A further consequence of our use of pure p-mode rotational kernels is that, as
can be seen in Fig. C1, the nonradial p-mode wavefunctions decay to zero at
small radius, and so these kernels are almost entirely insensitive to the
radiative core (whose boundary is situated in our best-fitting stellar
structure at $r/R\sim 0.03$). For rotational inferences made using mixed
modes, it is common to parameterise radial differential rotation in terms of a
phenomenological two-zone model, with the core and envelope treated as
separately rotating solid bodies (e.g. Klion & Quataert, 2017; Triana et al.,
2017; Mosser et al., 2018; Gehan et al., 2018; Eggenberger et al., 2019b;
Fellay et al., 2021; Ahlborn et al., 2022). By contrast, any constraints on
radial differential rotation that we might obtain using our observed p-modes,
through these p-mode rotational kernels, will be localised entirely within the
convective envelope.

###  C.2 Radial Differential Rotation

Our marginal posterior distribution for the difference between the dipole and
quadrupole rotational splittings, Fig. 3, suggests that Zvrk’s envelope is not
rotating as a solid body, and possibly exhibits radial differential rotation.
Given the preceding discussion, we might wish to investigate this differential
rotation using existing rotational inversion techniques. However, our attempts
to do so are hindered by the limitations of the existing TESS data set —
including its poorer photometric stability, redder bandpass, and presently
shorter duration compared to the nominal _Kepler_ mission — which resulted in
us having to pool measurements of the rotational splitting width between
different modes in Section 2.1. Even then, each of our pooled rotational
splittings is constrained with less precision than that of individual
multiplets in _Kepler_ data (e.g. as in Tayar et al., 2022). Were we to have
constrained these splittings for each multiplet individually, the resulting
uncertainties would be larger still — too large for the rotational inversion
techniques that we have described above to be feasibly prosecuted. To proceed
further, we must adapt existing methods to this present situation.

####  C.2.1 Optimally-Localised Averages

(a)

(b)

Figure C2: Raw asteroseismic constraints on sensitivity to rotational shear.
(a) Posterior distributions for differences between the pooled quadrupole and
dipole rotational splitting. The upper panel shows the posterior distribution
of the differences in estimated rotation rates, which are in turn the
rotational splittings divided by effective sensitivity constants
$\left<\beta_{\ell}\right>\sim 1$. The lower panel shows the posterior
distribution of the differences in acoustic radii of the centres of
sensitivity associated with the underlying averaged rotational kernels of each
degree, considered separately. The two quantities are statistically
independent. (b) Differential sensitivity kernels associated with
distributions in (a). Associated with each pooled rotational splitting
$\delta\nu_{\ell}$ is an effective rotational kernel $K_{\ell}$, which is a
weighted average of the rotational kernels associated with each mode of that
degree. Faint curves show samples from the posterior distribution, while the
solid coloured curves show the posterior median effective kernel for each
degree. The inner turning points at ${\nu_{\text{max}}}$ of the best fitting
stellar model from our optimisation procedure are shown with the vertical
dashed lines, while the centres of sensitivity for the median kernels are
indicated with the vertical dotted lines. The ordering of their positions can
be seen to reversed compared to the theoretical inner turning points. The
localisation widths of the median kernels are indicated by the horizontal
intervals placed around the centres of sensitivity.

