additional equation (e.g., describing wave propagation) is computed. This
allows us to not only perform three-dimensional numerical simulations, but
also to extend the simulation box to much larger distances from the star. For
this reason, I call these “global” wind models, or large-scale wind models.
These models usually consider rotation, allowing us to calculate angular
momentum losses (cf. Sect. 5.3). The obvious drawback of such models is that
they do not consider the physics of what happens within $0.003\,R_{\odot}$
above the photosphere, which is where the bulk of the heating takes place.
Therefore, the temperature, and also the density, at the wind base are free
parameters of polytropic wind models. These free parameters are usually
determined from complementary observations, such as X-ray observations, radio
observations and observationally-derived mass-loss rates (e.g., Holzwarth and
Jardine, 2007; Johnstone et al., 2015b, a; Réville et al., 2016; Ó Fionnagáin
and Vidotto, 2018; Vidotto et al., 2018b; O Fionnagain et al., 2021).

To reduce the uncertainty in these free parameters, some wind models have used
X-ray properties to constrain, for example, the density and temperature at the
wind base (e.g., Holzwarth and Jardine, 2007; Réville et al., 2016; Ó
Fionnagáin et al., 2019). In the Sun, coronal holes are X-ray dark, and the
closed field line region is X-ray bright, with the more active/denser region
showing higher temperatures (Peres et al., 2004). Given that the wind flows
through coronal holes, relating wind base properties with X-ray properties is
at the end an assumption that the thermal properties of coronal holes are
_proportional_ to the thermal properties of the closed corona. Such an
assumption implies that, given that the corona of a fast rotating star is
hotter (Güdel et al., 1997; Telleschi et al., 2005), their wind should be
proportionally hot (corona and wind temperatures are not necessarily the same
though). Likewise, stars with denser corona would have denser winds in these
models. These wind models thus predict that mass-loss rates are higher for
young and fast rotating stars.

An alternative approach that has been used to constrain the wind base
temperature in other stars is by assuming that the thermal speed (and thus the
temperature) at the base of the wind is a fixed fraction (about 22%) of the
surface escape velocity (Matt and Pudritz, 2008; Matt et al., 2012). Because
the activity level is not considered in this approach, two stars with same
mass and radius, but with different magnetic activity, would have winds with
the same temperature. One uncertainty is that, because the magnetic field flux
at the base of open-field regions increases with magnetic activity, the
heating efficiency may not be the same during the spin evolution. Given that
the surface escape velocity does not change considerably during the main
sequence, these wind temperatures change by a smaller amount during stellar
spin evolution than the approach using X-ray observations.

There are two advantages of global wind models that I would like to highlight
here. Firstly, because the numerical grid can extend out to large distances,
it is possible to characterise the stellar wind conditions around exoplanets
(Vidotto et al., 2009a, 2010b, 2012, 2015; Cohen et al., 2011a, b; Llama et
al., 2013; Nicholson et al., 2016; Vidotto and Donati, 2017; Strugarek et al.,
2019). The characterisation of the local conditions of the stellar wind is
important to quantify the wind (magnetic and particles) effects on exoplanets,
as I will discuss later on in this review. Secondly, the global wind models
can also incorporate more complex magnetic field topologies, including
synoptic maps derived from ZDI studies. The magnetic map is imposed as
boundary condition at the stellar wind base. Usually, the process involves
extrapolating the surface field into the computational domain initially
assuming the field is in its lowest energy state (e.g., a potential field).
After the interaction with the stellar wind flow, the magnetic field becomes
stressed. The self-consistent interaction between magnetic field lines and
stellar wind are allowed to evolve, until a relaxed solution is found (for
more details, see e.g., Vidotto et al., 2014b). Figure 20 illustrates the
initial (left panel) and final (right panel) conditions of a 3D
magnetohydrodynamics (MHD) stellar wind simulation, that adopts a polytropic
heating and incorporates observed surface magnetic fields at its inner
boundary.

Figure 20: (a) Magnetic field line extrapolations, assuming a potential field
source surface model. The surface magnetic field is derived from ZDI
observations (Donati et al., 2008). The potential field model assumes the
stellar magnetic field is in its minimum energy state. (b) The field lines are
stressed after interaction with stellar wind flow. Based on the simulations
presented in Vidotto et al. (2014b).

Being able to include the diversity of observed magnetic field geometries is
important. In the case of the Sun, observations of the solar wind with the
Ulysses spacecraft revealed that the geometry of the solar magnetic field
affects the velocity distribution of the solar wind (McComas et al., 2008).
When the Sun is in its activity minimum and has a simpler magnetic field
topology, close to a dipole, the wind structure is bi-modal, with larger wind
velocities in the (high-latitude) coronal holes than in the low-latitude
region. On the other hand, when the Sun is at its activity maximum, its
magnetic field geometry becomes more complex, which is then reflected in the
wind structure. Numerical simulations indeed show that the geometry of the
stellar magnetic field affects the wind velocity (e.g., Vidotto et al., 2009b;
Llama et al., 2013; Réville et al., 2016; Ó Fionnagáin et al., 2019). Figure
21 shows results of two 3D MHD simulations of winds of solar-like stars that
include the observed surface magnetic fields. The left panel of Fig. 21 shows
the solution of the stellar wind model of the planet-hosting star HD 189733
from Llama et al. (2013) that was computed using the observationally-derived
ZDI magnetic map from Fares et al. (2010). This panel has a similar format as
Fig. 10. The background shows the computed X-ray emission of the hot,
quiescent corona of the star due to thermal free-free radiation. Note that
coronal X-ray emission should come mainly from smaller scale active regions.
As the small-scale magnetic structure is not resolved in ZDI observations, the
X-ray emission computed in Llama et al. (2013) captures only the emission of
the background coronal wind and, as such, provide a lower limit for the
emission. Overlaid to the computed X-ray image is the velocity of the stellar
wind in km/s at the position of the orbit of the exoplanet HD 189733b. Regions
of fast wind correspond to X-ray dark regions in the corona. Slow wind regions
tend to lie over the largest helmet streamers in the coronal field. This is a
result of the nature of the magnetic force, which generates meridional flows
that bring wind from open-line regions (coronal holes) to the top of the
streamers (Vidotto et al., 2009b). Therefore the structure that the stellar
magnetic field imposes on the X-ray corona is correlated with the structure of
the stellar wind. The right panel of Fig. 21 shows a histogram of wind
velocities derived in another 3D MHD simulation of the wind of the young
solar-like star HII 296, where three velocity components can be identified
(Réville et al., 2016). A slow component around 250 km/s originates from wind
emanating from helmet-streamer structures of the magnetic field, a fast
component around 500 km/s originates due to magneto-centrifugal effects and
flux tube expansion (dashed magenta line), and an intermediate component at
400 km/s, which coincides with the non-magnetised polytropic wind solution
(dashed black line). Réville et al. (2016) showed that the result of the
interaction of the flow with strong, non-axisymmetric field generates a
complex wind-speed distribution, which is wider in the case of stars with
higher magnetisation and faster rotation (see also Figure 5 in Ó Fionnagáin et
al., 2019). The intermediate component, which is similar to the non-magnetised
polytropic solution, originates from regions of weak (and likely radial)
field, where the interaction of the wind flow with the magnetic field is the
weakest (Réville et al., 2016).

Figure 21: Left: Velocity of the simulated stellar wind (in km/s) of the
planet-hosting star HD 189733, at the position of the orbit of the exoplanet
HD 189733b. The blue circles denote an inward component of the magnetic field
and the green diamonds denote an outward component. The computed free-free
X-ray emission of the coronal wind is shown in the background. The structure
that the stellar magnetic field imposes on the X-ray corona is correlated with
the structure of the stellar wind. Figure from Llama et al. (2013). Right:
Histogram of wind velocities for 3D MHD simulations of the wind of the young
solar-like star HII 296. Three velocity components related to magnetic field
geometry can be identified: a slow one at 250 km/s emanating from helmet
streamers, a fast component at 500 km/s (dashed-magenta line) from expanded
flux tubes and an intermediate velocity of 400 km/s. Figure from Réville et
al. (2016).

##### (iii) Hybrid approach:

Recently, there have been efforts in developing a hybrid model that combines
the two approaches described above to study winds of low-mass stars. In these
hybrid models, a phenomenological approach of the (solar-based) wave heating
mechanism is implemented in three-dimensional simulations of solar/stellar
winds, starting from the upper chromosphere (van der Holst et al., 2010, 2014;
Sokolov et al., 2013; Garraffo et al., 2016; Alvarado-Gómez et al., 2016a, b;
Cohen, 2017; Boro Saikia et al., 2020; O Fionnagain et al., 2021). These
models present a step forward in the 3D modelling of winds of low-mass stars,
as they, for example, do not need to impose a polytropic index to mimic energy
deposition in the wind. However, the full computation of the heating process
from the photosphere is still not available.

