximation; see subsequent discussion) was even loosely considered,
however, it became immediately clear that models in evolutionary stages beyond
the pre-main sequence would not fit HD 139614 unless they were infeasibly
old—at least $0.8$ Gyr. Given the additional context of literature estimates
of the association’s age (10–20 Myr, Sec. 1; Pecaut & Mamajek 2016; Luhman &
Esplin 2020), we imposed a generous $30$ Myr age maximum on the tracks and
restricted further analysis to pre-main sequence models only.

Figure 3: Select tracks with varying mass are shown in the HR diagram. The
$1\sigma$ observational uncertainties are shown in the green shaded region.
All tracks were terminated at an age of $30$ Myr, with the exception of the
$1.50$-M⊙ model, shown in orange, to illustrate the main-sequence evolution.

Stellar ages prior to the ignition of hydrogen burning are measured from a
zero point defined by the choice of central temperature for the pre-main
sequence model. For our case, the default setting of pre_ms_T_c = 3d5 (i.e.
$3\times 10^{5}$ K) was used. This choice is arbitrary but is widely adopted
among MESA users and does not have a significant effect on the measured age
(see discussion in Sec. 6.3).

Fig. 4a demonstrates the rapid change in $\Delta\nu$ in the model’s first
$\sim$15 Myr of evolution (see also Suran et al. 2001). During this time, the
star undergoes catalytic hydrogen fusion by proton capture with
C${}^{12}_{6}$, but completion of the CNO cycle does not yet occur, and
C${}^{12}_{6}$ is exhausted over a $\sim$2-Myr period (Fig. 4b). The tracks in
Fig. 4 have the same parameters as those in Fig. 3, but demonstrate
sensitivity to mass for only $\sim$12 of the first 15 Myr. The sensitivity to
$\Delta\nu$, however, is present regardless of other parameters; likewise, our
full seismic models are also sensitive to the rapidly evolving structure
during this period. It is this sensitivity that allowed us to date our best-
fitting model with high precision.

\begin{overpic}[width=203.80193pt]{figs/dnu_C12.pdf}
\put(15.0,93.0){\large{\bf(a)}} \put(15.0,45.0){\large{\bf(b)}} \end{overpic}
Figure 4: (a): The evolution of $\Delta\nu$ over time is shown along the pre-
main sequence for six evolutionary tracks spanning a small range of masses.
The point corresponding to our best-fitting model is indicated via marker. We
note that the values of $\Delta\nu$ shown here are asymptotic values taken
from MESA directly and are therefore approximate, computed from the stellar
structure equations rather than from the oscillation equations. (b): Evolution
of the core carbon-12 content.

####  4.1.1 Consideration of Rotation

Since HD 139614 is an A-type star and its equatorial rotation rate is not
known, it is prudent to consider rotation as an input parameter in the models
where possible. Before ultimately restricting to the pre-main sequence, our
coarse grids included variations in surface rotation velocity: $v_{\rm
surf}=[0,20,80,160,250]$ km s-1. Ignoring age priors and $\Delta\nu$ entirely,
the preferred surface rotation rates (according to a weighting scheme
involving only agreement with temperature and luminosity) were found to be in
the range of 60–80 km s-1, consistent with calculations based on $v\sin i$ and
disc inclinations in Sec. 2.

To calculate the impact of oblateness, we computed the rotating models’ polar,
equatorial, and average values for effective temperature and luminosity. The
difference between the polar and equatorial temperature is 30 K for a MESA
model rotating at $v_{\rm eq}=80$ km s-1 that has the same $T_{\rm eff}$ and
$\log L$ as HD 139614. For the luminosity, the difference in the log is 0.012.
These values are small compared to the observational uncertainties.
Ultimately, we explored a wide parameter space surrounding our best seismic
fit (see discussion in Sec. 4.3), so oblateness had no effect on our search
for viable models. We discuss its (negligible) impact on the results in Sec.
5. This is despite the fact that we were unable to include rotation in our
models directly. Pre-built stellar models with rotation are already evolved to
an age older than the literature values for HD 139614, and as such, the best-
fitting rotating models all had ages $>$ 0.8 Gyr and asymptotic values of
$\Delta\nu$ were never within 20% of their expected value. We therefore had to
calculate the pre-main sequence evolution directly, but calculations that
included rotation never converged to hydrostatic models. As such, the pre-main
sequence grid ultimately did not include rotation.

