∎

11institutetext: A. A. Vidotto 22institutetext: School of Physics, Trinity
College Dublin, The University of Dublin, Dublin-2, Ireland
22email<EMAIL_ADDRESS>
# The evolution of the solar wind

Aline A. Vidotto

(Received: date / Accepted: date)

###### Abstract

How has the solar wind evolved to reach what it is today? In this review, I
discuss the long-term evolution of the solar wind, including the evolution of
observed properties that are intimately linked to the solar wind: rotation,
magnetism and activity. Given that we cannot access data from the solar wind 4
billion years ago, this review relies on stellar data, in an effort to better
place the Sun and the solar wind in a stellar context. I overview some clever
detection methods of winds of solar-like stars, and derive from these an
observed evolutionary sequence of solar wind mass-loss rates. I then link
these observational properties (including, rotation, magnetism and activity)
with stellar wind models. I conclude this review then by discussing
implications of the evolution of the solar wind on the evolving Earth and
other solar system planets. I argue that studying exoplanetary systems could
open up new avenues for progress to be made in our understanding of the
evolution of the solar wind.

###### Keywords:

Solar wind Stellar winds and outflows Stars: activity, magnetism, rotation
Stellar winds: observations and models

††journal: Living Reviews in Solar Physics

###### Contents

  1. 1 Introduction: placing the Sun in a stellar context
  2. 2 How to detect winds of solar-like stars
    1. 2.1 Detecting astrospheric absorption in Ly-$\alpha$
    2. 2.2 Detecting free-free radio emission from winds
    3. 2.3 Using exoplanets to probe stellar winds
    4. 2.4 Using prominences to probe stellar winds
    5. 2.5 Detecting coronal mass ejections (CMEs) through type II radio bursts
    6. 2.6 Propagation/suppression of radio emission from a point source (planet) embedded in a stellar wind
    7. 2.7 Constraining stellar winds (upper limits) though X-ray emission
    8. 2.8 Detecting charge-exchange induced X-ray emission
    9. 2.9 Accretion onto white dwarfs as probe of the wind of secondary companion
  3. 3 An observed evolutionary sequence for mass-loss rates?
  4. 4 Observed evolution of the main ingredients of the solar wind
    1. 4.1 Evolution of magnetism
    2. 4.2 Evolution of stellar rotation
    3. 4.3 Evolution of coronal activity
  5. 5 Where are we in terms of modelling?
    1. 5.1 Overview on the different treatments used in stellar wind models
    2. 5.2 Brief overview of MHD stellar wind theory
      1. 5.2.1 Thermally-driven winds
      2. 5.2.2 The magneto-rotator wind
    3. 5.3 Modelling the evolution of the Sun’s rotation
      1. 5.3.1 A simple model for the evolution of stellar rotation and its relation to gyrochronology
      2. 5.3.2 Magnetic braking laws: prescriptions for angular momentum-loss rate
  6. 6 Implications for the evolution of the solar system planets and exoplanets
    1. 6.1 The evolution of Earth’s magnetosphere
    2. 6.2 Implication for exoplanets and new lessons for the solar system
    3. 6.3 Evolution of the heliospheric size and implications for the penetration of Galactic cosmic rays
  7. 7 Final remarks and open questions

##  1 Introduction: placing the Sun in a stellar context

It is fair to say that the Sun is the best studied star in the whole Universe:
we can measure its rotation, magnetic activity, composition, size,
irradiation, and wind properties with accuracies like no other star in the
Universe. However, all this information just tells us about how the Sun looks
like _now_. To understand the past, and future, evolution of the Sun,
including its wind, magnetism, activity, rotation, and irradiation,
astronomers rely on information from other “suns” in the Universe at different
evolutionary stages. In a broad sense, these other “suns” belong to the group
of solar-like stars.111Usually, stars similar to the Sun are classified into
three groups: solar twins, which are a subset of so-called “solar analogues”,
which are in turn a subset of “solar-like stars” (Meléndez et al., 2010).
Solar twins are stars that are (almost) identical to the Sun, with
indistinguishable temperature, gravity, age and composition from solar
(Ramírez et al., 2009; Porto de Mello et al., 2014). Solar twins are extremely
rare. The second group, those of solar analogues, contains stars with spectral
types between G0 and G5, within a factor of 2–3 in metallicity from the Sun
(Meléndez et al., 2010). Finally, the largest group is formed by solar-type
stars, which are stars with spectral types from late F to early K (Ramírez et
al., 2009). The third group has the least stringent classification allowing
not only for different compositions, but also for different ages. When dealing
with the evolution of the solar wind, I will draw stars from the group of
solar-like stars. These are the definitions used in this review, but note that
they can differ in the literature.

Stellar evolution models predict that the Sun has evolved, since its pre-main
sequence phase, from a spectral type K to its current G2 classification. It
will leave the main sequence phase in another $\sim 4$ Gyr, when it will
become a red giant star, go through a planetary nebula phase, until it will
finish its days as a white dwarf that will cool down indefinitely. The large
variation in solar photospheric temperature, radius and luminosity during the
Sun’s evolution is also accompanied by variations in the properties of the
solar wind – this outflow of particles, embedded in the solar magnetic field,
that propagates throughout the solar system. Although it is unlikely that the
solar wind has been able to remove a significant amount of mass from the Sun,
the solar wind has been able to remove significant amounts of angular momentum
via magnetic field stresses. For this reason, the solar wind has played a
fundamental role in the Sun’s evolution, as it regulates solar rotation.

