# Internal rotation and inclinations of slowly pulsating B stars: Evidence of
interior angular momentum transport

May G. Pedersen Kavli Institute for Theoretical Physics, Kohn Hall, University
of California, Santa Barbara, CA 93106, USA

###### Abstract

One of the largest uncertainties in stellar structure and evolution theory is
the transport of angular momentum in the stellar interiors. Asteroseismology
offers a powerful tool for measuring the internal rotation frequencies of
pulsating stars, but the number of such measurements has remained few for
$\gtrsim 3\,{\rm M}_{\odot}$ main-sequence stars. In this work, we compile a
list of 52 slowly pulsating B stars for which the interior rotation has been
measured asteroseismically. The measurements of the spin parameters, which
describe the relative importance of rotation, for the gravito-inertial mode
oscillations show that for 40 of the stars the oscillations fall within the
sub-inertial regime. We find that the core rotation frequencies of the stars
decrease as a function of age, and show evidence of angular momentum transport
occurring on the main-sequence. Finally, we derive the inclination angles of
the stars, showing that they are generally consistent with the expectations
from surface cancellation effects for the given oscillation modes.

Stellar rotation(1629) — Stellar cores(1592) — Stellar interiors(1606) —
Stellar pulsations(1625) — Stellar evolution(1599)

##  1 Introduction

Asteroseismology, the study and interpretation of stellar oscillations, is not
only a powerful tool for studying the internal rotation profiles of stars but
also our only means of doing so. The advent of space telescopes providing
long-term high-precision photometric light curves has significantly increased
the resolution of our measured frequency spectra, allowing us to more easily
identify the effects of rotation on the oscillation frequencies. This has
opened up a unique avenue to studying angular momentum transport in stars, and
has revealed new surprising results, such as the need of much more efficient
angular momentum transport to explain the much slower than predicted rotation
of the cores of evolved subgiant and red giant stars (e.g. Marques et al.,
2013; Goupil et al., 2013; Cantiello et al., 2014; Ouazzani et al., 2019).
Current attempts at including additional mechanisms in the theory of angular
momentum transport are either still under debate or unable to fully solve this
issue (Cantiello et al., 2014; Fuller et al., 2014; Belkacem et al., 2015;
Pinçon et al., 2017; Mathis et al., 2018; Fuller et al., 2019).

While the number of stars for which measurements of the internal rotation has
increased, the number of intermediate- and high-mass stars on the main-
sequence with such measurements has remained small. This is partly caused by
the lack of suitable observational data for these stars, but also hindered by
the reality that the rotational frequencies of the stars are comparable to the
oscillation frequencies, requiring the Coriolis acceleration be taken into
account. Recent improvements have been achieved for the $\gamma$ Doradus
($\gamma$ Dor; $1.4\,{\rm M}_{\odot}\lesssim M\lesssim 2.0\,{\rm M}_{\odot}$,
e.g. Kaye et al., 1999) stars, where the number of stars with measured near-
core rotation frequencies has increased from $\approx 50$ stars (e.g. Van
Reeth et al., 2016) to $\geq 600$ (Li et al., 2020). In comparison, the number
of main-sequence stars with $\gtrsim 3\,{\rm M}_{\odot}$ which have such
measured near-core rotation frequencies from space photometry were $\leq 10$
(e.g. Aerts et al., 2019). We refer the reader to Aerts et al. (2019) and
references therein for a detailed review of the current inferences of internal
rotation and angular momentum transport from asteroseismology in general, and
discuss here the inferred rotation properties of a sample of 52 slowly
pulsating B (SPB) stars, for which internal rotation frequencies have been
derive asteroseismically. For an overview of asteroseismic inferences obtained
for early type stars observed by space telescopes see also Bowman (2020).

The SPB stars are main-sequence B-type stars ($3\,{\rm M}_{\odot}\lesssim
M\lesssim 10\,{\rm M}_{\odot}$) which oscillate in non-radial gravity (g)
modes, where bouyancy acts as the dominant restoring force (e.g. Aerts et al.,
2010). These modes have the highest probing power near the convective cores.
Some SPB stars are known to be hybrid pulsators, and also exhibit pressure (p)
mode oscillations where pressure acts as the dominant restoring force. Figure
1 illustrates the difference between the propagation cavities and rotational
kernels of p- and g-mode oscillations for a 5 M⊙ star at two different main-
sequence ages. For the remainder of this work we will focus on the g-mode
oscillations for the sake of homogeneity of the sample.

Figure 1: Hydrogen mass fraction (X), propagation diagrams, and rotational
kernels as a function of fractional radius for a 5 M⊙ SPB star at two
different stages of the main-sequence evolution (left = zero-age main-
sequence, right = mid main-sequence). Top: During the main-sequence evolution,
the core hydrogen mass fraction has decreased and a chemical gradient been
developed which gives rise to the spike in the Brunt-Vaisala frequency shown
in blue in the middle right panel. Middle: The blue curve shows the Brunt-
Vaisala frequency $N$, while the Lamb frequency for a dipole mode $S_{\ell=1}$
is shown by the orange curve. The blue and orange shaded regions correspond to
the propagation cavity of the g- and dipole p-modes, respectively. Bottom:
Rotational kernel $K_{n\ell}(r)$ for a dipole prograde g-mode of radial order
$n=-22$ (blue), and a $(\ell,m,n)=(1,1,5)$ p-mode (orange) calculated using
Eq. (3.356) in Aerts et al. (2010). The rotational kernel is made
dimensionless by scaling with the stellar radius $R$. The kernels show that
the g-modes mainly probe the rotation profile near the convective core, while
the p-modes mainly provide information about the rotation near the stellar
surface.

The oscillation frequencies are characterized by a set of three quantum
numbers: 1) the degree $\ell$ defining the number of surface nodes associated
with the oscillation, 2) the azimuthal order $m$ giving the number of the
surface nodes crossing the equator, and 3) the radial order $n$ which equals
the number of radial nodes of the oscillation. The process of assigning these
three numbers to an oscillation is referred to as mode identification. This
information is required for asteroseismic modeling, and can be obtained in
different ways, some of which also simultaneously provide information about
the internal rotation of the stars.

A property of the g-modes is that modes with the same $\ell$ and $m$ values
form period spacing patterns, which are the period differences $\Delta
P_{\ell,m}=P_{\ell,m,n}-P_{\ell,m,n-1}$ between oscillations consecutive in
$n$ plotted as the function of the oscillation period $P_{\ell,m,n}$. The
detection of these patterns is used to identify the modes as required for
asteroseismic modeling. For non-rotating, chemically homogeneous stars the
period differences are approximately the same across all radial orders and the
period spacing pattern is flat. The inclusion of rotation, however, introduces
a tilt in the period spacing patterns (Bouabid et al., 2013) and the slope of
the patterns is used to infer the near-core rotation frequency of the stars
(Van Reeth et al., 2016; Ouazzani et al., 2017). A challenge of using period
spacing patterns to simultaneously derive internal rotation frequencies and
mode identifications is the high density of g-mode oscillation frequencies in
the Fourier spectrum. A high frequency resolution is required in order to
identify individual pulsation modes. The high-precision, high-cadence, and
long time-base photometric data made available by space telescopes have been
crucial for studying period spacing patterns in SPB stars. A period spacing
pattern was detected for the first time for an SPB star (HD 50230 Degroote et
al., 2010) using data from the CoRoT space telescope (Auvergne et al., 2009;
Baglin et al., 2009).

The rotational frequency of the star can also be measured in two other ways.
The first is the case of a slowly rotating star, where the rotational
splitting of an oscillation frequency into a $2\ell+1$ multiplet allows one to
not only obtain a mode identification of the oscillations, but also measure
the rotation frequency based on the width of the splitting. We will return to
the subject of rotational frequency splitting in Sect. 3, and point here the
reader to the beautiful example of the lower mass hybrid pulsator KIC 11145123
(Kurtz et al., 2014, see e.g. their figure 3). As in the case of the period
spacing pattern, a high-frequency resolution is required in order to detect
rotationally split oscillation modes. Finally, in the absence of period
spacing patterns and rotational split oscillation modes, as is generally the
case for SPB stars with ground-based data only, one may attempt to derive the
internal rotation frequency when carrying out mode-identification from multi-
color photometry. Assuming uniform rotation, Daszyńska-Daszkiewicz et al.
(2007, 2008, 2015) demonstrated that including the effect of rotation in the
calculation of the amplitude ratios from different photometric passbands
allows one to simultaneously derive $\ell,m$, and the rotational velocity of
the star.

In this work we compile a list of 52 SPB stars for which their interior
rotation has been measured with asteroseismology using data from space
telescopes or multi-color ground-based observing campaigns. An overview of the
sample is provided in Sect. 2. We start by calculating the rotational spin
parameters in Sect. 3, and investigate the age dependence of the near core
rotation in Sect. 4. We look for evidence of angular momentum transport in
Sect. 5. The assumption of rigid rotation is then used to infer the
inclination angles and its relation to the mode identification of the
oscillation frequencies in Sect. 6. Finally, our conclusions are presented in
Sect. 7.

##  2 Overview of the SPB star sample

Figure 2: SPB instability strip and the locations of the sample of SPB stars
studied in this work for which derived luminosities are available. The shaded
region indicates the number of radial orders of degree $\ell=1,2$ that are
expected to be excited in a given region of the HR diagram (Moravveji, 2016).
Example evolutionary tracks for five initial stellar masses are shown in grey.
The dot-dashed lines indicate the boundaries of the main-sequence, and the
dashed black line shows the beginning of the $\beta$ Cephei instability strip.
The colors of the data points indicate if the rotational frequency was derived
based on space photometry (pink) or ground-based data (grey).

The list of 52 SPB stars considered in this work was constructed by carrying
out a literature search to identify known SPB stars, which have 1) an interior
rotation estimate derived using their g-mode oscillations, and 2) an
unambiguous mode identification for at least one of their g-mode oscillation
frequencies. The sample of stars is shown in the HR diagram in Fig. 2. We
group the stars in two separate samples, one for which the rotation frequency
measurements and mode identification was done based on space photometry (pink)
and the other where only ground-based data was available (grey). We discuss
each of the two samples separately below.

For all of the stars in the sample, we likewise collect three different types
of age indicators, which are used in Sect. 4 to investigate the main-sequence
evolution of the internal rotation. For the first age indicator we use the
ratio $X_{\rm c}/X_{\rm ini}$ of the current core hydrogen mass fraction
$X_{\rm c}$ to the initial hydrogen mass fraction $X_{\rm ini}$. These ratios
are generally available in the cases where an asteroseismic modeling has been
carried out for the star. Secondly, we use the ratio $t/t_{\rm MS}$ between
the current age of the star $t$ and the main-sequence lifetime $t_{\rm MS}$.
These literature values were estimated by comparing the position of the stars
in the HR diagram to stellar evolution tracks, and are generally available for
the SPB stars with ground-based data. Finally, the surface gravity $\log g$ is
used as an age indicator. For six of the stars in the sample, this was the
only available age indicator. A comparison between $X_{\rm c}/X_{\rm ini}$,
$t/t_{\rm MS}$, and $\log g$ is given in Fig. 11 in Appendix A. For stars
where either $X_{\rm c}/X_{\rm ini}$ or $t/t_{\rm MS}$ values are available,
the same figure is used to make a conversion between the two, see Appendix A
for details. For stars where stellar parameters and rotation rates have been
derived but no errors on these estimates were provided, we adopt the median
fractional errors111As an example, the error on the rotational frequency
$\sigma_{f_{\rm rot}}$ for an SPB star where $\sigma_{f_{\rm rot}}$ has not
been estimated is taken to be $\sigma_{f_{\rm rot}}^{\star}=f_{\rm
rot}^{\star}\times(\text{median}\,\sigma_{f_{\rm rot}}/f_{\rm rot}\text{ of
the sample})$. of either the sample with space photometry or ground-based
data. An overview of the sample is given in Table 1.

As discussed in Appendix A, both $X_{\rm c}/X_{\rm ini}$ and $t/t_{\rm MS}$
are better age indicators than $\log g$ as these ratios are less affected by
the mass and by the internal mixing history of the stars. For the same reason,
we recommend that future asteroseismic modeling efforts include both ratios as
output of the modeling rather than just $X_{\rm c}$ and $t$ values.

###  2.1 Space photometry

33 of the 52 SPB stars were observed by space telescopes and 31 of these have
undergone asteroseismic modeling. The largest sample was taken from Pedersen
et al. (2021), who recently carried out an asteroseismic modelling of the
dipole period spacing patterns in 26 SPB stars observed by the _Kepler_ space
telescope (Koch et al., 2010; Borucki et al., 2010). For 11 of these stars,
patterns had previously been identified (Pápics et al., 2014, 2015, 2017;
Szewczuk & Daszyńska-Daszkiewicz, 2018; Zhang et al., 2018) but only three of
them had been asteroseismically modelled (Moravveji et al., 2015, 2016;
Szewczuk & Daszyńska-Daszkiewicz, 2018; Michielsen et al., 2021; Bowman &
Michielsen, 2021). Pedersen et al. (2021) discovered period spacing patterns
in 15 new SPB stars and performed asteroseismic modelling of the entire sample
of 26 SPB stars. In this work, we adopt the updated parameter estimates from
Pedersen (2022), which take into account that different shapes of the internal
mixing profiles do equally well at explaining the observed period spacing
patterns for some of the stars222The parameter values from Table 4 and 5 in
Pedersen (2022) were used in this paper. For the five stars for which a single
shape of the internal mixing profile was found to explain the observed period
spacing patterns the best, the values from Tables 2 and 3 in Pedersen (2022)
were used instead..