###  4.2 Seismic Models

MESA estimates $\Delta\nu$ for each model by integrating the sound speed
across the star. According to the asymptotic theory of stellar oscillations,
this value approaches the actual spacing of p modes in the limit of high
radial orders. However, the modes in $\delta$ Sct stars have $n=1$ to
approximately 8, and so it is preferable to solve the oscillation equations
directly, which provides accurate frequencies at all $n$. We performed these
calculations with GYRE (Townsend & Teitler, 2013; Townsend et al., 2018).

GYRE is integrated with MESA but independently maintained and can be used as a
stand-alone package. It uses the Magnus Multiple Shooting scheme to solve the
oscillation equations (Kiehl, 1994; Gander & Vandewalle, 2007). We performed a
scan using a linear grid with alpha_osc = 10 and alpha_exp = 2 (Townsend,
2020) over the frequency range $0.5-75$ d-1. Our calculations were adiabatic,
we considered pressure and gravity modes with any $n$ for $l=0,1,2,3$, and $m$
was taken to be zero for all modes.

We computed a set of synthetic frequency spectra with GYRE for every model in
the pre-main sequence grid that was consistent with the classical constraints.
In total, over 13 300 synthetic frequency spectra were computed out of nearly
4 million total evolutionary points in the set of stellar tracks. We compared
the calculated frequencies to F1 through F11 under ten scenarios for the mode
identifications. The identifications given in Table 2 gave the best fit to the
models (see Sec 6.1 for details). Following Bedding et al. (2020), we did not
correct the model frequencies for near-surface effects in the way that is
usually done for solar-like oscillations (Kjeldsen et al., 2008; Ball & Gizon,
2014). It seems that the models fit without such corrections, probably because
A stars have thin surface convection zones. We return to the discussion of
atmospheric boundary conditions in Sec. 4.3.

###  4.3 Determining Global Best Fits

The quality of seismic fits was assessed by generating échelle diagrams, with
each theoretical frequency spectrum superimposed on the observations. Our
statistically preferred model is shown in Fig. 5, and we can see it has
excellent agreement. Formal, quantitative agreement was assessed using a
pseudo-$\chi^{2}$ cost statistic.

Figure 5: Échelle diagram showing the best asteroseismic model for HD 139614,
folded with the empirically derived $\Delta\nu$ of 6.83 d-1 found here and in
Murphy et al. (2020a).

We assigned uniform uncertainties of $0.1$ d-1 to each frequency because (i)
the reported observational uncertainties for the individual frequencies are
very small; (ii) uncertainty in mode ID is taken into account via the
consideration of multiple ID assignment scenarios; and (iii) there is no
reason to prioritize agreement with any particular mode over any other (of the
core 11 frequencies considered). This avoids the fit being dominated by the
strongest few modes. This uncertainty of 0.1 d-1 corresponds roughly to that
of the weakest identified modes and to the symbol sizes in our échelle
diagrams.

For each seismic model, we computed the normalized, pseudo-$\chi^{2}$
agreement statistic:555This is not a true $\chi^{2}$ because the underlying
distribution is not strictly normal, nor are the modeled frequencies
independent.

$\displaystyle\chi^{2}=\frac{1}{N}\sum_{i=1}^{N}\left(\left[\frac{F_{t,i}-F_{o,i}}{\sigma_{F,o_{i}}}\right]^{2}\right).$
(1)

In this statistic, which we refer to as the “seismic $\chi^{2}$,” $F$ are the
individual frequency values, $N$ is the number of frequencies considered, and
$o$ and $t$ subscripts refer to observed and theoretical values, respectively.
The seismic $\chi^{2}$ for our models ranged from roughly 0.3 to over 50 000.
Comparison with échelle diagrams confirmed that the statistically preferred
models also gave the best visual fits.

