and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 36: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/disk = DN_Tau /variables/name = DN Tau /variables/mol = CO
/variables/line = 12CO (2-1)

Figure 37: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 38: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = 13CO /variables/line = 13CO (2-1)

Figure 39: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 40: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = C18O /variables/line = C18O (2-1)

Figure 41: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 42: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/disk = IQ_Tau /variables/name = IQ Tau /variables/mol = CO
/variables/line = 12CO (2-1)

Figure 43: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 44: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = 13CO /variables/line = 13CO (2-1)

Figure 45: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 46: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = C18O /variables/line = C18O (2-1)

Figure 47: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 48: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/disk = UZ_Tau /variables/name = UZ Tau E /variables/mol = CO
/variables/line = 12CO (2-1)

Figure 49: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 50: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = 13CO /variables/line = 13CO (2-1)

Figure 51: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 52: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.

/variables/mol = C18O /variables/line = C18O (2-1)

Figure 53: Observations of /variables/line emission in the /variables/name
disk. Shown are the channel map of the observed line brightness distribution
(left), the moment-zero map (top right), and the integrated spectrum (bottom
right). The channel map shows 16 velocity channels with a step of $0.5$ km s-1
in the [-4.5, +4.5] km s-1 range around the systemic velocity. The contour
lines start at $2$ K, with a step of $2$ K. The color bar shows line
brightness temperatures (K). The contour lines in the moment-zero plot start
at 3, 6, and $9\sigma$, with a step of $6\sigma$ afterward. The $1\sigma$ rms
noise (mJy km s-1) is shown in the upper left corner of the moment-zero plot,
and the synthesized beam is depicted by the dark ellipse in the left bottom
corner. Figure 54: Observations of /variables/line emission in the
/variables/name disk. Shown is a pixel-deprojected Keplerian plot consisting
of the three panels: (left) the radial profile of the line brightness
temperature (K), with observations and errorbars in black, and profile derived
from the best-fit disk model in red, (top) the integrated spectrum (black
line) overlaid with the best-fit Gaussian profile (red line), and (bottom
right) aligned and stacked line intensity (K) as a function of disk radius
(Y-axis; au) and velocity (X-axis; km s-1). The color bar units are in Kelvin.


11institutetext: Observatoire de Genève, Université de Genève, 51 Ch. Pegasi,
CH-1290 Versoix, Suisse
email<EMAIL_ADDRESS>
# Angular momentum transport near convective-core boundaries of Gamma Doradus
stars

F.D. Moyano  P. Eggenberger  S.J.A.J. Salmon

(Received: 22 November 2023; accepted: 25 December 2023)

Recent asteroseismic studies have revealed that the convective core of
$\gamma$ Doradus stars rotates faster than their radiative interior. We study
the development of differential rotation near the convective core to test
angular momentum transport processes that are typically adopted in stellar
evolution models. Models that only include the advection of angular momentum
by meridional circulation and shear instabilities cannot reproduce current
rotational constraints, irrespective of the initial conditions. The latest
formulation of internal magnetic fields based on the Tayler instability is
indeed able to reproduce the internal rotation rate of post-main sequence
stars, however, it appears too efficient during the main sequence and has thus
been disfavoured. A less efficient version of the same transport process can
simultaneously reproduce the rotation rate of the convective core, the
rotation rate in radiative regions as probed by gravity-modes, and the surface
rotational velocities of $\gamma$ Doradus stars. Our work suggests that there
are additional physical processes apart from internal magnetic fields at work
in the stellar interiors of post-main sequence stars.