We first consider how the OLA method may be adapted to operate with respect to
our pooled rotational measurements. We observe that the pooled rotational
splitting for dipole modes must be related to those of the underlying
individual dipole multiplets through some kind of visibility-weighted
averaging procedure, not necessarily linear. If we approximate this with a
linear combination of mode-wise rotational splittings as
$\delta\omega_{\ell=1}\sim\sum_{n,\ell=1}w_{n}\delta\omega_{n,\ell}$, then the
corresponding linear combination of dipole-mode kernels effectively provides
us with a “pooled” sensitivity kernel, $K_{\text{eff},1}$, for the observed
dipole rotational splitting. We may also divide the averaged splitting by the
associated linear combination of sensitivity constants,
$\left<\beta_{1}\right>=\sum_{n,\ell=1}w_{n}\beta_{n\ell}$, to yield an
estimate of $\Omega(t_{1})$ as localised by this pooled kernel. Such an
estimate is localised, roughly speaking, at the centre of sensitivity for this
pooled kernel, $t_{1}=\int t\cdot K_{\text{eff},1}(t)\ \mathrm{d}t$. Since we
have two averaged rotational splittings at our disposal — one for dipole
modes, and one for quadrupole modes — we in principle then should be able to
constrain rotational shear in terms of the differences in localised rotation
rates, $\Delta\Omega\sim\Omega(t_{2})-\Omega(t_{1})$, as well as differences
in the locations at which these estimates are made, $\Delta t=t_{2}-t_{1}$.

It remains for us to determine how these coefficients $w_{i}$ are to be
related to the mode visibilities. By ansatz, we set $w_{i}\propto H_{i}$, the
observed mode heights appearing in Eq. 2. Through the above discussion, we may
then evaluate $\Delta\Omega$, $\Delta t$, and the per-degree pooled kernels in
closed form, given values for all of the nonradial mode heights and averaged
rotational splittings. Since our nested-sampling procedure in Section 2.1
provides us with a large number of samples from the joint distributions
between all of these quantities, applying this procedure to all of these
samples permits us to estimate the entire posterior distributions for
$\Delta\Omega$, $\Delta t$, and the pooled kernels themselves, constrained
directly by the power spectrum.

We examine these posterior distributions in Fig. C2. In Fig. C2a, we show the
marginal distributions for $\Delta\Omega$ (top) and $\Delta t$ (bottom), in
both cases marking out 0 with a dashed line, which is the value demanded by a
null hypothesis of solid-body rotation. We find that $p(\Delta\Omega>0)\gtrsim
0.85$, weakly suggesting that rotational shear is indeed present, although
unfortunately we are not able to rule out the null hypothesis altogether.
Since we have defined $\Delta t$ in the sense $t_{2}-t_{1}$, rudimentary JWKB
analysis would suggest that it be positive, as the inner turning points of the
quadrupole p-modes, set by where the Lamb frequency
$S_{\ell}=\sqrt{\ell(\ell+1)}c_{s}/r$ matches the mode frequency $\omega$,
lies outside of that of the dipole modes. Contrary to this expectation, the
posterior distribution for $\Delta t$ is more consistent instead with negative
values. This would indicate that the inner portions of the convective envelope
are rotating more quickly than the outer portions.

We examine these kernels in more detail in Fig. C2b, where we plot with the
faint thin curves 100 samples from the posterior distribution over both pooled
kernels, and with the thick solid curves the posterior median pooled kernels,
using different colours for each degree. Additionally, we mark out the
location of the JWKB inner turning points (evaluated at ${\nu_{\text{max}}}$)
from our best-fitting model with vertical dashed lines, in matching colours.
The quadrupole inner turning point can indeed be seen to lie exterior to the
dipole inner turning point. However, the shape of the quadrupole-mode pooled
kernel can also be seen to be more spread out than the dipole-mode kernel.
Ultimately, this is because the expressions for the rotational kernels contain
explicit dependence on $\ell$ through contributions from a term
$[\ell(\ell+1)-1]\xi_{h}^{2}$, in addition to the implicit dependences of the
eigenfunctions themselves on $\ell$. For low-degree p-modes, the displacement
eigenfunctions become approximately radial in the outer parts of the star,
where the approximation of plane-parallel stratification holds well. Thus,
this degree-dependent enhancement to the rotational kernel from the horizontal
component of the eigenfunction is most significant near the interior of the
star. Accordingly, the quadrupole-mode kernels feature a “leg” protruding
inwards from their first local maximum after the inner turning point, which is
not present in the dipole-mode kernels. This has the effect of rendering the
quadrupole-mode kernels more spread out compared to the dipole mode kernels,
and of moving their centres of sensitivity inwards (shown with the dotted
vertical lines). For Zvrk in particular, this adjustment to the centre of
sensitivity is comparable to, and slightly larger than, the differences in the
inner turning points. In any case, any differences in the locations of the
centres of sensitivities are minute compared to both the spread implied by the
posterior distribution on $\Delta t$, and the localisation widths of the
pooled kernels themselves (indicated with the horizontal intervals). Thus,
without access to the individual rotational splittings, our constrained
modification to the OLA techique is unfortunately unable to pin down the
location, or even direction, of any shear in the envelope.