In the sketch I show in Fig. 1, I exemplify the “life-cycle” of the wind of an
isolated solar-like star. I start with the text bubble #1: winds of solar-like
stars are magnetic in nature, and are thus able to carry away a significant
amount of angular momentum from the star. Consequently, the star spins down as
it evolves (#2). Because of this variation in surface rotation, there is a
redistribution of internal angular momentum transport, which changes the
interior properties of the star (#3). With a different internal structure, the
dynamo that is operating inside the star changes, changing therefore the
properties of the emerging magnetic fields (#4). With a new surface magnetism,
the stellar wind also changes (back to text bubble #1). This cycle repeats
itself during the entire main-sequence lifetime of solar-like stars.
Therefore, as an isolated solar-like star ages, its rotation decreases (Kraft,
1967; Skumanich, 1972) along with its chromospheric–corona activity (Ribas et
al., 2005; Mamajek and Hillenbrand, 2008), and magnetism (Vidotto et al.,
2014a). These three parameters are key ingredients in stellar wind theories.
As a consequence of their decrease with age, winds of main-sequence solar-like
stars are also expected to decrease with time (Wood, 2004; Jardine and Collier
Cameron, 2019).

Figure 1: The big picture: evolution of winds of cool dwarf stars. As the star
ages, its rotation and magnetism decrease, causing also a decrease in angular
momentum removal. In the images, I highlight some of the areas to which wind-
rotation-magnetism interplay is relevant. Image reproduced with permission
from Vidotto (2016a).

Two physical quantities often quoted in stellar wind studies are the mass-loss
rate $\dot{M}$ and terminal velocity $u_{\infty}$, which is the speed winds
asymptotically reach at large distances. These two quantities carry a lot
information about stellar wind driving. Thus, they are key to inform wind
models. Unfortunately, for solar-like stars, these physical quantities are
very challenging to measure, because these winds are very tenuous and do not
provide strong, detectable observational signatures, such as a P Cygni line
profile, that is common in stars with denser winds. Figure 2 shows an HR
diagram, colour-coded by $\dot{M}$, where we see that estimated mass-loss
rates in the lower main-sequence range between $\sim 10^{-16}$ to $\sim
10^{-12}\,M_{\odot}~{}{\rm yr}^{-1}$. For comparison, the solar wind mass-loss
rate, which is adopted throughout this review, is $\dot{M}_{\odot}=2\times
10^{-14}\,M_{\odot}~{}{\rm yr}^{-1}$. As these stars evolve off of the main
sequence, their winds become more massive, and mass-loss rates increase by
several orders of magnitudes. This increase in $\dot{M}$ is caused by a change
in wind driving, which likely has stronger contributions of thermal driving
forces in the lower main-sequence phase, as these stars show signs of hot
coronae (seen in X-rays). As the low-mass stars evolve to become red giants in
the post-main sequence phase, thermal driving becomes less important,
resulting in cooler winds (moving from the yellow region in Figure 2 to the
top red region). The winds of evolved low-mass stars that do not show signs of
coronae are likely driven by mechanical forces, such as for example pulsations
and wave-driven mechanisms. In the lower main sequence, cool stars have hotter
winds (on the order of $10^{6}$ K) and low mass-loss rates, while on the top
right corner of the HR diagram cool stars have colder winds that can reach
temperatures of $10^{4}$ K and maybe even lower, and high mass-loss rates.
There is a relatively smooth transition between these two groups, with stars
that show signs of weak/warm coronae belonging to an intermediate ‘hybrid’
group, and they perhaps have a combination of wind driving mechanisms (thermal
and mechanical, O Fionnagain et al. 2021).

Figure 2: An overview of mass-loss rates (colour coded) in the cool-star HR
diagram. Winds of cool stars evolve from hot ($\sim 10^{6}$ K) and tenuous
(‘hot corona’) to cold ($\sim 10^{4}$ K) and denser (‘no corona’). Evolved
low-mass stars that show weak or sporadic signatures of a hot corona are
denoted ‘warm/hybrid’. The zero-age main sequence is shown by the grey line.
Image reproduced with permission from Cranmer and Winebarger (2019), copyright
by Annual Reviews.

To derive an accurate picture of the solar wind evolution, a combined
observational-theoretical approach is more suitable. Ideally, we would like to
inform models by measuring winds of Sun-like stars at different evolutionary
stages, thus sampling the physical properties of winds from young to old suns.
In practice, however, this is not a simple task, as observing winds of cool
dwarfs is currently very challenging. I start this review with an overview of
observations of winds of solar-like stars (Sect. 2). Using these results, I
attempt to derive an evolutionary sequence of the solar wind mass-loss rate in
Sect. 3. One cannot talk about the solar wind evolution without talking about
evolution of the key physical ingredients of the solar wind: rotation,
magnetism and activity. Thus, in Sect. 4, I introduce some key observations
that have allowed us to infer the evolution of these three parameters. As we
will see, data sampling activity, rotation and even magnetism are more
abundant in the literature than observational data sampling wind evolution. In
Sect. 5, I present the results of some models investigating the solar wind
evolution. In Sect. 6, I discuss some implications of solar wind evolution on
the past of the solar system, as well as on other extra-solar systems. I
finish this review with a summary and a discussion of open questions in the
field (Sect. 7).