###### Key Words.:

asteroseismology – stars: rotation – stars: interiors – stars: evolution –
stars: variables: Gamma Doradus – methods: numerical

##  1 Introduction

A number of recent results point to a misunderstanding with respect to the
modelling of rotation in stellar interiors through evolution (e.g. Aerts et
al., 2019, and references therein). Since the theory of angular momentum (AM)
transport by meridional circulation and shear instabilities has failed to
account for observational constraints brought by asteroseismology (e.g.
Eggenberger et al., 2012), this has led to the conclusion that additional
physical processes would transport AM efficiently in stellar interiors.
Currently, the theory of internal magnetic fields generated by the Tayler-
Spruit dynamo (hereafter, TS; Spruit, 2002) in its revised form (Fuller et
al., 2019) is the closest and most widely tested solution to the AM transport
problem in stellar evolution theory. However, disagreements with regard to the
core rotation rates of subgiants have undermined this solution (e.g.
Eggenberger et al., 2019). The recent detection of radially differential
rotation between the convective core and the radiative interior of main
sequence (MS) stars (Saio et al., 2021) poses an additional, yet fundamental,
challenge to our understanding of AM transport in stellar interiors, since
asteroseismic studies of MS gravity-mode pulsators had provided the rotation
rate only in their radiative regions.

In MS stars, particularly Gamma Doradus stars (hereafter $\gamma$ Dor), an
almost uniform rotation is favoured along the radiative interior (e.g. Van
Reeth et al., 2018), although differential rotation in the near-core regions
could not be excluded. The rotation rate of the convective core was recently
measured for the first time in a sample of 16 $\gamma$ Dor stars (Saio et al.,
2021) by modelling features called ‘dips’ seen in the period spacing patterns
of gravity modes 111A period spacing pattern is the relation between the
difference in period, $\Delta P_{\rm n}=P_{\rm n+1}-P_{\rm n}$, between modes
of consecutive radial order and the pulsation period, $P_{\rm n}$, of
different radial modes.. As shown by Ouazzani et al. (2020) those dips are
caused by resonant coupling between inertial modes trapped in the convective
core and gravity modes propagating in the radiative interior (See also Tokuno
& Takata, 2022; Aerts & Mathis, 2023). What is crucial to glean from these
first constraints is that the convective core rotates slightly faster than the
radiative interior, with a difference of at most $\sim 20\%$. Although this
may appear to be a small difference, in combination with previous results, it
provides strong information on how the physical processes should operate in
stellar interiors. Moreover, these constraints allow us to study the AM
transport in regions with strong chemical gradients, since the mass of the
convective core in these stars is expected to decrease through the MS, thereby
leading to a gradual change in the chemical composition between the convective
core and the radiative interior. This makes it particularly important to carry
out detailed tests of the inherent physical processes, since several
hydrodynamic and magnetic instabilities in stellar interiors are inhibited by
chemical composition gradients. This is because the fluid motions are expected
to be able to counteract the buoyancy forces (e.g. Heger et al., 2000).

In this paper, we study the development of differential rotation between the
convective core and the radiative interior through the MS in models with
purely hydrodynamical processes, namely, the advection of angular momentum by
meridional currents and shear instabilities, as well as in models that also
include internal magnetic fields generated by the TS dynamo. We further test
whether these models are in agreement with the estimates provided by Saio et
al. (2021) and we consider their applicability to more advanced phases.

##  2 Input physics of models

We used the Geneva stellar evolution code (GENEC; Eggenberger et al., 2008) to
compute models of rotating stars during the MS. The code includes a detailed
treatment of AM transport following the formalism of Zahn (1992), which takes
into account AM advection by meridional currents and diffusion by turbulent
processes such as shear instabilities. We employed the formulation of Maeder
(1997) for the secular shear instability and the formulation presented by
Eggenberger et al. (2022) for the internal magnetic fields generated by the
Tayler instability. We assumed that convective overshooting mixes chemical
elements instantaneously, but the temperature gradient remains non-adiabatic
in the overshooting regions (i.e. the step-overshooting formalism). We used an
overshooting strength of $d=0.05H_{\rm p}$ (where $H_{\rm p}$ is the local
pressure scale height) for models with $M<1.7M_{\odot}$, and $d=0.1H_{\rm p}$
for models with $M\geq 1.7M_{\odot}$. The rest of the input parameters and the
physics included are the same as those given in Moyano et al. (2023), unless
otherwise specified, and we refer to this work for further details.