In the hybrid models, there are also free parameters that need to be
considered, such as the energy flux of waves at the inner boundary and its
dissipation length scale (Sokolov et al., 2013; O Fionnagain et al., 2021). In
the case of the solar wind, these free parameters can be constrained from
solar observations (Sokolov et al., 2013; van der Holst et al., 2014). In the
case of winds of other stars, it is less clear how to constrain these free
parameters. Of particular relevance is the wave energy flux, or Poynting flux,
at the inner boundary of these models (chromosphere). Alvarado-Gómez et al.
(2016b) and Cohen (2017), for example, followed the approach of Sokolov et al.
(2013) who used empirical relation between X-ray and magnetic fluxes to
constrain the wave input Poynting flux. While in a polytropic formalism the
heating is not directly related to the surface magnetic field, in their hybrid
approach, the input wave flux at the chromosphere is taken to be proportional
to the local surface magnetic field strength (Sokolov et al., 2013). By
assuming that the Alfvén wave turbulent dissipation is the main source of
heating of the solar wind/corona, Sokolov et al.’s approach implies that the
total heating power scales with the magnetic flux (Equation 1 in Sokolov et
al. 2013), with a fraction of this total power generating the observed X-ray
luminosity seen in scalings from Pevtsov et al. (2003). Sokolov et al. (2013)
derived the constant of proportionality between the Poynting flux and the
local surface magnetic field as $F_{A}/B\simeq 44.7(4\pi
R_{\star}^{2})^{0.1488}\simeq 1.1\times 10^{5}(R_{\star}/R_{\odot})^{0.2976}$
erg cm-2 s-1G-1. Note the stellar size dependence, which should be considered
when this implementation is applied to other stars (Cohen, 2017).

This is not the only approach to determine the input wave energy flux. An
alternate one was discussed in Boro Saikia et al. (2020), who constrained the
wave flux using far-ultraviolet spectral lines that are formed in the upper
chromosphere or transition region of stars. For completeness, I also highlight
alternative approaches for determining wave energy fluxes used in 1D models
(which start from the photosphere, contrary to the hybrid studies discussed in
this section, that start in the chromosphere). Cranmer and Saar (2011) and
Cranmer (2017) constrained the Poynting flux used in their models from the
magneto-convection models of Musielak and Ulmschneider (2002), who studied the
dependence on the emerging wave energy flux with stellar gravity under the
mixing-length theory. The gravity dependence can be particularly relevant for
stars that evolve off the main sequence (O Fionnagain et al., 2021).
Dependence with mass and metallicity of the emerging wave energy flux has also
been considered in the work of Suzuki (2018), when investigating winds of
population III stars. In this case, the amplitude of the wave velocity
fluctuations $\delta v$ (note that $F_{A}\propto\rho\langle\delta
v^{2}\rangle$) at the photosphere is determined from the surface convective
flux, which is proportional the stellar luminosity: $\rho\delta v^{3}\propto
L_{\star}$ (see their Eq. 15).

By considering a Poynting flux dependent on the local magnetic field strength,
the Poynting flux varies across the stellar surface. As a result, hybrid 3D
models show stronger contrast between the wind properties and the large-scale
magnetic field geometry than those based on a purely polytropic formalism.
This technique is recent in comparison to those discussed in (i) and (ii) and
I expect it will be further developed in the years to come.

###  5.2 Brief overview of MHD stellar wind theory

Each of the different treatments adopted in wind models have their advantages
and their drawbacks. When it comes to modelling rotational evolution, models
that explicitly treat heat deposition in the wind are less suitable, as most
of these models neglect rotation. To study angular momentum losses, it is more
common to adopt polytropic models (i.e., with implicit energy deposition). The
first polytropic model for the solar wind was developed by Parker (1958). This
model is, in fact, isothermal, which is a special case of a polytropic wind
with $\Gamma=1$. Here, I do not present the derivation of the MHD equations
that are often used in stellar wind theory – those can be found in textbooks
(e.g., Mestel, 1999; Goedbloed and Poedts, 2004). Here, I only present some
key MHD equations that I will use to discuss stellar wind models and angular
momentum losses.

####  5.2.1 Thermally-driven winds

The first important equation in wind models is that of mass continuity

$\frac{\partial\rho}{\partial t}+\boldsymbol{\nabla}\cdot(\rho\vec{u})=0\,,$
(28)

where $\rho$ is the mass density, and $\vec{u}$ is the wind velocity. In the
absence of sinks or sources, mass is conserved in a stellar wind.

For the momentum equation, it is useful to use the Lagrangian formalism. The
Lagrangian time derivative $D/Dt$, sometimes also known as the total
derivative, is given by

$\frac{DQ}{Dt}=\frac{\partial Q}{\partial
t}+\vec{u}\cdot\boldsymbol{\nabla}Q.$ (29)

The $D/Dt$ operator can be applied to scalar $Q$ or vector $\vec{Q}$
quantities. The first term on the right hand side is the rate of change of $Q$
computed at a fixed location (i.e., the Eulerian time derivative) and the
second term arises because the fluid element with velocity $\vec{u}$ has moved
to a new location where the quantity $Q$ has a different value. This term is
also known as the ‘convective derivative’. In the Lagrangian description,
Newton’s second law can be simply written as

$\rho\frac{D\vec{u}}{Dt}=\sum\frac{\vec{F}}{\rm volume}\,.$ (30)

The term of the right hand side represents the sum of all volumetric forces
acting on the wind. Once derived, however, the momentum equation is more
convenient to use from the Eulerian viewpoint. For a polytropic (or an
isothermal) stellar wind with only gravitational forces and pressure gradients
acting on the wind, the momentum equation, using the two previous equations,
is

$\rho\left(\frac{\partial\vec{u}}{\partial
t}+\vec{u}\cdot\boldsymbol{\nabla}\vec{u}\right)=-\frac{\rho
GM_{\star}}{r^{2}}\hat{\bf{r}}-\boldsymbol{\nabla}P\,,$ (31)

where $\hat{\bf{r}}$ is the unit vector along the radial direction. This
equation can be further simplified by assuming steady state
(${\partial/\partial t}=0$) and spherical symmetry
(${\partial/\partial\theta}=0$, ${\partial/\partial\varphi}=0$):

$\rho{u_{r}}\frac{du_{r}}{dr}=-\frac{\rho GM_{\star}}{r^{2}}-\frac{dP}{dr}\,.$
(32)

Under these conditions, the continuity equation (28) can be written as

$\frac{d}{dr}(\rho
r^{2}u_{r})=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\to\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\rho
r^{2}u_{r}={\rm constant.}$ (33)

Taking into account that the stellar wind emerges from the $4\pi$ steradians
of stellar surface, we then define the mass-loss rate as

$\dot{M}=4\pi\rho r^{2}u_{r}\,,$ (34)

which is constant with radial distance in a 1D stellar wind.

Equations (32) and (34) form the basis of the thermally-driven wind model
derived by Parker (1958). The thermal pressure can be written using the ideal
gas law

$P=\frac{\rho k_{B}T}{\mu m_{p}}=\rho a^{2}\,,$ (35)

where $k_{B}$ is the Boltzmann constant, $m_{p}$ the proton mass, $T$ the wind
temperature and $\mu$ is the mean “molecular” weight. In the case of hot
stellar winds, no molecules exist, and the mean molecular weight is simply the
average mass of the particles. For a purely hydrogen plasma, fully ionised
(50% protons, 50% electrons), the mean mass of the particles are $\mu
m_{p}=0.5m_{p}+0.5m_{e}\simeq 0.5m_{p}$, where $m_{e}$ is the electron mass,
so $\mu=0.5$. In the equation above, I wrote the isothermal sound speed as
$a=\sqrt{k_{B}T/(\mu m_{p})}$. Thus, differentiating the ideal gas law along
the radial coordinate, we get

$\frac{dP}{dr}=a^{2}\frac{d\rho}{dr}+\frac{\rho a^{2}}{T}\frac{dT}{dr}\,.$
(36)

In Parker’s model, the wind is considered isothermal, thus the terms with
$dT/dr$ are null in the previous equation. I am going to assume here that the
wind is polytropic, which implies that

$P\propto\rho^{\Gamma}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm
or}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T\propto\rho^{\Gamma-1}\,,$ (37)

where I used the ideal gas law to convert between the two equations. From
this, the temperature gradient is

$\frac{1}{T}\frac{dT}{dr}=(\Gamma-1)\frac{1}{\rho}\frac{d\rho}{dr}$ (38)

and Eq. (36) is written as

$\frac{dP}{dr}=a^{2}\frac{d\rho}{dr}+a^{2}(\Gamma-1)\frac{d\rho}{dr}=\Gamma
a^{2}\frac{d\rho}{dr}=c_{s}^{2}\left(-\frac{\rho}{u_{r}}\frac{du_{r}}{dr}-\frac{2\rho}{r}\right)\,,$
(39)

where I used the mass conservation Eq. (33) to write ${d\rho}/{dr}$ and wrote
the sound speed for a polytropic gas as

$c_{s}^{2}={\Gamma a^{2}}\,.$ (40)

Substituting the previous equation in the momentum Eq. (32) and rearranging,
we arrive at the momentum equation of a thermally-driven wind:

$\rho{u_{r}}\frac{du_{r}}{dr}=-\frac{\rho
GM_{\star}}{r^{2}}+c_{s}^{2}\left(\frac{\rho}{u_{r}}\frac{du_{r}}{dr}+\frac{2\rho}{r}\right)\,.$
(41)
$\frac{1}{u_{r}}\frac{du_{r}}{dr}=\left(-\frac{GM_{\star}}{r^{2}}+\frac{2c_{s}^{2}}{r}\right)/({u_{r}^{2}}-{c_{s}^{2}})\,.$
(42)

The equation above reduces to Parker’s solar wind equation if one assumes the
wind is isothermal and thus $\Gamma=1$ and hence $c_{s}=a$.

