11institutetext: Department of Astrophysics-AIM, University of Paris-Saclay
and University of Paris, CEA, CNRS, Gif-sur-Yvette Cedex 91191, France
11email<EMAIL_ADDRESS>
# Accounting for differential rotation in calculations of the Sun’s angular
momentum-loss rate

A. J. Finley 11 A. S. Brun 11

(Received December 30, 2022; accepted February 24, 2023)

###### Abstract

Context. Sun-like stars shed angular momentum due to the presence of
magnetised stellar winds. Magnetohydrodynamic models have been successful in
exploring the dependence of this “wind-braking torque” on various stellar
properties, however the influence of surface differential rotation is largely
unexplored. As the wind-braking torque depends on the rotation rate of the
escaping wind, the inclusion of differential rotation should effectively
modulate the angular momentum-loss rate based on the latitudinal variation of
wind source regions.

Aims. Here we aim to quantify the influence of surface differential rotation
on the angular momentum-loss rate of the Sun, in comparison to the typical
assumption of solid-body rotation.

Methods. To do this, we exploit the dependence of the wind-braking torque on
the effective rotation rate of the coronal magnetic field, which is known to
be vitally important in magnetohydrodynamic models. This quantity is evaluated
by tracing field lines through a Potential Field Source Surface (PFSS) model,
driven by ADAPT-GONG magnetograms. The surface rotation rates of the open
magnetic field lines are then used to construct an open-flux weighted rotation
rate, from which the influence on the wind-braking torque can be estimated.

Results. During solar minima, the rotation rate of the corona decreases with
respect to the typical solid-body rate (the Carrington rotation period is 25.4
days), as the sources of the solar wind are confined towards the slowly-
rotating poles. With increasing activity, more solar wind emerges from the
Sun’s active latitudes which enforces a Carrington-like rotation. Coronal
rotation often displays a north-south asymmetry driven by differences in
active region emergence rates (and consequently latitudinal connectivity) in
each hemisphere.

Conclusions. The effect of differential rotation on the Sun’s current wind-
braking torque is limited. The solar wind-braking torque is $\sim 10-15\%$
lower during solar minimum, (compared with the typical solid body rate), and a
few percent larger during solar maximum (as some field lines connect to more
rapidly rotating equatorial latitudes). For more rapidly-rotating Sun-like
stars, differential rotation may play a more significant role, depending on
the configuration of the large-scale magnetic field.

###### Key Words.:

Solar Wind – Solar Atmosphere – Solar Rotation

##  1 Introduction

The rotation periods of Sun-like stars slow systematically throughout the
main-sequence (Skumanich, 1972), which can in some cases be used to determine
the age of a star, or a population of stars (a technique known as
“Gyrochronology”; see Barnes, 2007). This is a consequence of magnetised
stellar wind-braking, which enables the relatively feeble mass-loss rates of
Sun-like stars to carry away significant amounts of angular momentum
(Schatzman, 1962; Weber & Davis, 1967; Mestel, 1984; Kawaler, 1988). The wind-
braking torque is often described as,

$\tau=\dot{M}\Omega_{*}\langle R_{A}\rangle^{2},$ (1)

where $\dot{M}$ is the mass-loss rate of the wind, $\Omega_{*}$ is the
rotation rate of the star, and $\langle R_{A}\rangle$ is the effective Alfvén
radius of the wind, which essentially measures how far the magnetic field can
exert a torque on the wind plasma before it becomes super-Alfvénic and
effectively ‘disconnected’ from the star (see studies of Réville et al.,
2015a; Finley & Matt, 2018).

The evolution of magnetism in Sun-like and low-mass stars, is largely
constrained by systematic studies (e.g. See et al., 2019b, and references
therein) using the Zeeman broadening (see review of Reiners, 2012) and Zeeman-
Doppler imaging (Semel, 1989; Donati et al., 2007) techniques. As is the
evolution of their rotation periods, recovered from their long-term
photometric variability (Curtis et al., 2019; Santos et al., 2021; Rampalli et
al., 2021). There are also a growing number of measured mass-loss rates
derived from astrospheric Lyman-alpha absorption (Wood et al., 2021),
slingshot prominences (Jardine & Collier Cameron, 2019), and planetary
transits (Vidotto & Bourrier, 2017). Though these observations are currently
unable to fully-constrain the evolution of stellar mass-loss rates during the
main sequence (Ó Fionnagáin & Vidotto, 2018), and subsequently the
effectiveness of stellar wind-braking (Brown, 2014; Matt et al., 2015;
Gondoin, 2017; Garraffo et al., 2018; Breimann et al., 2021).

Studying the Sun directly bypasses many observational issues, given that the
solar wind angular momentum flux can be measured in-situ (Lazarus & Goldstein,
1971; Pizzo et al., 1983; Marsch & Richter, 1984; Finley et al., 2019b).
However the scale of the solar wind and the locality of the in-situ
measurements often leads to issues of interpretation when computing a global
wind-braking torque. In addition, all modern observations of the solar wind
have been taken during a single epoch of its main-sequence lifetime (see the
review of Vidotto, 2021), whereas the typical wind-braking timescales for a
star like the Sun is 10 - 100 Myrs. Using proxies for solar activity stored in
natural archives, the longest reconstructions of solar activity are of the
order of 10,000 years (Beer et al., 1998; Usoskin, 2017; Wu et al., 2018).
From this, the likely variations of the wind-braking torque can be inferred
(Finley et al., 2019a), though this is still an order of magnitude away from
being sensitive to the braking timescale of the Sun. It has also been
suggested that the wind-braking of Sun-like stars begins to decrease
significantly around the age of the Sun (van Saders et al., 2016; Metcalfe et
al., 2016; Booth et al., 2017). Evidence for this has begun to grow thanks to
new asteroseismic observations (Hall et al., 2021), and models of the wind-
braking torque for stars crossing the so called transition (Metcalfe &
Egeland, 2019; Metcalfe et al., 2022).

Despite the difficulties in its interpretation, a reliable assessment of the
Sun’s wind-braking torque has the potential to provide insight on the Sun’s
evolution and that of other Sun-like stars. In order to achieve this, there is
a need to reconcile the wind-braking torques calculated using
magnetohydrodynamic (MHD) models (Réville & Brun, 2017; Finley et al., 2019c),
and those calculated from in-situ measurements (e.g. Finley et al., 2019b),
which to-date disagree by a factor of a few. This may relate to the ‘open flux
problem’ (Linker et al., 2017; Riley et al., 2019), in which the
interplanetary magnetic field modelled by extrapolation from the observed
photospheric magnetic field is systematically weaker than observed in-situ.
Models that reproduce the observed value of the open magnetic flux in the
solar wind tend to have better agreement with measurements of the in-situ
angular momentum flux at 1au. Yet the angular momentum flux of the solar wind
in the near-Sun environment, measured by Parker Solar Probe, has been shown to
vary significantly from all model predictions, with tangential flows as strong
as $\pm 50$ km/s (Kasper et al., 2019).

Interestingly, when these large scale variations in the solar wind angular
momentum flux are averaged over a solar rotation, they produce a wind-braking
torque similar to that of the MHD models (as shown for the first two
encounters of Parker Solar Probe by Finley et al., 2020). This suggests that
the structure in the solar wind angular momentum flux develops in the low to
middle corona (1 - 20 solar radii), either through the interaction of the
magnetic field with rotation, or due to the interaction between neighbouring
solar wind streams. An interaction-based mechanism for angular momentum
redistribution appears to be supported by the prevalence of the fast solar
wind carrying a negative (deflected) angular momentum flux, and the slow wind
containing the more dominant net positive angular momentum flux (Finley et
al., 2021; Verscharen et al., 2021), i.e. removing angular momentum from the
Sun. However the travel time from the solar surface to Parker Solar Probe
(during encounters) is seemingly too short for wind-stream interactions to
fully develop, perhaps indicating that the development of structure in the
near-Sun environment is linked to the rotational state of the corona. Coronal
rotation is also integral to the use of ballistic back-mapping for tracking
solar wind plasma back to its source (e.g. Macneil et al., 2022).