From now on, I will refer to winds of isolated, main-sequence, Sun-like stars
simply as winds or stellar winds. Otherwise, whenever I refer to winds of
other types of stars, or at different evolutionary phases, I will specify
(e.g., winds of red giants, or winds of pre-main sequence stars).

##  2 How to detect winds of solar-like stars

Although winds of low-mass stars are challenging to detect, astronomers have
come up with some clever methods to infer the presence of a wind as well as to
derive their physical properties. Table 1 summarises some proposed methods to
detect winds of Sun-like stars, which I will discuss further. As I will show
next, different methods perform better for different systems, so a range of
detection methods can help us probe the range of winds of Sun-like stars at
different ages.

Table 1: Some proposed methods to detect winds of Sun-like stars. Section number, Method | Requirement for detection | Key references
---|---|---
2.1 Astrospheric Ly-$\alpha$ | partially neutral ISM, $\lesssim 10$pc | Wood (2004)
2.2 Radio free-free emission | denser winds and/or radio flares | Lim and White (1996); Guedel (1992)
2.3 Exoplanets as probes | evaporating planet | Vidotto and Bourrier (2017)
2.4 Prominences in H-$\alpha$ | fast rotation | Jardine and Collier Cameron (2019)
2.5 Detection of CME-dominated winds | fast CME associated with a flare | Crosley and Osten (2018)
2.6 Propagation of radio emission | point source (planet?) within wind | Vidotto and Donati (2017)
2.7 X-ray emission from the stellar wind | hot coronal winds | Lim and White (1996); Llama et al. (2013)
2.8 Charge-exchange induced X-rays | partially neutral ISM | Wargelin and Drake (2001, 2002)
2.9 Accretion onto white dwarfs | close binary with cool dwarf secondary | Debes (2006)

###  2.1 Detecting astrospheric absorption in Ly-$\alpha$

A stellar wind propagates from the star all the way to the interstellar medium
(ISM), forming a similar structure as that observed around the solar wind,
known as an “astrosphere”, in analogy to the heliosphere around the Sun.
Figure 3 sketches an astrosphere, which has similar properties as the
heliosphere. The inner part of the astrosphere is surrounded by the
termination shock, where the stellar wind decelerates from supersonic to
subsonic velocities. Further out, the astropause separates the stellar wind
and the ISM flow. Surrounding the astrosphere, if the relative motion between
the stellar wind and the ISM is supersonic, a bow shock forms. The enhancement
in hydrogen density, known as the hydrogen wall, occurs between the astropause
and the bow shock. Therefore, after Ly-$\alpha$ photons are emitted from the
star, they have a long journey until they reach us. They first cross the
stellar wind and the hydrogen wall around the astrosphere. Then comes the long
journey through the ISM itself. Finally, they cross the hydrogen wall around
the heliosphere and traverse the interplanetary space (solar wind) until they
reach us and we can observe them (in space, above the Earth’s atmosphere).
During each stage of this journey, neutral hydrogen present in these different
regions absorb the Ly-$\alpha$ spectral line at different velocities. As a
consequence, the original Ly-$\alpha$ line is severely altered. The hydrogen
wall, in particular, is key for quantifying the stellar wind. In this region,
ISM and stellar wind materials mix together, which can lead to charge-exchange
between the ionised stellar wind material and the neutral component of the
ISM. As a result, stellar wind particles are neutralised, but they still
retain their high velocity and temperature. This hot neutral hydrogen, which
causes substantial absorption in the Ly-$\alpha$ line, provides the signature
required to indirectly quantify stellar winds.

Figure 3: Sketch of the ISM (left) interacting with a stellar wind (right) and
giving rise to an astrosphere, which is surrounded by a hydrogen wall and
possibly a bow shock. The hydrogen wall, which is an enhancement of hydrogen
density, is located between the bow shock and the astropause. Figure adapted
from Ó Fionnagáin (2020).

By modelling these different absorption components, one can then infer the
column density of neutral hydrogen in the wall surrounding an astrosphere
(Wood and Linsky, 1998). With assistance from hydrodynamical models of the
astrosphere-ISM interaction, it is then possible to estimate the wind ram
pressure at the site of the interaction (Wood et al., 2001). Given that the
wind ram pressure is linked to the wind mass-loss rate as $P_{\rm
ram}\propto\dot{M}u_{\infty}$, it is then possible to derive $\dot{M}$ if
$u_{\infty}$ is known. In the case of astrospheric measurements, a unique
$u_{\infty}=400$ km/s has been assumed for all observed stars, allowing the
mass-loss rate of the observed star to be inferred. For more details, I refer
the reader to the method review in Wood (2004).

The “astrosphere method” has been the most successful method for measuring
mass-loss rates of winds of solar-like stars, with nearly 20 measurements to
date (Wood, 2018), from stars ranging from spectral types M to F on the main
sequence, and a few more evolved cool stars as well. The reason this method
has not provided measurements for a larger sample of stars is that it has a
sweet spot for optimal performance. Firstly, the ISM needs to be neutral, or
at least partially neutral, for charge-exchange to take place, and thus the
astrospheric absorption signature to occur in the Ly-$\alpha$ line. This means
that once the ISM becomes fully ionised, beyond 10–15 pc, the method becomes
inapplicable (Redfield and Linsky, 2008). Secondly, if the stars are beyond
$\sim 10$ pc from us, the ISM column density can be too large and the ISM
absorption obscures the astrosphere absorption. For these reasons, Ly-$\alpha$
observations of nearly all stars beyond 10 pc have turned out to be non-
detections (Wood et al., 2005b).