There are two important points to highlight here. Firstly, note that the
density cancels out. Thus, in a thermally-driven wind, the acceleration of the
wind does not depend on the density of the wind, which leaves the density as a
scaling parameter that is usually adopted to match the required mass-loss rate
(Equation 34). Secondly, this equation has a singularity at $u_{r}=c_{s}$,
i.e., when the stellar wind radial velocity reaches sound speed, the
denominator goes to zero. The momentum equation for a polytropic wind is,
regardless of that, continuous, as long as the numerator and the denominator
go to zero simultaneously. For this to happen, the wind speed must reach the
sonic speed at $r=r_{c}$: $u_{r}(r_{c})=c_{s}$. At this critical point, known
as the sonic point, the numerator is written as

$-\frac{GM_{\star}}{r_{c}^{2}}+\frac{2c_{s}^{2}}{r_{c}}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\to\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r_{c}=\frac{GM_{\star}}{2c_{s}^{2}}\,.$
(43)

This is actually a very important point in the theory of stellar winds.
Equation (42) has an infinite number of mathematical solutions, but the only
physical solution for a stellar wind is the transonic solution, i.e., the one
that starts subsonic and becomes supersonic. Figure 22 illustrates solutions
for the wind radial velocity, assuming isothermal winds ($\Gamma=1$) at four
different temperatures: 0.75, 1.5, 3 and 4 MK (from bottom to top curves). I
assumed here a star with $1\,M_{\odot}$ and $1\,R_{\odot}$. The curves show
the typical wind radial velocity profile, which is characterised by a
relatively quick increase in the velocity at inner regions and then an
asymptotic solution, which tends to the terminal wind velocity, $u_{\infty}$.
Note how sensitive $u_{\infty}$ is to the temperature, going from 400 km/s for
a 0.75-MK wind to 1200 km/s for a 4-MK wind. The zoom-in panel in the right
shows the position of the sonic point. Because $r_{c}\propto c_{s}^{-2}\propto
1/T$, the sonic point happens closer to the star for higher temperature winds.
Likewise, because $c_{s}\propto T^{1/2}$, the speed at which the sonic point
is reached is increasingly larger for higher temperatures. As discussed in
page 5.1, the base temperature, along with the base density, are free
parameters in polytropic wind models.

Figure 22: Solutions to the momentum equation of an isothermal wind for
different wind temperatures. The sonic points are marked by the filled
circles, in the zoomed-in panel on the right.

Using the mass conservation equation and the solutions for $u_{r}$ in Fig. 22,
the density profiles can be found: $\rho=\dot{M}/(4\pi r^{2}u_{r})$. Note here
that the density depends on the _choice_ of mass-loss rate $\dot{M}$, which is
related to the discussion above on the density in thermally-driven wind being
a scaling factor. For larger distances, when the wind has reached asymptotic
speeds, $\rho\propto r^{-2}$. In this case, the ram pressure of the wind
$P_{\rm ram}=\rho u_{r}^{2}$ also decreases with $r^{-2}$. This is relevant
for calculating the size of the heliosphere, and how it evolved. I will
discuss this in Sect. 6.3.

####  5.2.2 The magneto-rotator wind

The thermally-driven wind discussed before does not consider two physical
parameters that are observed in cool stars: rotation and magnetic fields. I am
going to show now an important effect when considering magnetised, rotating
winds: the removal of angular momentum from the star. Consider the initial
setup shown in Fig. 23a: a split monopole magnetic field in the equator of the
star. The magnetic field lines are anchored on the stellar surface, and thus,
once stellar rotation is set, they must co-rotate with the star (i.e., at same
rotation rate $\Omega_{\star}$). I will consider the stellar rotation axis
pointing out of the page (rotation is counter-clockwise).

Figure 23: Left: initial setup of the split monopole. Right: the trailing
spiral structure that is created once stellar rotation is set.

##### Magnetic field geometry:

I will first derive the geometry of the magnetic field lines, using the
magnetic induction equation for a perfectly conducting plasma (null
resistivity)

$\frac{\partial\vec{B}}{\partial
t}=\boldsymbol{\nabla}\times(\vec{u}\times\vec{B})\,.$ (44)

I will keep assuming steady state (${\partial/\partial t}=0$) and axi-symmetry
(${\partial/\partial\varphi}=0$). Given that the equation is solved in the
equatorial plane, ${\partial/\partial\theta}=0$. Thus the magnetic induction
equation becomes

$\boldsymbol{\nabla}\times(\vec{u}\times\vec{B})=0\,.$ (45)

The wind velocity vector in spherical coordinates is
$\vec{u}=[u_{r},0,u_{\varphi}]$ and similarly the magnetic field is
$\vec{B}=[B_{r},0,B_{\varphi}]$. Notice that even though I presented in Fig.
23a a split monopole ($B_{\varphi}=0$), the rotating star will generate an
azimuthal component for the magnetic field in the wind (Fig. 23b). Using the
curl in spherical coordinates, and the fact that
$\vec{u}\times\vec{B}=[B_{r}u_{\varphi}-u_{r}B_{\varphi}]\hat{\theta}$, the
last equation is written as

$\frac{1}{r}\frac{\partial}{\partial
r}(r[B_{r}u_{\varphi}-u_{r}B_{\varphi}]\hat{\varphi})=0\,.$ (46)

This implies that the term inside the parenthesis is constant for different
radial distances: $r(B_{r}u_{\varphi}-u_{r}B_{\varphi})=\rm constant$. To find
this constant, I analyse the wind physical properties at the wind base, where
$r=R_{\star}$. There, the wind radial velocity is very small
$u_{r}(R_{\star})\simeq 0$ and the azimuthal velocity is the rotational
velocity of the star, $u_{\varphi}(R_{\star})=\Omega_{\star}R_{\star}$. The
magnetic field anchored on the surface of the star is still mostly radial,
thus $B_{\varphi}(R_{\star})\simeq 0$. Hence, the constant computed at
$R_{\star}$ is: $B_{r}(R_{\star})\Omega_{\star}R_{\star}^{2}$. This can be
further generalised by using magnetic flux conservation:
$B_{r}r^{2}=B_{r}(R_{\star})R_{\star}^{2}$. The magnetic induction equation
then becomes

$r(B_{r}u_{\varphi}-u_{r}B_{\varphi})=\Omega_{\star}B_{r}r^{2}\,.$ (47)

Rewriting this equation, I arrive at the equation describing the magnetic
field geometry of the wind of a rotating magnetised star

$\frac{B_{\varphi}}{B_{r}}=\frac{u_{\varphi}-\Omega_{\star}r}{u_{r}}\,.$ (48)

Close to the star (small $r$), the rotational velocity is
$u_{\varphi}(R_{\star})=\Omega_{\star}R_{\star}$ and thus $B_{\varphi}\simeq
0$. Far from the star (large $r$), $u_{\varphi}\ll\Omega_{\star}R_{\star}$ and
thus $B_{\varphi}/B_{r}\simeq-\Omega_{\star}R_{\star}/u_{r}$. This implies
that not only an azimuthal component of the magnetic field is generated at
large distances, but that this component has opposite sign to the radial
field: $B_{\varphi}/B_{r}<0$. This is illustrated in the sketch on the right
panel of Fig. 23. As a result, the rotating wind generates a trailing spiral:
if $B_{r}>0$ (pointing outward), then $B_{\varphi}<0$ (clockwise). On the
contrary, if $B_{r}<0$ (pointing inward), then $B_{\varphi}>0$ (counter-
clockwise).

##### The momentum equation of a magneto rotator wind:

The momentum equation of a magnetic rotator is similar to the equation used
before, except that now the Lorentz force must also be considered

$\rho\vec{u}\cdot\boldsymbol{\nabla}\vec{u}=-\frac{\rho
GM_{\star}}{r^{2}}\hat{\bf{r}}-\boldsymbol{\nabla}P+\vec{F}_{B}\,,$ (49)

where I assumed steady state and $\vec{F}_{B}$ is the Lorentz force per unit
volume

$\vec{F}_{B}=\frac{1}{c}\vec{J}\times\vec{B}=\frac{1}{4\pi}(\boldsymbol{\nabla}\times\vec{B})\times\vec{B}\,,$
(50)

where $\vec{J}$ is the current density and I used the Ampère’s law (${\bf
J}={c}/({4\pi})({\bf\nabla}\times{\bf B})$) in the second equality. Using the
curl in spherical coordinates, it is straightforward to demonstrate that
$\boldsymbol{\nabla}\times\vec{B}=-r^{-1}\partial(rB_{\varphi})/\partial
r\hat{\theta}$ and thus

$\vec{F}_{B}=\frac{1}{4\pi
r}\left[-B_{\varphi}\frac{\partial(rB_{\varphi})}{\partial
r}\hat{r}+B_{r}\frac{\partial(rB_{\varphi})}{\partial
r}\hat{\varphi}\right]\,.$ (51)

Before substituting Eq. (51) in the momentum equation (49), it is necessary to
write the convective derivative of $\vec{u}$ in spherical coordinates:

$\vec{u}\cdot\boldsymbol{\nabla}\vec{u}=\left[u_{r}\frac{\partial
u_{r}}{\partial
r}-\frac{u_{\varphi}^{2}}{r}\right]\hat{r}+\left[u_{r}\frac{\partial
u_{\varphi}}{\partial r}+\frac{u_{r}u_{\varphi}}{r}\right]\hat{\varphi}\,.$
(52)

Substituting Eqs. (52) and (51) into (49), I finally arrive at the momentum
equation of a magneto-rotator wind. It is convenient to split it into the
radial and azimuthal components, which are respectively

$\rho\left(u_{r}\frac{\partial u_{r}}{\partial
r}-\frac{u_{\varphi}^{2}}{r}\right)=-\frac{\rho
GM_{\star}}{r^{2}}-\frac{\partial P}{\partial r}-\frac{B_{\varphi}}{4\pi
r}\frac{\partial(rB_{\varphi})}{\partial r}$ (53)

and

$\rho\left(u_{r}\frac{\partial u_{\varphi}}{\partial
r}+\frac{u_{r}u_{\varphi}}{r}\right)=\frac{B_{r}}{4\pi
r}\frac{\partial(rB_{\varphi})}{\partial r}\ \,.$ (54)

These are the equations derived by Weber and Davis (1967).