In addition to these 26 stars, two more SPB stars observed by _Kepler_ (KIC
8264293 and KIC 8324483) have period spacing patterns which have been modeled
asteroseismically. The period spacing pattern of KIC 8264293 was identified by
Szewczuk et al. (2021), and the subsequent asteroseismic modeling carried out
by Szewczuk et al. (2022) revealed that the star is close to the zero-age
main-sequence. Zhang et al. (2018) identified a period spacing pattern in KIC
8324483. The initial modeling of the pattern (Deng et al., 2018) resulted in a
mode identification of $(\ell,m)=(1,0)$ and was later improved by Wu et al.
(2020), revealing that the star is $\approx 33\%$ through its main-sequence
evolution.

Two of the stars in the sample with space photometry (HD 50230 and HD 43317)
were observed by the CoRoT space telescope. Degroote et al. (2010) identified
a period spacing pattern in HD 50230 consisting of dipole zonal modes
$(\ell,m)=(1,0)$. The star was later identified to be a spectroscopic binary,
exhibiting both g-mode pulsations and pressure (p) modes (Degroote et al.,
2012), the latter of which showing a frequency splitting $\Delta f_{\rm
p}=0.044\pm 0.007$ d-1 likely caused by rotation. The authenticity of the
pattern has later been questioned by Szewczuk & Daszyńska-Daszkiewicz (2015a).
Using the observed frequency splitting of the p-modes, Wu & Li (2019) derived
the rotational frequency of the star to be $f_{\rm rot}=0.007\pm 0.001$ d-1
and carried out an asteroseismic modeling of the star based on the g-mode
period spacing pattern. Fitting the slope of the period spacing pattern
following the method by Van Reeth et al. (2016) results in a rotation
frequency of $0.0955^{+0.0955}_{-0.0637}$ d-1 assuming $(\ell,m)=(1,0)$.

HD 43317 is a magnetic (Briquet et al., 2013; Buysschaert et al., 2017) B-type
star, and is one out of two stars in the sample with space photometry without
an identified period spacing pattern. The star was initially thought to be a
hybrid pulsator, with p- and g-mode oscillations detected in the CoRoT data of
HD 43317 by Pápics et al. (2012), who identified one of the g-mode
oscillations to be a prograde quadropole $(\ell,m)=(2,2)$ mode based on
spectroscopic mode identification. Buysschaert et al. (2018) reanalyzed the
CoRoT data of the star and carried out an asteroseismic modelling, in which
they identified ten additional $(\ell,m)=(1,-1)$ g-modes and five
$(\ell,m)=(2,-1)$ g-modes and found the star to be an SPB star rather than a
hybrid pulsator333The previously identified p-modes turned out to be
rotationally shifted g-modes.. As the only exception to the rest of the 51 SPB
stars, the rotation frequency of HD 43317 was derived from a magnetometric
analysis of the star rather than from asteroseismology (Buysschaert et al.,
2017), and then adopted for the asteroseismic modeling of the star assuming
rigid rotation.

One star (HD 201433) in the sample has been asteroseismically modeled and had
its internal rotation profile derived based on data from the BRITE satellite
constellation (Weiss et al., 2014). HD 201433 is a triple system with one
short period and one long period binary component, and has been studied in
detail by Kallinger et al. (2017) who also carried out an asteroseismic
modeling of the star. Aside from KIC 10526294, it is the only star in the
sample rotating slowly enough to exhibit clear rotationally split dipole
g-modes, which were used to infer the internal rotation profile of the star,
indicating that its radiative envelope is rigidly rotating up to $\sim 90\%$
of the stellar radius.

The last two stars (HD 261810 and HD 48012) in the sample of SPB stars with
space photometry were observed by the MOST space telescope (Walker et al.,
2003) and lie in the field of the open cluster NGC 2264. Zwintz et al. (2017)
identified a period spacing pattern in both stars based on the data from MOST,
which were asteroseismically modelled to show that both stars are on or just
approaching the zero-age main-sequence (ZAMS). For HD 261810, Zwintz et al.
(2017) found the period spacing pattern to consists of either $(\ell,m)=(1,1)$
or $(2,0)$ modes, meaning that the mode identification was not determined
unambiguously. We nevertheless choose to include the star in our sample, as it
is one on only few young SPB stars with a rotational frequency estimate
available. For remainder of this work, we will assume that the pattern
consists of $(\ell,m)=(1,1)$ modes.

###  2.2 Ground-based data

For the remaining 19 out of our sample of 52 SPB stars, the internal rotation
frequency was derived based on data from ground-based observing campaigns.
Szewczuk & Daszyńska-Daszkiewicz (2015b) performed mode identification and
estimated rotational velocities assuming uniform rotation of 31 SPB stars for
which multi-colour UBV Geneva photometry was available from ground-based
observing campaigns (using the oscillation frequencies and amplitudes provided
by North & Paltani, 1994; De Cat & Aerts, 2002; De Cat, 2002, 2007; Szewczuk &
Daszyńska-Daszkiewicz, 2015b). For 16 of the 31 SPB stars, Szewczuk &
Daszyńska-Daszkiewicz (2015b) obtained an unambiguously mode identification.
These 16 stars have therefore been included in our sample of SPB stars.
Asteroseismic modeling was previously achieved for one of the stars, HD 21071,
but unfortunately the main-sequence age was not provided (Szewczuk &
Daszyńska-Daszkiewicz, 2015a).

For the remaining three stars in the sample (HD 36999, HD 66181, and HD
97895), the internal rotation was derived by combining photometric data from
five different ground based surveys (Fedurco et al., 2020). The mode
identification of their oscillation frequencies was obtained by comparing the
observed oscillation frequencies to a grid of theoretical oscillation
frequencies computed under the assumption that the stars are slowly rotating.
While this modeling procedure also provided an absolute age estimate for each
of the stars, a fractional main-sequence age is not available.

We note that in this sample of 19 SPB stars there is a lack of fast-rotators.
All but two stars have projected equatorial velocities below 60 km s-s, while
the fastest rotating star in the sample rotates more than twice as slow than
the fastest rotating star in the sample of SPB stars with space photometry,
see Table 2. This lack of fast rotators was also previously noted by De Cat &
Aerts (2002) and Szewczuk & Daszyńska-Daszkiewicz (2015b).

=2.05in

Table 1: Overview of the sample of 52 SPB stars and their three different age
indicators.

Star ID | Data source | Known binary | $M$ | $X_{\rm c}/X_{\rm ini}$ | $t/t_{\rm MS}$ | $\log g$ | Reference
---|---|---|---|---|---|---|---
|  |  | (M⊙) | (%) | (%) |  |
KIC 3240411 | Kepler | N | 5.34${}^{+0.04}_{-0.04}$ | 68.33${}^{+0.23}_{-0.23}$ | 51.56$\pm 0.28$ | 4.09${}^{+0.05}_{-0.02}$ | 1
KIC 3459297 | Kepler | N | 3.63${}^{+0.66}_{-0.66}$ | 17.98${}^{+13.94}_{-13.94}$ | 92.66$\pm 7.16$ | 3.79${}^{+0.12}_{-0.12}$ | 1
KIC 3865742 | Kepler | N | 5.81${}^{+0.15}_{-0.15}$ | 28.13${}^{+6.19}_{-6.19}$ | 86.98$\pm 3.80$ | 3.69${}^{+0.13}_{-0.13}$ | 1
KIC 4930889 | Kepler | Y | 4.06${}^{+0.31}_{-0.31}$ | 36.21${}^{+0.73}_{-0.73}$ | 81.67$\pm 0.52$ | 3.91${}^{+0.00}_{-0.00}$ | 1
KIC 4936089 | Kepler | N | 3.76${}^{+0.24}_{-0.24}$ | 36.55${}^{+13.41}_{-13.41}$ | 81.42$\pm 9.58$ | 3.83${}^{+0.09}_{-0.09}$ | 1
KIC 4939281 | Kepler | N | 5.38${}^{+0.01}_{-0.01}$ | 48.01${}^{+0.09}_{-0.09}$ | 72.45$\pm 0.08$ | 3.70${}^{+0.05}_{-0.03}$ | 1
KIC 5309849 | Kepler | N | 5.67${}^{+0.07}_{-0.07}$ | 15.42${}^{+0.04}_{-0.04}$ | 93.93$\pm 0.02$ | 3.70${}^{+0.05}_{-0.05}$ | 1
KIC 5941844 | Kepler | N | 3.66${}^{+0.09}_{-0.09}$ | 93.99${}^{+0.44}_{-0.44}$ | 12.62$\pm 0.82$ | 4.30${}^{+0.02}_{-0.02}$ | 1
KIC 6352430 | Kepler | Y | 3.18${}^{+0.23}_{-0.23}$ | 53.23${}^{+24.90}_{-24.90}$ | 67.75$\pm 24.13$ | 4.04${}^{+0.15}_{-0.15}$ | 1
KIC 6462033 | Kepler | N | 7.27${}^{+0.17}_{-0.17}$ | 14.92${}^{+1.86}_{-1.86}$ | 94.17$\pm 0.90$ | 3.67${}^{+0.03}_{-0.03}$ | 1
KIC 6780397 | Kepler | N | 4.87${}^{+1.08}_{-1.08}$ | 28.44${}^{+23.80}_{-23.80}$ | 86.78$\pm 14.99$ | 3.64${}^{+0.08}_{-0.08}$ | 1
KIC 7630417 | Kepler | N | 7.01${}^{+0.09}_{-0.09}$ | 15.33${}^{+0.25}_{-0.25}$ | 93.97$\pm 0.12$ | 3.68${}^{+0.01}_{-0.01}$ | 1
KIC 7760680 | Kepler | N | 3.43${}^{+0.06}_{-0.06}$ | 60.63${}^{+7.94}_{-7.94}$ | 60.34$\pm 8.50$ | 4.01${}^{+0.02}_{-0.02}$ | 1
KIC 8057661 | Kepler | N | 9.14${}^{+0.59}_{-0.59}$ | 17.11${}^{+1.69}_{-1.69}$ | 93.10$\pm 0.85$ | 3.77${}^{+0.00}_{-0.00}$ | 1
KIC 8255796 | Kepler | N | 5.73${}^{+0.01}_{-0.01}$ | 2.01${}^{+0.04}_{-0.04}$ | 99.64$\pm 0.02$ | 3.64${}^{+0.05}_{-0.03}$ | 1
KIC 8264293 | Kepler | N | 3.54${}^{+0.06}_{-0.12}$ | 98.59${}^{+1.41}_{-3.11}$ | 3.73$\pm 6.97$ | 4.37${}^{+0.05}_{-0.06}$ | 2
KIC 8324482 | Kepler | N | 6.08${}^{+0.14}_{-0.25}$ | 71.07${}^{+0.25}_{-1.04}$ | 48.14$\pm 0.82$ | 4.25${}^{+0.03}_{-0.04}$ | 3
KIC 8381949 | Kepler | N | 7.69${}^{+1.06}_{-1.06}$ | 43.35${}^{+9.34}_{-9.34}$ | 76.32$\pm 7.46$ | 3.84${}^{+0.09}_{-0.09}$ | 1
KIC 8459899 | Kepler | Y | 3.94${}^{+0.61}_{-0.61}$ | 26.38${}^{+7.89}_{-7.89}$ | 88.03$\pm 4.69$ | 3.86${}^{+0.10}_{-0.10}$ | 1
KIC 8714886 | Kepler | N | 6.09${}^{+0.24}_{-0.24}$ | 29.56${}^{+0.37}_{-0.37}$ | 86.09$\pm 0.23$ | 3.86${}^{+0.01}_{-0.01}$ | 1
KIC 8766405 | Kepler | N | 4.40${}^{+0.48}_{-0.48}$ | 12.58${}^{+6.52}_{-6.52}$ | 95.27$\pm 3.01$ | 3.59${}^{+0.07}_{-0.07}$ | 1
KIC 9020774 | Kepler | N | 3.47${}^{+0.04}_{-0.04}$ | 60.20${}^{+15.63}_{-15.63}$ | 60.79$\pm 16.75$ | 4.15${}^{+0.08}_{-0.08}$ | 1
KIC 9715425 | Kepler | N | 4.66${}^{+0.38}_{-0.38}$ | 32.11${}^{+13.61}_{-13.61}$ | 84.45$\pm 9.00$ | 3.77${}^{+0.06}_{-0.06}$ | 1
KIC 10526294 | Kepler | N | 3.64${}^{+0.05}_{-0.15}$ | 23.29${}^{+3.95}_{-3.95}$ | 89.81$\pm 2.22$ | 3.74${}^{+0.13}_{-0.07}$ | 1
KIC 10536147 | Kepler | N | 6.51${}^{+0.51}_{-0.51}$ | 74.28${}^{+11.73}_{-11.73}$ | 43.93$\pm 15.86$ | 4.11${}^{+0.07}_{-0.07}$ | 1
KIC 11360704 | Kepler | N | 5.13${}^{+0.75}_{-0.75}$ | 45.13${}^{+10.78}_{-10.78}$ | 74.88$\pm 8.89$ | 3.90${}^{+0.09}_{-0.09}$ | 1
KIC 11971405 | Kepler | N | 3.59${}^{+0.08}_{-0.08}$ | 44.13${}^{+2.31}_{-2.31}$ | 75.70$\pm 1.87$ | 3.99${}^{+0.00}_{-0.00}$ | 1
KIC 12258330 | Kepler | N | 3.62${}^{+0.09}_{-0.09}$ | 74.28${}^{+6.62}_{-6.62}$ | 43.93$\pm 8.91$ | 4.28${}^{+0.02}_{-0.02}$ | 1
HD 43317 | CoRoT | N | 5.80${}^{+0.10}_{-0.20}$ | 77.14${}^{+1.43}_{-2.86}$ | 40.00$\pm 2.98$ | 4.14${}^{+0.01}_{-0.02}$ | 4
HD 50230 | CoRoT | Y | 6.21${}^{+0.06}_{-0.06}$ | 43.10${}^{+1.41}_{-1.13}$ | 76.52$\pm 1.01$ | 3.72${}^{+0.00}_{-0.01}$ | 5
HD 48012 | MOST | N | 4.00${}^{+0.07}_{-0.14}$ | 100.00${}^{+0.00}_{-3.16}$ | 0.88$\pm 5.75$ | 4.30${}^{+0.15}_{-0.15}$ | 6
HD 261810 | MOST | N | 5.50${}^{+0.09}_{-0.19}$ | 100.00${}^{+0.00}_{-3.16}$ | 0.88$\pm 5.75$ | 4.25${}^{+0.10}_{-0.10}$ | 6
HD 201433 | BRITE | Y | 3.05${}^{+0.05}_{-0.11}$ | 57.29${}^{+9.00}_{-9.00}$ | 63.80$\pm 9.11$ | 4.15${}^{+0.07}_{-0.07}$ | 7
HD 1976 | ground based | Y | 5.10${}^{+1.00}_{-1.00}$ | 43.75${}^{+28.37}_{-28.37}$ | 76.00$\pm 21.00$ | 3.97${}^{+0.20}_{-0.20}$ | 8
HD 21071 | ground based | N | 4.00${}^{+0.60}_{-0.60}$ | 77.85${}^{+22.01}_{-22.01}$ | 39.00$\pm 30.00$ | 4.09${}^{+0.05}_{-0.05}$ | 8, 9
HD 24587 | ground based | Y | 3.70${}^{+0.50}_{-0.50}$ | 79.92${}^{+20.43}_{-20.43}$ | 36.00$\pm 29.00$ | 4.26${}^{+0.20}_{-0.20}$ | 10
HD 25558 | ground based | Y | … | … | … | 4.20${}^{+0.20}_{-0.20}$ | 11
HD 26326 | ground based | N | 4.50${}^{+0.70}_{-0.70}$ | 70.39${}^{+27.23}_{-27.23}$ | 49.00$\pm 32.00$ | 4.14${}^{+0.20}_{-0.20}$ | 11
HD 36999 | ground based | N | 3.60${}^{+0.06}_{-0.12}$ | … | … | 4.31${}^{+0.05}_{-0.06}$ | 12
HD 53921 | ground based | Y | 3.70${}^{+0.50}_{-0.50}$ | 77.85${}^{+22.01}_{-22.01}$ | 39.00$\pm 30.00$ | 4.23${}^{+0.20}_{-0.20}$ | 10
HD 66181 | ground based | N | … | … | … | 4.35${}^{+0.05}_{-0.06}$ | 12
HD 74195 | ground based | N | 5.50${}^{+1.00}_{-1.00}$ | 35.73${}^{+28.04}_{-28.04}$ | 82.00$\pm 18.00$ | 3.91${}^{+0.20}_{-0.20}$ | 10
HD 85953 | ground based | N | 6.80${}^{+1.40}_{-1.40}$ | 35.73${}^{+28.04}_{-28.04}$ | 82.00$\pm 18.00$ | 3.91${}^{+0.20}_{-0.20}$ | 13
HD 92287 | ground based | Y | 5.40${}^{+1.00}_{-1.00}$ | 50.80${}^{+27.33}_{-27.33}$ | 70.00$\pm 23.00$ | 4.00${}^{+0.20}_{-0.20}$ | 10
HD 97895 | ground based | N | 4.80${}^{+0.08}_{-0.17}$ | … | … | 4.39${}^{+0.05}_{-0.06}$ | 12
HD 121190 | ground based | N | 2.80${}^{+0.30}_{-0.30}$ | … | … | 4.43${}^{+0.09}_{-0.09}$ | 13
HD 123515 | ground based | Y | 3.00${}^{+0.40}_{-0.40}$ | 79.92${}^{+19.67}_{-19.67}$ | 36.00$\pm 28.00$ | 4.27${}^{+0.20}_{-0.20}$ | 10
HD 160762 | ground based | Y | 6.14${}^{+0.22}_{-0.22}$ | … | … | 3.82${}^{+0.07}_{-0.07}$ | 14
HD 179588 | ground based | Y | 3.10${}^{+0.40}_{-0.40}$ | 81.26${}^{+19.20}_{-19.20}$ | 34.00$\pm 28.00$ | 4.28${}^{+0.20}_{-0.20}$ | 15
HD 182255 | ground based | Y | 3.90${}^{+0.60}_{-0.60}$ | 77.85${}^{+22.82}_{-22.82}$ | 39.00$\pm 31.00$ | 4.23${}^{+0.20}_{-0.20}$ | 10
HD 208057 | ground based | N | 5.10${}^{+0.80}_{-0.80}$ | 72.72${}^{+26.96}_{-26.96}$ | 46.00$\pm 33.00$ | 4.15${}^{+0.20}_{-0.20}$ | 10
HD 215573 | ground based | N | 4.00${}^{+0.70}_{-0.70}$ | 62.78${}^{+27.08}_{-27.08}$ | 58.00$\pm 28.00$ | 4.09${}^{+0.20}_{-0.20}$ | 10