Figure 6: The grid of evolutionary tracks considered is shown in terms of mass
and metallicity. Each point may represent multiple models with different
parameters (e.g. ages), in which case the lowest seismic $\chi^{2}$ is shown.
Unequal grid spacing is due to the use of an adaptive search method (Joyce &
Chaboyer, 2018a).

Models were initially calculated in broad a grid surrounding the classical
constraints and that grid was expanded adaptively to find the global minimum
seismic $\chi^{2}$ by reference to heatmaps (Fig. 6), ensuring the best
solution was not bounded by the edge of the computational grid. Consistency
with the classical parameters was then re-evaluated when inferring the best-
fitting stellar parameters (Sec. 5).

Once the global minimum in seismic $\chi^{2}$ was found, we tested the effects
of different stellar atmospheres on the $\chi^{2}$ value, calculating new
evolutionary tracks with the same mass, mixing length, metallicity, and helium
abundance. The age was resampled using the new tracks. All of
simple_photosphere, Krishna_Swamy, photosphere_tables and grey_and_kap
atmospheres gave similar $\chi^{2}$ minima at ages within 0.25 Myr of each
other (i.e. to well within the 1$\sigma$ uncertainties we derive in Sec. 5).
Similar conclusions have been reached for other classes of pulsators even more
sensitive to atmosphere boundary conditions (Yıldız, 2007; Joyce & Chaboyer,
2018b; Nsamba et al., 2018; Viani et al., 2018). We tested different nuclear
nets similarly. For a 1.5-M⊙ star, most deuterium burning occurs in the first
1 Myr of the stellar evolution and by 10 Myr the effects of this burning on
the evolution are negligible (Krumholz, 2011, 2014). We recalculated
evolutionary and seismic models for our best-fitting model, adding the
pp_extras nuclear net and adopting a deuterium-to-hydrogen mass fraction of
$2\times 10^{-5}$ (Stahler et al., 1980; Linsky, 1998). This nuclear net more
accurately represents nuclear reactions in young stars but the resulting
change in the stellar parameters of HD 139614 was insignificant. The
$\chi^{2}$ minimum was of comparable depth at an age of 10.73 Myr, i.e. adding
deuterium burning resulted in an age difference of only 0.02 Myr, much smaller
than our quoted 0.77 Myr uncertainty.

##  5 Results

We evaluated the 13 000 models with calculated pulsation spectra to find the
global minimum in our seismic $\chi^{2}$ statistic and thereby find the best-
fitting stellar parameters. Note that the seismic $\chi^{2}$ is not normalised
(it is set by the adopted uncertainties in frequencies used for modelling). We
found that the global minimum could be isolated by setting a threshold of
$\chi^{2}<1$, which we adopted in our analysis. A broad overview of the
investigated models and the dependence of the seismic $\chi^{2}$ on the
various stellar parameters is shown in Fig. 7. The best models are well
localised for all parameters except $\alpha_{\rm MLT}$, from which we conclude
that the results do not depend on the mixing length.

Table 3: Modelling results for HD 139614. The best model (lowest seismic $\chi^{2}$ statistic, 0.3862) also coincides with the medians of the parameter distributions, quoted here. Uncertainties are the standard deviations of samples with seismic $\chi^{2}\leq 1.0$. †This asymptotic value of $\Delta\nu$ exceeds the empirical value for reasons explained in Sec. 4.2. Parameter | Units | Value | Uncertainty
---|---|---|---
mass | M☉ | 1.520 | 0.018
$Z$ |  | 0.0100 | 0.0010
age | Myr | 10.75 | 0.77
$\log L/{\rm L}_{\sun}$ (avg) |  | 0.829 | 0.016
$T_{\rm eff}$ (avg)666We quote the precision determined from modelling, only. Absolute accuracy is limited to 2% by the calibration of the $T_{\rm eff}$ scale by interferometry (Casagrande et al., 2014; White et al., 2018) | K | 7615 | 64
$\alpha_{\rm MLT}$777No dependence found. The best-fitting model happened to have $\alpha_{\rm MLT}=1.9$. |  | 1.9 | —
$\Omega$ | km s-1 | 0.0000 | (fixed)
$Y$ |  | 0.29 | (fixed)
$\Delta\nu$ (MESA, asymptotic†) |  $\upmu$Hz | 87.0670 |
$\Delta\nu$ (MESA, asymptotic†) | d-1 | 7.523 |