The radial component of the momentum equation of the magneto-rotator wind
(53), compared to that of the thermally-driven wind (Equation 32) has two
additional terms: the slinging effect of the rotating magnetic field
($-{B_{\varphi}}/[{4\pi r}]\,\partial[rB_{\varphi}]/{\partial r}$) and the
centrifugal force ($\rho{u_{\varphi}^{2}}/{r}$). These two terms originate
from the rotation of the star. Similar to the thermally-driven wind, the
magneto-rotator wind also has critical points. One important critical point
for angular momentum losses is the Alfvén radius888Note that I discussed about
the Alfvén surface, which is the generalisation of the Alfvén point in three
dimensions, when I discussed about prominences in winds in Sect. 2.4.,
$r_{A}$, which is defined as the position where the wind radial velocity
reaches Alfvén velocity, $v_{A}$: $u_{r}(r_{A})=v_{A}$. The Alfvén velocity is
given by $v_{A}=B/\sqrt{4\pi\rho}$.

This means that the wind of a magnetised, rotating star is trans-Alfvénic. In
the sub-Alfvénic part of the stellar wind (closer to the star), the kinetic
energy is smaller than the magnetic energy: $\rho u_{r}^{2}/2\ll
B^{2}/(8\pi)$. In this case, the wind is too weak to modify the structure of
magnetic field and the wind is forced to flow along the magnetic field lines.
In the super-Alfvénic part of the wind, the opposite is true and $\rho
u_{r}^{2}/2\gg B^{2}/(8\pi)$. In this case, the magnetic field lines are
dragged with the flow.

##### Angular momentum losses:

The angular momentum carried away by the stellar wind can be found from the
azimuthal component of the momentum equation of the magneto-rotator (54),
which can be written as

$\rho u_{r}\frac{\partial(ru_{\varphi})}{\partial
r}=\frac{B_{r}}{4\pi}\frac{\partial(rB_{\varphi})}{\partial r}\ \,.$ (55)

Rearranging this equation, one has

$\frac{\partial(ru_{\varphi})}{\partial r}=\frac{B_{r}}{4\pi\rho
u_{r}}\frac{\partial(rB_{\varphi})}{\partial
r}=\frac{B_{r}r^{2}}{\dot{M}}\frac{\partial(rB_{\varphi})}{\partial
r}=\frac{\partial}{\partial
r}\left(\frac{B_{r}r^{2}rB_{\varphi}}{\dot{M}}\right)=\frac{\partial}{\partial
r}\left(\frac{rB_{r}B_{\varphi}}{4\pi\rho u_{r}}\right)\,,$ (56)

where I use the facts that the mass-loss rate ($\dot{M}=4\pi r^{2}\rho u_{r}$,
Equation 34) and the magnetic flux ($B_{r}r^{2}$) do not depend on $r$. Thus,

$\frac{\partial}{\partial
r}\left(ru_{\varphi}-\frac{rB_{r}B_{\varphi}}{4\pi\rho u_{r}}\right)=0\,,$
(57)

which implies that the term in parenthesis is constant with radial distance

$ru_{\varphi}-\frac{rB_{r}B_{\varphi}}{4\pi\rho u_{r}}={\rm constant}={\cal
L}\,,$ (58)

where ${\cal L}$ is the specific angular momentum (angular momentum per unit
mass). Fig. 24 shows the two terms as computed for the present-day solar wind.
The first term on the left hand side is the specific angular momentum carried
by the gas (the one that we are most familiar with) and the second term is the
angular momentum carried by magnetic stresses. We notice from Fig. 24 that
even a star that has a modest magnetic field like the Sun and a relatively
weak wind, the magnetic term dominates the angular momentum losses. This can
be understood in light of the size of the Alfvén surface. Rewriting Eq. (58)
with the help of Eq. (48), one finds

${\cal L}=ru_{\varphi}-\frac{rB_{r}^{2}}{4\pi\rho
u_{r}^{2}}(u_{\varphi}-\Omega_{\star}r)=ru_{\varphi}-\frac{v_{A}^{2}}{u_{r}^{2}}(ru_{\varphi}-\Omega_{\star}r^{2})\,.$
(59)

Given that $\cal{L}$ is a constant, we can calculate the previous equation at
$r=r_{A}$, in which case $u_{r}=v_{A}$ and the equation simplifies to

${\cal L}=\Omega_{\star}r_{A}^{2}\,.$ (60)

This means that, if we know where the Alfvén radius is, then we can calculate
the specific angular momentum of the wind by using the previous equation. I
will show later on that this is however not an easy task. It is curious to see
that Eq. (60) resembles so much the angular momentum of a solid body with a
radius $r_{A}$ rotating at the angular speed of the star. Because of this,
sometimes this equation is incorrectly interpreted as a wind rotating as a
solid body out to $r_{A}$. In fact, the wind is forced to co-rotate with the
magnetic field (which is anchored on the star) out to a distance that is much
smaller than $r_{A}$.

Figure 24: The specific angular momentum of the solar wind calculated by Weber
and Davis (1967). The sum of these two terms is constant with distance and
simplifies to ${\cal L}=\Omega_{\star}r_{A}^{2}$ (see Equations 58 and 60),
where $r_{A}$ is the distance to the Alfvén point.

Finally, I can now calculate the angular momentum loss rate. The flux of
angular momentum is the product of the specific angular momentum ${\cal L}$ by
the mass flux $\rho\vec{u}$. Integrating the flux of angular momentum over a
surface area $S$, one finds the angular momentum loss rate

$\dot{J}=\oint{\cal L}\rho\vec{u}\cdot
d\vec{S}=\oint\Omega_{\star}r_{A}^{2}\rho_{A}v_{A}d{S_{A}}\,,$ (61)

where the subscript ‘A’ indicates values computed at the Alfvén radius. Note
that all this calculation was done for the equatorial plane, but in reality,
the wind is permeating the whole volume surrounding the star. To consider
this, one needs to take into account a $\sin^{2}\theta$ in the equation above.
This comes from the fact that the ‘lever arm’ $r_{A}$ should be the distance
from a point in the wind where the torque is applied to the spin axis, i.e.,
it should be a cylindrical radius $r_{A}\sin\theta$. In the equatorial plane,
$\sin\theta=1$, returning to the equation above. Inserting the
$\sin^{2}\theta$ in the equation above and integrating it, one arrives at the
angular momentum loss rate (or torque) of a stellar wind

$\dot{J}=\oint\Omega_{\star}r_{A}^{2}\rho_{A}v_{A}\sin^{2}\theta
d{S_{A}}=\Omega_{\star}\frac{\dot{M}}{4\pi}r_{A}^{2}\oint\sin^{3}\theta
d\theta d\varphi\,,$ $\dot{J}=\frac{2}{3}\dot{M}\Omega_{\star}r_{A}^{2}\,.$
(62)

###  5.3 Modelling the evolution of the Sun’s rotation

####  5.3.1 A simple model for the evolution of stellar rotation and its
relation to gyrochronology

Here I show that using the magneto-rotator wind model, it is possible to
explain the relation $\Omega_{\star}\propto t^{-1/2}$, which forms the basis
of gyrochronology for stars older than after $\sim 600$ Myr.

Using magnetic flux conservation, one can write
$B_{r,\star}R_{\star}^{2}=B_{r,A}r_{A}^{2}$, where the subscripts ‘$\star$’
and ‘$A$’ relate to the stellar surface and Alfvén surface, respectively. With
this, Eq. (62) becomes

$\dot{J}=\frac{2}{3}(4\pi\rho_{A}v_{A})\Omega_{\star}(B_{r,\star}R_{\star}^{2}/B_{r,A})^{2}=\frac{2}{3}\frac{\Omega_{\star}}{v_{A}}(B_{r,\star}R_{\star}^{2})^{2}\,,$
(63)

where I substituted for the Alfvén velocity in the second equality.