This study makes a simplified assessment of the impact of differential
rotation at the base of the solar wind on the resulting wind-braking torque.
This is a first step towards future works examining the influence of more
complex coronal rotation on the solar wind angular momentum flux. Section 2
sets out the data and methodology of the study, in which the Potential Field
Source Surface (PFSS) model is used to rapidly realise the connectivity of the
solar corona from a series of ADAPT-GONG magnetograms spanning one solar
activity cycle (2007-2022). Section 3 presents the results of the inclusion of
differential rotation in the calculation of the Sun’s wind-braking torque.
Finally, Section 4 puts our findings in context with current Heliospheric
missions, and the winds of other Sun-like stars.

##  2 Data and methodology

###  2.1 Rotation rate at the ‘coronal base’

Figure 1: Rotation rate versus heliographic latitude. A typical solar differential rotation pattern is plotted with a solid black line. The rotation rate at $\theta\approx 26.5^{\circ}$, otherwise referred to as the Carrington rotation rate (of 25.4 days or $\sim 456$ nHz), is indicated with a blue dashed horizontal line. A less-extreme differential rotation pattern, taken from fitting the apparent motion of coronal holes in AIA-193A synoptic images (see Appendix A), is plotted with a red dotted line. At the top and bottom of the figure, the latitudinal distribution of the source regions of the open magnetic field during the maximum activity and minimum activity periods of solar cycle 24, respectively, are indicated (further explored in Figure 4). During periods of low activity, the sources of the open magnetic field are confined to the slowly rotating poles. In periods of higher activity, the open field emerges more frequently from low-latitude features. During solar cycle 24, a north-south asymmetry is also observed during solar maximum. Table 1: Rotation Rate Parameters. Profile | $\Omega_{eq}$ | $\alpha_{2}$ | $\alpha_{4}$ | Source
---|---|---|---|---
| [nHz] | [nHz] | [nHz] |
Surface | 472.6 | -73.9 | -52.1 | Snodgrass (1983)
Coronal Hole | 463.0 | -26.7 | -33.1 | Appendix A
Carrington | 455.7 | - | - | —

The rotation of the solar surface is typically written in the form,

$\Omega_{*}(\theta)=\Omega_{eq}+\alpha_{2}\cos^{2}\theta+\alpha_{4}\cos^{4}\theta,$
(2)

where $\Omega_{eq}$ is the equatorial rotation rate, and the values of
$\alpha_{2}$ and $\alpha_{4}$ describe the north-south symmetric differential
rotation profile. Typical values describing the Sun’s surface rotation rate
are given in Table 1, taken from Snodgrass (1983). Many studies have attempted
to constrain the Sun’s differential rotation profile (Newton & Nunn, 1951;
Wilcox & Howard, 1970; Howard et al., 1984; Beck, 2000; Lamb, 2017; Beljan et
al., 2017; Jha et al., 2021), however the profile from Snodgrass (1983)
remains representative and is still frequently used in the literature. A
schematic of this rotation rate profile versus latitude is shown in Figure 1.
Solar-like differential rotation has the equatorial plasma (and features
embedded there) rotating faster than the polar regions. The equator rotates
once around every 24.5 days, whereas the poles rotate once around every 33.4
days. Some strong magnetic features, however, appear to rotate around every
25.4 days (otherwise referred to as the Carrington rotation period). Whether
this relates to the anchoring of magnetic field in the interior (Gilman, 1983;
Miesch et al., 2008; Brun et al., 2004; Nelson et al., 2013; Dikpati et al.,
2021; Käpylä, 2022; Brun et al., 2022), or the role of the near surface shear
layer in sculpting the toroidal flux before emergence (as discussed in
Brandenburg, 2005), is unclear. Generally, magnetic features at the top of the
convection zone are still subject to differential rotation, but this can be
less that would be expected from the observed rate at the photosphere (see
Gigolashvili et al., 2013), and depends on their field strength and surface
area (Imada & Fujiyama, 2018). For some strong active regions, their observed
shearing can be entirely independent of the global differential rotation
pattern (Yan et al., 2018).

Another diagnostic of rotation is the evolution of coronal holes in extreme
ultraviolet (EUV) imagery, which appear dark as they are losing energy/mass
directly to the (fast) solar wind. Previous authors have derived rotation
rates from coronal holes in EUV (see Heinemann et al., 2018, for a study
combining remote-sensing and in-situ observations). Again, these regions often
appear to be less influenced by the surface differential rotation (Timothy et
al., 1975; Insley et al., 1995), with some evidence for nearly rigid rotation
in the low corona (Hiremath & Hegde, 2013). Authors performing statistical
studies of coronal hole rotation generally find a reduced amplitude of
differential rotation (Bagashvili et al., 2017; Oghrapishvili et al., 2018).
However it is unclear how these techniques are influenced by the latitudinal
distribution of coronal holes, their appearance, evolution, and decay
timescales, coupled with our limited ability to track them. In Appendix A, a
retrieval of the rotation profile from the apparent deformation of trans-
equatorial coronal holes is shown through the comparison of multiple synoptic
AIA-193$\AA$ charts. By maximising the coronal hole overlap from chart to
chart, a reduced amplitude of differential rotation is recovered in each case.
The average fit values are given in Table 1, and are comparable to those found
in Bagashvili et al. (2017) and Oghrapishvili et al. (2018). As the solar wind
is accelerated over a large range of heights in the solar atmosphere, it is
unclear what rotation rate should apply to the base of the wind (otherwise
referred to as the coronal base). Therefore in Section 3, the assessment of
the impact of differential rotation on calculations of the wind-torque is
performed using both the surface and coronal hole motivated differential
rotation profiles.

###  2.2 Magnetic field extrapolation

The use of PFSS models to infer the source locations of the solar wind
(otherwise referred to as ‘connectivity’) has become wide-spread due to the
efficiency and simplicity of the model (Altschuler & Newkirk, 1969; Schrijver
& DeRosa, 2003). Despite only taking information from the radial magnetic
field at the surface, and having a single free parameter (the source surface
radius, $R_{ss}$), this model has been shown to work well (Badman et al.,
2020; Panasenco et al., 2020), typically in advance of computing the more
resource-intensive MHD models (see comparison in Riley et al., 2006). For this
study, our aim is to quantify the long-term variation in the field line
mapping to different latitudes throughout the solar cycle. In this case, PFSS
modelling represents a useful and reliable method to achieve this (see similar
work by Stansby et al., 2021). The PFSS magnetic field is constructed based on
the following equations,

$\displaystyle
B_{r}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\alpha_{lm}(r)Y_{lm}(\theta,\phi),$
(3) $\displaystyle
B_{\theta}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\beta_{lm}(r)Z_{lm}(\theta,\phi),$
(4) $\displaystyle
B_{\phi}(r,\theta,\phi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\beta_{lm}(r)X_{lm}(\theta,\phi),$
(5)

where $r$ denotes radial distance from the origin, $\theta$ the latitude from
the rotation pole, $\phi$ the Carrington longitude, and the typical $l$-degree
and $m$-order spherical harmonic functions, using the legendre polynomial
functions $P_{lm}(\cos\theta)$, are,

$\displaystyle Y_{lm}$ $\displaystyle=$ $\displaystyle
c_{lm}P_{lm}(\cos\theta)e^{im\phi},$ (6) $\displaystyle Z_{lm}$
$\displaystyle=$
$\displaystyle\frac{c_{lm}}{l+1}\frac{dP_{lm}(\cos\theta)}{d\theta}e^{im\phi},$
(7) $\displaystyle X_{lm}$ $\displaystyle=$
$\displaystyle\frac{c_{lm}}{l+1}P_{lm}(\cos\theta)\frac{im}{\sin\theta}e^{im\phi},$
(8)

with the normalisation of,

$\displaystyle c_{lm}=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}.$ (9)