Note. — The type of data used for the asteroseismic analysis is provided in
the second column, the third column gives the mass of the star, while the
fourth column indicates if the star is a known/suspected binary (Y) or not
(N). Column five to seven lists the three age indicators available or derived
from literature values (see Appendix A), while the last column gives the
source of the age indicator and mass.

References: (1) Pedersen (2022), (2) Szewczuk et al. (2022), (3) Wu et al.
(2020), (4) Buysschaert et al. (2018), (5) Wu & Li (2019), (6) Zwintz et al.
(2017), (7) Kallinger et al. (2017), (8) De Cat & Aerts (2002), (9) Gebruers
et al. (2021), (10) Hubrig et al. (2006), (11) Sódor et al. (2014), (12)
Fedurco et al. (2020), (13) Aerts & Kolenberg (2005), (14) Briquet et al.
(2007), (15) De Cat et al. (2007).

##  3 Rotational spin parameters

Figure 3: Spin parameter as a function of oscillation frequency in the co-
rotating frame of reference. Stars for which inferences were made based on
space photometry are shown in the left panel, while stars with ground based
data are shown on the right. For stars with period spacing patterns, the
lowest (circle) and highest (triangle) frequency in the period spacing pattern
are connected by lines. Open symbols correspond to stars without period
spacing patterns. The colors indicate the derived rotation frequencies of the
stars. The super-inertial regime is indicated by the hatched region, while the
light-grey shaded region shows part of the sub-inertial regime.

In the absence of rotation, all oscillation modes with $m=-\ell,\dots,\ell$
and the same radial order $n$ share the same oscillation frequency. For a
rotating star, however, this degeneracy is lifted, and a rotational splitting
of the oscillation frequency into a $2\ell+1$ multiplet occurs due to the
effects of the Coriolis force (Aerts et al., 2010). The faster the rotation,
the larger the observed splitting will be until the regularity of the
multiplets are no longer clearly visible. Depending on how fast the star is
rotating, the contribution from the Coriolis force can be treated as a
perturbation to the oscillation equations. However, if the rotation frequency
becomes comparable to or larger than the oscillation frequency this is no
longer the case, and the g-mode oscillations occur in the sub-inertial instead
of the super-inertial regime. In both regimes, the oscillations behave as
gravito-inertial modes in radiative regions, but either propagate as inertial
modes (sub-inertial regime) or are evanescent (super-inertial regime) in
convective zones (e.g. Mathis et al., 2014). To quantify these regimes, we use
the spin parameter $s$, which is also the inverse of the Rossby number and
describes the relative importance of rotation

$s=\frac{2f_{\rm rot}}{f_{\rm co}}.$ (1)

Here $f_{\rm rot}$ is the rotation frequency of the star, and $f_{\rm co}$ is
the frequency of the oscillation in the co-rotating frame. It is related to
the frequency in the inertial frame of reference $f_{\rm in}$, i.e. the
observed oscillation frequency, through the transformation between the two
reference frames

$f_{\rm in}=f_{\rm co}+mf_{\rm rot}.$ (2)

For four of the 52 SPB stars (KIC 8264293, KIC 8322482, HD 48012, and HD
24587), an equatorial rotational velocity $V_{\rm eq}$ was provided instead of
$f_{\rm rot}$ from asteroseismology. Common for these four stars is that the
$V_{\rm eq}$ values were derived under the assumption of rigid rotation, and
we therefore convert them to rotation frequencies using $f_{\rm rot}=V_{\rm
eq}/(2\pi R)$, where $R$ is the radius of the star. For stars without a known
radius we calculate it, when possible, using either
$R=\sqrt{L/(4\pi\sigma_{\rm SB}T_{\rm eff}^{4})}$ or $R=\sqrt{GM/g}$ depending
on what information is available.

In Fig. 3 we show the spin parameters calculated for the SPB stars with an
asteroseismic rotation frequency measured from space photometry (left) and
ground-based data (right). For stars with period spacing patterns the highest
(triangle) and lowest (circle) oscillation frequency in the pattern is
connected by a line to show the range in spin parameters for the pattern. For
stars without period spacing patterns, we instead calculate the spin parameter
for the highest amplitude oscillation mode (open circles). The colors indicate
how fast the star is rotating near the convective core, where the g-modes have
their dominant probing power. As seen in the figure, 25 of the 32 stars with
period spacing patterns have at least a part of their pattern falling inside
the sub-inertial regime. For 15 of the 20 SPB stars without period spacing
patterns, the dominant oscillation mode likewise fall within the sub-inertial
regime. This is in line with previous results for SPB stars and their lower
mass counterpart the $\gamma$ Dor stars (Van Reeth et al., 2016; Aerts et al.,
2017; Li et al., 2020), which likewise oscillate in g-modes and have period
spacing patterns. We list the derived spin parameters shown in Fig. 3 in Table
2 along with the parameters used to derive them.

Pedersen et al. (2021) took the effects of the Coriolis force on the
oscillation frequencies into account when calculating the theoretical
oscillation frequencies with the stellar oscillation code GYRE (Townsend &
Teitler, 2013; Townsend et al., 2018; Goldstein & Townsend, 2020) and when
deriving the observed rotational frequency from the slope of the period
spacing patterns by adopting the Traditional Approximation of Rotation (TAR,
Eckart, 1960; Lee & Saio, 1987; Bildsten et al., 1996; Townsend, 2005). In
this approximation, the horizontal component of the rotation vector is
ignored, allowing the pulsation equations to be separated into their spherical
coordinates $(r,\theta,\phi)$, while also assuming that the star is
sufficiently slowly rotating such that the centrifugal force can be ignored
(Lee & Saio, 1987; Bildsten et al., 1996; Lee & Saio, 1997). The TAR was
previously adopted in the estimation of the rotation frequency and/or the
asteroseismic modeling for 46 out of the 52 SPB stars discussed in this work,
whereas the rotation was considered small enough for the Coriolis force to be
treated as a perturbation to the oscillation equations for six of the stars.
Under the assumption of rigid rotation, five of these six stars rotate less
than 6 % of their Keplerian breakup velocity, in which case 1st order
perturbation methods are sufficient for the frequency range of the observed
oscillations, see Fig. 5 of Ballot et al. (2010). For the sixth star, the
rotation rate with respect to the breakup velocity could not be estimated
because the mass of the star is unknown.

For increasingly faster rotating stars, the TAR will start to break down as
the importance of the centrifugal force increases and the star becomes more
and more deform. The effects on the deformation of the star are smaller near
the convective cores where the gravity and gravito-inertial modes are most
sensitive, implying that the TAR may still be adequate even in the cases of
rapid rotation for some of the oscillation modes. One study comparing the
period spacings obtained with the TAR to those derived when the 2D effects of
rotation are accounted for was carried out by Ouazzani et al. (2017),
following a previous similar study by Ballot et al. (2012). Figure 1 of
Ouazzani et al. (2017) shows that the largest deviations between the TAR and
the 2D treatment are found for the retrograde ($m<0$) modes, while the
differences are smaller than $\approx 4\%$ and $\approx 0.8\%$ for zonal
($m=0$) and prograde ($m>0$) modes, respectively, for $f_{\rm rot}<2\,{\rm
d}^{-1}$. For rotation frequencies smaller than 1 d-1, the differences remain
smaller than $0.2\%$ for all azimuthal orders. For 40 out of the 52 SPB stars
the derived rotation frequencies are smaller than 1 d-1, and only two of the
stars have $f_{\rm rot}\approx 2$ d-1. We therefore consider the TAR to be a
good assumption for the majority of the 52 SPB stars considered in this work.

=0.5in

Table 2: Additional asteroseismic information on the SPB star sample.