The angular momentum of the star (assuming the moment of inertia $\cal{I}$ of
a spherical solid body) is

$J={\cal I}\Omega_{\star}=\frac{2}{5}M_{\star}R_{\star}^{2}\Omega_{\star}\,.$
(64)

Differentiating this equation with respect to time, a second way to express
$\dot{J}$ becomes

$\dot{J}=\frac{2}{5}M_{\star}R_{\star}^{2}\frac{d\Omega_{\star}}{dt}\,.$ (65)

Now here I am making the assumption that $M_{\star}$ and $R_{\star}$ remains
constant after $\sim 600$ Myr. Equating Equations (63) and (65), I then find:

$\frac{2}{3}\frac{\Omega_{\star}}{v_{A}}(B_{r,\star}R_{\star}^{2})^{2}=\frac{2}{5}M_{\star}R_{\star}^{2}\frac{d\Omega_{\star}}{dt}\,,$
(66)

which can be rewritten as

$dt=\frac{3}{5}\frac{M_{\star}}{R_{\star}^{2}}\frac{v_{A}}{B_{r,\star}^{2}}\frac{d\Omega_{\star}}{\Omega_{\star}}\,.$
(67)

The previous equation has to be integrated to derive $\Omega_{\star}(t)$,
i.e., the rotational evolution of the star. I already assumed that $M_{\star}$
and $R_{\star}$ are independent of age, and thus, independent of rotation. Two
further assumptions need to be made before integrating the equation above. The
first one is related on how $B_{\star}$ depends on $\Omega_{\star}$. I will
assume they are related to each other in a linear way (i.e., a linear-type
dynamo). This is not too far from what is observed, as I showed in Fig. 13b.
The second assumption, and the most tricky one, is on how $v_{A}$ depends on
$\Omega_{\star}$. The reason why this is tricky is that
$v_{A}=B/\sqrt{4\pi\rho}$ depends on the magnetic field and density at the
Alfvén point, which in turn depends on the thermal and magnetic properties of
the wind. Thus, there is no way to derive the location of the Alfvén point,
and its velocity, before computing a wind model. I will assume, for the sake
of simplicity, that $v_{A}$ is independent of $\Omega_{\star}$ in the main
sequence for $t\gtrsim 600$ Myr, but I will come back to this in the next
section. With these assumptions, the previous equation can be written as

$\int
dt=\frac{3}{5}\frac{M_{\star}v_{A}}{R_{\star}^{2}}\int\frac{1}{B_{r,\star}^{2}}\frac{d\Omega_{\star}}{\Omega_{\star}}\propto\int\frac{1}{\Omega_{\star}^{2}}\frac{d\Omega_{\star}}{\Omega_{\star}}\,.$
(68)

Integrating the previous expression results in $t\propto\Omega_{\star}^{-2}$,
arriving at the following relation

$\Omega_{\star}\propto t^{-1/2}\,,$

which surprisingly agrees with observations!

In summary, two main conclusions can be extracted from this simple
evolutionary model. First, it demonstrates that rotating, magnetised stellar
winds carry away angular momentum from the star. As angular momentum leaves
the star, the star spins down, explaining the observed trend that stars spin
down as the square-root of age. Second, angular momentum extraction is
enhanced by the magnetic field: even for a star with a $\sim$ weak magnetic
field as the Sun, the angular momentum carried away by magnetic stresses is
the dominant contributor to the loss rate.

In a more sophisticated treatment, one should not use the moment of inertia of
a spherical solid body nor the assumption of constant mass and radius through
the main sequence. These quantities can be extracted from stellar evolution
models and thus coupling the wind angular momentum loss prescription to
stellar evolution models would provide a more sophisticated modelling for
$\Omega_{\star}(t)$. The other naive assumption made above refers to the
velocity at the Alfvén point (or surface). I assumed that the Alfvén velocity
is independent of stellar rotation. However, this is a very rough
oversimplification of the problem, as $v_{A}$ depends on the structure of the
wind, which depends on the magnetic, thermal and rotational properties of the
wind. Thus, there is no way to derive the location of the Alfvén surface, and
its velocity, without computing a wind model. I will explore in the next
Section how this can be achieved.

####  5.3.2 Magnetic braking laws: prescriptions for angular momentum-loss
rate

One of the biggest challenge in stellar rotational evolution studies is on
determining the size of the Alfvén surface, so that the angular momentum loss
rate $\dot{J}$ can be calculated. Equation (62) looks deceptively simple! The
Alfvén radius, in a 1D geometry, or surface, in 3D, can be interpreted as the
lever arm of the wind torque – the “position” where the torque is applied in
order to change the angular rotation of the star. In fact, there is not a
unique radius where the torque is applied in a stellar wind, as magnetic field
stresses, particle motion and even thermal pressure (not discussed here, see
Appendix in Vidotto et al. 2014b) contribute to the stress tensor throughout
the wind. Figure 25 shows an example of an Alfvén surface, computed in a 3D
simulation. The surface is far from being a smooth, spherical surface as the
1D model that I derived before would suggest. To makes matter worse, both the
thermally driven wind or the magneto-rotator winds depends on free parameters
that directly affect the size of the Alfvén surface. These free parameters are
related to the thermal properties of the wind (density and temperature), which
are not directly observed (though they can be inferred to some level).

Figure 25: In 3D, the Alfvén radius takes the form of an Alfvén surface (grey
surface). Because of the complex stellar surface magnetic field geometry
(shown by the colour bar), the Alfvén surface can be highly asymmetric. Figure
from Vidotto et al. (2014b).

Provided that one can determine the size of the Alfvén surface and thus
calculate $\dot{J}$, then $\dot{J}$ can be implemented in stellar evolution
codes so that the stellar structure and its rotation can be evolved forward in
time. This has been done with different levels of complexity in stellar
evolution models (e.g., Kawaler, 1988; Pinsonneault et al., 1989; Bouvier et
al., 1997; Spada et al., 2011; Johnstone et al., 2015a; Matt et al., 2015;
Amard et al., 2019). What these models all have in common is that they rely on
a parameterisation for $\dot{J}$, i.e., an analytical relation that summarises
how $\dot{J}$ varies with different physical properties of the star, such as
its mass, radius, rotation, surface magnetic fields, etc. Such analytical
relation can be derived from analytical considerations (Kawaler, 1988) or
numerical simulations (Matt et al., 2012). Kawaler (1988), for example,
prescribed two power-laws, one relating surface magnetism and rotation
($B_{\star}\propto\Omega_{\star}^{a}$, where $a=1$ represents a linear-type
dynamo) and another relating how $\dot{J}$ varies with magnetic field geometry

$\dot{J}=\frac{2}{3}\dot{M}\Omega_{\star}R_{\star}^{2}\left(\frac{r_{A}}{R_{\star}}\right)^{n},$
(69)

in which $n=2$ corresponds to a radial field geometry (as I derived before)
and $n=3/7$ to a dipolar field geometry. Combining these two properties,
Kawaler (1988) obtained an angular-momentum loss prescription of the form

$\dot{J}\propto\Omega_{\star}^{1+4an/3}R_{\star}^{2-n}\dot{M}^{1-2n/3}M_{\star}^{-n/3}\,.$
(70)

To find the proportionality constant (omitted) in the previous equation,
models are usually calibrated to the solar values, i.e., for a given $n$
(field geometry) such a constant is found when the solar rotational velocity
is obtained at the solar age and radius. Similarly, $n$ is adjusted to find
the best fit to observational data. Kawaler (1988) found that $n=1.5$
reproduces the Skumanich relation ($\Omega_{\star}$$\propto t^{-1/2}$). Also
very illuminating, but not discussed in this review, is that once the solar
calibration and calibration for solar-mass stars in clusters are done, the
model can be applied to low-mass stars with a range of masses.

The power-law indices and constants in the analytical description of $\dot{J}$
(e.g., from Eq. (70)) hide our ignorance of the physics that is driving winds
of Sun-like stars (Kawaler, 1988). For example, different surface magnetic
field geometries and thermal properties of the wind change the size of the
Alfvén surface. By sweeping a large parameter space in, for example, magnetic
field strengths, geometry, and wind thermal properties, numerical simulations
can provide a more general view of $\dot{J}$, and thus constants that were
previously assumed (e.g., $a$, $n$, etc) can be derived from numerical
studies. Matt et al. (2012) performed a range of 2D simulations for solar-type
stars with different rotation rates $\Omega_{\star}$ and dipolar magnetic
field strengths $B_{{\rm dip},\star}$. The authors found a generalised fit for
their simulations that can be written in terms of constants $k_{1}$, $k_{2}$
and $m$ as

$\dot{J}=k_{1}^{2}B_{{\rm
dip},\star}^{4m}\dot{M}^{1-2m}R_{\star}^{2+4m}\frac{\Omega_{\star}}{(k_{2}^{2}v_{{\rm
esc},\star}^{2}+\Omega_{\star}^{2}R_{\star}^{2})^{m}},$ (71)

with $k_{1}=1.30$, $k_{2}=0.0506$, and $m=0.2177$. The surface escape velocity
is $v_{{\rm esc},\star}=(2GM_{\star}/R_{\star})^{1/2}$. With all values given
in cgs units, then the angular momentum loss rate has units of g cm2 s-2,
which is an energy unit and thus $\dot{J}$ is frequently expressed in erg.

The angular-momentum loss prescriptions shown, for example, in Eqs. (70) and
(71) can then be implemented in stellar evolution codes. Using Eq. (71) is an
advantage over the use of Eq. (70), as discussed in Johnstone et al. (2015a).
However, stellar evolution models still need to adjust the value of $k_{1}$ so
that they can be properly calibrated for the Sun (Gallet and Bouvier, 2013;
Johnstone et al., 2015a; See et al., 2019a). One challenge in such
prescriptions for $\dot{J}$ is that they require a value of $\dot{M}$, which,
as discussed in Sect. 2, is difficult to observationally derive. In
particular, the mass-loss rates are dependent of the thermal properties of the
wind. In wind models that implicitly assume heating, the thermal properties at
the wind base are free parameters (see page 5.1). Models that assume, for
example, that wind temperatures increase with stellar activity (e.g., Ó
Fionnagáin and Vidotto, 2018) predict that mass-loss rates are higher for
young and fast rotating stars.