The AM transport by internal magnetic fields in our models relies on a
generalised prescription of the Tayler instability as given by Eggenberger et
al. (2022). In this prescription, the uncertainties on the timescale needed to
damp the instability on the azimuthal magnetic field is parameterised through
a variable $C_{\rm T}$. The general expressions that determine the efficiency
of AM transport are hereafter given by Eqs. 1 and 2. The former determines the
condition needed for the dynamo to operate which requires a minimum degree of
local differential rotation, termed $q_{\rm
min}\equiv|\partial\log\Omega/\partial\log r|_{\rm min}$, and given by:

$q_{\rm min}=\frac{1}{C_{\rm T}}\left(\frac{N_{\rm
eff}}{\Omega}\right)^{(n+2)/2}\left(\frac{\eta}{r^{2}\Omega}\right)^{n/4},$
(1)

where $N_{\rm eff}$ is the effective Brunt-Väisälä frequency, $\eta$ is the
magnetic diffusivity, $\Omega$ is the angular velocity, $r$ is the radial
distance from the centre of the star, and $n$ is a parameter that determines
the damping timescale of the instability. And the latter sets the efficiency
of the AM transport which is regulated by the magnetic viscosity ($\nu_{\rm
mag}$) and is computed as

$\nu_{\rm mag}=\frac{\Omega r^{2}}{q}\left(C_{\rm T}q\frac{\Omega}{N_{\rm
eff}}\right)^{3/n}\left(\frac{\Omega}{N_{\rm eff}}\right).$ (2)

We recall that with $C_{\rm T}=1$ and $n=1$ the formalism reduces to that
originally proposed by Spruit (2002) and $n=3$ to the one proposed by Fuller
et al. (2019). In this work, we use $n=1$ and different values of $C_{\rm T}$
to study the degree of differential rotation between the convective core, and
the near-core rotation rate in the gravity-mode cavity probed with
asteroseismic techniques, since it sets the efficiency of AM transport; a
lower(higher) value of $C_{\rm T}$ leads to less(more) efficient AM transport.

We compute the near-core rotation rate ($\Omega_{\rm nc}/2\pi$) as an average
rotation rate of the gravity-mode (g-mode, hereafter) cavity from our models
as

$\frac{\Omega_{\rm nc}}{2\pi}=\frac{1}{2\pi}\frac{\int_{r_{\rm in}}^{r_{\rm
out}}{\Omega Ndr/r}}{\int_{r_{\rm in}}^{r_{\rm out}}{Ndr/r}},$ (3)

where $N$ is the Brunt-Väisälä frequency, and $r_{\rm in}$ and $r_{\rm out}$
are the inner and outer limits of the g-mode cavity, respectively. The
rotation rate of the convective core ($\Omega_{\rm cc}/2\pi$) is simply taken
as the rotation rate of the centre of the model since we assume rigid rotation
in convective regions. In the following sections, all the models presented
have an initial mass of $M=1.5M_{\odot}$, an initial metallicity of $Z=0.01$,
and an initial rotation rate of $\Omega/2\pi=20\mu$Hz at the zero-age main
sequence (ZAMS) assuming initial solid body rotation, unless otherwise
specified.

Figure 1:  Ratio of rotation rate between the convective core and the near-
core radiative interior probed by gravity modes, as a function of the central
hydrogen abundance in mass fraction for models without internal magnetic
fields. The effect of initial rotation rates (top), metallicity (middle), and
initial mass (bottom) are illustrated in each panel. All models are computed
assuming uniform rotation as an initial condition, except for the model in
light-blue dashed line in the top panel where we assumed that initially the
convective core rotates faster than the near-core regions.

##  3 Models with no internal magnetic fields develop too much differential
rotation

Models without internal magnetic fields always develop differential rotation
between the convective core and the near-core radiative interior (hereafter
inner differential rotation) early during the MS. In Fig. 1 we show the degree
of inner differential rotation as measured by the ratio of the rotation rate
between the convective core and the near-core radiative region ($\Omega_{\rm
cc}/\Omega_{\rm nc}$) during the MS. The initial rotation rate (and hence the
rotational velocity) of the models affects the degree of inner differential
rotation more strongly than the initial mass or the metallicity. While
metallicity has a negligible effect, the initial mass has a mild effect,
leading to a difference of at most 5% in the inner differential rotation
during the early MS. Low initial velocities lead to higher degree of inner
differential rotation during the late MS (central hydrogen mass fraction of
$X\lesssim 0.35$), whereas high initial velocities lead to a rapid decoupling
between the convective core and the radiative regions as a result of the
higher efficiency of advection by meridional currents at higher rotation
rates. We also verify whether it is possible to progressively decrease the
degree of inner differential rotation by starting with $\Omega_{\rm
cc}/\Omega_{\rm nc}\simeq 1.25$ at the ZAMS (shown by the light-blue dashed
line in Fig. 1). This behaviour was suggested by Saio et al. (2021) due to the
correlation that these authors found between the inner differential rotation
and the central hydrogen abundance. We find that this scenario is not
achievable in our models with purely hydrodynamical processes.