KIC | $\ell$ | $m$ | $f_{\rm rot}$ | $f_{\rm in}^{\rm 1}$ | $f_{\rm in}^{\rm 2}$ | $s^{\rm 1}$ | $s^{\rm 2}$ | $R$ | $V_{\rm eq}\sin i$ | $i$ | References
---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (d-1) | (d-1) | (d-1) |  |  | ($R_{\odot}$) | (km s-1) | (o) |
KIC 3240411 | 1 | 0 | 1.166${}^{+0.008}_{-0.008}$ | 0.61554$\pm$0.00003 | 1.45306$\pm$0.00002 | 3.79${}^{+0.03}_{-0.03}$ | 1.60${}^{+0.01}_{-0.01}$ | 3.31${}^{+0.18}_{-0.18}$ | 22 $\pm$ 5 | 6.3${}^{+1.5}_{-1.5}$ | 1,2,3
KIC 3459297 | 1 | 1 | 0.623${}^{+0.025}_{-0.025}$ | 0.89350$\pm$0.00002 | 1.12832$\pm$0.00003 | 4.61${}^{+1.25}_{-1.25}$ | 2.47${}^{+0.42}_{-0.42}$ | 4.00${}^{+0.29}_{-0.29}$ | 106 $\pm$ 14 | 55.9${}^{+11.8}_{-11.8}$ | 1,2,4
KIC 3865742 | 1 | 1 | 0.625${}^{+0.022}_{-0.022}$ | 0.97001$\pm$0.00003 | 1.27767$\pm$0.00007 | 3.63${}^{+1.51}_{-1.51}$ | 1.92${}^{+0.53}_{-0.53}$ | 5.78${}^{+0.83}_{-0.83}$ | 129 $\pm$ 26 | 42.3${}^{+13.7}_{-13.7}$ | 1,2,5
KIC 4930889 | 1 | 1 | 0.740${}^{+0.008}_{-0.008}$ | 1.02072$\pm$0.00007 | 1.37295$\pm$0.00006 | 5.27${}^{+0.16}_{-0.16}$ | 2.34${}^{+0.04}_{-0.04}$ | 3.64${}^{+0.15}_{-0.15}$ | 116 $\pm$ 6 | 58.2${}^{+6.6}_{-6.6}$ | 1,2,3
KIC 4936089 | 1 | 0 | 0.144${}^{+0.025}_{-0.025}$ | 0.78792$\pm$0.00006 | 1.89420$\pm$0.00006 | 0.37${}^{+0.12}_{-0.12}$ | 0.15${}^{+0.05}_{-0.05}$ | 3.92${}^{+0.51}_{-0.51}$ | 48 $\pm$ 7 | … | 1,2,6
KIC 4939281 | 1 | 1 | 0.772${}^{+0.001}_{-0.001}$ | 1.22779$\pm$0.00013 | 2.23421$\pm$0.00004 | 3.38${}^{+0.01}_{-0.01}$ | 1.06${}^{+0.00}_{-0.00}$ | 5.38${}^{+0.38}_{-0.38}$ | 115 $\pm$ 20 | 33.1${}^{+7.3}_{-7.3}$ | 1,2,4
KIC 5309849 | 1 | 1 | 0.462${}^{+0.005}_{-0.005}$ | 0.87635$\pm$0.00002 | 1.64622$\pm$0.00001 | 2.23${}^{+0.04}_{-0.04}$ | 0.78${}^{+0.01}_{-0.01}$ | 5.55${}^{+0.75}_{-0.75}$ | … | … | 1,2,7
KIC 5941844 | 1 | 0 | 0.952${}^{+0.037}_{-0.037}$ | 0.97725$\pm$0.00007 | 2.48880$\pm$0.00013 | 1.95${}^{+0.15}_{-0.15}$ | 0.76${}^{+0.06}_{-0.06}$ | 2.25${}^{+0.07}_{-0.07}$ | 34 $\pm$ 4 | 18.2${}^{+3.0}_{-3.0}$ | 1,2,5
KIC 6352430 | 1 | 1 | 0.659${}^{+0.062}_{-0.062}$ | 0.99772$\pm$0.00002 | 1.75446$\pm$0.00002 | 3.90${}^{+2.76}_{-2.76}$ | 1.20${}^{+0.45}_{-0.45}$ | 2.89${}^{+0.48}_{-0.48}$ | 70 $\pm$ 2 | 41.9${}^{+13.1}_{-13.1}$ | 1,2,8
KIC 6462033 | 1 | 0 | 0.634${}^{+0.012}_{-0.012}$ | 0.92520$\pm$0.00001 | 3.43849$\pm$0.00002 | 1.37${}^{+0.12}_{-0.12}$ | 0.37${}^{+0.03}_{-0.03}$ | 6.66${}^{+0.24}_{-0.24}$ | 75 $\pm$ 19 | 20.3${}^{+6.1}_{-6.1}$ | 1,2,5
KIC 6780397 | 1 | 0 | 0.465${}^{+0.250}_{-0.250}$ | 0.91610$\pm$0.00003 | 2.35034$\pm$0.00006 | 1.02${}^{+0.45}_{-0.45}$ | 0.40${}^{+0.18}_{-0.18}$ | 5.56${}^{+1.00}_{-1.00}$ | 120 $\pm$ 0 | 67.9${}^{+14.0}_{-14.0}$ | 1,2,9
KIC 7630417 | 1 | 1 | 0.451${}^{+0.000}_{-0.000}$ | 0.81281$\pm$0.00001 | 1.35478$\pm$0.00003 | 2.49${}^{+0.11}_{-0.11}$ | 1.00${}^{+0.03}_{-0.03}$ | 6.31${}^{+0.08}_{-0.08}$ | 135 $\pm$ 16 | 69.4${}^{+10.4}_{-10.4}$ | 1,2,6
KIC 7760680 | 1 | 1 | 0.453${}^{+0.022}_{-0.022}$ | 0.68838$\pm$0.00005 | 1.15034$\pm$0.00008 | 3.86${}^{+0.71}_{-0.71}$ | 1.30${}^{+0.13}_{-0.13}$ | 2.89${}^{+0.05}_{-0.05}$ | 62 $\pm$ 5 | 68.9${}^{+9.9}_{-9.9}$ | 1,2,10
KIC 8057661 | 1 | 0 | 0.426${}^{+0.142}_{-0.142}$ | 0.55151$\pm$0.00002 | 2.04747$\pm$0.00007 | 1.54${}^{+0.54}_{-0.54}$ | 0.42${}^{+0.15}_{-0.15}$ | 6.57${}^{+0.30}_{-0.30}$ | 35 $\pm$ 8 | 14.1${}^{+9.0}_{-9.0}$ | 1,2,5
KIC 8255796 | 1 | 1 | 0.146${}^{+0.006}_{-0.006}$ | 0.35248$\pm$0.00003 | 0.39565$\pm$0.00003 | 1.41${}^{+0.07}_{-0.07}$ | 1.17${}^{+0.06}_{-0.06}$ | 5.93${}^{+0.47}_{-0.47}$ | … | … | 1,2
KIC 8264293 | 1 | 1 | 2.412${}^{+0.171}_{-0.159}$ | 2.82140$\pm$0.00009 | 3.17810$\pm$0.00010 | 11.77${}^{+5.75}_{-5.35}$ | 6.29${}^{+1.85}_{-1.72}$ | 2.03${}^{+0.06}_{-0.04}$ | 284 $\pm$ 13 | … | 11,6
KIC 8324482 | 1 | 0 | 0.017${}^{+0.001}_{-0.001}$ | 0.63845$\pm$0.00017 | 2.88685$\pm$0.00017 | 0.05${}^{+0.00}_{-0.00}$ | 0.01${}^{+0.00}_{-0.00}$ | 3.06${}^{+0.09}_{-0.07}$ | … | … | 12
KIC 8381949 | 1 | 1 | 0.817${}^{+0.182}_{-0.182}$ | 1.29847$\pm$0.00004 | 1.44268$\pm$0.00003 | 3.39${}^{+0.21}_{-0.21}$ | 2.61${}^{+0.13}_{-0.13}$ | 5.58${}^{+0.91}_{-0.91}$ | 245 $\pm$ 21 | 90.0${}^{+10.3}_{-10.3}$ | 1,2,6
KIC 8459899 | 1 | 0 | 0.140${}^{+0.075}_{-0.075}$ | 0.36985$\pm$0.00006 | 0.52630$\pm$0.00008 | 0.76${}^{+0.30}_{-0.30}$ | 0.53${}^{+0.21}_{-0.21}$ | 3.88${}^{+0.30}_{-0.30}$ | 53 $\pm$ 4 | … | 1,2,3
KIC 8714886 | 1 | -1 | 0.246${}^{+0.024}_{-0.024}$ | 0.57215$\pm$0.00002 | 1.63250$\pm$0.00003 | 0.60${}^{+0.06}_{-0.06}$ | 0.26${}^{+0.02}_{-0.02}$ | 4.78${}^{+0.04}_{-0.04}$ | 52 $\pm$ 9 | 61.0${}^{+12.6}_{-12.6}$ | 1,2,6
KIC 8766405 | 1 | 1 | 0.563${}^{+0.042}_{-0.042}$ | 0.80955$\pm$0.00008 | 1.11239$\pm$0.00004 | 4.56${}^{+1.81}_{-1.81}$ | 2.05${}^{+0.47}_{-0.47}$ | 5.59${}^{+0.27}_{-0.27}$ | 240 $\pm$ 12 | … | 1,2,3
KIC 9020774 | 1 | 1 | 1.030${}^{+0.024}_{-0.024}$ | 1.50818$\pm$0.00002 | 2.29514$\pm$0.00005 | 4.31${}^{+1.44}_{-1.44}$ | 1.63${}^{+0.30}_{-0.30}$ | 2.60${}^{+0.23}_{-0.23}$ | 129 $\pm$ 28 | 68.8${}^{+14.3}_{-14.3}$ | 1,2,4
KIC 9715425 | 1 | 1 | 0.826${}^{+0.190}_{-0.190}$ | 1.40955$\pm$0.00013 | 1.73851$\pm$0.00016 | 2.83${}^{+0.79}_{-0.79}$ | 1.81${}^{+0.39}_{-0.39}$ | 4.63${}^{+0.42}_{-0.42}$ | 122 $\pm$ 12 | 39.4${}^{+10.6}_{-10.6}$ | 1,2,6
KIC 10526294 | 1 | 0 | 0.103${}^{+0.005}_{-0.057}$ | 0.47222$\pm$0.00006 | 1.01341$\pm$0.00004 | 0.44${}^{+0.02}_{-0.24}$ | 0.20${}^{+0.01}_{-0.11}$ | 4.37${}^{+1.01}_{-1.23}$ | 18 $\pm$ 4 | 52.1${}^{+15.2}_{-15.2}$ | 1,2,13
KIC 10536147 | 1 | 1 | 1.150${}^{+0.006}_{-0.006}$ | 1.67156$\pm$0.00013 | 1.86423$\pm$0.00014 | 4.41${}^{+1.09}_{-1.09}$ | 3.22${}^{+0.62}_{-0.62}$ | 3.70${}^{+0.17}_{-0.17}$ | 163 $\pm$ 18 | 48.5${}^{+10.3}_{-10.3}$ | 1,2,5
KIC 11360704 | 1 | 1 | 1.189${}^{+0.119}_{-0.119}$ | 2.04329$\pm$0.00006 | 2.46320$\pm$0.00001 | 2.78${}^{+0.19}_{-0.19}$ | 1.87${}^{+0.10}_{-0.10}$ | 4.15${}^{+0.33}_{-0.33}$ | 303 $\pm$ 12 | … | 1,2,6
KIC 11971405 | 1 | 1 | 1.535${}^{+0.002}_{-0.002}$ | 1.85641$\pm$0.00008 | 2.01955$\pm$0.00016 | 9.55${}^{+0.46}_{-0.46}$ | 6.33${}^{+0.21}_{-0.21}$ | 3.14${}^{+0.01}_{-0.01}$ | 242 $\pm$ 14 | 82.9${}^{+7.0}_{-7.0}$ | 1,2,4
KIC 12258330 | 1 | 0 | 2.065${}^{+0.002}_{-0.002}$ | 1.30679$\pm$0.00009 | 3.35279$\pm$0.00018 | 3.16${}^{+0.08}_{-0.08}$ | 1.23${}^{+0.03}_{-0.03}$ | 2.29${}^{+0.02}_{-0.02}$ | 130 $\pm$ 8 | 32.9${}^{+2.5}_{-2.5}$ | 1,2,3
HD 43317 | 1 | -1 | 1.114${}^{+0.000}_{-0.000}$ | 0.99541$\pm$0.00003 | … | 1.06${}^{+0.00}_{-0.00}$ | … | 3.39${}^{+0.08}_{-0.03}$ | 115 $\pm$ 9 | 37.0${}^{+3.6}_{-3.4}$ | 14,15
HD 50230 | 1 | 0 | 0.096${}^{+0.096}_{-0.064}$ | 0.63683$\pm$0.00005 | 1.23685$\pm$0.00004 | 0.30${}^{+0.30}_{-0.20}$ | 0.15${}^{+0.15}_{-0.10}$ | 5.68${}^{+0.18}_{-0.13}$ | 7 $\pm$ 2 | 14.6${}^{+12.5}_{-12.1}$ | 16,17,18
HD 48012 | 1 | 1 | 1.830${}^{+0.130}_{-0.121}$ | 2.79000$\pm$0.00200 | 3.05400$\pm$0.00800 | 3.81${}^{+0.79}_{-0.73}$ | 2.99${}^{+0.53}_{-0.49}$ | 2.79${}^{+0.08}_{-0.06}$ | 225 $\pm$ 20 | 60.7${}^{+9.7}_{-9.7}$ | 19
HD 261810 | 1 | 1 | 1.427${}^{+0.101}_{-0.094}$ | 2.30500$\pm$0.00500 | 2.72300$\pm$0.00300 | 3.25${}^{+0.61}_{-0.56}$ | 2.20${}^{+0.33}_{-0.31}$ | 3.27${}^{+0.10}_{-0.07}$ | 180 $\pm$ 15 | 49.7${}^{+8.2}_{-7.9}$ | 19
HD 201433 | 1 | 1 | 0.006${}^{+0.000}_{-0.000}$ | 0.42340$\pm$0.00140 | 0.98431$\pm$0.00029 | 0.03${}^{+0.00}_{-0.00}$ | 0.01${}^{+0.00}_{-0.00}$ | 2.77${}^{+0.18}_{-0.18}$ | 8 $\pm$ 2 | ($68\pm 5$) | 20
HD 1976 | 1 | 1 | 0.644${}^{+0.100}_{-0.100}$ | 0.93895$\pm$0.00004 | … | 4.36${}^{+2.16}_{-2.16}$ | … | 4.00${}^{+1.30}_{-1.30}$ | 143 $\pm$ 9 | 90.0${}^{+0.0}_{-12.6}$ | 21,22
HD 21071 | 1 | 0 | 1.594${}^{+0.004}_{-0.004}$ | 1.18843$\pm$0.00001 | … | 2.68${}^{+0.01}_{-0.01}$ | … | 2.70${}^{+0.70}_{-0.70}$ | 51 $\pm$ 4 | 13.5${}^{+5.5}_{-5.6}$ | 21,22,5
HD 24587 | 1 | 1 | 0.731${}^{+0.036}_{-0.036}$ | 1.15690$\pm$0.00060 | … | 3.44${}^{+0.46}_{-0.46}$ | … | 2.50${}^{+0.60}_{-0.60}$ | 28 $\pm$ 4 | 17.6${}^{+6.8}_{-6.8}$ | 21,23,24
HD 25558 | 1 | 0 | 0.731${}^{+0.036}_{-0.036}$ | 0.65265$\pm$0.00002 | … | 2.24${}^{+0.11}_{-0.11}$ | … | 2.79${}^{+0.97}_{-0.97}$ | 28 $\pm$ 2 | 15.7${}^{+8.8}_{-8.8}$ | 21,22
HD 26326 | 1 | 0 | 0.419${}^{+0.054}_{-0.054}$ | 0.53380$\pm$0.00080 | … | 1.57${}^{+0.20}_{-0.20}$ | … | 3.10${}^{+1.00}_{-1.00}$ | 11 $\pm$ 6 | 9.6${}^{+8.5}_{-8.5}$ | 21,23,24
HD 36999 | 1 | -1 | 0.036${}^{+0.003}_{-0.002}$ | 1.43227$\pm$0.00002 | … | 0.05${}^{+0.00}_{-0.00}$ | … | 2.24${}^{+0.07}_{-0.05}$ | … | … | 25
HD 53921 | 2 | -1 | 0.334${}^{+0.213}_{-0.213}$ | 0.60540$\pm$0.00060 | … | 0.71${}^{+0.29}_{-0.29}$ | … | 2.60${}^{+0.70}_{-0.70}$ | 17 $\pm$ 10 | 22.7${}^{+15.9}_{-15.7}$ | 21,23,24
HD 66181 | 1 | 1 | 0.046${}^{+0.001}_{-0.001}$ | 0.88522$\pm$0.00002 | … | 0.11${}^{+0.00}_{-0.00}$ | … | 4.32${}^{+0.13}_{-0.10}$ | … | … | 25
HD 74195 | 1 | 0 | 0.134${}^{+0.036}_{-0.036}$ | 0.35033$\pm$0.00009 | … | 0.77${}^{+0.20}_{-0.20}$ | … | 4.30${}^{+1.40}_{-1.40}$ | 9 $\pm$ 5 | 18.0${}^{+14.0}_{-13.9}$ | 21,23,24
HD 85953 | 1 | -1 | 0.511${}^{+0.003}_{-0.003}$ | 0.26630$\pm$0.00060 | … | 1.32${}^{+0.00}_{-0.00}$ | … | 4.90${}^{+1.70}_{-1.70}$ | 18 $\pm$ 10 | 8.2${}^{+7.6}_{-7.5}$ | 21,23,24
HD 92287 | 1 | -1 | 0.525${}^{+0.025}_{-0.025}$ | 0.21480$\pm$0.00007 | … | 1.42${}^{+0.02}_{-0.02}$ | … | 3.90${}^{+1.30}_{-1.30}$ | 41 $\pm$ 16 | 23.3${}^{+14.0}_{-13.6}$ | 21,23,24
HD 97895 | 1 | 0 | 0.062${}^{+0.000}_{-0.000}$ | 0.92263$\pm$0.00001 | … | 0.13${}^{+0.00}_{-0.00}$ | … | 3.34${}^{+0.10}_{-0.07}$ | … | … | 25
HD 121190 | 1 | 1 | 1.876${}^{+0.152}_{-0.152}$ | 2.68310$\pm$0.00040 | … | 4.65${}^{+1.26}_{-1.26}$ | … | 1.70${}^{+0.30}_{-0.30}$ | 118 $\pm$ 3 | 47.0${}^{+11.0}_{-11.0}$ | 21,26,27
HD 123515 | 1 | 0 | 0.654${}^{+0.079}_{-0.079}$ | 0.68528$\pm$0.00010 | … | 1.91${}^{+0.23}_{-0.23}$ | … | 2.20${}^{+0.50}_{-0.50}$ | 6 $\pm$ 3 | 4.7${}^{+3.1}_{-3.2}$ | 21,23,24
HD 160762 | 1 | -1 | 0.457${}^{+0.016}_{-0.016}$ | 0.28671$\pm$0.00027 | … | 1.23${}^{+0.02}_{-0.02}$ | … | 5.06${}^{+0.50}_{-0.50}$ | 6 $\pm$ 1 | 2.9${}^{+0.6}_{-0.6}$ | 28
HD 179588 | 1 | 0 | 0.855${}^{+0.207}_{-0.207}$ | 0.85654$\pm$0.00081 | … | 2.00${}^{+0.48}_{-0.48}$ | … | 2.20${}^{+0.50}_{-0.50}$ | 52 $\pm$ 14 | 33.1${}^{+14.4}_{-14.3}$ | 21,22
HD 182255 | 1 | 1 | 0.468${}^{+0.076}_{-0.076}$ | 0.97185$\pm$0.00003 | … | 1.86${}^{+0.58}_{-0.58}$ | … | 2.60${}^{+0.70}_{-0.70}$ | 10 $\pm$ 6 | 9.3${}^{+8.1}_{-8.1}$ | 21,23,24
HD 208057 | 1 | 0 | 0.779${}^{+0.096}_{-0.096}$ | 0.80213$\pm$0.00010 | … | 1.94${}^{+0.24}_{-0.24}$ | … | 3.20${}^{+1.00}_{-1.00}$ | 104 $\pm$ 6 | 55.5${}^{+13.3}_{-13.2}$ | 21,22,24
HD 215573 | 1 | 0 | 0.508${}^{+0.054}_{-0.054}$ | 0.56540$\pm$0.00060 | … | 1.80${}^{+0.19}_{-0.19}$ | … | 3.00${}^{+1.00}_{-1.00}$ | 5 $\pm$ 2 | 3.7${}^{+3.7}_{-3.9}$ | 21,23,24