Would then mass-loss rates increase indefinitely with rotation? As I presented
in Sect. 3, although it is believed that younger stars should have higher
$\dot{M}$, there is still no clear picture from observations (see Fig. 11).
Recent observations, using prominences to probe stellar winds, suggest that
mass-loss rates seem to keep increasing towards young ages (Figs. 8 and 11) –
AB Dor, for instance, is a $\sim 120$ Myr-old star, with 350 times the
present-day solar mass-loss rate, or the even younger LQ Lup ($\sim 25$ Myr),
with $\dot{M}$$\sim$ 4500 times the solar value (Jardine and Collier Cameron,
2019). These findings challenge the results from the astrosphere method, which
indicated that mass-loss rate might actually reduce at ages below $\sim 600$
Myr, with $\pi^{1}$ UMa ($\sim 500$ Myr) showing half the present-day solar
mass-loss rate (Wood et al., 2014). I believe that the answer to this should
come from a combined approach between modelling and observations.

##  6 Implications for the evolution of the solar system planets and
exoplanets

In this Section, I briefly discuss possible implications of the solar wind
evolution on planets in the solar system and on exoplanets orbiting Sun-like
stars.

###  6.1 The evolution of Earth’s magnetosphere

One frequent argument in favour of the larger mass-loss rates of the young Sun
is the lack of a substantial atmosphere in Mars. Given that Mars lacks a
large-scale protective magnetic field similar to that of Earth, the
interaction between the planet and the solar wind is not shielded by the
planet’s magnetic field. As a result, the young Martian atmosphere is believed
to have been eroded by a stronger solar wind (e.g., Kulikov et al., 2007).
This reasoning has been recently challenged in light of ion escape
measurements in terrestrial planets (Brain et al., 2013). Escape rates of O+
in Earth, Mars and Venus are reasonably similar, yet these terrestrial planets
have very different properties, such as gravity, orbital distances and the
presence of a large-scale magnetic field. The fact that there are not two
planets in the solar system that are identical, except for their magnetic
fields, makes it very difficult to isolate how, or if, a large-scale magnetic
field can be an effective atmospheric shield (Brain et al., 2013).

There are two arguments relating to whether a planetary magnetic field might,
or not, protect planetary atmospheres.

•

In favour: The magnetosphere of a planet acts to deflect the stellar wind
particles, preventing its direct interaction with planetary atmospheres and,
therefore, potential atmospheric erosion. If the magnetosphere is too small,
part of the atmosphere of the planet becomes exposed to the interaction with
stellar wind particles. Lammer et al. (2007) suggested that, for a
magnetosphere to protect the atmosphere of planets, planets are required to
have magnetospheric sizes $\gtrsim 2$ times the planet’s radius. This can have
important effects on the habitability of exoplanets, including terrestrial
type planets orbiting inside the habitable zones of their host stars (e.g.,
Lammer et al., 2007; Vidotto et al., 2013; See et al., 2014; Ribas et al.,
2016).

•

Against: However, it is not only the size of the magnetosphere that matters.
Depending on the orientation of the stellar wind (i.e., interplanetary)
magnetic field relative to the planetary one, magnetic reconnection can occur
and the interplanetary magnetic field lines can connect to auroral zones
around the planet’s magnetic pole. Because these field lines are open, stellar
wind plasma can be channeled towards the polar caps, causing local heating
that could ultimately increase atmospheric losses through polar/auroral flows
(Moore and Horwitz 2007. This is potentially more important for heavier
species like O+ than for lighter ones like H+, Brain et al. 2013.) The amount
of stellar wind plasma that is channeled towards the polar caps depends on the
collecting area of the magnetosphere (i.e., its cross section $\sim\pi
r_{M}^{2}$), which is larger for higher magnetic fields (equations will be
derived below).

Thus, while a magnetosphere larger than the extent of the upper atmosphere can
protect atmospheric losses, larger magnetospheres also have greater
‘collecting area’ for stellar wind plasma, which could lead to enhanced
atmospheric losses. It is likely that these two processes coexist and that
their contributions differ throughout the life of the planet. Blackman and
Tarduno (2018) suggested that the competition between the strength of the
stellar wind and the large collecting areas of magnetospheres indicates
whether a planetary magnetic field has a protective effect on the planetary
atmosphere or not. In the case of Earth, they suggested that our planet’s
magnetic field has played a crucial role in protecting our atmosphere.
However, different solar wind characteristic in the future could reverse this
protective effect, where a reduced inflow speed (related to the speed of
magnetic reconnection) and larger planetary magnetospheres would conspire to
generate more atmospheric losses than what an unmagnetised Earth would have
presented.

In a first order approximation, we can estimate the magnetospheric stand-off
distance by setting pressure balance between the incoming stellar wind and the
planetary magnetic pressures. At the planet–stellar wind interaction zone,
pressure balance between the stellar wind (left-hand side) and planetary
magnetosphere (right-hand side) can be written as

$\rho_{w}u_{w}^{2}\simeq\frac{B_{{p},r_{M}}^{2}}{8\pi},$ (72)

where $\rho_{w}$ and $u_{w}$ are the density and speed of the stellar wind at
the planetary orbit, and $B_{{p},r_{M}}$ is the planetary magnetic field
intensity at a distance $r_{M}$ from the planet centre. Assuming that the
planet’s magnetic field is dipolar, with $B_{p,{\rm eq}}$ its surface magnetic
field at the equator, the previous equation can be rewritten so that
magnetospheric size of the planet is given approximately by

$\frac{r_{M}}{R_{p}}\simeq\left[\frac{B_{p,{\rm
eq}}^{2}}{8\pi\rho_{w}u_{w}^{2}}\right]^{1/6}.$ (73)

Equation (73) shows that a large stellar wind pressure acts to reduce the size
of planetary magnetospheres for a given planetary magnetisation. This is the
Chapman–Ferraro equation firstly derived for Earth’s magnetosphere. Note that
this equation needs to be modified for exoplanets in closer orbits, as in
those cases, the magnetic pressure and thermal pressure of the stellar wind,
as well as ram pressure of the planetary atmosphere, might have to be
incorporated in Equations (72) and (73) (Vidotto et al., 2013; Vidotto and
Cleary, 2020).

Recently, Carolan et al. (2019) studied the evolution of Earth’s magnetosphere
during the solar main sequence. For that, they first performed simulations of
the evolution of the solar wind using a magneto-rotator wind model (1.5D MHD).
With that, they were able to model the conditions of the evolving solar wind
at the orbit of the Earth. These conditions were then implemented as external
boundary conditions into 3D MHD simulations of the Earth’s magnetosphere.
Carolan et al. (2019) showed that, as the Sun spun down and the solar wind
density and velocity values at Earth’s orbit evolved, the Earth’s
magnetosphere gradually increased following a power-law
$r_{M}\propto\Omega_{\star}^{-0.27}$ up until the Sun reached a rotation of
$\Omega_{\star}\sim 1.4\Omega_{\odot}$. After that (i.e., for lower rotations
$\Omega_{\star}\lesssim 1.4\Omega_{\odot}$), their model predicted a more
rapid increase of the Earth’s magnetosphere with
$r_{M}\propto\Omega_{\star}^{-2}$. The piece-wise evolution they predict for
the Earth’s magnetosphere lies on assumptions for the temperature of their
stellar wind models.

Carolan et al. (2019) used a simplified stellar wind model, which allowed them
to predict the evolution of the solar wind for a large interval of time. More
sophisticated wind models have opted for a more focused approach. For example,
Sterenborg et al. (2011) used modified solar magnetograms as input for their
3D MHD simulations for how the solar wind was 3.5 Gyr ago. Their wind results
were then implemented in simulations of Earth’s “paleomagnetosphere”, in which
the strength and orientation of Earth’s magnetic field were varied. Depending
on the specific conditions of the Earth’s magnetic field, Sterenborg et al.
(2011) showed that the paleomagnetosphere stand-off distance could have been
as small as 4.25 Earth radii.

As the solar wind interacts with Earth’s magnetosphere, not only the
magnetospheric stand-off distance changes, but also does the area of Earth’s
polar cap containing open magnetic field lines. Sterenborg et al. (2011)
showed that at 3.5Ga the Earth’s open-field polar cap area was larger for
several conditions they explored. The only exception was for when Earth’s
dipolar field had a substantial tilt, or when the solar wind magnetic field
was parallel to the Earth’s magnetic field at the interaction zone. In this
case, the Earth’s magnetosphere was closed, without any open-field polar cap
area. Appendix B in Carolan et al. (2019) discusses the differences between
open and closed planetary magnetospheres, which can more clearly be seen in
their Fig. B1.