The evolution of the rotation rate in the stellar interior (i.e. the rotation
profiles) through the MS without internal magnetic fields is shown in Fig. 2.
The central regions correspond to the convective core, so the rotation profile
is flat there since we assume uniform rotation in convective regions, while
the rotation rate in the outer regions decreases mainly as a consequence of
the radial expansion through the MS. The rotation profile reaches a pseudo-
equilibrium state early during the MS where the convective core rotates faster
than the radiative interior as a result of the meridional circulation and the
shear instability. Thus, its morphology is relatively unaffected until the end
of the MS.

Figure 2: Rotation rate as a function of the radial distance from the centre
of the star in units of solar radii, for models without internal magnetic
fields. The points over the lines indicate the mean rotation rate of the
gravity-mode cavity computed with Eq. 3 for each model. The models are shown
at different stages during the MS, as indicated by the mass fraction of
central hydrogen in the figure.

##  4 Internal magnetic fields allow for small inner differential rotation

Models with internal magnetic fields develop less inner differential rotation
than models with purely hydrodynamical processes. In Fig. 3 we show the
evolution through the MS of the degree of inner differential rotation in
models with internal magnetic fields and its sensitivity to the constant
$C_{\rm T}$ (see Eqs. 1 and 2). Since the constant $C_{\rm T}$ sets the
efficiency of the AM transport, low values of such a constant (e.g. $C_{\rm
T}=0.001$) lead to similar evolution and degree of inner differential rotation
as in models without magnetic fields, while high values lead to smaller inner
differential rotation. Models with a weak AM transport efficiency converge to
the same degree of inner differential rotation towards the end of the MS
because the receding convective core leads to an extended region of relatively
strong chemical gradients, which suppress the TS dynamo and so the convective
core can decouple from the radiative interior as in models with only
hydrodynamical processes. The model computed with $C_{\rm T}=1$ corresponds to
the original prescription provided by Spruit (2002). Models computed with the
prescription proposed by Fuller et al. (2019) lead to rigid rotation near
convective-core boundaries.

Figure 3: Same as Fig. 1, but for models including internal magnetic fields.
The efficiency of angular momentum transport is proportional to $C_{\rm T}$
(See Sect. 2). Models without magnetic fields and models computed with the
original prescriptions proposed by Spruit (2002) and Fuller et al. (2019) are
also included for reference.

In Fig. 4, we show an example of the kind of rotation profiles during the MS
of our models with internal magnetic fields using a value of $C_{\rm T}=0.125$
in Eqs. 1 and 2. As the stars evolve through the MS, the convective core
detaches from the radiative interior because the convective core retreats;
thus, the gradients of chemical composition develop close to the convective
boundaries. Furthermore, since the TS dynamo is inhibited by chemical
composition gradients, thus differential rotation close to the boundaries of
the convective core can develop progressively through the MS. In the outer
radiative interior and up to the surface, the rotation profile is nearly
uniform since in chemically homogeneous regions the dynamo can operate
efficiently and even more far from the centre (see Eq. 2). However, this kind
of profile is only possible when adopting values of $C_{\rm T}\lesssim 0.5$. A
value of $C_{\rm T}=216$, as needed to reproduce the core rotation rate of red
giants (Eggenberger et al., 2022), leads to almost rigid rotation during the
MS. The same conclusion is reached when using the prescription provided by
Fuller et al. (2019).

Figure 4: Same as Fig. 2, but for models including internal magnetic fields
following a modified and less efficient version of the Tayler-Spruit dynamo,
which corresponds to models computed with $C_{\rm T}=0.125$ shown in Fig. 3.