Note. — List of mode identifications ($\ell,m$), rotational frequencies
$f_{\rm rot}$, and spin parameters $s$ derived from the minimum ($f_{\rm
in}^{\rm 1}$, $s^{\rm 1}$) and maximum ($f_{\rm in}^{\rm 2}$, $s^{\rm 2}$)
frequency of the period spacing patterns in the in inertial frame of
reference. For stars without period spacing patterns, the frequency and spin
parameter of the dominant oscillation mode is provided instead. Column 9-11
list the stellar radii, the projected equatorial velocity $V_{\rm eq}\sin i$,
and the derived inclination angles achieved under the assumption of rigid
rotation, respectively. The last column lists the source of the mode
identification, rotation frequency, oscillation frequencies, radii, and
$V_{\rm eq}\sin i$ values. Spin parameters and inclination angles were derived
in this work. Inclination angles taken from the literature are listed in
parenthesis.

References: (1) Pedersen et al. (2021), (2) Pedersen (2022), (3) Lehmann et
al. (2011), (4) Pápics et al. (2017), (5) Gebruers et al. (2021), (6) Hanes et
al. (2019), (7) Zhang et al. (2018), (8) Pápics et al. (2013), (9) Frasca et
al. (2016), (10) Pápics et al. (2015), (11) Szewczuk et al. (2022), (12) Wu et
al. (2020), (13) Pápics et al. (2014), (14) Buysschaert et al. (2017), (15)
Buysschaert et al. (2018), (16) Degroote et al. (2010), (17) Degroote et al.
(2012), (18) Wu & Li (2019), (19) Zwintz et al. (2017), (20) Kallinger et al.
(2017), (21) Szewczuk & Daszyńska-Daszkiewicz (2015b), (22) De Cat (2007),
(23) De Cat & Aerts (2002), (24) Hubrig et al. (2006), (25) Fedurco et al.
(2020), (26) North & Paltani (1994), (27) Aerts & Kolenberg (2005), (28) De
Cat (2002)

##  4 Age dependence of rotation

Figure 4: Core rotational frequency as a function of main-sequence age (top
and middle panels) and surface gravity (bottom panels). The panels on the
left-hand side show the results for the 33 SPB stars with space photometry,
while the stars with ground based data are shown on the right. Open symbols
indicate known or suspected binaries. The four SPB stars with ground-based
data for which the inclination angle is found to be inconsistent with the mode
identification as discussed in Sect. 6 are marked in purple. The results are
compared to the core rotation frequencies of $\approx 300$ $\gamma$ Dor stars
derived by Li et al. (2020) in grey. The corresponding Spearman’s rank
correlation coefficients $r_{s}$ and their $p$-values are indicated in the
subplots.

Aerts et al. (2019) visualized the evolution of the core and surface rotation
derived from asteroseismology by collecting the core rotation rates for 1210
stars in the literature, out of which 45 also had measured surface rotation
rates. These stars cover masses between 0.72-7.9 M⊙ and various stages of
stellar evolution from main-sequence stars to white dwarfs, but majorly
consists of stars on the red giant branch. In their Fig. 4, Aerts et al.
(2019) show the variation of the core rotation as a function of the
asteroseismic $\log g$, which is used as a proxy for the age of the stars.
Since the generation of this figure, the previously limited sample of main-
sequence stars has been significantly increased with the derivation of core
rotation rates of more than 600 $\gamma$ Dor stars (Li et al., 2020), while
the number of B-stars has remained small. An updated version of Fig. 4 in
Aerts et al. (2019) including this new sample of $\gamma$ Dor stars is
available in Aerts (2021) (Fig. 6), resulting in core-to-surface rotation
rates now being available for 111 stars. Here we take a closer look at the
core rotation frequency of the 52 SPB stars considered in this work and study
its dependence on the age of the star. We note that by core rotation, we mean
in fact the rotation just outside of the convective core where the rotational
kernel for the g-modes is largest, see bottom panels of Fig. 1.

Figure 4 shows the evolution of the near-core rotation of the SPB stars with
space photometry (left) and ground-based data (right) as a function of main-
sequence age measured as the ratio of the current to the initial core hydrogen
mass fraction of the star $X_{\rm c}/X_{\rm ini}$ (panels (a) and (d)), and as
a fraction of the current age to the main-sequence lifetime $t/t_{\rm MS}$
(panels (b) and (e)). Panels (c) and (f) show the corresponding evolution as a
function of the surface gravity444When available, we adopt the asteroseismic
$\log g$ instead of the spectroscopic one. $\log g$, and the near-core
rotation is compared to those of the $\approx 300$ $\gamma$ Dor stars (grey)
from Li et al. (2020) for which spectroscopic $\log g$ values were available
from Frasca et al. (2016). As seen in Fig. 4, the errors on the age indicators
for the sample of SPB stars for which the rotation frequency was estimated
using data from ground-based telescopes are generally much larger than for the
stars with space photometry. This is mainly a result of the lack of an
asteroseismic modelling of the stars, in which case a much more precise
estimate of the age can be obtained.

For the SPB stars with space photometry we see a trend that the core rotation
decreases as a function of age, independently of whether $X_{\rm c}/X_{\rm
ini}$, $t/t_{\rm MS}$, or $\log g$ is used as an age indicator. A similar
trend is seen for the sample of 19 SPB stars with asteroseismology based on
ground-based data. To quantify this, we calculate the Spearman’s rank
correlation coefficient $r_{s}$, which takes values between -1 and 1,
corresponding to a strong negative or positive correlation, respectively.

As indicated in panels (a) and (b) of Fig. 4, the Spearman’s rank correlation
coefficient shows a positive correlation between $f_{\rm rot}$ and both
$X_{\rm c}/X_{\rm ini}$ ($r_{s}=0.58$, $p<0.001$) and $t/t_{\rm MS}$
($r_{s}=-0.58$, $p<0.001$) for the SPB stars with space photometry, confirming
that $f_{\rm rot}$ does indeed decrease with main-sequence age. A similar
trend is seen for the SPB stars with ground-based data, but the higher
p-values ($p=0.07$) indicate that there is only weak evidence for rejecting
the null-hypothesis that $f_{\rm rot}$ is uncorrelated with $X_{\rm c}/X_{\rm
ini}$ and $t/t_{\rm MS}$, compared to the very strong evidence found for the
33 SPB stars with space photometry. Strong evidence is found for a correlation
between $f_{\rm rot}$ and $\log g$ for the SPB stars with space photometry
($r_{s}=0.49$, $p=0.04$), whereas no such correlation is found for the 19 SPB
stars with ground-based data.

One important difference between the top two panels on the right-hand side of
Fig. 4 and panel (f) is that $\log g$ is the only age indicator available for
six of the SPB stars with ground-based data. The three slowest rotating stars
are among these six stars, and all three of them had their rotation period
determined based on data collected from five different ground-based
photometric surveys of varying observing cadence not specifically designed for
asteroseismology (Fedurco et al., 2020). Nevertheless, while the correlation
between the near-core rotation frequency and the three age indicators in
panels (a) to (d) have different levels of significance, the overall lack of
stars in the upper right corners of these panels show that there are currently
no known old main-sequence SPB stars with a high near-core rotation frequency.