In the PFSS model, the coefficients $\alpha_{lm}$ and $\beta_{lm}$ are given
by,

$\displaystyle\alpha_{lm}(r)=\epsilon_{lm}\frac{l(R_{*}/R_{ss})^{2l+1}(r/R_{*})^{l-1}+(l+1)(r/R_{*})^{-(l+2)}}{l(R_{*}/R_{ss})^{2l+1}+(l+1)},$
(10)
$\displaystyle\beta_{lm}(r)=(l+1)\epsilon_{lm}\frac{(R_{*}/R_{ss})^{2l+1}(r/R_{*})^{l-1}+(r/R_{*})^{-(l+2)}}{l(R_{*}/R_{ss})^{2l+1}+(l+1)},$
(11)

where $\epsilon_{lm}$ represent the strength of each spherical harmonic mode.
The $\epsilon_{lm}$ coefficients are extracted from the input magnetogram of
the photospheric magnetic field by evaluating111To compute these coefficients
the pySHTOOLS python package is used, which provides access to the Fortran-95
SHTOOLS library.,

$\displaystyle\epsilon_{lm}=\frac{1}{c_{lm}}\int_{\phi}\int_{\theta}B_{r}(\theta,\phi)P_{lm}(\cos\theta)\cos(m\phi)\sin\theta
d\theta d\phi.$ (12)

In this study, the PFSS model is driven by spherical harmonic decomposition of
the ADAPT-GONG magnetograms222Data accessed Feb 2022:
https://nso.edu/data/nisp-data/adapt-maps/ (Arge et al., 2010). These
magnetograms are produced using a combination of data assimilation and forward
modelling, which accounts for the effects of differential rotation, meridional
circulation, and diffusion on older observations (further discussed in
Hickmann et al., 2015). In general, the Sun’s polar magnetic fields are
difficult to capture due to their proximity to the limb and weak line-of-sight
strength. The ADAPT-GONG magnetograms leverage the underlying flux transport
model to reproduce the time evolution of the polar fields based on data
assimilated at lower latitudes. This has been shown to remain consistent with
direct observations of the polar field (Arge et al., 2011), and so reduces the
potential variation in the reconstructed coronal magnetic field structure
caused by the varying visibility of the Sun’s poles. Magnetograms are taken at
a monthly cadence from Jan 2007 to Feb 2022 ($\sim 200$ Carrington Rotations),
always using the first realisation of the magnetogram. To reduce computational
cost, the magnetic field is reconstructed up to a spherical harmonic degree of
$l_{max}\approx 30$ using $R_{ss}=2.5R_{\odot}$ on an equally-spaced grid of
$r\times\theta\times\phi$ resolution of $12\times 92\times 184$ points. This
set-up is found to reliably recovers structures based on the resolution of the
ADAPT-GONG magnetogram inputs.

###  2.3 Effective rotation rate

Figure 2: Schematic depiction of the methodology used in this study. The
surface rotation rate is extrapolated into the corona via a PFSS model driven
by ADAPT-GONG magnetograms. In three dimensions this produces a longitude-
latitude grid of rotation rates at the height of the source surface
($R_{ss}=2.5R_{\odot}$), the height at which the coronal magnetic field opens
into the solar wind. The open flux weighted rotation rate is then calculated
from the longitude-latitude grid of rotation rates, however the angular
momentum flux in the solar wind is expected to vary with distance from the
rotation axis. Therefore a factor of $\sin\theta$ is introduced into open flux
weighted averaging process, giving additional weight to the solar wind close
to the equator (represented by the magnitude of the black arrows).  Figure 3:
Summary of five models from different phases of solar cycle 24. The first
column shows 3D renderings of the PFSS models, with open magnetic field lines
coloured by surface rotation rate (see Figure 1). Closed magnetic field lines
are shown in grey. The solar surface is coloured with red and blue
representing the radial magnetic field used in each PFSS model, and the
location of the Heliospheric Current Sheet (HCS) is indicated in black at the
source surface. The center column displays the same information projected on a
latitude-longitude grid. The final column shows the effective rotation rate of
field lines at the source surface ($R_{ss}=2.5R_{\odot}$). These values are
acquired by tracing field lines down to the surface, and returning the value
of the surface rotation rate (this is the same as for the colour on the field
line renderings). The mean value of the effective rotation rate
$\langle\Omega_{wind}\rangle$ is listed with each model (the Carrington rate
is $\sim 456$ nHz).

In ideal MHD models of solar/stellar wind, the footpoints of the magnetic
field should remain anchored to the surface (or inner boundary condition).
This requires, for a perfectly conducting rigidly rotating boundary with a
frozen-in magnetic field, that the electric field at the surface in the
rotating frame be zero. In the case of a steady state solution, i.e.
axisymmetric or rigid rotation, this condition produces a scalar quantity
which is constant along magnetic field lines, the effective rotation rate
(Mestel, 1968; Sakurai, 1985). This quantity in CGS units is,

$\Omega_{eff}=\frac{1}{r\sin\theta}\bigg{(}v_{\phi}-\frac{BB_{\phi}}{4\pi\rho
v}\bigg{)},$ (13)

where $r$ is the radius, $\theta$ is the latitude from the rotation pole, $v$
is the fluid velocity, $B$ is the magnetic field vector, $\rho$ is the fluid
density, and quantities with subscript $\phi$ are taken in the azimuthal
direction. This quantity is typically set at the lower boundary to enforce a
given surface rotation rate $\Omega_{eff}=\Omega_{*}$ (e.g. Zanni & Ferreira,
2009).

In the recent work of Ireland et al. (2022), the value of $\Omega_{eff}$ was
allowed to vary in latitude in order to model the influence of differential
rotation on the wind-braking torques from their 2.5D stellar wind models (see
also Pinto et al., 2021). This has the effect of anchoring the field lines at
different latitudes to different rotation rates. As this model is
axisymmetric, the effective rotation rate remained constant along magnetic
field lines (the conservation of this quantity is also used to validate the
performance of the numerical methods). The authors tested varying degrees of
solar-like differential rotation, whilst also altering the stellar magnetic
field strength. The resulting wind-braking torques were shown to be well-
behaved, and described by a correction factor that accounted for the implicit
change in the rotation rate $\langle\Omega_{wind}\rangle$ of the simulation in
comparison to solid-body $\Omega_{*}$. As the stellar magnetic field in this
case was dipolar, it was shown that the rotation rate of the wind scaled with
the latitude of the last open magnetic field line.

In the case of a 3D non-axisymmetric magnetic field with differential
rotation, shearing in the corona creates a time-dependant solution which is
sensitive to the degree of non-axisymmetry in the magnetic field, and the
contrast in rotation rate between the footpoints of closed coronal loops. A
steady-state solution is unlikely to be reached in this case. Irrespective of
this, the rotation rate of the open magnetic field will likely still be
strongly influenced by the anchoring speed of the footpoints. Given the
relatively slow rotation of the Sun, in this study these interactions are
assumed to be weak, such that the effective rotation rate is conserved along
each field line. This allows for the effective rotation rate to be propagated
into the corona.

Figure 4: Histogram of the source latitudes of magnetic field lines traced
down from the source surface for each magnetogram in our time-series. The top
panel indicates, for each model, the percentage of field lines that connect to
a given latitude (bin width $\sim 2^{\circ}$) from a homogeneous sampling of
the source surface. The bottom panel blends this information with that of the
rotation rate at those latitudes, i.e. the more vivid the colours the larger
the fraction of field lines connecting to that latitude (as in the panel
above). Arrows indicate the major pole-ward surges of magnetic flux during
solar cycle 24. The snapshots shown in Figure 3 are identified with green
vertical dot-dashed lines in both panels. The mean latitude of connectivity
(including a factor of $\sin\theta$ to match $\langle\Omega_{wind}\rangle$) is
shown with red solid lines. The same calculation is repeated for the northern
and southern hemispheres individually, plotted in black dashed and dotted
lines respectively; highlighting the degree of asymmetry.