Figure 4 summarises the detections of winds of solar-like stars (filled red
circles) using the astrosphere method. This figure shows that the mass-loss
rate per unit surface area ($\dot{M}/R_{\star}^{2}$) varies as a function of
X-ray flux $F_{X}$ as $\dot{M}/R_{\star}^{2}\propto F_{X}^{1.34}$ (shaded
line). Because X-ray flux is a measure of stellar activity, $F_{X}$ can be
used as a rough proxy for age – stars with relatively large X-ray fluxes tend
to be younger than stars with lower X-ray fluxes. For stars with $F_{X}\gtrsim
10^{6}\mathrm{\ erg\ s^{-1}\ cm^{-2}}$, Wood (2004) proposed the existence of
a wind dividing line, beyond which the power-law fit (shaded line) would cease
to be valid. For solar-like stars, this X-ray flux of $10^{6}\mathrm{\ erg\
s^{-1}\ cm^{-2}}$ roughly corresponds to an age of 600 Myr. Would there be
something happening at around this age that makes the wind mass-loss rate drop
more than 2 orders of magnitude? It has been suggested that the magnetic
topology of stars with $F_{X}\gtrsim 10^{6}\mathrm{\ erg\ s^{-1}\ cm^{-2}}$
suffers an abrupt change that could affect their winds (Wood et al., 2005a),
an idea that is backed-up by solar wind observations (see Sect. 3 for the
solar wind analogy). However, magnetic field reconstructions of young stars do
not show abrupt changes in their field topology, but rather a smooth
transition from a magnetic field with an important toroidal component at young
ages (fast rotation) to a topology that is dominated by a poloidal field at
older ages (Petit et al., 2008; Folsom et al., 2018; See et al., 2015; Vidotto
et al., 2016).

Figure 4: Summary of mass-loss rates for low-mass stars derived from the
astrosphere method. For GK dwarfs (red filled circles), the mass-loss rate per
unit surface area varies as a function of X-ray flux as $\propto F_{X}^{1.34}$
(shaded line). It has been suggested that active (and overall younger) stars
with $F_{X}\gtrsim 10^{6}\mathrm{\ erg\ s^{-1}\ cm^{-2}}$ would have reduced
mass-loss rates, thus giving rise to a ‘wind dividing line’. Some new mass-
loss rates measurements indicate that mass-loss rates of young Suns could
actually remain large (cf. Sect. 3). Figure from Wood (2018).

$\pi^{1}$ UMa, a young solar-analogue star, is at the centre of the discussion
on the existence of the wind dividing line. As can be seen in Figure 4, the
derived astrospheric mass-loss rate is quite low, and yet, the star has a high
$F_{X}$ (Wood et al., 2014). $\pi^{1}$ UMa would, therefore, support the idea
of the existence of a wind dividing line. However, an alternative explanation
is that $\pi^{1}$ UMa might be in a strongly ionised ISM (the star is at a
distance of 14 pc) and the measured Ly-$\alpha$ absorption could be taking
place in a cloud between the star and the observer instead of in the hydrogen
wall (Wood et al., 2014). Indeed, spin down models (see Sect. 5.3) predict
that solar analogues with $\pi^{1}$ UMa’s rotation period and the age should
have a mass loss rate that is about one order of magnitude higher than the
present-day solar wind mass-loss rate (Johnstone et al., 2015a). One way to
clarify this contradiction is to use multiple wind-detection methods applied
to the same star. In fact, radio observations to detect the wind of $\pi^{1}$
UMa have been conducted (Fichtinger et al., 2017). In the next section, I will
discuss this method in more details.

###  2.2 Detecting free-free radio emission from winds

What is unfortunate about the astrosphere method is that the non-detections do
not allow us to derive upper limits for mass-loss rates. Although a non-
detection could indeed be due to very low wind mass-loss rates, it could also
be due to a fully ionised ISM around the astrosphere or an ISM that causes too
much absorption, rendering the Ly-$\alpha$ astrospheric signature
undetectable. On the other hand, the method that I will discuss now, based on
thermal radio emission of winds, is a method that can extract meaningful
information even in cases of non-detection of stellar winds. The main current
disadvantage of this method is that no wind of low-mass star has ever been
detected and, thus, all it has provided so far are upper limits on mass-loss
rates. With the advance of radio technologies and construction of more
sensitive radio telescopes, this could change in the very near future.

The stellar wind is a thermal, ionised plasma and, as such, it emits continuum
bremsstrahlung (free-free) radiation across the electromagnetic spectrum. In
particular, the densest region in a stellar wind (its innermost region) can
emit at radio wavelengths, thus providing a way to directly detect the wind in
radio (Güdel, 2002). For large enough densities, the innermost regions can
become optically thick to radio wavelengths, creating a radio photosphere (Ó
Fionnagáin et al., 2019; Kavanagh et al., 2019), that, if detected, can allow
us to quantify the mass-loss rate of the wind. In this case, the underlying
non-thermal radio emission from the star cannot be seen. On the other hand, a
low-density wind would be radio transparent, and radio (non-thermal) emission
from the stellar surface can pass through the wind unattenuated. Non-thermal
radio emission, such as radio flares, has been detected in M dwarfs and solar-
like stars (e.g., Lim and White, 1996; Fichtinger et al., 2017). These radio-
transparent winds can nonetheless provide important upper limits to wind mass
loss rates. Regardless of whether the wind is optically thin or thick, the
wind itself produces radio emission and that may provide a direct detection
method.