Figure 5: Core rotational frequency as a function of main-sequence age (top)
and surface gravity (bottom) for 34 $\gamma$ Dor stars (pink), which have been
modeled asteroseismically. The $\approx 300$ $\gamma$ Dor stars from Li et al.
(2020) are once again shown in grey, and the Spearman’s rank correlation
coefficients $r_{s}$ and their $p$-values are indicated in the subplots.

For the $\gamma$ Dor sample shown in grey in panels (c) and (f), the high
$p$-value means that we cannot reject the null-hypothesis that $r_{s}=0$, i.e.
that $f_{\rm rot}$ and $\log g$ are uncorrelated. In an attempt to see if this
picture changes if we make use of stellar models derived from asteroseismology
for $\gamma$ Dor stars, we carry out a similar comparison for 34 $\gamma$ Dor
stars in Fig. 5, which were recently modeled asterosesimically by Mombarg et
al. (2021). Their core rotation frequencies were taken from Van Reeth et al.
(2016) and their spectroscopic $\log g$ values from Van Reeth et al. (2015).
For this smaller sample of $\gamma$ Dor stars, no correlation is found between
$f_{\rm rot}$ and $X_{\rm c}/X_{\rm ini}$, which is consistent with the
results for the larger sample from Li et al. (2020). However, the Spearman’s
rank correlation coefficient does indicate a positive correlation between
$f_{\rm rot}$ and $\log g$ for the 34 modeled $\gamma$ Dor stars, with the
associated $p<0.05$ value indicating that this correlation is significant at
the 95% confidence level.

##  5 Angular momentum transport

In order to investigate if the decrease in the near-core rotation frequencies
of the sample of SPB stars as a whole as a function of time is evidence for
efficient angular momentum transport, we consider the two extreme cases of 1)
no angular momentum transport between adjacent cells in the stellar model, and
2) the angular momentum transport occurs instantaneously and enforces rigid
rotation throughout the main-sequence evolution. In both cases we assume that
the stars start out as rigidly rotating on the zero-age main-sequence.

In the case of no angular momentum transport, we assume that the specific
angular momentum $j_{1}(r)=\Omega_{1}\left(r\right)i_{1}\left(r\right)$ at a
time $t_{1}$ and later time $t_{2}$ is the same

$j_{1}=j_{2}.$ (3)

Here $r$ is the central radius of the spherical cell, and $\Omega(r)$ and
$i(r)$ are the angular rotation frequency and specific moment of inertia of
the star at $r$, respectively. We thus obtain the rotation profile
$\Omega_{2}\left(r\right)$ at time $t_{2}$ using

$\Omega_{2}\left(r\right)=\Omega_{1}\left(r\right)\frac{i_{1}\left(r\right)}{i_{2}\left(r\right)}.$
(4)

For the second case of instantaneous angular momentum transport, we assume
that the total angular momentum $J$ is conserved between different time steps
$t_{1}$ and $t_{2}$.

$J_{1}=J_{2}.$ (5)

The total AM is calculated as

$J=\frac{2}{3}\int_{0}^{R}r^{2}\Omega(r){\rm d}m,$ (6)

where ${\rm d}m$ is the enclosed mass of a cell. If we assume that the stars
remain rigidly rotating throughout their main-sequence evolution, it simply
follows that

$\Omega_{2}=\Omega_{1}\frac{I_{1}}{I_{2}},$ (7)

where $I_{1}$ and $I_{2}$ are the total moment of inertia at time $t_{1}$ and
$t_{2}$, respectively.

Figure 6: Evolution of the average rotation frequencies experienced by the
g-modes (left) and the surface equatorial velocities (right) as a function of
the two age indicators $X_{\rm c}/X_{\rm ini}$ (top) and $t/t_{\rm MS}$
(bottom). The evolution of the rotation rates are given with respect to the
initial rotation rate at the zero-age main-sequence. Different line styles
correspond to different stellar masses: 2.5 M⊙ (full), 5 M⊙ (dashed), 7.5 M⊙
(dash-dotted), and 10 M⊙ (dotted). Results for rigid rotation and no angular
momentum transport are shown in blue and orange, respectively. Models with
angular momentum transport from the Spruit-Taylor dynamo assuming an initial
rotation rate of $10\,\%\,\Omega_{\rm crit}$ are shown in green for
comparison.

As shown in the bottom panels of Fig. 1, the g-modes mainly probe the near
core rotation profile of the star and do not provide information about the
rotation frequency of the convective core itself. Following Ouazzani et al.
(2019), we therefore calculate the average rotational frequency experienced by
the g-modes over their propagation cavity weighted by the Brunt-Vaisala
frequency $N$ as

$\langle\Omega\rangle=\frac{\int_{\rm gc}\Omega\left(r\right)N(r)\frac{{\rm
d}r}{r}}{\int_{\rm gc}N(r)\frac{{\rm d}r}{r}},$ (8)

and will use this average for comparison with the observed g-mode rotation
frequencies rather than a derived central rotation frequency of the convective
core. For comparison, we also calculate the corresponding surface equatorial
velocities

$V_{\rm eq}=\Omega_{\rm surf}R,$ (9)

where $\Omega_{\rm surf}=\Omega(R)$.

###  5.1 Evolution of the near-core and surface rotation

As a first step, we derive the evolution of the near-core and surface rotation
rates for four different stellar masses 2.5, 5, 7.5, and 10 M⊙ covering the
SPB instability strip. The models were computed with the stellar structure and
evolution code MESA version r12115 (Paxton et al., 2011, 2013, 2015, 2018,
2019) following the same setup as Pedersen et al. (2021) but excluding winds
to ensure conservation of angular momentum, see also Appendix A for more
information. The results of these calculations are shown in Fig. 6. The
different line styles corresponds to different stellar masses, while the
evolution of the rotation rates assuming rigid rotation and no angular
momentum transport are shown in blue and orange, respectively. For comparison,
we also show in green what the evolution of the near-core and surface rotation
is expected to be for stellar models assuming angular momentum transport from
the Spruit-Taylor dynamo (Spruit, 2002; Paxton et al., 2013) assuming an
initial rotation rate of $10\%\,\Omega_{\rm crit}$. The corresponding MESA
setup is discussed in Appendix B. Increasing the initial rotation rate causes
the evolution of the average rotation frequency and surface velocity assuming
angular momentum transport from the Spruit-Taylor dynamo to approach the case
of solid body rotation, as illustrated in Fig. 12 in Appendix B.

The left-hand side of Fig. 6 demonstrates that irrespective of whether the SPB
stars experience no or very efficient angular momentum transport, the average
rotation frequency decreases with age except at the final stages of the
evolution towards hydrogen depletion. The rate of decrease stays nearly the
same for the two cases for the first half on the main-sequence evolution until
$X_{\rm c}/X_{\rm ini}\approx 0.6$ and $t/t_{\rm MS}\approx 0.4$, and then
starts to slow down for the case of no angular momentum transport for the
remainder of the evolution. In other words, older SPB stars are expected to
have smaller average interior rotation frequencies measured by the g-mode
oscillations if the stars are rigid rotators, compared to if no angular
momentum transport is happening during the main-sequence evolution. The
smallest average rotation frequency
$\langle\Omega\rangle/\langle\Omega\rangle_{\rm ZAMS}\approx 0.35$ is obtained
at $X_{\rm c}/X_{\rm ini}\approx 0.05$ and $t/t_{\rm MS}\approx 0.098$ in the
case of rigid rotation, compared to
$\langle\Omega\rangle/\langle\Omega\rangle_{\rm ZAMS}\approx 0.64$ at $X_{\rm
c}/X_{\rm ini}\approx 0.27$ and $t/t_{\rm MS}\approx 0.87$ for the case of no
angular momentum transport.

The evolution of the equatorial velocities shown on the right-hand side of
Fig. 6 display much larger differences between the two cases of no and highly
efficient angular momentum transport with the surface rotation slowing down
much faster for the first case. For the solid body rotation scenario, the
equatorial velocity remains nearly constant for the first half of the main-
sequence evolution and then reaches a minimum $V_{\rm eq}/V_{\rm
eq,ZAMS}\approx 0.77$ at $X_{\rm c}/X_{\rm ini}\approx 0.06$ and $t/t_{\rm
MS}\approx 0.97$. The age at the which the minimum is achieved is the same for
the computations without angular momentum transport but the surface velocities
are lower ($V_{\rm eq}/V_{\rm eq,ZAMS}\approx 0.46$). We note that the
evolution of the rotation rates is only weakly dependent on the stellar mass
for the SPB stars, particularly so for $M\geq 5\,$M⊙. The largest mass
dependence is seen for the surface rotational velocities assuming rigid
rotation.

###  5.2 Comparison to the SPB star sample

To study angular momentum transport on the main-sequence, ideally one would
know the starting rotational velocity of the stars on the ZAMS. In their
analysis of the evolution of the near-core rotation frequencies of 37 $\gamma$
Dor stars, Ouazzani et al. (2019) accomplished this by combining rotational
velocity distributions of three young stellar clusters with disc locking
models for the interaction between pre-main-sequence stars and their accretion
disks. Here we instead start by comparing the average rotational frequencies
of the four youngest SPB stars with space photometry with the rotation
distribution of the remaining 29 stars. For this first part we exclude the
sample of SPB stars with ground-based data because of their smaller sample
size and larger errors on their main-sequence ages.

Figure 7: Distribution of near-core rotation frequencies with age for the 33
SPB stars with space photometry (grey). Binary stars are indicated by open
symbols. The vertical dotted lines show the dividing ages for the four
different age groups G1-G4, and the black data points give the position of the
average rotation and ages of each age group. The evolution of the near-core
rotation for the case of rigid rotation (blue) and no angular momentum
transport (orange) has been scaled to the average rotation and age in group
G1. The same results assuming models with angular momentum transport from the
Spruit-Taylor dynamo are shown in green for comparison.

We split the 33 SPB stars with space photometry into four age groups and
calculate the average near-core rotation frequency and age for the sample in
each group. For this analysis we focus on the two age indicators $X_{\rm
c}/X_{\rm ini}$ and $t/t_{\rm MS}$. The four different age groups are:

  * G1:

$X_{\rm c}/X_{\rm ini}>0.9$ and $t/t_{\rm MS}<0.3$,

  * G2:

$0.6<X_{\rm c}/X_{\rm ini}<0.9$ and $0.3<t/t_{\rm MS}<0.6$,

  * G3:

$0.3<X_{\rm c}/X_{\rm ini}<0.6$ and $0.6<t/t_{\rm MS}<0.8$,

  * G4:

$X_{\rm c}/X_{\rm ini}<0.3$ and $t/t_{\rm MS}>0.8$.

We show again the distribution of rotational frequencies as a function of age
in Fig. 7, where the vertical dotted lines separate the four age groups. The
averages of the age groups and their $1\sigma_{\rm std}$ are shown in black.

Using the curves for the evolution of the near-core rotation frequency shown
for the 5 M⊙ star in the left-hand side of Fig. 6, we scale the curves to the
average rotation frequency of the four youngest SPB stars and compare the
expected evolution of the near-core rotation with the averages in each of the
three older age groups. For both G2 and G3 we find that the observed averages
are consistent with the predicted evolution of the near-core rotation
frequency for both the case of no and highly efficient angular momentum
transport. However, for the oldest age group G4, the predicted rotation rate
for the sample is 1.72 and 1.66 times higher than the observed average for the
case of highly efficient angular momentum transport for $X_{\rm c}/X_{\rm
ini}$ and $t/t_{\rm MS}$, respectively, but within $1.54\,\sigma_{\rm std}$
and $1.28\,\sigma_{\rm std}$ of the average. The discrepancy is even higher
for the case of no angular momentum transport. Here the predicted rotation
frequency is 2.7 and 2.5 times higher for $X_{\rm c}/X_{\rm ini}$ and
$t/t_{\rm MS}$, respectively, corresponding to a $3.65\,\sigma_{\rm std}$ and
$2.89\,\sigma_{\rm std}$ deviation from the observed average. Furthermore, we
note that none of the SPB stars are found within the gap between the two
curves in the age range of G4. This is seen most clearly for the $X_{\rm
c}/X_{\rm ini}$ age indicator.

Figure 8: Expected ranges of observed near-core rotation frequencies as
predicted for the sample of SPB stars assuming rigid rotation (blue) and no
angular momentum transport (hatched orange). The predicted regions are shown
separately for the 33 stars with space photometry (left) and 13 stars with
ground based data (right) which have measured $X_{\rm c}/X_{\rm ini}$ and
$t/t_{\rm MS}$ values. SPB stars which show and inconsistency between the
derived inclination angles and mode identification are shown in purple. See
Fig. 7 for further explanations of the figure details.

Next we take a closer look at this gap and derive the regions for the
distribution of the near-core rotation with age spanned by the sample of SPB
stars assuming no and highly efficient angular momentum transport. We do these
calculations separately for the sample with space photometry and ground-based
data. The regions are derived by scaling the same curves for the 5 M⊙ star in
Fig. 6 to each of the 46 SPB stars with a known value for $X_{\rm c}/X_{\rm
ini}$ and $t/t_{\rm MS}$, and then taking the minimum and maximum predicted
rotation at each age step. The resulting regions are shown in Fig. 8. As seen
in the figure, we end up with an area spanned by the calculations without
angular momentum transport (hatched orange) where old SPB stars are expected
to be found, but are absent in the case of solid body rotation (full blue).
None of the 46 SPB stars are found in this area of the diagrams, implying that
the sample shows evidence of efficient angular momentum transport on the main-
sequence.