In addition to the significance of the field line footpoint rotation, field
lines closer to the equator will carry a larger angular momentum-flux (e.g.
Keppens & Goedbloed, 1999), due to the geometrical lever arm from the rotation
axis. Thus the rotation rate of field lines nearer the equator will have a
stronger influence on the mean rotation rate that is needed to describe the
wind-braking torque with equation (1). The mean rotation rate of the wind, is
therefore calculated via an open-flux weighted average in the magnetically-
open corona including a $\sin\theta$ dependence,

$\langle\Omega_{wind}\rangle=\frac{\oint_{A}\Omega(r,\theta)\sin\theta|{\boldsymbol{B}}\cdot
d{\boldsymbol{A}}|}{\oint_{A}\sin\theta|{\boldsymbol{B}}\cdot
d{\boldsymbol{A}}|},$ (14)

where $\Omega(r,\theta)=\Omega_{*}(R_{*},\theta_{*})$ is the value of the
surface differential rotation rate mapped along the magnetic field from
$(R_{*},\theta_{*})$ to $(r,\theta)$, and the closed integral over the area
$A$ of the magnetic field vector $\boldsymbol{B}$ returns the unsigned
magnetic flux in the wind. The radius $r$ should therefore be larger than the
last closed magnetic field loop. The dependence of the angular momentum flux
on latitude from the rotation axis is further discussed in Finley et al.
(2019b).

A schematic depiction of this calculation is shown in Figure 2. The technique
described here could also be used to correct the wind-braking torques from MHD
wind simulations performed with solid-body rotation. The form of
$\langle\Omega_{wind}\rangle$ is motivated by insights from the scaling of the
wind-braking torque in Ireland et al. (2022). It is left for future work to
truly validate this relation. In addition, throughout this work, the effective
rotation rate of the magnetic field lines and that of the solar wind are
assumed to be interchangeable, however this is an oversimplification of
equation (13). The development of stress in the magnetic field can also modify
the rotation rate of the wind. This does not affect the calculation of
$\langle\Omega_{wind}\rangle$, but caution should be used in the
interpretation of the extrapolated coronal rotation rates. To self-
consistently model coronal rotation, time-dependent simulations which are
continuously driven, such as magnetofrictional models (e.g. Yeates, 2013;
Hoeksema et al., 2020), may be better suited (though more computationally
expensive).

##  3 Results

###  3.1 Global connectivity and rotation rate

Figure 3 displays PFSS models for five different ADAPT-GONG magnetograms
(2009-06-01, 2010-10-01, 2014-01-01, 2017-10-01, and 2020-01-01), each
representing a different phase of solar cycle 24. 3D renderings are shown in
the first column with magnetic field lines coloured by the surface rotation
rate, following equation (2). Closed field line are coloured grey. This
information is re-projected onto a latitude-longitude grid in the center
column. The final column shows a latitude-longitude map of the effective
rotation rate at $2.5R_{\odot}$ created by tracing field lines down from an
equally-spaced grid of $48\times 96$ points at the source surface ($r=R_{ss}$)
to sample the corresponding surface rotation rate (see Figure 2). From this
map, equation (14) is evaluated and the value of $\langle\Omega_{wind}\rangle$
is noted with each model. This calculation is repeated for each magnetogram in
our sample.

Figure 3 shows some clear trends during the solar cycle. At minima of solar
activity (top and bottom rows), the open magnetic field emerges primarily from
the slowly rotating polar coronal holes. There are some additional
contributions of open magnetic field from small active regions or equator-ward
coronal holes which create pockets of fast rotation near to the Heliospheric
Current Sheet (HCS). In more active phases (middle rows), the HCS becomes
increasingly warped by underlying active regions, and equatorial coronal
holes. These low-latitude sources of the solar wind increases the proportion
of fast rotation in the corona. At solar maximum, the dipole axis is
completely tilted, closing off most of the field at the poles, leading to
faster rotation throughout the entire corona. There are subtle differences
between the rising and decay phases of the cycle, with surges of magnetic flux
towards the poles during the decay phase leading to more extended polar
coronal holes in latitude and accordingly coronal rotation rates that are
slightly elevated with respect to the rising phase. Throughout the cycle, the
degree of warping of the HCS appears an indirect indicator of the rotational
state of the corona, with deviation from a perfectly flat HCS in the equator
most-likely due to source regions at low-latitude, anchored at more rapidly
rotating latitudes.

The variation of solar wind source regions during solar cycle 24 is shown more
clearly in Figure 4. Histograms of the open magnetic field footpoint latitudes
are shown for each model in our time-series of $\sim 200$ magnetograms. Open
field lines are traced down from the source surface with seeds distributed
homogeneously over all latitudes and longitudes. Values in the top panel are
given as a percentage of field lines from the total seeds that connected to a
given latitude bin (width $\sim 2^{\circ}$), irrespective of longitude. During
solar minimum, the open magnetic field is mostly confined to the rotational
poles, which is in contrast to solar maximum where the majority of the open
magnetic field emerges from lower-latitudes. The bottom panel of Figure 4 is
similar, but with result coloured by the surface rotation rate. The presence
of multiple pole-ward rushes in magnetic flux is identifiable in the field
line connectivity (highlighted with dashed arrows).

The value of $\langle\Omega_{wind}\rangle$ calculated from equation (14) using
the surface rotation rate and the coronal hole rotation rate is shown in
Figure 5 versus time, along with the sunspot number. During solar minimum,
$\langle\Omega_{wind}\rangle$ is smaller than the Carrington rate, however
does not reach the polar rotation rate due to the volume-filling patches of
low-latitude connectivity. These regions are close to the equator and are
therefore more heavily-weighted by the additional $\sin\theta$ dependence in
equation (14). With increasing solar activity, the value of
$\langle\Omega_{wind}\rangle$ increases to be close to, but slightly larger
than, the Carrington rotation rate, as more open magnetic field is emerging
from the active latitudes. Naturally, this leads to a correlation between
$\langle\Omega_{wind}\rangle$ and the amount of open magnetic flux in the
wind. As typically an increase in open magnetic flux results from increased
flux emergence around the active latitude (which are evidently rotating faster
than the poles).

Figure 5: Effective rotation rate versus solar cycle, calculated with equation
(14). Solid black line represents the ADAPT-GONG magnetograms using
$R_{ss}=2.5R_{\odot}$, and the observed surface rotation rate. Dashed lines
indicate the equatorial, polar and Carrington rotation rate. The solid grey
line instead uses the less-extreme coronal hole rotation profile (see Appendix
A). PFSS models shown in Figure 3 are highlighted with black circles. The
daily, monthly, and monthly-smoothed sunspot number from the Sunspot Index and
Long-term Solar Observations (SILSO) are displayed in the background of the
figure with a solid green lines of varying opacity. Figure 6: Same as Figure
5, but now showing the expected change in wind-braking torque between solid
body (SB) and differentially rotating (DR) models on the vertical axis.
Figure 7: Azimuthally-averaged rotation rates from the PFSS models shown in
Figure 3. Averaging is peformed on the open field regions, grey regions
indicate closed field at all longitudes (hence no value returned). The limited
number of open magnetic field lines over the north pole during the ‘Activity
Maximum’ model, lead to some oddity in the north-most latitude bin that should
be disregarded. The lower panels show the latitudinal profiles coloured by
radial distance from the surface (yellow being the furthest). The rising and
declining phases show a clear north-south asymmetry which is also observed in
the systematic differences in $\langle\theta\rangle$ between the two
hemispheres in Figure 4.