The idea that winds of early-type (hot) stars could emit in radio was proposed
in the seminal works of Panagia and Felli (1975), Wright and Barlow (1975) and
Olnon (1975), which initially evolved from the theory of continuum emission in
HII regions. The main modification to the previous theory was that, contrary
to a constant density approximation adopted in HII regions, stellar winds were
assumed to have a spherically-symmetric density profile of the form
$\rho\propto r^{-2}$, where $r$ is the radial coordinate (this implicitly
assumes a constant wind speed, which, as I will show next, is incorrect in the
innermost region of stellar winds). This modification led to a change in the
shape of the spectrum. In the asymptotic limit of an optically thick plasma
($\tau_{\nu}\gg 1$ at low frequencies $\nu$), the flux density of an HII
region behaves as $S_{\nu}\propto\nu^{2}$, while for a stellar wind, it was
then found that $S_{\nu}\propto\nu^{0.6}$. In the optically thin asymptotic
limit ($\tau_{\nu}\ll 1$ at high $\nu$), the radio flux density of a stellar
wind has the same frequency-dependence as that of an HII region:
$S_{\nu}\propto\nu^{-0.1}$. Later-on, Reynolds (1986) dropped the assumption
of spherically symmetric winds with constant speeds and demonstrated that
spectral indices could range between $-0.1$ and $2$. Indeed, in the low-
frequency, optically thick regime, wind models predict a range of slopes – Ó
Fionnagáin et al. (2019) derive indices from 1.2 to 1.6 in 3D wind
simulations.

Although currently there has been no detection of free-free emission
originating in winds of solar-type stars, radio observations have provided
upper limits of mass-loss rates for a number of objects (Guedel, 1992; Lim and
White, 1996; Lim et al., 1996; Gaidos et al., 2000; Villadsen et al., 2014;
Fichtinger et al., 2017; Vidotto and Donati, 2017). Even for the pre-main
sequence phase, when stellar winds are expected to have higher mass-loss
rates, only upper limits have been derived (e.g., $3\times
10^{-9}\,M_{\odot}~{}{\rm yr}^{-1}$ for the weak-lined TTauri star V830 Tau,
Vidotto and Donati 2017), indicating that this is indeed a tricky detection
with current instrumentation. For stars identified as good solar analogues,
like $\pi^{1}$ UMa, $\beta$ Com, $\kappa$ Ceti, EK Dra and $\xi^{1}$ Ori,
upper limits of mass-loss rates are as low as $3\times
10^{-12}\,M_{\odot}~{}{\rm yr}^{-1}$ and as high as $1.3\times
10^{-10}\,M_{\odot}~{}{\rm yr}^{-1}$ (Gaidos et al., 2000; Fichtinger et al.,
2017). Another complication is that, if the wind has a small mass-loss rate,
its weak emission competes against the expected thermal free-free radio
emission from the chromosphere and gyromagnetic emission from active regions.
For such weak detections, the distinction between a wind signature and the
thermal component from close to the stellar surface requires extremely
accurate radio spectra (Drake et al., 1993; Villadsen et al., 2014).

In a number of simulations of winds of solar-like stars, Ó Fionnagáin et al.
(2019) predicted that the radio flux density should increase for higher
stellar rotation rates $\Omega_{\star}$ (Fig. 5). In particular, at $\nu=1$
GHz, the radio flux density

$\displaystyle S_{\rm 1GHz}$ $\displaystyle\simeq$ $\displaystyle 0.7\rm{\mu
Jy}\,\left[\frac{\Omega_{\star}}{\Omega_{\odot}}\right]^{0.7}\left[\frac{10\,\rm
pc}{d}\right]^{2}$ $\displaystyle\simeq$ $\displaystyle 0.7\rm{\mu
Jy}\,\left[\frac{4.6\,\rm Gyr}{t}\right]^{0.39}\left[\frac{10\,\rm
pc}{d}\right]^{2}$

where $d$ is the distance, $t$ the age, and $\Omega_{\odot}$ is the present-
day solar rotation rate. In the equation above, I used the fact that
$\Omega_{\star}\propto t^{-0.56}$, which applies to ages $\gtrsim 600$ Myr
(Delorme et al., 2011, see Sect. 4.2), i.e., younger stars rotate faster. This
implies that an analogue of our present-day Sun, when placed a 10 pc, would
emit a flux density of $\sim 0.7\mu$Jy at 1GHz.222From Fig. 5, we see that the
solar magnetic cycle has little influence in the predicted radio spectra, due
to small variations in the solar wind properties in a cycle timescale. This is
below detection limits of current radio telescopes. In fact, the distance-
square decay in Eq. (2.2) plays a more important role than the increase in
rotation rate (or decrease in age) of the stars. Young suns, such as $\kappa$
Ceti and $\pi^{1}$ UMa, are also the closest to us in the simulated sample of
Ó Fionnagáin et al. (2019) – for these objects, the predicted 1-GHz flux
density is still only a few $\mu$Jy. At the time of writing (Spring 2020),
there exist plans to upgrade the existing VLA system, which would increase
instrument sensitivity by a factor of 10 (Osten et al., 2018). The expected
sub-$\mu$Jy sensitivity level of the future Square Kilometre Array (SKA) means
that winds of close-by young suns (such as $\kappa$ Ceti and $\pi^{1}$ UMa)
are potentially detectable below 1 GHz (see also discussion in the previous
paragraph).