We determined the best-fitting values by calculating the median of each
parameter distribution, using the models inside the classical error box with
seismic $\chi^{2}<1$. We quote 1$\sigma$ uncertainties from the standard
deviation of those models. With the exception of mode IDs, we used only a
single set of fixed modelling assumptions, so the uncertainties do not take
into account modelling systematics or theoretical uncertainties in the
physical prescriptions themselves. The results are given in Table 3. The best
fitting model, with $\chi^{2}=0.3862$, happens to have $Z$, age, and mass
equal to those median values. This is the model shown in Fig. 5.

In the following sections we describe key results for individual parameters,
starting with metallicity, and discuss our $\chi^{2}<1$ threshold.

Figure 7: Corner plot of the modelling results. Grey circles show all models
for which a seismic $\chi^{2}$ was calculated, and coloured circles show the
125 models that have seismic $\chi^{2}<1$, with darker colours representing
lower values. The shaded red area encloses the subset of those 125 models that
lie within the $1\sigma$ error box in $T_{\rm eff}$ and $\log L$. Histograms
are calculated on that 47-model subset, the median and standard deviation of
which are written.

###  5.1 Metallicity

Figure 8: The seismic $\chi^{2}$ of the best models as a function of the
metallicity, $Z$. Models within the classical box are shown in blue, whereas
models outside this box are smaller and grey. The black line is our threshold
at $\chi^{2}=1$. The red circle and error bar show the median and 1$\sigma$
values determined.

Fig. 8 shows the seismic $\chi^{2}$ values as a function of metallicity, which
have a well-defined minimum at $Z=0.0100\pm 0.0010$. Comparison with Fig. 7
shows that the right-hand edge in the metallicity distribution in Fig. 8 is a
real drop-off in seismic agreement and does not represent an edge of the
computational grid. Similarly, metal weak models having $Z\lesssim 0.007$ are
strongly ruled out by seismology alone. We have made the first asteroseismic
determination of the interior metallicity of a $\lambda$ Boo star, and we
found it is not metal weak.

####  5.1.1 Present-day surface abundance vs global composition

Since the chemical composition of any star is stratified and varies with time,
the $Z$ we have determined is not necessarily representative of the observed
$Z$, even in the absence of chemical peculiarities. Choices for model
parameters governing mixing processes, especially diffusion, will therefore
affect the evolution of the surface abundance (Thoul et al., 1994; Dotter et
al., 2017). Given the young age of HD 139614, however, the timescale over
which diffusion acts is small. Nonetheless, we emphasize that our models do
not include diffusion and that the metallicities discussed in our results
represent the initial, global metal abundance and should not be conflated with
a spectroscopic surface abundance.

###  5.2 Age

Figure 9: (a): The seismic $\chi^{2}$ of the top models as a function of the
stellar age. Models within the classical box are shown in blue, whereas models
outside this box are smaller and grey. The black line is our threshold at
$\chi^{2}=1$. The red circle and error bar show the median and 1$\sigma$
values determined. (b): The mass and metallicity of those models in the shaded
region of panel a. More than one model can occupy each point in this plane.

Fig. 9a shows the seismic $\chi^{2}$ values as a function of age, which also
have a well-defined minimum ($10.75\pm 0.77$ Myr) separated by the threshold
of $\chi^{2}$ = 1. However, there is a division between models inside and
outside the classical box among the models with $\chi^{2}<1$, which occurs at
9.5 Myr. Fig. 9b shows that this division corresponds to an edge in mass and
metallicity: the more massive and lower metallicity models fall outside the
classical box. This division also extends to models of higher $\chi^{2}$, and
is visible in Fig. 7.

Fig. 9a also shows a sharp edge at 8.8 Myr where seismic agreement rapidly
falls off. Fig. 7 shows that this is not an edge effect in our computational
grid: models with younger ages were calculated (down to $\sim$5 Myr), but
these do not provide good seismic matches. Had the models between 8.8 and 9.5
Myr not been excluded by the classical error box, the inferred age would
differ by less than $1\sigma$ because of the strong seismic constraints.