The open magnetic flux is also increased by lowering the source surface
height. The results presented so far use a fixed value of the source surface
radius ($R_{ss}=2.5R_{\odot}$), however this value is likely to vary during
the solar cycle (Arden et al., 2014; Pinto et al., 2011; Perri et al., 2018;
Hazra et al., 2021). Our analysis is repeated in Appendix B with source
surface radii of 2 $R_{\odot}$ and 3 $R_{\odot}$, to investigate the
dependence of $\langle\Omega_{wind}\rangle$ on the source surface height (see
Figure B). As expected, the smaller the source surface the more higher-order
magnetic field is opened which increases the open magnetic flux. Once again,
this shifts connectivity towards smaller active regions. During solar minimum,
this means a smaller source surface can more-easily connect to the faster
equator-ward latitudes (increasing $\langle\Omega_{wind}\rangle$ by around
$20$ nHz), with the opposite effect for larger source surfaces (decreasing by
$10$ nHz). This is further detailed in Appendix B.

###  3.2 Solar wind angular momentum-loss rate

MHD models have been used to explore the dependence of the wind-braking torque
on various configurations of the coronal magnetic field, under different
coronal heating scenarios, and rotation rates (Matt et al., 2012; Réville et
al., 2015a; Pantolmos & Matt, 2017; Finley & Matt, 2018; Hazra et al., 2021;
Ireland et al., 2022). In the slowly rotating regime, where centrifugal
effects on the wind acceleration are negligible, the mass-loss rate and Alfvén
radius are unaffected by the inclusion of differential rotation. This results
in a linear dependence between the wind-braking torque and the effective
rotation rate, as in equation (1). Provided that equation (14) is a reasonable
approximation for the rotation rate of the wind, changes to the wind-braking
torque $\tau_{DR}$ are then given by,

$\tau_{DR}=\frac{\langle\Omega_{wind}\rangle}{\Omega_{*}}\tau_{SB},$ (15)

where $\tau_{SB}$ is the wind-braking torque calculated using the solid-body
rotation value $\Omega_{*}$ (which is taken to be the Carrington rotation
rate). For the Sun, $\tau_{SB}$ has been calculated by many authors using a
variety of models and semi-analytic relations. In general, the wind-braking
torque is largest during periods of increased solar activity as the amount of
open magnetic flux in the solar wind is increased. Here, the semi-analytic
relation for the solar wind-braking torque from Finley et al. (2019a) is
adopted. This relation is derived from a parameter study of 2.5D MHD
simulations in Finley & Matt (2017, 2018). The wind-braking torque is given
by,

$\displaystyle\tau_{SB}=(2.3\times
10^{30}[\text{erg}])\bigg{(}\frac{\dot{M}}{1.1\times
10^{12}[\text{g/s}]}\bigg{)}^{0.26}$
$\displaystyle\times\bigg{(}\frac{\phi_{open}}{8.0\times
10^{22}[\text{Mx}]}\bigg{)}^{1.48},$ (16)

where $\dot{M}$ is the solar mass-loss rate, and $\phi_{open}$ is the open
magnetic flux in the solar wind. Both of these variables are estimated from
in-situ measurements of the solar wind from the Wind spacecraft, as done in
Finley et al. (2019a). In-situ measurements from the equatorial solar wind are
averaged on the timescale of a Carrington rotation ($\sim 27$ days as viewed
from Earth), in order to remove longitudinal structures. Latitudinal
variations in the mean mass flux and magnetic flux are assumed to be small at
1au, such that the averaged equatorial values can be used to create global
estimates of $\dot{M}$, and $\phi_{open}$. From these values, the solar wind-
braking torque computed with equation (16) is plotted in Figure 6. Applying
the value of $\langle\Omega_{wind}\rangle/\Omega_{*}$ to the solar wind-
braking torque, produces the corrected torque $\tau_{DR}$ for the surface and
coronal hole profiles (plotted in Figure 6 with red and blue lines
respectively).

The percentage change in the wind-braking torque ($\tau_{DR}/\tau_{SB}\times
100\%$) during the solar cycle varies from 10-15$\%$ during solar minima, to a
few percent at solar maximum (see inset of Figure 6). In times of increased
solar activity, the equatorial solar wind can emerge from sources closer to
the equator than $\theta\approx\pm 26.5^{\circ}$ (e.g. the Carrington rate
latitude). This results in a more rapidly rotating corona than the typical
solid body value, and hence a slightly larger wind-braking torque. However, as
previously discussed, equatorial coronal holes visible in EUV imagery do not
always show this rapid rotation, and so in reality, the equatorial rotation
rate may be closer to that of the coronal hole motivated profile. Using this
rotation profile instead of the typical differential rotation profile reduces
the effect at solar minima (to around 5$\%$) and results in a negligible
change during solar maximum. In either case, the effect of differential
rotation tends to reinforce the pre-existing variation of the Sun’s angular
momentum-loss rate during the solar cycle. With the strongest influence of
differential rotation occurring when the solar wind-braking torque is
smallest, the overall impact on the long-term angular momentum-loss rate is
minimised.

The weak dependence of the wind-braking torque on the Sun’s differential
rotation profile is easily explained by considering the extreme values that
$\langle\Omega_{wind}\rangle$ could take. These being the polar and equatorial
rotation rates, i.e. all the wind rotates either at the slowest or fastest
possible rotation rate. In this case, taking the solid-body rotation rate to
be that of the poles, around 33.4 days, will produce a solar wind-braking
torque that is $\sim 24\%$ smaller than using the Carrington rotation rate.
Similarly, by using the equatorial rotation rate of 24.5 days the wind-braking
torque increases by 4$\%$. Given the small variation in wind-braking torques
between these maximum and minimum values of the effective rotation rate, the
use of solid-body rotation in steady-state MHD models of the solar wind
appears to be justified to first order.

Figure 8: The potential bias from detecting coronal rotation from white light
streamer structures. Panel a) depicts the azimuthally-averaged rotation
profile versus latitude at the source surface ($R_{ss}=2.5R_{\odot}$)
throughout the timeseries of magnetograms. Panel b) charts the average
brightness of scattered white light from streamers at three solar radii
observed by LASCO-C2 onboard Solar and Heliospheric Observatory. Streamers are
confined to the equator near minimum activity and spread to all latitude
during maximum. Given that the apparent motion of white light streamers is
used to measure coronal rotation, this technique may be biased towards
recovering the Carrington rotation rate at most available latitudes.

##  4 Discussion

###  4.1 Asymmetric Rotation of the Corona

The mean latitude of connectivity during the solar cycle (weighted by a factor
of $\sin\theta$ to be consistent with the definition of
$\langle\Omega_{wind}\rangle$) is plotted in Figure 4 with symmetric solid red
lines in each hemisphere. Performing this calculation independently for the
northern and southern hemispheres produces the black dashed and dotted lines
respectively. Deviation from the symmetric solid red lines indicates asymmetry
between the two hemispheres.

As solar cycle 24 progresses, active regions appear in the north and south
following the typical butterfly-pattern (e.g. Hathaway, 2015). At first,
active latitudes in the north are more frequently sources of the solar wind
than in the south, which pulls the mean latitude closer to the equator in the
north than in the south. This leads to an asymmetric increase of coronal
rotation in the northen hemisphere. After a pole-ward surge in the northern
hemisphere (starting in 2013), the connectivity begins to favour the southern
active latitudes. This briefly reverses the asymmetry in the coronal rotation,
producing a faster southern hemisphere. A pole-ward surge in the southern
hemisphere (starting in 2014) then reverses the situation, leaving the
northern active latitudes more frequent connected to the solar wind and
driving-up the mean rotation rate in the northern hemisphere. Balance is
restored at the end of the declining phase of activity with a final pole-ward
surge in the northern hemisphere (starting in 2016) returning the source-
latitude distribution to a near-dipolar configuration. This sequence shows
that changes in the distribution of northern and southern active regions can
drive asymmetry in the resulting coronal rotation rate.