Figure 5: Predicted evolution of the radio flux with rotation, used as proxy
for stellar age, for a number of solar-like stars. The spectra of all these
stars have been normalised to 10 pc. Young stars, close-by, are the targets
which would present the strongest radio flux. Figure adapted from Ó Fionnagáin
et al. (2019).

One consequence of the free-free emission of stellar winds is the creation of
a radio photosphere, which is paraboloidal in shape (Kavanagh et al., 2019).
Radio emission produced from sources inside the photosphere might be absorbed
by the stellar wind. An expected source of low-frequency radio emission are
magnetised close-in exoplanets (Farrell et al., 1999). If a planet is embedded
in the radio photosphere of their host star, then it is possible that most of
the planetary radio emission gets absorbed and does not escape (Vidotto and
Donati, 2017; Kavanagh and Vidotto, 2020). Although this might be problematic
for planet detection in radio, close-in exoplanets can be a useful tool to
probe the inner regions of stellar winds, as I explain next.

###  2.3 Using exoplanets to probe stellar winds

Close-in, giant planets form the majority of the exoplanet population known
nowadays. With orbital distances significantly smaller than Mercury’s orbit
(and as small as $\lesssim 0.02$ au), these exoplanets are embedded in a
stellar wind regime that is unprecedented for solar system planets. For
comparison, the innermost planet in the solar system is Mercury, with a semi-
major axis of 0.4 au. Although the solar wind speed at Mercury is similar to
the wind speed at Earth’s orbit (roughly 400 km/s), the local solar wind
density at Mercury is about $1/0.4^{2}$ times larger. This is a consequence of
the $r^{2}$-decay of density for winds at asymptotic terminal speeds. However,
going even closer to the Sun, one reaches the acceleration zone of the solar
wind, in which the wind speed is still increasing with distance. Due to mass
conservation, densities increase nearly exponentially towards small
heliocentric distances.

Not only the local stellar wind densities around close-in planets are orders
of magnitude larger than the $\sim 5~{}{\rm cm}^{-3}$ at Earth’s orbit, the
magnetic field embedded in the stellar wind is also expected to be several
orders of magnitude larger (Vidotto et al., 2015). Additionally, the close
orbital distances imply that close-in planets have high Keplerian velocities
($v_{\rm kep}\propto r^{-1/2}$). This means that, in the planet’s reference
frame, the stellar wind particles could still arrive at large velocities, even
though the local stellar wind itself might still not have reached terminal
speed (Vidotto et al., 2010a, 2011). Altogether, this shows that close-in
planets face extreme wind environments (Vidotto et al., 2015).

Although an extreme stellar wind environment could be detrimental for life
formation (Lammer et al., 2007; Khodachenko et al., 2007; Scalo et al., 2007;
Vidotto et al., 2013), it is precisely the extreme wind conditions around
close-in planets that amplify signatures of the wind-planet interaction,
potentially leading to detection of such signatures. Signatures of wind-planet
interaction have been detected in the hot-Jupiters HD209458b, HD189733b and
the warm-Neptune GJ436b, that orbit main-sequence stars of spectral types F8,
K2 and M3, respectively. These planets show strong atmospheric escape, which
can only be interpreted when considering the interaction with the winds of
their host stars (e.g., Holmström et al., 2008; Bourrier et al., 2013, 2016;
Villarreal D’Angelo et al., 2014; Kislyakova et al., 2014). Given that the
escaping atmospheres of these gas giants are hydrogen dominated, their
outflows are detected in the Ly-$\alpha$ line during planetary transits, in a
technique known as transmission spectroscopy or spectroscopic transit
observations (for a review on the topic see, e.g., Kreidberg, 2018).

By modelling the stellar Ly-$\alpha$ line profile that is transmitted through
the planetary atmosphere, one can derive the conditions of the stellar wind
surrounding the exoplanet. The stellar wind has two effects that leave their
fingerprints in the Ly-$\alpha$ line. Firstly, the wind shapes the outflow of
the planet, in a similar way as the ISM shapes the astrosphere. This causes an
asymmetry in the planetary outflow, as the interaction occurs preferentially
on one side of the planet, causing lightcurve asymmetries (Villarreal D’Angelo
et al., 2018a; McCann et al., 2019). Secondly, the ionised wind exchanges
charge with the neutral hydrogen escaping the planetary atmosphere, creating a
population of high-velocity neutrals (previously, these were high velocity
ions from the stellar wind that became neutralised during the charge-exchange
process). This leads to a high-velocity, blue-shifted component of the stellar
Ly-$\alpha$ line (e.g., Holmström et al., 2008). Therefore, modelling these
fingerprints left in the Ly-$\alpha$ line, one can then obtain the local
densities and speeds of the stellar wind (Bourrier et al., 2016), and,
consequently, the wind mass-loss rates as well (Kislyakova et al. 2014,
Vidotto and Bourrier 2017).