(a)(b)(c)(d) Figure C3: Regularised asteroseismic constraints on rotational
shear. (a) Joint posterior distributions for the stellar inclination, inner
and outer rotation rates, location, and thickness of the rotational shear
layer, which parameterise our regularised two-zone description of the
rotational profile, Eq. C3. We show in blue the posterior distributions
obtained from the power spectrum with the flat priors described in the main
text; with orange a conditional distribution obtained by demanding that the
outer rotation rate matches the value provided by photometry; and with gray a
conditional distribution from our null-hypothesis scenario in which solid-body
rotation is enforced. (b) Samples from the posterior distribution over
rotational profiles from the main exercise are shown with the thin faint
lines. These are coloured blue if they are consistent with differential
rotation in the sense of the rotation rate decreasing towards the surface, and
red for the converse sense. The median rotational profile is shown with the
thick black curve, and the $\pm 1\sigma$ posterior credible region around it
is demarcated with the dashed curves. (c) The same as (b), but for our
restricted scenario where the outer rotation rate is matched to the
photometric value. Draws from the posterior distribution with the rotation
rate increasing outwards are shown in orange, while those with the converse
sense of differential rotation are shown in gray. (d) A closer look at the
conditional posterior distribution for the rotation rate under a solid-body
rotation scenario.

####  C.2.2 Regularised Least-Squares

The preceding discussion has implicitly relied on a two-zone model of radial
differential rotation, where we have used the separation between pooled dipole
and quadrupole modes to provide us both the averaged rotation rate in these
two zones, as well as the locations of the zones themselves. Since the
underlying rotational kernels have proven uncooperative in allowing us to
distinguish the locations of these two zones, we may instead make this
assumption explicit by parameterising $\Omega(t)$ to contain two physically
separated regions of solid-body rotation, with a shear layer between the two.
More concretely, we describe the rotational profile as

$\Omega(t;\Omega_{\text{in}},\Omega_{\text{out}},t_{\text{shear}},\Delta
t)=\Omega_{\text{in}}+\left(\Omega_{\text{out}}-\Omega_{\text{in}}\over
2\right)\left(\tanh\left(t-t_{\text{shear}}\over\Delta t\right)+1\right),$
(C3)

which we use to generate predictions for the rotational splittings through Eq.
C1, as done in the RLS technique. Rather than comparing these predictions
against measured rotational splittings (which we do not have in hand), we may
instead constrain the parameters
$(\Omega_{\text{in}},\Omega_{\text{out}},t_{\text{shear}},\Delta t)$ of this
description directly against the power spectrum, by inserting these
predictions into our parametric model of the power spectrum, Eq. 2, in lieu of
the pooled rotational splittings, and evaluating likelihoods using the same
$\chi^{2}$-2-degree-of-freedom likelihood function (Eq. 3) as we had earlier
used for the pooled rotational splittings. This allows us to derive posterior
distributions for these parameters of Eq. C3 (and the other existing
parameters of the model) through the same nested sampling procedure as we had
used in Section 2.1. For this purpose, we place the same uniform priors on
$\Omega_{\text{bulk}}=(\Omega_{\text{in}}+\Omega_{\text{out}})/2$ as we did on
the pooled rotational splittings in Section 2.1 (up to factor of $2\pi$), and
we also place very wide uniform priors on
$\Delta\Omega=\Omega_{\text{out}}-\Omega_{\text{in}}$, permitting it to vary
between $-0.016\ \mu$Hz and $+0.016\ \mu$Hz — i.e. our prior is chosen
explicitly so as not to favour either sense of the rotational shear.