##  5 Comparison with asteroseismic constraints

We compare the degree of inner differential rotation and near-core rotation
rates obtained in our models with the estimates provided by Saio et al. (2021)
and Li et al. (2020). We chose these quantities because they are not strongly
sensitive to stellar parameters: the near-core rotation rate derived from the
slope of the period spacing patterns by Li et al. (2020) is robust to changes
in mass, metallicity, and evolutionary age (Ouazzani et al., 2017). In
addition, the inner differential rotation as measured by the ratio
$\Omega_{\rm cc}/\Omega_{\rm nc}$ is robust compared to changes in stellar
parameters extracted from the best-fit models presented by Saio et al. (2021),
such as central hydrogen abundance or stellar mass. For the comparison with
the models, we take the rotation rate of the convective core estimated from
the best-fit models without overshooting presented by Saio et al. (2021) and
the near-core rotation rates provided by Li et al. (2020). The uncertainty on
the rotation rate of the convective core is simply taken as the difference
between the estimates obtained with models using a different overshooting
strength by Saio et al. (2021).

Figure 5: Near-core rotation rate ($\Omega_{\rm nc}/2\pi$) as a function of
the ratio of rotation rate between the convective core and the near-core
radiative region ($\Omega_{\rm cc}/\Omega_{\rm nc}$). The models include
angular momentum transport by meridional currents and shear instabilities, but
do not include internal magnetic fields. The models are computed from the zero
age main sequence assuming initial solid-body rotation $(\Omega_{\rm
cc}/\Omega_{\rm nc})_{\rm ini}=1$, with different initial rotation rates as
indicated in the figure. The triangles over the evolutionary tracks indicate
the time the central hydrogen content reaches $X=0.7,0.5,0.35,$ and $0.1$
(from left to right, respectively). The data points correspond to the
constraints provided by Saio et al. (2021).

In Fig. 5, we show that the degree of inner differential rotation in models
without internal magnetic fields is in disagreement with the current
constraints, since (as shown in Fig. 2) the convective core decouples from the
radiative interior early during the MS. This is illustrated by the points over
the different curves located when the central abundance of hydrogen reaches
$X=0.7,0.5,0.35,$ and 0.1. For the model with an initial near-core rotation
rate of $\Omega_{\rm nc}/2\pi=29\mu$Hz, as soon as the central hydrogen
content reaches $X\simeq 0.7,$ the convective core already rotates
approximately 20% faster than the radiative interior. For the lowest initial
velocities, the points at $X=0.7$ are located progressively to the left
because the convective core decouples slower at low initial velocities (as
shown in Fig. 1). This is because the efficiency of meridional circulation is
proportional to the rotation rate. While, in principle, these models can
potentially reproduce the constraints, we do not deem it very likely as the
phase of readjustment of the rotation profile occurs relatively fast.

In Fig. 6, we show that models with internal magnetic fields following the
original prescription by Spruit (2002) cannot reproduce the degree of inner
differential rotation and the near-core rotation rate simultaneously. These
kinds of models cannot reproduce the observational constraints for any initial
rotational velocity, due to the fact that the TS dynamo acts more efficiently
at higher rotational velocities (see Eqs. 1 and 2). Because of this, the
degrees of inner differential rotation displayed by the data cannot be
reproduced by choosing high initial velocities for the models. In fact, our
model with an initial rotation rate of $\Omega/2\pi=33\mu$Hz corresponds to
$80$% of its critical velocity and achieves a critical value at roughly the
middle of the MS; this is also why the behaviour of this model is different
towards the end of the MS, resulting from AM removal due to mechanical mass-
loss. More efficient formulations, such as that proposed by Fuller et al.
(2019), lead to quasi-rigid rotation and, thus, they are not favoured by the
data. As a result, we find that a less efficient AM transport process is
needed.

Figure 6: Same as Fig. 5, but for models including internal magnetic fields
following the original prescription of Spruit (2002).