###  5.3 Mass

Figure 10: The seismic $\chi^{2}$ of the top models as a function of the
stellar mass. Models within the classical box are shown in blue, whereas
models outside this box are smaller and grey. The black line is our threshold
at $\chi^{2}=1$. The red circle and error bar show the median and 1$\sigma$
values determined.

The mass range of the global minimum in seismic $\chi^{2}$ occurs at 1.5–1.6
M⊙ (Fig. 10), but the low-$\chi^{2}$ models above 1.55 M⊙ are not consistent
with classical constraints. Using all models inside the classical box with
seismic $\chi^{2}<1$, we determine the stellar mass as $1.520\pm 0.018$ M⊙. We
checked the impact of neglecting oblateness, which equates to a maximum change
of 0.012 in $\log L$ (Sec. 4.1.1): extending the classical box by this amount
does not cause any additional models to fall inside it.

##  6 Discussion

###  6.1 Mode identification ambiguity

HD 139614 is on the pre-MS, where pulsation properties change rapidly (Fig.
4). Our mode identification is based on lessons learned in Bedding et al.
(2020), where stars were all thought to be on the MS. We must therefore be
careful in extrapolating to the pre-MS regime, and look carefully at our
assumptions.

Our final mode identification used GYRE outputs to refine an initial mode
identification based on ridges in the echelle diagram. While these are clear
and well-separated at high frequencies, there is some ambiguity at lower
frequencies, in the orders with $n=1$ or 2. We initially assumed that the
radial and dipole modes were the strongest peaks lying near to each ridge.
This has physical motivation: mode visibility on an unresolved stellar disc
diminishes rapidly from $\ell\geq 2$ (Aerts et al., 2010), so one does not
generally expect $\ell\geq 2$ modes to have higher amplitudes than $\ell\leq
1$ modes. However, the latter are not necessarily driven as strongly as the
former, which also affects the observed amplitudes.

An example of a lower-amplitude peak being a better fit to the ridge is the
$n=2$ dipole mode. If the stronger peak here is adopted instead (F12 instead
of F2), the dipole ridge is made less straight, in conflict with the models.
The $n=2$ radial mode is also slightly ambiguous. We considered that the peak
lying at slightly higher frequency (F13) could be the radial mode instead of
F8, but this caused disagreement in the frequency ratio of the $n=2$ and $n=3$
radial modes, as well as the frequency difference with the $n=2$ dipole mode.

At $n=1$, there are two strong peaks that we might identify: F1 and F14. F1
lies at the base of the dipole ridge, hence we identified it as the $n=1$
dipole mode. We considered the possibility that this could be the fundamental
radial mode instead (in accordance with the identification in Murphy et al.
2020a), but this was a much worse fit in all models. Ultimately, the
fundamental radial mode is not assigned to any mode, as it does not lie close
to F14 either. This could be considered a weakness of our analysis, but it is
also possible that the fundamental mode is simply not excited in this star.
None of the seven peaks (F14–F20) at frequencies lower than the $n=1$ dipole
mode were identified, including among g-mode frequencies we calculated. Some
of these unidentified modes could be f modes (e.g. Cowling 1941; Stello et al.
2014) or island modes (e.g. Lignières & Georgeot 2009; Bouchaud et al. 2020).
Other observed peaks that remain unidentified (F21–F35) could belong to modes
of higher degree, but there are many more peaks at radial orders of $n=1$ and
2 than can be explained simply by including $\ell=2$ and $\ell=3$ modes. They
could also be modes of different azimuthal order that our non-rotating models
are unable to identify.

In our modelling, we have placed importance on some high-frequency modes of
low SNR. Pulsation models lack the ability to predict observed amplitudes, but
the mode frequencies remain relevant for constraining the internal structure
of the star, regardless of the amplitude to which they are driven. The fact
that these low SNR modes lie on ridges in the échelle and are reproduced by
the models is evidence that the identifications are correct. One of our mode
ID scenarios used only the higher-frequency modes that have the same ID in
every scenario (i.e. with none of the ambiguity mentioned above). The top
models ($\chi^{2}<1$) under that scenario had median values of mass, age and
metallicity within the 1$\sigma$ values quoted in Table 3.