Another focused work on the young Sun-Earth study was presented in do
Nascimento et al. (2016), who developed a 3D study of the magnetohydrodynamic
environment surrounding $\kappa^{1}$ Cet, which has been recognised as a good
proxy for the young Sun at the age when life arose on Earth. According to the
authors, the mass-loss rate of $\kappa^{1}$ Cet is about 50 times larger than
the present-day Sun (see also Airapetian and Usmanov, 2016). Due to this
larger mass-loss rate, the ram pressure of the young solar wind impacting on
the magnetosphere of the young Earth was larger than the present-day values.
do Nascimento et al. (2016) showed that the magnetospheric sizes of the young
Earth should have been reduced to approximately half to a third of the value
it is today ($\sim 11$ Earth radii). The relatively large size of the early
Earth’s magnetosphere may have been the reason that prevented the volatile
losses from Earth and created conditions to support life (Tarduno et al.,
2010; do Nascimento et al., 2016). This is opposed to the fate of a young,
non-magnetised Mars, which could have lost its atmosphere due to a stronger
younger solar wind (Kulikov et al., 2007).

###  6.2 Implication for exoplanets and new lessons for the solar system

There are several ways in which the solar wind evolution, and the linked
activity evolution, can affect planets. One way, already mentioned above,
refers to the direct interaction between the evolving solar wind and the
planetary magnetosphere, in the case of magnetised planets. In the case of
unmagnetised or weakly magnetised planets, the magnetic pressure on the right-
hand side of Eq. (72) should be replaced by the thermal and ram pressure of
the planetary atmosphere. In particular, planets that receive a large amount
of high-energy irradiation can have evaporating atmospheres. This happens in
either planets that orbit old stars at close distances, or planets with larger
semi-major axis orbiting more active (young) stars. These two conditions lead
to enhanced XUV energy flux deposition on the top layers of planet
atmospheres, which leads to an increased heating, causing atmospheres to
inflate and to more easily evaporate (e.g., Lammer et al., 2003; Baraffe et
al., 2004; Sanz-Forcada et al., 2011). Since the XUV emission of solar-type
stars is observed to evolve (Sect. 4.3), exoplanetary evaporation is also
expected to evolve in time.

This implies that the history of the host star plays an essential role in the
evolution of atmospheric evaporation of their planets. For example, a young
host star rotating more slowly than other stars at the same age would generate
less EUV radiation and thus their planets would evaporate less and end up with
a higher atmospheric mass fraction. On the contrary, a young host star that is
a fast rotator would emit more EUV radiation, which could potentially lead to
a planet with a reduced atmosphere. Tu et al. (2015) showed that a 0.5-Earth-
mass planet orbiting at 1 au around a solar-mass star will present an
atmosphere with a very different hydrogen content at 4.5 Gyr, depending on
whether the host star was a slow or fast rotator during its youth. Starting
with an initial hydrogen atmosphere of $0.005$ Earth mass, they showed that
the atmosphere of the terrestrial planet is entirely lost before an age of 100
Myr if the host star begins its main-sequence evolution as a rapid rotator,
while if the host star was a slow rotator, at 4.5 Gyr, the planet would still
have about 45% of its initial atmosphere.

Because the ‘final’ (i.e., observed) planetary atmospheric fraction depends on
the EUV history of the star, Kubyshkina et al. (2019a) suggested that
detecting the atmospheres of planets today can provide a way to probe the
evolutionary path of the host star. Kubyshkina et al. (2019b) applied this
model to the Kepler-11 system, formed by a star that is very similar to the
Sun (in mass and age) and that has six known transiting planets with orbital
distances between 0.09 and 0.5 au. Given that these six close-in planets seem
to have retained hydrogen-dominated atmospheres, these authors concluded that
this Sun-like star was a slow rotator at its youth. In a similar reasoning,
models using instead the Earth and Venus atmospheric composition were
conducted by Lammer et al. (2020), who showed that one possible explanation
for the noble gas composition in Earth’s and Venus’ atmospheres could be due
to the Sun being a slow to moderate rotator at young ages.

Works like these demonstrate that modelling star-exoplanet interactions could
open up new avenues for us to progress in our understanding of the solar
system, and the Sun and solar wind evolutions.

Another important consideration for the young solar system is that the wind of
the young Sun might have had an increased rate of CMEs and even might have
been CME-dominated at an early point in time (see Sect. 2.5). Although CME
rates decrease with age, due to geometrical effects, close-in planets interact
much more frequently with stellar CMEs than planets at larger orbits (Kay et
al., 2016). The CME itself might also be highly affected by the presence of a
close-in planet, given that the CME does not have enough time to expand before
it encounters the close-in planet (Cohen et al., 2011b). CMEs cause
perturbations in stellar wind conditions, which can then lead to changes in
the magnetospheres and extended atmospheres of exoplanets (Khodachenko et al.,
2007; Cherenkov et al., 2017). CME-planet interaction might be more common in
planets orbiting M dwarfs, as these stars remain active for a long part of
their lives (West et al., 2008; Irwin et al., 2011) and have high flare and
CME rates (Vida et al., 2016, 2017). M dwarf stars are the prime targets for
detecting terrestrial planets in their habitable zone. However, the
consequence of their high rate of energetic events might affect habitability
of M dwarf planets (Airapetian et al., 2020).

Although stellar winds constantly interact with exoplanets, exoplanets do not
interact with every stellar CME, because the planet may not lie along certain
CME trajectories. In the Sun, CMEs are observed within $\pm 30^{o}$ latitude,
corresponding to the active region belt (Gopalswamy et al., 2019). However, in
young, fast rotating stars, active regions are often located at high, polar-
region latitudes (Marsden et al., 2006; Jeffers et al., 2007). If CMEs
originate in active regions, it is therefore more likely that CMEs in young
stars could emerge at high latitudes and be directed above/below the
equatorial plane. Considering planet formation in the equatorial plane, this
could mean that young planets might at the end not be strongly affected by
CMEs, even though their host stars are expected to have higher CME rates.
Future research in this area might shed more light into the effects of CMEs in
young planetary systems.

Regardless of whether exoplanets lie along CME trajectories, they can
nevertheless be affected by the almost-instantaneous increase in irradiation
caused by an eventual CME-associated flare, since this emission covers a wider
solid angle. An increase in irradiation input caused by a flare can lead to
increase in evaporation in close-in gas giants (Cherenkov et al., 2017;
Bisikalo et al., 2018; Hazra et al., 2020). Transit observations of the hot
Jupiter HD189733b showed an enhancement of atmospheric evaporation that took
place 8h after the host star flared. Lecavelier des Etangs et al. (2012)
suggested that the violent event and the enhanced evaporation are potentially
related to each other. The remaining open question is whether it is the
increase in XUV emission caused by the flare or the interaction between an
unseen associated CME and the exoplanet (or even both) that caused the
increase in the observed evaporation rate.

###  6.3 Evolution of the heliospheric size and implications for the
penetration of Galactic cosmic rays

The solar wind expands into the interstellar medium (ISM), giving rise to the
heliosphere. Analogously to the Chapman–Ferraro equation for calculating the
Earth’s magnetospheric stand-off distance, one can use pressure balance
between the solar wind and ISM to derive the size of the heliosphere. The ram
pressure of the solar wind can be rewritten as $P_{\rm ram}=\rho
u^{2}=\dot{M}u_{\infty}/(4\pi r^{2})$, where I assumed that for larger $r$,
the wind has reached terminal speed $u_{\infty}$. Thus, pressure balance leads
to

$\frac{\dot{M}u_{\infty}}{4\pi r^{2}}=\rho_{\rm ISM}u_{\rm
ISM}^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\to\,\,\,\,\,\,\,\,\,\,\,\,\,r=\left(\frac{\dot{M}u_{\infty}}{4\pi\rho_{\rm
ISM}u_{\rm ISM}^{2}}\right)^{1/2}$ (74)

where the subscript ‘ISM’ refer to ISM ‘wind’ quantities. If one assumes that
the ISM density and velocity have not changed significantly over the past 4.5
Gyr, the larger mass-loss rates of the young Sun implies that the heliosphere
was larger in the past. Rodgers-Lee et al. (2020) estimated that heliospheric
size of the solar wind extended out to 950 au when the Sun had an age of 1 Gyr
and could have ranged between 1300 and 1700 au at $t=600$ Myr, i.e., more than
a factor of ten larger than the present-day value of 122 au.

The size of the heliosphere, as well as the solar wind conditions, have
consequences for the propagation of cosmic rays. Galactic cosmic rays are
suppressed as they travel through the solar wind all the way to the Earth –
this “suppression” is usually referred to as cosmic ray “modulation”. I refer
the reader to Potgieter (2013) for a comprehensive review on the modulation of
cosmic rays in the heliosphere. In the framework of a diffusive transport
equation for the Galactic cosmic rays, there are three main competing
processes in place: the diffusion of cosmic rays into the solar system, and
their advection, in momentum and in space, which acts to suppress the flux of
Galactic cosmic rays reaching Earth. These processes depend on the level of
turbulence present in the solar wind magnetic field and on the wind velocity,
both of which evolve in time.

Using the solar wind evolution model of Carolan et al. (2019), Rodgers-Lee et
al. (2020) showed that, at the Earth’s orbit, the advective processes (pushing
cosmic rays out) are relatively more important than diffusive processes
(allowing them in) in the Sun’s past, for GeV and MeV cosmic rays. This means
that cosmic rays of these energies are more suppressed and thus fewer of those
have reached Earth in the past, compared to present-day values. This is seen
in Fig. 26, which shows the modelled evolution of Galactic cosmic ray spectrum
at Earth for a range of solar rotation rates. The black solid line is the
cosmic ray spectrum at the local interstellar medium (Vos and Potgieter,
2015). The black dashed-line is the present-day value (which was calibrated to
present-day measurements) at Earth’s orbit, at approximately solar minimum.
The red lines and shaded area represent a range of rotation values that the
Sun could have had at 600 Myr, i.e., the predicted flux of Galactic cosmic
rays can change by more than one order of magnitude depending on whether the
Sun was a fast/slow rotator. Rodgers-Lee et al. (2020) considered a 1D
propagation model for cosmic rays. An interesting future study would be to use
a 3D cosmic ray propagation model, in 3D wind simulations that include ZDI
observations, similar to the approach adopted in Cohen et al. (2012).