##  6 Inclinations

Under the assumption of uniform rotation, we can estimate the inclination
angles $i$ of the stars if their $V_{\rm eq}\sin i$ values have been measured
spectroscopically. This is the case for 46 out of the 52 SPB stars. As shown
in Eq. (9) the measured rotation frequencies and stellar radii directly give
us the equatorial velocity of a spherical symmetric star. The adopted $V_{\rm
eq}\sin i$ values and stellar radii $R$ are listed in Table 2 along with the
derived inclination angles. We show the results in Figs. 9 and 10 for the
dipole modes of the SPB stars with space photometry and ground-based data,
respectively, and note that the assumption of uniform rotation matches well
with what has previously been observed for single main-sequence stars with
convective cores for which both asteroseismic core and envelope rotation
frequencies have been measured, as illustrated in Fig. 4 of Aerts et al.
(2019).

Figure 9: _Left:_ Spectroscopic $V_{\rm eq}\sin i$ as a function of the
asteroseismic $V_{\rm eq}$ derived under the assumption of rigid rotation for
the sample of SPB stars with space photometry. The size of the data points
scale with the amplitude of the dominant dipole mode, while the symbols
indicate whether the oscillations are retrograde (square), zonal (triangle),
or prograde (circles) modes. Orange data points correspond to stars for which
the $V_{\rm eq}\sin i$ cannot be reconciled with the $V_{\rm eq}$ values
within the estimated errors. _Right:_ Derived inclination angles as a function
of the azimuthal order. The data points have been slightly offset from the
measured $m$ value for the sake of visibility. The black horizontal line marks
where $i=90^{\rm o}$, i.e. the maximum inclination angle. Binary stars are
indicated with open symbols. Figure 10: Same as Fig. 9 but for SPB stars
observed from the ground. Stars with $i<20^{\rm o}$ are indicated in purple.

The left hand sides of Figs. 9 and 10 compare the observed spectroscopic
$V_{\rm eq}\sin i$ to the asteroseismically derived $V_{\rm eq}$. As the
$V_{\rm eq}\sin i$ value is only a lower estimate of the true $V_{\rm eq}$
value, we would expect to always find $V_{\rm eq}\sin i\leq V_{\rm eq}$.
However, as highlighted by the orange symbols there are six stars (KIC
4936089, KIC 8264293, KIC 8459899, KIC 8766405, KIC 11360704, HD 201433) in
Fig. 9 for which the asteroseismic equatorial velocity cannot be reconciled
with the measured $V_{\rm eq}\sin i$ values within the $1\sigma$ errors. There
are a few reasons why this would be the case. For one, we have assumed rigid
rotation for the derivation of $V_{\rm eq}$, which might not be valid for
these stars. In this case, the envelopes of the stars would have to spin
faster than the cores in order to be able to reconcile the spectroscopic
$V_{\rm eq}\sin i$ with the asteroseismic $V_{\rm eq}$. This is the case for
the SPB star HD 201433, which is a known single-lined spectroscopic triple
system. Using nine observed rotationally split dipole g-modes, Kallinger et
al. (2017) derived the internal rotation profile of the star showing that it
is rigidly rotating up to $\sim 90\%$ of the stellar radius, but the surface
(i.e. outer $\sim 4\%$) has spun up likely due to tidal interactions with its
inner companion. Based on visibility arguments of the components of the
rotationally split g-modes, Kallinger et al. (2017) arrived at an inclination
of $68\pm 5^{\rm o}$ for HD 201433. For this to be consistent with the
observed $V_{\rm eq}\sin i$ would require $V_{\rm eq}\approx 8.6$ km s-1
corresponding to $f_{\rm rot}\approx 0.0615$ d-1 at the surface.

On the other hand, an envelope rotating faster than the core has previously
been measured for a main-sequence $\gamma$ Dor star showing both g- and p-mode
oscillations (Kurtz et al., 2014). Similarly, the internal rotation profile of
KIC 10526294 has previously been inferred by Triana et al. (2015), who showed
that not only is the envelope rotating faster than the core of the star but it
is also rotating in the opposite direction. Internal gravity waves (IGWs)
generated by the convective core have been proposed as an efficient mechanism
to transport angular momentum (Rogers et al., 2013), possibly explaining both
of these observed rotation profiles (Rogers, 2015).

Another possible explanation is related to the fact that the total broadening
of the spectral lines used to measure the $V_{\rm eq}\sin i$ value is not only
affected by the rotation of the stars, but also by stellar oscillations which
give rise to a time dependent pulsational broadening of the line-profiles
(Aerts et al., 2014). As a result, the $V_{\rm eq}\sin i$ will vary as a
function of time, and its measured value is a sum of the contribution from the
rotational and pulsational broadening. In other words, the measured
spectroscopic $V_{\rm eq}\sin i$ for an oscillating star is an upper limit of
the true $V_{\rm eq}\sin i$ value. The effects of pulsational broadening on
the estimated $V_{\rm eq}\sin i$ is expected to be larger for slowly rotating
stars (Aerts et al., 2014).

To investigate any possible correlations with the pulsations on the measured
$V_{\rm eq}\sin i$, the symbol sizes in Figs. 9 and 10 have been scaled
according to the amplitude of the dominant g-mode with a known mode
identification for each star. For stars with period spacing patterns, the
dominant mode of the pattern used to derive the near-core rotation frequency
was used. As the photometric data come from a variety of different telescopes,
we converted all amplitudes to the Johnson V band and units of parts-per-
million as described in detail in Appendix C. As seen in Figs. 9 and 10, the
pulsations of the slowest rotating stars tend to have higher amplitudes. This
is also seen in Fig. 10, where the considered SPB stars with ground-based data
cover a smaller range in surface velocities and have generally higher
amplitudes than the sample of SPB stars with space photometry. This could
explain the lack of observed fast rotating SPB stars from ground-based
observations (e.g. De Cat & Aerts, 2002).

The right hand panels in Figs. 9 and 10 show the derived inclination angles
$i$ for the 40 SPB stars with measured $V_{\rm eq}\sin i<V_{\rm eq}$, as a
function of the azimuthal order including a small offset for the sake of
clarity. Focusing first on Fig. 9, we see that the stars with zonal modes tend
to have smaller inclination angles than the stars with either retrograde or
prograde modes. This is entirely in line with the theoretical expectations
from surface cancellation effects should there be equipartition among the
components of multiplets, as for solar-like oscillators (Gizon & Solanki,
2003, see Fig. 2). Although there is no equipartition of mode energy among
multiplet components of heat-driven modes, this is exactly what is seen in
Fig. 9.

The results are slightly different for the sample of SPB stars with ground-
based data as shown in Fig. 10. Here four stars (HD 24587, HD 85953, HD
160762, HD 182255) with either prograde or retrograde modes are found to have
$i<20^{\rm o}$. There are a few reasons why such low inclination angles might
be obtained. The first is that the estimated stellar parameters might be
wrong. As an example, changing the stellar radius of HD 85953 from $R=4.9\pm
1.7$ R⊙ (Hubrig et al., 2006, adapted in this work) to $R=4.7\pm 1.0$ R⊙ (De
Cat & Aerts, 2002) increases the inclination of the star to $44\pm 17^{\rm o}$
and in line with the trend that is seen for the SPB stars with space
photometry. Given that the stellar parameters of the sample of SPB stars with
ground based data included in Fig. 10 have been derived by comparing the
position of the stars in the HR diagram with evolutionary tracks, much better
parameter estimates could be obtained in the future if asteroseismic modeling
is accomplished for the stars. Another possibility is that mode identification
is wrong, and that the dipole prograde/retrograde oscillations with $i<20^{\rm
o}$ are actually zonal modes.

Finally, a third possibility is that the assumption of rigid rotation is
inaccurate and the stars are differentially rotating in their radiative
envelopes. In this case, the surface of the stars would have to be rotating
slower than their near-core regions to explain the inclination angles.
Increasing the inclination above $20^{\rm o}$ would require the surface to be
rotating at least 1.1, 2.4, 6.6, and 2.1 times slower than the near-core
regions for HD 24587, HD 85953, HD 160762 and HD 182255, respectively,
assuming all other relevant parameters stay the same. Future asteroseismic
modeling of these stars combining ground-based data with data from the TESS
space telescope hold the promise of bringing answers to these questions as
well as significantly improving the age estimates for the sample of SPB stars
with ground-based data.

##  7 Conclusions

In this work, we took a closer look at the internal rotation properties of 52
SPB stars which have an asteroseismic measurement of the internal rotation
based on their g-mode oscillations. The spin parameters of the stars reveal
that most of the g-modes are located in the sub-inertial regime, where the
effects of the Coriolis force cannot be treated as a simple perturbation to
the oscillation equations (Ballot et al., 2010). This was taken into account
in the initial calculation of the rotation frequency for the majority of the
stars in the sample through the use of the TAR, in which the centrifugal force
is ignored. Henneco et al. (2021) demonstrated that the detectability of the
effect of the centrifugal force, following the extension of the TAR by Mathis
& Prat (2019), increases with increasing rotation rate but decreases as the
stars become older. As the frequency resolution is equal to the inverse of the
length of the light curve, they also showed that the effects are easier to
detect for stars observed by _Kepler_ with its four years of continuous
observations, compared to stars observed in the continuous viewing zone (351
d) of the _TESS_ space telescope (Ricker et al., 2015).

Independent of the considered age indicator, the near-core rotation
frequencies are found to decrease as a function of the main-sequence evolution
for the 52 SPB stars. The only exception is seen for the sample of stars whose
asteroseismic analysis was based on data from ground-based telescopes, where
no correlation is found between $f_{\rm rot}$ and $\log g$. We speculate that
this might be caused by differences in the adopted method for deriving the
internal rotation for three of the slowest rotating stars in the sample.

Next, we tested if the decrease in near-core rotation rate is consistent with
the expectations for the evolution of the internal rotation profile in the two
limiting cases of no and highly efficient angular momentum transport. We find
that both descriptions predict that the near-core rotation frequency decreases
during the main-sequence evolution but at different rates. Calculations
without angular momentum transport predict higher rotation frequencies towards
the end of the main-sequence evolution than the case of highly efficient
angular momentum transport corresponding to rigid rotation, which is
inconsistent with the observations for the 52 SPB star sample. Therefore, the
lack of old SPB stars with rapid near-core rotation rates is evidence of
angular momentum transport occurring throughout the main-sequence evolution of
the stars.

Finally, we use the assumption of rigid rotation to derive the inclination
angles of the stars for which spectroscopic $V_{\rm eq}\sin i$ values are
available. For the sample of SPB stars with space photometry, we find that the
stars with zonal modes tend to have smaller inclination angles ($i\leq 70^{\rm
o}$), while stars with prograde or retrograde modes have higher inclination
angles ($i\geq 30^{\rm o}$). This is consistent with the expectations from
surface cancellation effects. Four stars with ground-based data, however, show
$m=-1$ and $m=1$ modes at low inclination angles ($i\leq 20^{\rm o}$). Future
asteroseismic modeling of the stars will be crucial for revealing the reason
for these low inclination angles.

We note that the results presented here are currently limited by the sample
size. Most importantly, the sample has a low fraction of young SPB stars with
only two out of the 52 SPB stars being known to be zero-age main-sequence
stars (Zwintz et al., 2017). Increasing the number of young SPB stars with
near-core rotation measured from asteroseismology will be crucial for
anchoring and testing models with different levels of angular momentum
transport such as previously done for $\gamma$ Dor stars by Ouazzani et al.
(2019).

Furthermore, while the current sample of SPB stars show evidence of efficient
angular momentum transport occurring throughout the main-sequence evolution,
it does not currently provide information on the level of internal
differential rotation in the stars. The only exceptions are two of slowly
rotating SPB stars in the sample, where inversion techniques were previously
applied to the g-mode oscillations to derive the internal rotation profile
(Triana et al., 2015; Kallinger et al., 2017). Improvements can be made by
combining near-core rotation measurements with surface rotation measurements
from either rotationally split p-modes or rotational spot modulation. However,
period spacing patterns on their own can provide important information about
the internal rotation profile of the star. Van Reeth et al. (2018) showed that
the level of internal differential rotation can be measured if more than one
period spacing pattern is detected in the stars, which is the case for several
of the SPB stars in the current sample (see e.g. Pedersen, 2020; Szewczuk et
al., 2021). Finally, Ouazzani et al. (2020) demonstrated that the coupling of
pure inertial modes in the convective core with gravito-inertial modes in the
radiative envelope give rise to unique features in the period spacing
patterns, which can be used to derive the rotation frequency of the convective
core. This was recently accomplished for 16 $\gamma$ Dor stars by Saio et al.
(2021), who found the majority of the stars to be rigidly rotating while a few
of the less evolved $\gamma$ Dor stars have convective cores rotating slightly
faster than the envelope in the g-mode propagation cavity. Such measurements
have yet to be accomplished for SPB stars.

The author is thankful to Conny Aerts, Lars Bildsten, Jim Fuller, and Jamie
Tayar for providing interesting discussions and useful comments at different
stages of this work. The author is also thankful to the anonymous referee
whose comments improved the manuscript. This research was supported in part by
the National Science Foundation under Grant No. NSF PHY-1748958. It was
performed in part at Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-1607611, and was partially supported by a grant
from the Simons Foundation. This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France, and of the VizieR catalogue access tool,
CDS, Strasbourg, France (DOI: 10.26093/cds/vizier). The original description
of the VizieR service was published in A&AS 143, 23. This research has made
use of the Spanish Virtual Observatory (https://svo.cab.inta-csic.es) project
funded by MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00

##  Appendix A Age indicators

This work considers three different age indicators to study the evolution of
the near-core rotation as a function of age on the main-sequence. As a minimum
we require that the surface gravity $\log g$ has to be known, which is the age
parameter previously used by Aerts et al. (2019); Aerts (2021) to study the
evolution of the near-core rotation frequency from the main-sequence to the
white dwarf evolutionary stage. We argue, however, that this is the least
accurate age parameter out of the three that are considered in this work as
the surface gravity not only depends on age, but also the mass, internal
mixing, and metallicity of the stars. The two remaining age indicators are the
ratio of the current to initial core hydrogen mass fraction $X_{\rm c}/X_{\rm
ini}$, and the ratio of the current age to the main-sequence lifetime
$t/t_{\rm MS}$. By considering the ratios instead of $X_{\rm c}$ and $t$, we
minimize the dependence of mass, metallicity, and internal mixing on the age
indicators as we will see below.