###  6.2 Comparing pre-ZAMS and post-ZAMS mode frequencies

For the final 10–20 Myr before the ZAMS, the evolutionary tracks of pre-MS
stars lie very close to or cross their post-ZAMS counterparts (Fig. 3). Suran
et al. (2001) compared the pulsations of 1.8-M⊙ pre-MS and post-ZAMS models,
finding that the $\ell=0$ and high-frequency $\ell=1$ and $\ell=2$ modes were
indistinguishable near where the tracks cross. This is because the models have
similar mean densities and outer layers. In contrast, Suran et al. (2001) also
found that there were significant differences in low-frequency $\ell=1$ and
$\ell=2$ modes. These are g modes or, in some cases, mixed modes, which are
sensitive to the deep internal structure. Observations of these low frequency
non-radial modes would therefore allow pre-MS and post-ZAMS stars to be
distinguished from seismology alone. We now discuss whether we are able to do
this for HD 139614.

Fig. 11 shows an échelle diagram for a 1.6-M⊙ model at the pre-MS and post-
ZAMS ages where the model has the same $\Delta\nu$ as HD 139614. As expected,
major differences between the models are seen at low frequencies. The modes
below the frequency of the radial fundamental mode are g modes, which are only
sometimes observed alongside $\delta$ Sct pulsations (e.g. Zhang et al. 2020),
and we have not detected them in HD 139614. These models also have mixed
$\ell=2$ modes in the region where unidentified modes have been detected in HD
139614. However, with the number of unidentified modes exceeding the expected
number of $\ell=2$ mixed modes, and no ridge of high-frequency $\ell=2$
p-modes to use as guides, we are unable to reliably identify any of these
peaks as $\ell=2$ mixed modes. Finally, while there are also subtle
differences in the pulsation frequencies on the $\ell=0$ and $\ell=1$ ridges,
these are observationally indistinguishable at this level. Therefore, external
age constraints such as our $<$30-Myr age prior from UCL membership are
necessary to distinguish between the two evolutionary phases for HD 139614.
This then casts doubt on how reliably pre-MS and MS stars can be distinguished
by their pulsation spectra alone in cases similar to this where no g modes are
detected, and no $\ell=2$ modes are identified.

Fig. 12 shows the regions in the HR diagram where there is substantial overlap
in the $\ell=0$ and $\ell=1$ frequencies of pre-MS and post-ZAMS models. This
is confined to a small area where the pre-MS crosses the main sequence just
prior to the ZAMS, and near the ZAMS itself. However, depending on the stellar
mass, the duration of this near-overlapping phase of the post-ZAMS track can
be a few-hundred Myr, up to about 20 per cent of the MS lifetime. If an age
prior is available, such as via membership in a stellar association, precise
asteroseismic masses, ages and metallicities may be determinable for young
stars over a wide range of masses.

Similar crossings of pre-MS and MS tracks occur further from the ZAMS. In
these instances the MS star will be more evolved and its $\ell=1$ frequencies
will exhibit bumping of mixed modes and other effects from chemical
discontinuities near the core. Since these will be absent in the homogeneous
pre-MS star, evolutionary phases of stars at these other crossings, which
include those modelled by Guenther et al. (2007), should therefore be
distinguishable.

Figure 11: Pre-MS models (blue) can be very similar to post-ZAMS models
(orange), shown here for a 1.6 M⊙ star and $\Delta\nu=6.83$ d-1. Circles,
triangles and squares show $\ell=0,1$ and 2 modes, respectively. Figure 12: HR
diagram indicating regions where it may not be possible to distinguish between
pre-MS and post-ZAMS models from seismology alone. The colour scale indicates
the largest absolute frequency difference of any $\ell=0$ or 1 p mode with
$n\leq 9$ between closest matching pre-MS and post-ZAMS models. The summed
difference over many frequencies may be much greater, and significant under a
$\chi^{2}$-like statistic. The dashed lines indicate the empirical boundaries
of the $\delta$ Sct instability strip from Murphy et al. (2019).