We show the resulting joint posterior distribution for these parameters,
together with the inclination, using blue points, contours, and histograms, in
Fig. C3a. As we encountered in our earlier discussion of the rotational
kernels, even this regularised construction is unable to constrain the
location, or the sharpness, of the rotational shear. It appears only to weakly
favour a sense of the rotational shear where the interior of the convective
envelope is rotating faster than the exterior. To better illustrate this, we
show in Fig. C3b several random samples for the posterior distribution over
possible shapes of $\Omega(r)$, using the thin faint curves. These curves are
coloured blue if $\Omega_{\text{in}}>\Omega_{\text{out}}$, and red otherwise.
The overall median rotational profile, which we show with the solid black
curve, is such that the rotation rate decreases outwards. Rotational shear in
this sense is also implied by the maximum-likelihood estimator for this
parameterisation of radial differential rotation. However, this technique does
not rule out the opposite sense of differential rotation, as some rotational
profiles in the converse sense (i.e. increasing outwards, shown in red) are
encompassed within the $1\sigma$-credible region, shown with the dashed black
curves.

To assess the statistical support behind this result, we perform a likelihood-
ratio model-comparison test against a solid-body rotation scenario, in which
$\Omega_{\text{out}}$ is constrained to be equal to $\Omega_{\text{in}}$, such
that $t_{\text{shear}}$ and $\Delta t$ are entirely unconstrained. For
illustrative purposes, we also estimate, again using nested sampling, the full
posterior distribution for all parameters under this solid-body scenario. We
show the joint distributions of the inclination $i$ and rotation rate $\Omega$
with the gray contours in Fig. C3a, and show the marginal posterior
distribution for $\Omega$ in more detail in Fig. C3d. The above two-zone
differential rotation scenario yields a likelihood-ratio test statistic of
$\Lambda=3.832$ against this solid-body rotation scenario: it is more
consistent with the power spectrum. However, this comes at the expense of
introducing 3 additional free parameters; accordingly, there is a fairly high
probability of $p=0.280$ that this improvement in likelihood can be explained
purely by coincidence. Differential rotation in the sense of $\Omega$
decreasing outwards is thus more likely than solid-body rotation, but not
unambiguously so.

Differential rotation in this direction would possibly require a second shear
layer to exist near the surface, not described by Eq. C3, for consistency with
the photometric rotation rate. This shear layer would also have to be located
very near the surface, in order not to overly affect values returned from the
integral expression in Eq. C1 from the low-degree kernels that we have access
to. Such an asteroseismically inaccessible (at least at low degree) near-
surface shear layer is not strictly speaking impossible. Indeed, it is the
preferred explanation adopted by Tayar et al. (2022) to explain the rotational
configuration of the red giant considered in that work, whose surface rotation
rate, obtained from spectroscopy, is similarly faster than a pooled estimate
of bulk rotation in the envelope recovered from quadrupole and octupole
p-modes. A near-surface shear layer (albeit with rotational shear in the
opposite sense, and at different optical depth than demanded here) is also
known to exist in the Sun, where its presence is confirmed by the
observational availability of intermediate and high-degree modes, whose
rotational kernels, being confined almost entirely to the near-surface layers,
would not find it similarly invisible (e.g. Schou et al., 1998; Howe, 2009).