To make a comparison between the three age indicators at different main-
sequence ages and stellar masses, we adopt the same MESA setup as in Pedersen
et al. (2021) and compute evolutionary tracks for four different stellar
masses: $2.5$ M⊙, $5$ M⊙, $7.5$ M⊙, and $10$ M⊙. The initial metallicity is
set to $Z=0.014$ and we adopt exponential diffusive overshoot and constant
envelope mixing for the internal mixing profile, corresponding to mixing
profile labeled $\boldsymbol{\psi_{1}}$ by Pedersen et al. (2021). For the
mixing parameters we choose $f_{\rm ov}=0.01$ and $D_{\rm env,0}=10$ cm2 s-1,
where the former sets the extent of the convective core boundary mixing region
and the second the level of mixing in the radiative envelope (see Pedersen et
al., 2021, for further details). We use the same set of computed stellar
models for the calculations of the evolution of the near-core rotation with
age in Sect. 5. To make sure that the total angular momentum is conserved for
these calculations, we excluded stellar winds from the MESA calculations. We
find that doing so does not significantly impact the results presented in this
work.

Figure 11: Comparison between four different age indicators $t/t_{\rm MS}$,
$X_{\rm c}/X_{\rm ini}$, $\log g$, and $\Pi_{0}$ for different masses and
exponential diffusive overshooting. Panels (a), (c), and (g) show $t/t_{\rm
MS}$ as a function of $X_{\rm c}/X_{\rm ini}$, $\log g$, and $\Pi_{0}$,
respectively. Different masses correspond to different line styles: 2.5 M⊙
(full), 5 M⊙ (dashed), 7.5 M⊙ (dot-dashed), and 10 M⊙ (dotted). Black lines
correspond to models with $f_{\rm ov}=0.01$, whereas models with $f_{\rm
ov}=0.04$ are shown in grey. Panels (b), (e), and (h) show the differences in
$X_{\rm c}/X_{\rm ini}$, $\log g$, and $\Pi_{0}$ obtained for a given
$t/t_{\rm MS}$ between the highest and lowest mass model, e.g.
$\Delta\left(X_{\rm c}/X_{\rm ini}\right)=\left.X_{\rm c}/X_{\rm
ini}\right|_{10\,{\rm M}_{\odot}}-\left.X_{\rm c}/X_{\rm
ini}\right|_{2.5\,{\rm M}_{\odot}}$. Panels (c), (f), and (i) gives the same
differences but between same mass models with different overshooting
parameters, e.g. $\Delta\left(X_{\rm c}/X_{\rm ini}\right)=\left.X_{\rm
c}/X_{\rm ini}\right|_{f_{\rm ov}=0.01}-\left.X_{\rm c}/X_{\rm
ini}\right|_{f_{\rm ov}=0.04}$.

The top two rows of Fig. 11 show $t/t_{\rm MS}$ as a function of $X_{\rm
c}/X_{\rm ini}$ (panel a) and $\log g$ (panel d), respectively, for the main-
sequence evolution of the stars. The differences in line styles corresponds to
different stellar masses, whereas the grey curves show the same calculations
for a larger convective core boundary mixing region ($f_{\rm ov}=0.04$).
Focusing first on panel (a), the relation between $t/t_{\rm MS}$ and $X_{\rm
c}/X_{\rm ini}$ is not purely linear, as the stellar evolution speeds up
towards hydrogen depletion. The relations are nearly the same independently of
the mass of the star and the extent of the convective core boundary mixing
region, but models with higher overshooting tend to have higher $t/t_{\rm MS}$
values for the same $X_{\rm c}/X_{\rm ini}$. This is seen more clearly in
panels (b) and (c), which show the difference in $X_{\rm c}/X_{\rm ini}$ as a
function of $t/t_{\rm MS}$ obtained between the highest and lowest mass models
(panel b) and the models with the largest and smallest extent of the
convective core boundary mixing region (panel c). Increasing the overshooting
parameter leads to larger differences in $X_{\rm c}/X_{\rm ini}$ compared to
those obtained by increasing the stellar mass, but in both cases the
differences remain smaller than $\approx 0.05$.

In comparison, increasing the stellar mass from 2.5 M⊙ to 10 M⊙ gives rise to
larger differences in $t/t_{\rm MS}$ for the same value of $\log g$. Panel (d)
shows that increasing the stellar mass causes the $t/t_{\rm MS}$ versus $\log
g$ curves to shift towards higher $\log g$ values, as also implied in panel
(e). Furthermore, models with higher overshooting starts to deviate from those
with small convective core boundary mixing regions for $t/t_{\rm MS}\gtrsim
0.2$ for the same stellar mass. The differences increases for increasing
$t/t_{\rm MS}$ and become larger than $\approx$0.4 dex at the end of the main-
sequence evolution (see panel (f)). This is why we consider $\log g$ to be the
least accurate age indicator out the three considered in this work.

In their study of internal angular momentum transport using a sample of 37
$\gamma$ Dor stars, Ouazzani et al. (2019) used the bouyancy radius, also
known as the asymptotic period spacing

$\Pi_{0}=2\pi^{2}\left(\int_{\rm gc}\frac{N}{r}{\rm d}r\right)^{-1},$ (A1)

as the age indicator. They demonstrated that $\Pi_{0}$ is monotonously
decreasing as a function of $X_{\rm c}/X_{\rm ini}$ for their considered mass
range of $1.4-1.8$ M⊙ and differs by up to $\approx 800\,{\rm s}$ for a given
$X_{\rm c}/X_{\rm ini}$ (Ouazzani et al., 2019, see their Fig. 2) . In
comparison, $\Pi_{0}$ on its own is an unreliable age indicator for SPB stars
as demonstrated in panels (g)-(i) in Fig. 11. There are multiple reasons for
this. First of all, $\Pi_{0}$ is largely dependent of the stellar mass. As the
SPB stars span a much larger range in stellar masses than the $\gamma$ Dor
stars, this correspondingly leads to a much larger difference in $\Pi_{0}$ for
the same value of $t/t_{\rm MS}$ as seen in panel (h) when comparing the 2.5
M⊙ to the 10 M⊙ models which results in $\Delta(\Pi_{0})>10000$ s. This
difference is generally much larger than the intrinsic variation in $\Pi_{0}$
for a given mass throughout the main-sequence evolution. Secondly, for SPB
stars the $\Pi_{0}$ values are no longer monotonously decreasing with age.
Instead, $\Pi_{0}$ is increasing in the beginning of the main-sequence
evolution and later starts to decrease for the rest of the evolution.
Increasing the extent of the convective core boundary mixing region causes the
$\Pi_{0}$ values to stay nearly the same or increasing for a larger part of
the main-sequence evolution before rapidly decreasing at $t/t_{\rm MS}\gtrsim
0.8$, see also panel (i). For these reasons, we chose not to consider the
evolution of $f_{\rm rot}$ as a function of $\Pi_{0}$ in this work.

While all of the considered 52 SPB stars have a known $\log g$ value either
from spectroscopy, multi-color Strömgren photometry, or asteroseismic
modeling, the $X_{\rm c}/X_{\rm ini}$ and $t/t_{\rm MS}$ values tend not to be
given simultaneously for the same star. Therefore, we choose to do a
conversion between $X_{\rm c}/X_{\rm ini}$ and $t/t_{\rm MS}$ and vise versa
by interpolating onto the curve of the 5 M⊙ star with $f_{\rm ov}=0.01$ in
Fig. 11 (a).

##  Appendix B MESA models including the Spruit-Taylor dynamo

In oder to compare the expected evolution of the near-core and surface
rotation assuming rigid rotation throughout the main-sequence evolution and no
angular momentum transport to the predictions when angular momentum transport
occurs through the Spruit-Taylor dynamo (shown in green in Figs. 6 and 7), we
use the same MESA setup discussed in Appendix A but with a few modifications.

The evolution of the stellar models is split into two, with the first being
the pre-main-sequence evolution where no angular momentum transport is
enforced. The evolution is stopped close to the zero-age main-sequence once
$L_{\rm nuc}/L\geq 0.95$ is fulfilled, where $L_{\rm nuc}$ is the total
luminosity from nuclear reactions, and a starting model for the main-sequence
evolution is saved. In the second step, the saved model is loaded into MESA
and evolved until the core hydrogen mass fraction drops below 0.001. For the
main-sequence evolution, the starting model is assumed to be rigidly rotating
at $10\,\%$ of the critical rotation rate for the green curves shown in Figs.
6 and 7. The Spruit-Taylor dynamo is assumed to be the only contributor to the
angular momentum transport and internal element mixing from rotation is turned
off. The latter is to make sure that the adopted angular momentum transport
mechanism is the only difference between these models and the computed models
described in Appendix A. The corresponding additional MESA inlist parameters
adopted for the main-sequence evolution are

change_v_flag = .true.
new_v_flag = .true.

change_rotation_flag= .true.
new_rotation_flag = .true.

new_omega_div_omega_crit = 0.1
relax_omega_div_omega_crit = .true.
relax_initial_omega_div_omega_crit = .true.

change_D_omega_flag = .true.
new_D_omega_flag = .true.

for the &star_job section, and

omega_function_weight = 20

D_ST_factor = 1
D_visc_factor = 0.0

am_nu_visc_factor = 1.0
am_nu_ST_factor = 1.0
am_D_mix_factor = 0.0
am_gradmu_factor = 0.1d0

for the &controls section, respectively.

Figure 12: Evolution of the average rotation frequencies (left) and the
surface equatorial velocities (right) for a 5 M⊙ star as a function of the two
age indicators $X_{\rm c}/X_{\rm ini}$ (top) and $t/t_{\rm MS}$ (bottom),
assuming angular momentum transport from the Spruit-Taylor dynamo (green). The
intensity of the green lines for the Spruit-Taylor dynamo depends on the
initial rotation rate as indicated in the legend. The two cases of no angular
momentum transport (orange) and solid body rotation (blue) are shown for
comparison.

For comparison, we show in Fig. 12 how varying the initial rotation rate from
$1-50\%\,\Omega_{\rm crit}$ for a 5 M⊙ star changes the evolution of average
rotation frequency experienced by the g-modes and the surface equatorial
velocity, when angular momentum transport occurs through the Spruit-Taylor
dynamo. As seen in the figure, the relative evolution starts to converge for
$>30\%\,\Omega_{\rm crit}$ and approaches the curve for the solid body
rotation.

##  Appendix C Amplitude corrections

The pulsation properties of the sample of 52 SPB stars considered in this work
were studied using a variety of different telescopes, see e.g. second column
of Tabel 1, and therefore the recorded dominant pulsation amplitudes were
obtained using different photometric passbands. Therefore, when comparing
these amplitudes as done in Figs. 9 and 10 it is important to make sure that
the differences in passbands have been accounted for. Furthermore, for the
sample of SPB stars where the asteroseismic analysis was done using ground-
based data, the amplitudes in the literature are given in milli-magnitudes
(mmag) whereas they are given in parts-per-million (ppm) for the SPB stars
with space photometry. As a first step, we convert all amplitudes in mmag to
ppm using

${\rm Amplitude\ (ppm)}=10^{3}\times\frac{{\rm Amplitude\
(mmag)}}{\log_{10}e}.$ (C1)

Next, we follow the methodology outlined by Lund (2019) to convert all
amplitudes to the Johnson V band. Here the conversion factors are obtained by
first deriving the conversion factor555${\rm Amplitude}_{\rm bol}=C_{P-{\rm
bol}}\times{\rm Amplitude}_{P}$ $C_{P-{\rm bol}}$ between the bolometric
amplitudes (${\rm Amplitude}_{\rm bol}$) and the amplitudes (${\rm
Amplitude}_{P}$) obtained for a given passband $P$ using Eq. (2) in Lund
(2019) and assuming black-body radiation. The the amplitude conversion factor
between the Johnson V band and a different passband $P$ is calculated as

$C_{P-V}=\frac{C_{P-{\rm bol}}(T_{\rm eff})}{C_{V-{\rm bol}}(T_{\rm eff})},$
(C2)

such that the amplitudes in the Johnson V band can be derived using

${\rm Amplitude}_{V}=C_{P-V}\times{\rm Amplitude}_{P}.$ (C3)

The photometric passbands used for deriving the conversion factors between
different passbands were obtained from the SVO filter
service666http://svo2.cab.inta-csic.es/theory/fps/ (Rodrigo et al., 2012;
Rodrigo & Solano, 2020). For the 19 SPB stars with ground-based data, the
amplitudes are already given in the Johnson V band. Figure 13 shows $C_{P-V}$
as a function of the effective temperature. For the photometric passband of
the BRITE space telescope the SDSS.r photometric passband was used (Weiss et
al., 2014). We note that the derived corrections are an estimate of the true
$C_{P-V}$ values, which also depend on the metallicity, surface gravity, and
the reddening of the stars.

Figure 13: Amplitude conversion factors derived using Eq. C2 as a function of
effective temperature.

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