###  6.3 Systematic age uncertainty

The dominant sources of age uncertainty in stellar modelling are, in no
particular order, (i) heavy element diffusion, which can deliver fresh nuclear
fuel into the star’s core; (ii) the size of the convective core, as influenced
in models by the degree of convective overshoot (pressure scale heights) and
the prescription for convective overshoot (exponential vs linear); (iii) bulk
material transport, which can also deliver fresh fuel to the core; and (iv)
the mixing length $\alpha_{\rm MLT}$, which determines the efficiency of the
energy transport mechanisms throughout the model and the ratio of flux
transported by each of them, by changing the size of the convection zone(s).
None of these four sources contributes substantial age uncertainty in our
model of HD 139614, primarily because of its young age. The degree of mixing
of nuclear fuel into the star’s core is irrelevant when the star’s main energy
source is gravitational contraction and the measurement of its age hinges on
the degree of that contraction to the ZAMS, rather than the amount of its
nuclear fuel that remains. We have already shown in Sec. 5 that $\alpha_{\rm
MLT}$ has no impact on our results. On the other hand, episodic accretion is
known to contribute a substantial fractional age uncertainty for young low-
mass stars (Baraffe et al., 2009; Vorobyov & Elbakyan, 2020), but this is not
well quantified for intermediate-mass stars, which are modelled with different
atmospheres and opacities. With the exception of atmospheric boundary
conditions, we have not investigated systematic modelling uncertainties in
this work, and do not include them in our quoted random uncertainties.
However, we have considered the inaccuracy between different age=0
definitions, which is of order $10^{4}$ yr (Soderblom, 2010) and therefore
insignificant relative to our uncertainty of $\sim$$10^{6}$ yr (Sec. 4.1; see
also Zwintz 2020; Steindl & Zwintz 2020).

###  6.4 Uncertainty from rotation

A potential source of systematic error in this work is the lack of inclusion
of rotation into our models. Rotation breaks the spherical symmetry of the
different azimuthal orders of non-radial pulsation, which leads to rotational
splittings when the star is observed from moderate inclinations. We see the
star nearly pole-on, so no splittings are observed, but asphericity has other
effects. One is gravity brightening at the poles, making the observed stellar
temperature and luminosity a function of viewing angle. This has already been
discussed in Sec. 4.1.1, and the magnitude of its effect has been calculated
and deemed insignificant to the methodology and results. Another effect of
asphericity is to shift pulsation frequencies. For an equatorial velocity of
80 km s-1, the frequency shifts are expected to be no more than 0.5% for
radial orders $3\leq n\leq 10$, which equates to little more than what our
uncertainties already encompass. For modestly larger inclinations
($30^{\circ}$ instead of 20∘), $v_{\rm eq}=50$ km/s-1 and the frequency shifts
are an order of magnitude smaller (Di Criscienzo et al., 2008). In summary, we
expect that our use of non-rotating models could potentially affect our
modelled frequencies by $\sim 1\sigma$.

###  6.5 Implications of the determined age of HD 139614

Our determined age of $10.75\pm 0.77$ Myr for HD 139614 is considerably
younger than literature estimates of the age of UCL ($16\pm 2$ Myr, Pecaut &
Mamajek 2016; 20 Myr, Luhman & Esplin 2020). Understanding the origin of this
difference is important for understanding the star formation history of the
association, the evolution of stars within UCL, and their interactions with
their discs.

UCL is so expansive that the assumption of a single age for the entire
association is overly simplistic. Pecaut & Mamajek (2016) showed that there
are clear age gradients in each subgroup of Sco-Cen, which for UCL amounts to
an age range spanning 10–26 Myr. In addition to this, any star-forming region
can be expected to have an intrinsic age spread of $\sim$2–5 Myr (Soderblom,
2010; Da Rio et al., 2016; Beccari et al., 2017; Krumholz et al., 2019), and
episodic accretion can also cause apparent age spreading (Baraffe et al.,
2009, 2012). After accounting for probable age spreads, our age measurement is
consistent with association membership, given the specific location of HD
139614 within UCL. The age precision achieved in this work makes HD 139614 a
useful benchmark within the association.