Examining more closely the PFSS models shown in Figure 3, the azimuthally
averaged rotation rate of the open magnetic field is plotted in Figure 7 along
with 1D cuts of rotation at various radial distances (yellow being the source
surface, and darker colours moving down towards the solar surface). Grey
regions indicate completely closed latitudes, i.e when field lines are traced
from all longitudes at this latitude, they are all closed therefore there is
no value to return. The snapshots from activity minima show rotation profiles
that are roughly north-south symmetric, with the exception of the location of
the closed field in grey. These cases are dominated by the slowly rotating
polar sources, with some faster equatorial connectivity associated with small
active regions (visible in Figure 3).

The models of rising (2nd panel) and declining (4th panel) activity both have
a clear north-south asymmetry in rotation, with the corona rotating
systematically slower in the southern hemisphere. As discussed, this is due to
the imbalance of source regions between the two hemispheres. This is clear
from Figure 4, where the histogram shows a higher density of source regions in
the low-latitude north than the south during these snapshots. This imbalance
persists throughout most of the active phase of Cycle 24, except for during
activity maximum in 2014 (in-between the pole-ward surges). Here the situation
is reversed with more sources in the low-latitude south, leading to faster
coronal rotation in the south. During this time, the dipole component of the
Sun’s magnetic field is weak or highly inclined, and so in Figure 7 the closed
regions (in grey) appear over the poles.

###  4.2 Apparent rotation from coronal streamers

Given the wealth of coronal observations in scattered white light, recent
works have begun to reconstruct the rotation of the corona based on the
apparent motion of streamer structures. Most recently, Edwards et al. (2022),
following the methodology of Morgan (2011), who measured the rotation rate of
long-lived streamer structures in LASCO C2 white light images from 2008 -
2020. While there are many local deviations in the measured rotation rate from
the surface rate, which may be of interest in the discussion of angular
momentum transport in the low-corona. The mean coronal rotation rate from this
method often deviates systematically from the surface rotation rate, with a
flatter rotation profile that is closer to the Carrington rate (similar to
that motivated in Appendix A, see Figure 1). However, these measurements come
with challenges in interpretation, as streamers do not form at all latitudes
during the solar cycle (discussed in Morgan, 2011). At solar minimum streamers
are confined to the equator, whereas during solar maximum streamers can be
found at all latitudes (with challenges in accurately reconstructing their
motion over the rotational poles).

From the PFSS modelling performed in this study, the potential bias of using
white light streamers to measure the overall rotation of the corona is
assessed based on the available streamer latitudes during the cycle and the
evolving rotation rate versus latitude of the corona. Figure 8 displays the
azimuthally-averaged rotation rate at the source surface throughout the
timeseries of magnetograms (panel a), along with the averaged scattered white
light emission at three solar radii observed by LASCO C2 onboard the Solar and
Heliospheric Observatory (panel b). The latitudinal variation of the mean
coronal rotation and the streamer structures have an almost identical
morphology. This is not unexpected as both quantities are driven by the
underlying reconfiguration of the Sun’s magnetic field. At solar minima,
coronal rotation is dominated by the slowly rotating flows from the polar
coronal holes and the streamers confined to the equator following the dipolar
configuration of the large-scale magnetic field. With increasing activity the
Sun’s large-scale magnetic field becomes more complex, and so the sources of
the solar wind move to the active latitudes and consequently faster rotating
areas. The evolution of the Sun’s large-scale magnetic field then allows for
streamer structures that traverse a much broader range of latitudes.

The presence of white light streamers is essential for inferring coronal
rotation at a given latitude and time in the cycle. From Figure 8 it is clear
that streamers typically appear at latitudes with faster rotation rates. It
might then be expected that the rotation profiles derived from white light
streamers should systematically differ from the surface rotation profile. In
our model, the rotation rate of coronal streamers is directly linked with the
rotation rates at the source of the streamer structure. In which case, given
that streamer structures are often anchored to the active latitudes, white
light observations are more likely to produce a mean rotation rate that is
flattened towards the Carrington rotation rate. This may explain some of the
findings from these previous works (i.e. Morgan, 2011; Edwards et al., 2022).

Figure 9: Same as the lower panel of Figure 4, but now field lines are traced
down from the source surface only around the equator ($\pm 15$ degrees). The
histogram density is coloured by surface rotation rate.

###  4.3 Equatorial connectivity and rotation rate

With the exception of the Ulysses spacecraft (an overview of observations is
presented in McComas et al., 2008), contemporary in-situ measurements of the
solar wind are limited to the ecliptic plane of the solar system (this will
change in future as ESA/NASA’s Solar Orbiter mission begins higher inclination
orbits in 2025). When trying to connect the theoretical results presented here
to in-situ measurements, it is important to consider the sources of equatorial
solar wind and their associated rotation rates. Figure 9 displays a similar
analysis to that of Figure 4, but now with only field lines from $\pm 15$
degrees of the equator at the source surface being traced down to their
sources. The histogram is coloured by the rotation rate at the surface,
assuming the rotation profile from Snodgrass (1983).

Low-latitude sources of the solar wind are present throughout the entire solar
cycle. In-situ measurements may then be expected to find slightly larger
tangential speeds (no more than a few km/s). The tangential speeds measured
in-situ depend strongly on the magnetic stresses that support the rotation of
coronal plasma out to larger distances than the source surface (the tangential
speed is equivalent to the rotation rate multiplied by the radial distance).
In this regard, most of these low-latitude sources will have strong magnetic
fields, and so these features could play a more significant role than
indicated by the PFSS modelling in this study, enforcing their rotation higher
up in the solar corona. In which case, the tangential speed of the solar wind
may be more strongly influenced (rigid-rotation up to $1.5R_{\odot}$ results
in a $\sim 7\%$ increase in the Sun’s wind-braking torque). Measurements of
the middle-corona are needed in order to evaluate impact of strong active
regions on coronal rotation (investigations into this area have become more
frequent West et al., 2022; Chitta et al., 2022).

Large tangential solar wind speeds are frequently measured by Parker Solar
Probe in the near-Sun environment (up to 50km/s Kasper et al., 2019), which
indicates that angular momentum transport in the corona is more complicated
than MHD models predicts (the expectation is nearer to 5-10km/s; Réville et
al., 2020). Finley et al. (2020) found that these tangential flows could be
made consistent with the expected angular momentum-loss rate of the Sun if
averaged together with regions of slow and retrograde rotation, also detected
in the equatorial solar wind by Parker Solar Probe. The model presented here
does not allow for absolute retrograde rotation in the corona, given that the
surface rotation profile is always prograde. Stream interactions between fast
and slow solar wind sources are frequently employed to explain the existence
of strong positive and negative tangential flow deflections in the solar wind
at 1au (Yermolaev et al., 2018). However, with Parker Solar Probe observing
these flows so close to the Sun, it seems unlikely that wind-interactions will
have had enough time to develop. To explain this, a strong contrast in the
effective rotation rate between fast and slow solar wind streams (anchored at
different latitudes) may be required to increase the frequency of collisions
during the wind acceleration process.