Figure 26: Evolution of the Galactic cosmic ray spectrum at Earth’s orbit. The
figure above shows Galactic cosmic ray differential intensity as a function of
their kinetic energies. The different curves refer to different solar rotation
rates normalised to present-day value (here denoted by $\Omega_{0}$). Each
curve thus represents a different age, which ranges from about 6 to 1 Gyr from
the top to the bottom dashed lines. The black dashed line is thus for the
present-day value (1$\Omega_{0}$) at Earth’s orbit. The black solid line is
the cosmic ray spectrum at the local interstellar medium. Cosmic rays are
suppressed (i.e., modulated) as they travel through the solar wind and, given
the solar wind evolution, the spectrum of Galactic cosmic rays at Earth
changes with time. The red lines and red shaded area represent a range of
rotation values that the Sun could have had at 600 Myr. Figure from Rodgers-
Lee et al. (2020)

With the knowledge of the spectrum of cosmic rays arriving at the top of
Earth’s atmosphere, further modelling can then be conducted to investigate
variations of high-energy particle fluxes into the young Earth’s atmosphere
(e.g., Grenfell et al., 2007; Stadelmann et al., 2010; Rimmer and Helling,
2013; Grießmeier et al., 2016). Interestingly, it has been suggested that the
variation of cosmic ray flux over the past 3 billion years coincide with
periods of glaciations at Earth (Svensmark 2006, see also Shaviv 2002).
Another interesting application, not discussed in this review, is to
investigate cosmic ray propagation through the winds of M dwarf stars
(Grießmeier et al., 2005, 2015). These stars are currently among the main
targets in search for habitable planets.

##  7 Final remarks and open questions

Over the years, we have collected a large amount of information on the solar
wind, in an incredible level of detail. All this information, however, can
only tell us about how the solar wind looks like now. To understand the past,
and future, evolution of the solar wind, we rely on observations from other
suns in the Universe, i.e., stars that resemble the Sun and are at different
evolutionary stages. In theory, connecting the observations of winds of
different solar-like stars under one unique evolutionary sequence would tell
us how the wind of our own star has evolved. In practice, however, this is not
straightforward.

One reason for this is that detecting winds of solar-like stars is a very
challenging task. Direct detection of winds of other Suns has not been
possible and the Sun still remains the only star for which the wind has been
probed in situ. However, over the years, a small number of clever techniques
have been developed to indirectly derive stellar wind properties, such as
mass-loss rates and speeds. Overall, these techniques have shown that the
solar wind had a higher mass-loss rate for the past $\sim$3.5–4 billion years.
Before that, from the time the Sun started its evolution in the main sequence
until about 600–800 Myr, clues point in different directions.

It has been suggested that the young solar wind had a lower mass loss rate,
probed in the lower-than-expected mass-loss rate of $\pi^{1}$ UMa, a 500-Myr-
old solar analogue, which has only half the present-day solar mass-loss rate
$\dot{M}_{\odot}$ (Wood et al., 2014). In particular, $\pi^{1}$ UMa does not
fit the trend of $\dot{M}$–$F_{X}$ relation built from the astrosphere wind
detection method, that older solar-like stars seem to fall on. However, a
recent method for detecting winds of rapidly rotating stars, using prominences
to probe the wind mass-loss rate, showed that young Suns could have much
higher mass-loss rates, with the solar analogue AB Dor ($\sim 120$ Myr)
presenting a mass-loss rate of $350\,\dot{M}_{\odot}$ and the even younger LQ
Lup ($\sim 25$ Myr), with $4500\,\dot{M}_{\odot}$ (Jardine and Collier
Cameron, 2019).

One way to shed light on the solar wind evolution is to look at physical
characteristics known to be intimately related to winds of solar-like stars:
rotation, magnetism and activity. These three ‘ingredients’ are related to the
evolution of the solar wind in a feedback loop that I summarised in Fig. 1.
The stellar wind carries away angular momentum, which leads stars to spin
down. A decrease in rotation rate implies that the dynamo weakens, generating
weaker magnetic fields. This affects the stellar wind and the amount of
angular momentum it can carry away. Then, the loop restarts. As a result,
rotation, magnetism and its proxy in the form of stellar activity decrease
with time.

These ingredients, which are better observationally constrained than wind
properties, can be implemented in theoretical models. Models of winds of
solar-like stars have been mainly developed in the magnetohydrodynamics (MHD)
framework, which treats the wind as a magnetised fluid. Within this framework,
models can be broadly divided into ones that calculate explicitly heating
deposition (due to dissipation of MHD waves mostly) and those that assume the
plasma has already been heated to MK temperatures, and thus energy deposition
is treated implicitly. Each model set has its advantages and drawbacks. The
first group has a more sophisticated treatment of the physical wind driving
mechanism, but is usually computed along a magnetic flux tube, neglects
rotation, and extend only out to a few solar radii. The second group has a
less sophisticated treatment of the physics, which implies that they can be
computed over the entire stellar surface, extend out to larger stellar
distances and include rotation.

For studying the evolution of angular momentum, the second group of models are
usually adopted. In that framework, it is fascinating to see that the early
model of Weber and Davis (1967), which considers a simple split-monopole
magnetic field embed in a thermally-driven wind of a rotating Sun already does
a good job in reproducing the observed relation $\Omega_{\star}\propto
t^{-1/2}$ for older stars (Sect. 5.3). In this sense, stellar evolution
models, calibrated to the observed rotation distribution of stars in open
clusters, have become an important aid for constraining the solar wind
evolution.

In this review, I focused on the solar wind evolution from the early main
sequence until today. That does not mean that what lies ahead is less
interesting! The Sun is about half-way through its main-sequence journey, and
will have thus another $\sim 4$ Gyr worth of evolution in a relatively calm
way. When the hydrogen fuel in its core is finally exhausted, the Sun will
evolve off the main sequence and become a red giant. The timescales of this
phase and subsequent ones become shorter and thus the Sun will pass through
these phases very quickly in comparison to the long evolution during its main
sequence. Eventually, the Sun will end its life as a white dwarf and cool down
indefinitely. Certainly, the solar wind will change radically. In the red
giant phase alone, the solar wind will likely be cooler ($\lesssim 10^{4}$ K),
with lower velocities, but higher densities, leading to mass-loss rate that
will be several magnitudes higher than today (Wood et al., 2016). Solar
rotation will change considerably due to its radius expansion. Given that
rotation also seems to be related to magnetism in evolved stars (Aurière et
al., 2015), the Sun’s magnetic field will change as well.

The evolution of the solar wind has applications that goes beyond
understanding stellar rotational evolution – it affects planets and
exoplanets, which at the end of the day are embedded in the outward-streaming
plasma from their host stars. A strong solar wind in the past is often cited
to explain the lack of a substantial atmosphere in Mars today – planetary
atmospheric erosion can be enhanced under strong stellar wind conditions. Such
strong stellar wind conditions are also faced by close-in exoplanets, even
when orbiting old and less active stars. Additionally, the Earth’s
magnetosphere is believed to have been smaller than it is today, as a result
of a strong stellar wind. Similarly, the strong solar wind in the past was
able to push further the boundary with the ISM – the heliosphere was likely
larger in the past. A different solar wind condition and much more expanded
heliosphere would then have likely reduced the amount of Galactic cosmic rays
reaching Earth. Both of these (Earth’s magnetosphere and cosmic ray flux) are
believed to be important factors for life formation and sustainability in this
planet.

I hope to have shown in this review that significant progress in understanding
the evolution of the solar wind is being made. Nevertheless this is currently
a “living” area of research and, as such, still many open questions remain.
Understanding the physical processes heating and driving the present-day solar
wind could provide a step further in modelling the solar wind at different
ages. Missions like NASA’s Parker Solar Probe and ESA’s Solar Orbiter will
obtain in situ measurements of the solar wind at unprecedented close
heliospheric distances. These regions are where the bulk of the acceleration
of the solar wind takes place, and thus these missions are expected to soon
shed light on the physical mechanisms that accelerate the solar wind. As an
added bonus, these close heliospheric distances coincide with the regions
where many close-in exoplanets are located – these missions will thus provide
information about the environment at the orbits of close-in exoplanets, which
will aid in further understanding how extreme wind conditions can affect
exoplanets, and potentially, link this to how the stronger solar wind in the
past interacted with the young solar system planets.

###### Acknowledgements.

I thank the editors for inviting me to write this review and the reviewers
(Manuel Güdel and anonymous) for their very thorough and constructive
feedback. There are several colleagues I would like to warmly thank for
fruitful discussions over the years: Moira Jardine, Jean-Francois Donati, Rim
Fares, Pascal Petit, Victor See, Julien Morin, Luca Fossati, Vincent Bourrier,
Sandra Jeffers, Colin Folsom, Lisa Lehmann, Joe Llama, Scott Gregory, Andrew
Cameron, Stephen Marsden. I thank Gopal Hazra for providing the data used Fig.
18. I acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 817540, ASTROFLOW).

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