In the case of GJ436b, a warm Neptune orbiting an M dwarf, models of the
spectroscopic transit in Ly-$\alpha$ predict local wind speeds of 85 km/s and
proton densities of $2\times 10^{3}~{}{\rm cm}^{-3}$ (Bourrier et al., 2016),
which translates to wind mass-loss rates of $1.2\times
10^{-15}\,M_{\odot}~{}{\rm yr}^{-1}$ (Vidotto and Bourrier, 2017). For the
solar-like star HD209458, Kislyakova et al. (2014) were able to model the
Ly-$\alpha$ line with a wind that resembles that of the present-day Sun: with
the same mass-loss rate, but with a scaled-up density at the orbit of
HD209458b. This results in local wind speeds of $400$ km/s and wind densities
of $5\times 10^{3}~{}{\rm cm}^{-3}$. For HD189733, the derived local stellar
wind speed at the orbit of the planet of $190$ km/s and density of $3\times
10^{3}~{}{\rm cm}^{-3}$ (Bourrier and Lecavelier des Etangs, 2013) yielded a
mass-loss rate of approximately $4\times 10^{-15}\,M_{\odot}~{}{\rm yr}^{-1}$.
One caveat to keep in mind, though, is that the values derived above are
likely not unique, and some degeneracy might exist (Mesquita and Vidotto,
2020).

Although the wind properties can be derived as a by-product of a method to
detect escape in exoplanets, there are disadvantages to conduct transmission
spectroscopy in the Ly-$\alpha$ line. The biggest of which is that the line
falls in the ultraviolet part of the electromagnetic spectrum, whose
observations require space instrumentation, which are very expensive.
Currently, only the Hubble Space Telescope is able to observe in the
ultraviolet. For this reason, the recent detection of escaping atmospheres in
lines that can be detected with ground-based instrumentation, such as
H-$\alpha$ or the HeI triplet at 10830Å (Yan and Henning, 2018; Nortmann et
al., 2018; Spake et al., 2018; Allart et al., 2019), can present new
opportunities for probing escaping atmospheres, and likely the interaction
region with stellar winds (Oklopčić and Hirata, 2018; Villarreal D’Angelo et
al., 2021).

The typical wind conditions around exoplanets are soon to become better
understood, with experiments that are taking place, right now, in our own
solar system. Until very recently, no in-situ measurements of the solar wind
plasma at close distances to the Sun existed. Measurements of the first
encounter of the NASA’s Parker Solar Probe (PSP) were recently released,
probing the solar wind at distances as close as $36\,R_{\odot}\simeq 0.17$ au
(Bale et al., 2019). PSP is set to remain in the heliospheric equator and,
after a series of Venus flybys, it will get closer and closer to the Sun,
reaching a highly elliptical orbit with a perihelion of 0.046 au – typical of
hot Jupiters. A second complementary spacecraft, ESA’s Solar Orbiter, was
launched in February 2020 and will also study the solar wind at close
distances, down to 0.29 au, albeit at high heliospheric latitudes (polar
regions). Together, these two spacecrafts will allow in situ measurements of
the solar wind at unprecedented close heliospheric distances. They will
provide more information about the physical mechanism that accelerates the
solar wind and will provide information about the environment at the orbits of
close-in exoplanets.

###  2.4 Using prominences to probe stellar winds

The last method for detecting winds of solar-like stars that I would like to
discuss involves using observations of slingshot prominences to derive wind
mass-loss rates. Slingshot prominences occur on fast rotating stars, and are
very extended (Jardine and van Ballegooijen, 2005), in contrast to solar
prominences. They are detected as absorption features that travel in the
H-$\alpha$ stellar line profile as the star rotates. Their observed velocities
indicate that slingshot prominences occur at or beyond the co-rotation radius,
which are several radii above the stellar surface (Collier Cameron and
Robinson, 1989a, b). Jardine and van Ballegooijen (2005) suggested that
prominences are formed at the top of long magnetic loops, which are filled
with mass from the stellar wind. The material in the prominence cools to a few
$10^{4}$ K – thus, if the material has enough optical depth, slingshot
prominences are seen in H-$\alpha$ Doppler maps in absorption, when they pass
in front of the stellar disc.

The dynamical support and lifetime of slingshot prominences depend on the
relative position between where they are formed (i.e., the co-rotation radius)
and the Alfvén and sonic surfaces of a stellar wind (Fig. 6). The Alfvén and
sonic surfaces are defined as the surface where the wind speed reaches the
Alfvén and sound speeds, respectively (more about this will be discussed in
the modelling Sect. 5). Two main conditions should be kept in mind with
regards to the existence of slingshot prominences. Firstly, given that these
prominences occur at loop tops, they can only exist when the co-rotation
radius lies below the Alfvén surface, as beyond the Alfvén surface, magnetic
field lines are open (Jardine and Collier Cameron, 2019). Villarreal D’Angelo
et al. (2018b) demonstrated that for older solar-like stars, the co-rotation
radius is above the Alfvén radius, and these stars cannot support these types
of prominences. Young, fast rotating stars, on the other hand, belong to the
other group, in which prominences occur within the Alfvén surface.

The second important condition to consider is the relative location between a
prominence (co-rotation point) and the sonic point. Information about what is
happening in the wind cannot be passed back to the star for any event that
takes place above the sonic point (Del Zanna et al., 1998; Vidotto and Cleary,
2020). Thus, if the prominence, formed at the co-rotation radius, is formed
above the sonic point, the star keeps loading the prominence with stellar wind
material and the loop top becomes denser and denser, until it eventually
erupts, and the cycle starts again. This represents the ‘limit-cycle regime’
proposed in Jardine and Collier Cameron (2019). If the site of prominence
formation at the co-rotation radius, on the other hand, occurs below the sonic
point, the mass-loading from the surface gets readjusted – although
prominences could still be formed in the ‘hydrostatic regime’, they would
erupt on an occasional basis only.