###  4.4 Potential impacts for other Sun-like stars

In this study, the Sun’s current differential rotation has been shown to have
little influence on its wind-braking torque when averaged over a solar cycle.
This results from the configuration of the Sun’s magnetic field and its
relatively weak differential rotation profile. During the lifetime of the Sun,
however, differential rotation could have had a larger impact, depending on
its initial rotation rate and subsequent degree of differential rotation
thereafter. In both observations and numerical simulation, the amplitude of
differential rotation $\Delta\Omega_{*}$ scales roughly with rotation rate
$\Omega_{*}$ to a power $n$ (e.g. Barnes et al., 2005). Taking $n=0.46$ (from
the global MHD simulations of Brun et al., 2022), and assuming a self-similar
solar differential rotation profile, for a young rapidly rotating Sun-like
star with $\Omega_{*}/\Omega_{\odot}=5$ ($\Omega_{*}$ = 2278 nHz), the
differential rotation contrast would be $5^{0.46}=2.1$ times stronger than
that of the current Sun ($\Delta\Omega_{*}\approx 265nHz$). With a similar
variation in wind source latitudes as the Sun, the value of
$\langle\Omega_{wind}\rangle$ would vary from $\sim 1842$ nHz to $\sim 2326$
nHz, meaning the wind-braking torque could decrease from the solid body value
by $20\%$ during a solar minimum-like field configuration or increase by $2\%$
during a solar maximum-like configuration.

The scaling of stellar magnetic field strengths, and their large-scale
magnetic field components continues to be constrained by Zeeman-Doppler
imaging (Vidotto et al., 2014; See et al., 2019a). Given that very young Sun-
like stars tend to possess stronger, and more dipolar field configurations,
$\langle\Omega_{wind}\rangle$ may be systematically lower than $\Omega_{*}$
during the early main sequence. This can be further investigated by examining
the differential rotation self-consistently produced in 3D MHD simulations of
stellar interiors (e.g. Brun et al., 2022). More precision than this requires
additional information on the heating and structuring of the stellar coronae,
which is needed to determine the latitudinal distribution of the stellar wind
sources, and thus the true impact of surface differential rotation.

For other low-mass stars (0.2 to 1.3 solar masses), it is possible that the
source latitudes of their stellar winds could produce effective rotation rates
that differ significantly from the rotation period of the star recovered from
their spot-modulated light curves. Asteroseismic inversions from Benomar et
al. (2018) recover latitudinal differential rotation profiles for Sun-like
stars which are much stronger than the solar case. The mean amplitude of the
differential rotation in their sample is six times stronger than that of the
current Sun, which means that variation of the stellar wind source latitudes
would have a much more pronounced effect on the wind-braking torque.
Interestingly, the strength of differential rotation has been observed to vary
with spectral type (Reiners, 2006), and in some cases shown to be
significantly enhanced in F-type stars (Marsden et al., 2006). This may lead
to a systematic divergence between physics-based rotation-evolution models and
the observed rotation period distributions.

In old, slowly rotating main sequence stars, differential rotation may have an
interesting effect during the (as of yet undetected) transition to anti-solar
differential rotation (the search for anti-solar rotators is detailed in Noraz
et al., 2022a). As the stellar magnetic field weakens and potentially loses
its cyclic nature (Brun et al., 2022; Noraz et al., 2022b; Käpylä, 2022), the
footpoints of the resulting axisymmetric dipole would be confined to the more
rapidly rotating poles, resisting the decrease in wind-braking torque with
age. This would challenge the current hypothesis of weakened magnetic braking
(van Saders et al., 2016), with angular momentum being more efficiently lost
due to the favourable configuration of the stellar magnetic field and
differential rotation pattern. It seems more likely that something prevents
Sun-like stars from entering this configuration, or that a decreasing mass-
loss rate could counteract this effect, producing the expected weakening of
the wind-braking torque (discussed in Metcalfe et al., 2022).

The recent study of Tokuno et al. (2022) investigates the influence of
differential rotation on the rotation-evolution of Sun-like stars. Their model
allows for the amplitude of the differential rotation at the stellar surface
to evolve in time, as a function of the Rossby number (rotation period
normalised by the convective turnover timescale). The effective rotation rate
used in the wind-braking torque of Matt et al. (2015) is then taken from a
low-latitude region, with the authors finding a weakening of the wind-braking
torque at late-ages as this rotation rate is smaller than the underlying
solid-body value. This study shows that the effective rotation rate used in
the wind-braking torque depends on the dominant source latitudes of the
stellar wind (during the wind-braking timescale), which is largely governed by
the stellar magnetic field.

##  5 Conclusion

This study provides evidence to suggest that the current Sun’s mean angular
momentum-loss rate is not strongly influenced by the observed surface
differential rotation (with respect to adopting a solid-body rotation rate).
The equatorial solar wind, in which the majority of the solar wind angular
momentum flux is transported, is supplied by a range of low-latitude sources,
along with flows from the equator-ward edges of polar coronal holes. Thus the
effective rotation rate of the solar wind remains close to the Carrington
rotation rate during most of solar cycle, except during solar minima. At these
times, when there are only a few small low-latitude source regions, the mean
rotation rate of the corona decreases by 50-60 nHz with respect to the
Carrington rate. This coincides with the cyclic minimum of the Sun’s angular
momentum-loss rate, and so the net impact, when averaged over the solar cycle,
is strongly limited.

Differential rotation could have a stronger influence on the wind-braking of
other Sun-like stars with larger contrasts in rotation between their equator
and poles (typically observed in the younger rapidly rotating stars). The
degree to which this differential rotation will impact their wind-braking
torque is dependent on the long-term latitudinal distribution of their stellar
wind sources, which remains uncertain. This may change in future when better
observational constraints on the latitudinal distribution of starspots
throughout the main-sequence of Sun-like stars become available (e.g.
Berdyugina, 2005; Shapiro et al., 2014; Morris et al., 2017; Işık et al.,
2018). It is left for future work to ascertain the importance of differential
rotation on the wind-braking torque on evolutionary timescales.

The PFSS modelling adopted in this work produces a static, force-free model of
the corona. This is a rapid and computationally inexpensive method for
assessing the likely variation of connectivity throughout the time-series of
magnetograms used in this work (see also the work of Badman et al., 2020;
Stansby et al., 2021). Our study assumes a direct relation between the
observed surface rotation rate and that of the solar wind above, however many
open questions still surround the rotation of the corona. It is likely that
the act of differential rotation on the overlying coronal magnetic field will
generate currents that modify the balance of forces in the corona. This effect
can be found in the magnetofrictional models of Yeates et al. (2008), who
utilise a time-varying photospheric magnetic field boundary condition with a
finite magnetic field relaxation timescale (see also van Ballegooijen et al.,
2000; Hoeksema et al., 2020). This allows for complexity to develop in the
corona, which is otherwise missing in force-free models. Observed coronal
features are often better matched by this kind of modelling (Meyer et al.,
2020). Magnetofrictional modelling has also been applied to other Sun-like
stars (see Gibb et al., 2016). Any potential hysteresis of the coronal
magnetic field will likely change the source latitudes of the solar wind, and
the degree of which the magnetic field enforces the surface rotation rate.

Coronal rotation has impacts in many areas of active research, such as the
accuracy of ballistic back-mapping of the solar wind when identifying
photospheric sources (e.g. Macneil et al., 2022), the production of accurate
models of the inner heliosphere, and the overall forecasting of space weather.
Thus, in the coming decade, studies of coronal rotation ranging from the
distortion of coronal hole boundaries, up to the variation in white light
streamers, and in-situ measurements of solar wind deflections, will be
required to understand the evolution of angular momentum from the solar
surface out into the solar wind.