In Fig. 7, we show that our models with internal magnetic fields following a
modified and less efficient version of the TS dynamo computed with $C_{\rm
T}=0.125$ can reproduce the qualitative trends in a reasonable way. These
models spend more than half of their MS lifetime with degrees of inner
differential rotation in agreement with asteroseismic constraints, as
illustrated by the symbols over the different tracks. This occurs because the
convective core decouples from the radiative interior progressively due to the
development of chemical gradients in the near-core regions and so, the inner
differential rotation increases progressively as the convective core recedes.
In the radiative regions above the core, the AM is redistributed efficiently,
leading to a decreasing $\Omega_{\rm nc}/2\pi$. We note that the evolution of
the rotation profile for this kind of model is shown in Fig. 4. The latter is
needed to satisfy the constraints on the apparently quasi-rigid rotation from
the surface to the near-core regions (Van Reeth et al., 2018; Li et al., 2020)
and to reproduce their surface rotational velocities (Moyano et al., 2023).
Moreover, this conclusion is largely independent of the rotational history
during the pre-MS, since before the hydrogen starts burning in the core,
little to no chemical composition gradients are present in the radiative
regions. This allows for the TS dynamo to operate efficiently, since buoyancy
forces only work to weakly stabilise the radial motions associated to the
Tayler instability when we are dealing with a chemically homogeneous medium.
Therefore, our models with internal magnetic fields always converge to a
uniform rotation profile by the time they arrive at the ZAMS.

We note that our interpretation of the data is different from that of Saio et
al. (2021), who suggested that the convective core re-couples with the
radiative interior through evolution; so, the degree of inner differential
rotation decreases progressively, instead of increasing as it is shown to do
in our models. However, that would imply that the near-core regions should
rotate faster as the stars evolve through the MS, which we do not support
since our previous studies have demonstrated that to reproduce the surface
rotational velocities of these kinds of stars, the AM from the near-core
regions should be transported to the surface (Moyano et al., 2023). Moreover,
evidence from previous studies using gravity modes and the modulation seen in
the light curves (potentially caused by magnetic spots) suggest quasi-rigid
rotation in radiative regions (Van Reeth et al., 2018; Li et al., 2020). This
should lead to decreasing near-core rotation rates, as the AM is extracted
from the near-core regions and redistributed in the whole radiative interior.

Figure 7: Same as Fig. 5, but for models including internal magnetic fields
following a modified and less efficient version of the Tayler-Spruit dynamo,
computed with $C_{\rm T}=0.125$ in Eqs. 1 and 2.

##  6 Implications for the angular momentum transport problem

Internal magnetic fields generated by the TS dynamo either in its standard
(Spruit, 2002), revised (Fuller et al., 2019), or calibrated forms
(Eggenberger et al., 2022) lead to quasi-rigid rotation and are rejected as a
result. Purely solid-body rotation or our models with purely hydrodynamical
processes are not able to reproduce the qualitative trends seen in the data.
Our models with a modified and less efficient version of the TS dynamo are in
better agreement with the rotational constraints on the convective core and
the radiative interior. However, this less efficient version leads to weaker
transport during the post-MS phase, which, in turn, leads to core rotation
rates that are in disagreement with asteroseismic constraints (e.g. those from
Mosser et al., 2012; Gehan et al., 2018). Moreover, Deheuvels et al. (2020)
found that two young subgiants just beginning to expand towards the red giant
phase, rotate roughly as solid bodies. We find that our models cannot account
for these constraints. This raises the question of whether the same physical
process should dominate the AM transport during the MS and the subgiant or red
giant phase. It is possible that other processes are at work during the post-
MS phase efficient enough to slow down the cores. Some of these processes
might include internal gravity waves (e.g. Pinçon et al., 2017), mixed modes
(Belkacem et al., 2015a, b) that remain as-yet unexplored in stellar evolution
calculations, or another form of magnetic fields or magnetic instabilities,
such as the azimuthal magneto-rotational instability (e.g. Meduri et al.,
2023) or large-scale internal magnetic fields (e.g. Takahashi & Langer, 2021).

###### Acknowledgements.

FDM thanks Masao Takata for useful discussions during the MIAPbP workshop held
during August of 2023. The authors have received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 833925, project STAREX). The
computations were performed at University of Geneva using Baobab HPC service.

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