However, without measurements of modes of intermediate or high degree
available, we cannot claim that this explanation is unique. For one, although
we have so far discussed differential rotation in only the radial direction,
it is possible that it also exists latitudinally. While the spherical
harmonics of the sectoral modes through which we observe the rotational
splitting are sensitive mostly to an equatorial band (i.e.
$P_{\ell}^{\ell}(\cos\theta)$ can be expressed in terms of only $\sin\theta$:
Arfken & Weber, 2005), the latitudes of the surface features producing our
photometric rotational signal are entirely unconstrained. Should they be
located at high latitude, and should Zvrk additionally exhibit latitudinal
differential rotation in an anti-solar sense, then we would expect its equator
to be rotating more slowly than the photometric modulations would imply; this
would resolve our discrepancy from purely geometrical considerations. However,
such an explanation would be fairly ad-hoc, as it would require Zvrk’s surface
features to persist at much higher latitudes than the spots on magnetically
active stars. In any case, latitudinal differential rotation in general, and
on stars that are not Sun-like in particular, is very poorly understood.

We can also explore alternative solutions to this issue by further
regularising our asteroseismic inferences. To demonstrate this, we repeat our
nested-sampling exercise, but further demand that $\Omega_{\text{out}}$ match
our photometric surface rotational period of 99 days (i.e. we place a
$\delta$-function prior on $\Omega_{\text{out}}$). We place a uniform prior on
$\Omega_{\text{in}}$ centered on this value, extending $0.016\ \mu$Hz in both
directions — again, so as not to favour either sense of the rotational shear a
priori. In this fashion, we recover a conditional posterior distribution for
the scenario where the outer rotation rate of the convective envelope matches
the surface. We show the results of this procedure in Fig. C3a using the
orange points, contours, and histograms. Again, the maximum-likelihood point
estimate under this scenario is more likely than that of solid-body rotation,
with a likelihood-ratio test statistic of $\Lambda=3.213$. While this is a
lesser improvement than the unconstrained two-zone model, such an improvement
in likelihood comes with fewer additional free parameters (only 2), yielding a
lower test probability of $p=0.201$.

Under this more restrictive scenario, the inferred conditional posterior
distribution on $\Omega_{\text{in}}$ can now be seen to favour differential
rotation in the sense of $\Omega$ increasing outwards. This can also be seen
when we examine the implied conditional distribution over the rotational
profiles themselves, shown in Fig. C3c, where sampled profiles which increase
outwards are shown in orange, and those which do not are shown in gray: the
latter can be seen to be exceedingly rare. However, we see also that the
imposition of this hard prior on $\Omega_{\text{out}}$ also requires
$t_{\text{shear}}$ to be situated much farther away from the surface, as can
be seen from the strong peak in its marginal distribution in Fig. C3a, at
roughly $t/T\sim 0.1$ (i.e. $r/R\sim 0.3$ in our best-fitting stellar
structure), as opposed to naturally producing a near-surface shear layer as we
originally posited. This does not, however, contraindicate the existence of
such a near-surface shear layer, as we have little a priori justification for
adopting such a $\delta$-function prior on $\Omega_{\text{out}}$ in the first
place.

We compare these rotational constraints against those from photometry and
spectroscopy in Fig. 8. While both of our direct measurements for the surface
rotation rate sit slightly higher than our unconstrained regularised
predictions for the surface rotation rate from asteroseismology (blue shaded
region), we note that the discrepancy between the two direct measurements is
greater than between either of the two, and our asteroseismic prediction. As
we discuss above, an ad-hoc combination of anti-solar latitudinal differential
rotation and high-latitude spotting could bring the equatorial surface
rotation rate implied by photometry into agreement with our asteroseismic
measurements of the bulk rotation rate in the envelope. Conversely, in the
more likely scenario of low-latitude features and solar-like latitudinal
differential rotation, our photometric value would be more representative of
an equatorial rotation rate than the value produced by Doppler line-
broadening, which is integrated over the visible disk.

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