###### Acknowledgements.

This research has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 810218 WHOLESUN), in addition to funding by the Centre
National d’Etudes Spatiales (CNES) Solar Orbiter, and the Institut National
des Sciences de l’Univers (INSU) via the Programme National Soleil-Terre
(PNST). This work utilizes data produced collaboratively between Air Force
Research Laboratory (AFRL) and the National Solar Observatory (NSO). The ADAPT
model development is supported by AFRL. The input data utilized by ADAPT is
obtained by NSO/NISP (NSO Integrated Synoptic Program). NSO is operated by the
Association of Universities for Research in Astronomy (AURA), Inc., under a
cooperative agreement with the National Science Foundation (NSF). The sunspot
number used in this work are from WDC-SILSO, Royal Observatory of Belgium,
Brussels. Data supplied courtesy of the SDO/HMI and SDO/AIA consortia. SDO is
the first mission to be launched for NASA’s Living With a Star (LWS) Program.
The SOHO/LASCO data used here are produced by a consortium of the Naval
Research Laboratory (USA), Max-Planck-Institut fuer Aeronomie (Germany),
Laboratoire d’Astronomie (France), and the University of Birmingham (UK). SOHO
is a project of international cooperation between ESA and NASA. Data
manipulation was performed using the numpy (Harris et al., 2020), scipy
(Virtanen et al., 2020), and pySHTOOLS (Wieczorek & Meschede, 2018) python
packages. Figures in this work are produced using the python packages
matplotlib (Hunter, 2007), and Mayavi (Ramachandran & Varoquaux, 2011).

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##  Appendix A Fitting the differential rotation of coronal holes

Figure 10: Fitting the differential rotation profile of the coronal holes in
SDO/AIA-193A images during CR2183-CR2186. Each row repeats the following
sequence. The left column depicts a given synoptic EUV chart in the sequence
CR2183 - CR2185 with a red contour identifying the coronal hole area using a
brightness threshold. The second column shows the next synoptic chart (CR2184
- CR2186) with a blue contour identifying the coronal hole area in the same
manner as the leftmost column. The area inside these contours is assumed to be
representative of the coronal hole area for each synoptic chart. The red
contour from the first column is then distorted using equation (18) by varying
the $\Omega_{eq}$, $\alpha_{2}$, and $\alpha_{4}$ parameters, in order to find
a maximum overlap of the area between the two contours. The ‘best fit’
distorted contour is over-plotted in the second column in red. The third
column depicts the quality of the overlap between the red distorted coronal
hole area and the blue coronal hole area, with green representing a match and
red/blue indicating that there is area in the red/blue contour that does not
match. The final column shows the surface rotation rate versus latitude using
in this study in black, with the best fit rotation profile in red. The average
fit parameters from all three retrievals are $\Omega_{eq}$= 463.0 nHz,
$\alpha_{2}$= -26.7 nHz, and $\alpha_{4}$= -33.1 nHz.

In order to provide an illustrative rotation profile to contrast with the
well-known surface differential rotation pattern, the evolution of two trans-
equatorial coronal holes in 2016 - 2017 are examined. Synoptic AIA 193$\AA$
charts from Carrington Rotations (CRs) 2183, 2184, and 2185 are shown in the
leftmost column of Figure 10. The coronal hole boundaries are highlighted in
red by selecting a threshold on the brightness of the EUV emission (around the
30th percentile). This process is repeated in the center column for CR 2184,
2185, and 2186, with the same threshold brightness shown in blue. An algorithm
is employed to distort the red contours from the leftmost column following the
form of solar-like differential rotation described by equation (2). The
longitudinal shift applied to the contour follows the form,

$\displaystyle\delta\phi_{CR}(\theta)$ $\displaystyle=$ $\displaystyle
t_{CR}[\Omega_{*}(\theta)-\Omega_{CR}],$ (17) $\displaystyle=$ $\displaystyle
t_{CR}\bigg{[}\Omega_{eq}+\alpha_{2}\cos^{2}\theta+\alpha_{4}\cos^{4}\theta-\frac{360^{\circ}}{t_{CR}}\bigg{]}.$
(18)

where $t_{CR}=25.4$ days is the Carrington rotation period, with the values of
$\Omega_{eq}$, $\alpha_{2}$, and $\alpha_{4}$ are each allowed to vary in an
attempt to match the coronal hole boundaries from the following CR. The
algorithm attempts to maximise the area of agreement between the two contours.
The best fit distorted contour in each case is over-plotted in the centre
column in red. The two right panels show the agreement or disagreement between
the area inside the contours, along with the fit rotation profile in
comparison to the typical surface differential rotation rate. Agreement
between contours is filled in green, area where the coronal hole exists in the
distorted coronal hole but not in the chart attempting to be fit is filled in
red, and the contrary in blue.

The observed trans-equatorial coronal holes in EUV are clearly distorted
during the course of four solar rotations, however this appears to be less
than would be predicted by the typical surface differential rotation profile.
In particular, there are a few equatorial coronal holes in this timeseries
that remain at an almost fixed longitude, i.e. rotating at the Carrington
rotation rate. Thus the rotation rate at base of the solar wind may not
correspond exactly to that of the photosphere. The mean fit parameters from
Figure 10 are used as an illustrative example for comparison with the typical
surface differential rotation rate throughout this work.

##  Appendix B Influence of the Source Surface Radius

Figure 11: Same as Figure 3, now with a source surface radius of 2.0 $R_{\odot}$. Figure 12: Same as Figure 3, now with a source surface radius of 3.0 $R_{\odot}$. Table 2: Effective Rotation Rates with Varying Source Surface Radius (the Carrington rate is $\sim 456$ nHz). Phase | Date |  $\langle\Omega_{wind}\rangle$ [nHz] |  |
---|---|---|---|---
|  | $R_{ss}=2.0R_{\odot}$ | $R_{ss}=2.5R_{\odot}$ | $R_{ss}=3.0R_{\odot}$
Activity Minimum | 2009-05-01 | 412.0 | 392.5 | 382.0
Rising Phase | 2010-10-01 | 445.4 | 437.8 | 429.9
Activity Maximum | 2014-01-01 | 463.3 | 464.7 | 466.6
Declining Activity | 2017-10-01 | 436.0 | 426.0 | 417.1
Activity Minimum | 2020-01-01 | 410.0 | 387.9 | 379.0

The source surface radius $R_{ss}$ sets the distance at which the coronal
magnetic field becomes purely radial in the PFSS model. In general, reducing
the size of the source surface allows more magnetic field to open into the
solar wind which comes from smaller scale magnetic features at the
photosphere. Increasing the source surface radius allows for larger closed
magnetic field loops and so the open magnetic field at the source surface is
reduced. In this study, for the PFSS modelling the source surface radius is
fixed to $2.5R_{\odot}$, which is a typical value from the literature.
However, it is known that the size of the source surface radius often needs to
be changed in order to match in-situ observations (Panasenco et al., 2020;
Badman et al., 2020), or observations of the coronal hole areas (Linker et
al., 2017), and even when making direct comparisons between PFSS and MHD
models (Réville et al., 2015b). To test the dependence of our result on our
chosen source surface radius, the calculations from Section 3 are performed
again for the source surface radii of 2 and 3 $R_{\odot}$. Figures 11 and 12
contrast the difference in connectivity of the magnetic field models. The
resulting values of $\langle\Omega_{wind}\rangle$ for each of these epochs
with varying source surface radii are compiled in Table 2. Notably, during
solar minimum increasing the source surface radius closes off some of the low-
latitude source of the solar wind found in the model with
$R_{ss}=2.5R_{\odot}$ in Figure 3, whereas decreasing the source surface
radius makes these regions more dominant in the calculation of the effective
rotation rate.

Figure 13: Same as Figure 5, now with showing the influence of varying the
PFSS source surface radius from 2 - 3 $R_{\odot}$. The PFSS models shown in
Figures 11, and 12 are highlighted.

The variation in the effective rotation rate of the solar wind between the two
extreme source surface radii is explored in Figure 13. There is very little
difference between the result from the two models during solar maximum, due to
the more effectively-homogeneous distribution of photospheric sources in
latitude. Here the larger source surface model can produce a slightly larger
effective rotation rate, which is a result of closed field over the rotation
poles being opened which has a slower rotation rate. The largest differences
come at solar minima where the source latitudes of the solar wind is highly
sensitive to the size of the source surface. The smaller source surface
connecting to more low-latitude sources of the solar wind and so have a larger
effective rotation rate, and vice versa.