HD 139614 is rather old for an object with a protoplanetary disc (Ribas et
al., 2015). Our precise age and bulk metallicity will be useful in
understanding the retention and eventual dissipation time of a massive dust
disc, which likely goes hand-in-hand with dust trapping (Pinilla et al.,
2018).

###  6.6 Future work

Our model and seismology is a starting point for a more detailed analysis that
includes rotation and specific prescriptions of diffusion. That might include
a two-zone model with an outer zone with recent accretion, and an inner zone
comprising the majority of the stellar mass whose primordial composition has
not been modified by selective accretion or mixing. Such an analysis would
become even more valuable if a similar but chemically normal $\delta$ Sct star
in UCL could be used for a differential analysis.

Similar analyses could be applied to other stars in UCL in order to construct
an asteroseismic age map across the association, or to stars in other
associations to fine-tune their ages. Bedding et al. (2020) have already used
this approach to resolve the age debate surrounding the recently-discovered
Pisces-Eridanus stream (Curtis et al., 2019; Meingast et al., 2019). In
particular, the ages of many associations are established by the lithium
depletion boundary, with which quantitative ages can be difficult to ascertain
(Soderblom, 2010). Asteroseismology would be very welcome to check these Li
depletion ages (Zuckerman, priv. comm.).

##  7 Conclusions

HD 139614 is in a particular evolutionary stage where the asteroseismic large
spacing, $\Delta\nu$, has good sensitivity to mass. We used MESA and GYRE to
leverage this sensitivity and determine a precise asteroseismic age ($10.75\pm
0.77$ Myr) and metallicity ($Z=0.0100\pm 0.0010$) without degeneracy with mass
($M=1.52\pm 0.02$). We demonstrated that similar analyses may be possible for
young stars over a wide range of masses, especially if external constraints on
age are available to distinguish whether a given young star lies just before
or after the ZAMS (Sec. 6.2). Environments such as young stellar associations
are therefore particularly well suited to further analyses of this kind,
raising the exciting prospect of asteroseismic age benchmarking at the 10%
level for a variety of stellar associations and clusters to help calibrate
other age determination methods.

We have made the first asteroseismic determination of the interior metallicity
of a $\lambda$ Boo star, and we found it is not metal weak. The precise bulk
stellar metallicity and age determined in this work provide an opportunity to
model gas–dust separation and planet formation more precisely, unencumbered by
large uncertainties imposed by chemical peculiarities befouling the stellar
spectroscopy and by age dispersion and episodic accretion affecting cluster
ages. Our snapshot of the evolution of HD 139614 and its planet-forming disc
may also hold clues as to how some stars are able to retain discs for a
mysteriously long time, perhaps aided by embedded planets.

Association ages are often determined by main-sequence turn-off of high-mass
stars, and/or by study of the lower-mass members, usually revealing mass
dependence in the age measurements. The availability of a calibrator at
intermediate mass will be helpful to this problem. Similarly, understanding
the formation, evolution, and occurrence rates of planets as a function of
stellar mass is an active area of research that will benefit from having more
precise masses for intermediate-mass stars, where the planet occurrence rate
drops precipitously.

## Acknowledgements

The authors thank Lorenzo Spina, Przemesław Walczak, Adam Jermyn, Mark
Krumholz, and Jonathan Gagné for discussions. This research made use of
Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve
Collaboration et al., 2018), and echelle (Hey & Ball, 2020). SJM was supported
by the Australian Research Council through DECRA DE180101104. MJ was supported
the Research School of Astronomy and Astrophysics at the Australian National
University and funding from Australian Research Council grant No. DP150100250.
MJ wishes to thank the MESA developers’ team and especially Richard Townsend
for advice and maintenance of the GYRE forums. MK gratefully acknowledges
funding by the University of Tartu ASTRA project 2014-2020.4.01.16-0029
KOMEET, financed by the EU European Regional Development Fund.

## Data Availability

TESS light curves are publicly available at MAST888https://mast.stsci.edu/.
The inlists corresponding to our best-fitting model (mesa and gyre) are
available at the mesa hub on Zenodo: https://doi.org/10.5281/zenodo.4291277.

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