Figure 6: The formation of slingshot prominences occurs when in the ‘limit-
cycle regime’. In this case, the site of prominence formation, i.e., on loop-
tops at the corotation radius, occurs above the sonic point (the star is
“unaware” that the prominence has formed and thus keeps loading it with
stellar wind material) and below the Alfvén radius (beyond the Alfvén radius
all magnetic field lines will be open). By observing slingshot prominences,
one can estimate the rate at which mass is loaded into the loop tops and thus
derive mass-loss rates of stellar winds. Figure from Jardine and Collier
Cameron (2019).

Altogether, these conditions imply that the formation of slingshot prominence
occur in the ‘limit-cycle regime’ (Figure 6). Such a condition is more easily
met in faster rotators, as these stars have smaller co-rotation radii and thus
more easily to be formed in the sub-Alfvénic regime. Their winds need to be
relatively hotter, so that the sonic point occurs at lower heights. As I will
discuss later on, we expect that active stars have hotter winds. These
conditions are more easily met in fast and ultrafast rotators, in agreement
with extended prominences seen in the ultrafast rotators such as Speedy Mic
(Jeffries, 1993; Dunstone et al., 2006), HQ Lup (Donati et al., 2000) and,
more recently, in V530 Per (Cang et al., 2020), all of which with rotation
periods $<0.4$ d. Slingshot prominences have been predicted to occur in stars
that are rotating at less extreme rates as well, such as in the young Sun HIP
12545 (Villarreal D’Angelo et al., 2018b), which shows a rotation period of
nearly 5 days.

From H-$\alpha$ observations, one can measure the prominence lifetime and the
amount of mass contained in the prominence, giving the rate of mass that
upflows into the prominence. To convert this mass-loading rate, which occurs
in a localised region of the stellar surface, to a wind mass-loss rate, the
prominence surface coverage is needed. Jardine and Collier Cameron (2019)
estimated that about 1% of the surface of a fast rotating star would be
covered in prominences. These authors then derived mass-loss rates of 350,
4500, and 130 times the solar-wind mass loss rate for the young solar-type
stars AB Dor, LQ Lup and Speedy Mic, respectively, all of which rotate with a
period shorter than half a day.333The sample in Jardine and Collier Cameron
(2019) also contains a couple of active M dwarf stars, namely V374 Peg and HK
Aqr, whose derived mass-loss rates are $4\times 10^{-12}$ and
$10^{-12}~{}M_{\odot}~{}{\rm yr}^{-1}$, respectively. These mass-loss rates
are substantially larger than those derived for M dwarfs with the astrosphere
method, such as in the case of the active M dwarf EV Lac and the less active
Proxima Cen ($2\times 10^{-14}$ and $<4\times 10^{-15}M_{\odot}~{}{\rm
yr}^{-1}$, respectively Wood 2004).

Due to their youth, these three solar-type stars are X-ray luminous. When
placing them together in the diagram shown in Fig. 4, we notice that the trend
of mass-loss rates with X-ray flux extends for a larger dynamical range in
X-ray fluxes, compared to the work of Wood (2018). In Sect. 3, I will discuss
how the outcomes of these detection methods can be used to derive an observed
evolutionary sequence for mass-loss rates.

###  2.5 Detecting coronal mass ejections (CMEs) through type II radio bursts

Although not all solar CMEs have a flare counterpart (and vice-versa), solar
CMEs are often associated to flares. Aarnio et al. (2011) showed that stronger
flares lead to more massive CMEs and that the CME mass scales with the X-ray
flare flux as $M_{\rm CME}\propto F_{\rm flare}^{0.7}$ or, in terms of the
flare energy, as $M_{\rm CME}\propto E_{\rm flare}^{0.68}$ (Aarnio et al.,
2012). If one were to extend these solar empirical relations to active stars,
which show higher flare energies, one would expect that the mass contained in
stellar CMEs could become substantially large. As a result, active stars, due
to their higher flare rates, could have CME-dominate winds with mass-loss
rates that could be substantially larger than solar.

In the Sun, CMEs contribute, on average, to a mass-loss rate of $\simeq
4\times 10^{-16}~{}M_{\odot}~{}{\rm yr}^{-1}$ (Vourlidas et al., 2010), which
is about only a few percent of the present-day solar wind mass-loss rate. In
contrast are the predictions for young stars – using flare–CME empirical
scalings, Aarnio et al. (2012), Drake et al. (2013) and Osten and Wolk (2015)
estimated CME mass-loss rates for T Tauri stars in the range $\sim
10^{-12}$–$10^{-9}~{}M_{\odot}~{}{\rm yr}^{-1}$, which are several orders of
magnitude larger than the present-day solar wind. However, for CME-dominated
winds with rates $\gtrsim 10^{-10}~{}M_{\odot}~{}{\rm yr}^{-1}$, Drake et al.
(2013) argued that they would have kinetic energies that could amount to 10%
of the stellar bolometric luminosity, which is a rather substantial energy
budget associated to CMEs. These authors questioned that instead the solar
flare–CME relation could not be extrapolated indefinitely to higher flare
energies. For example, the relationship might flatten out towards higher flare
energies or that observed stellar flares might not always be accompanied by a
CME. This could happen if, for example, CMEs on active stars were more
strongly confined, and so fewer of them would be produced for any given number
of X-ray flares.

This stronger confinement could be caused by the noticeable differences
between the solar and stellar magnetic field characteristics. Indeed, more
active stars seem to have more toroidal large-scale magnetic field topologies
(Petit et al., 2008; See et al., 2015; Vidotto et al., 2016), which could
indeed result in more confined CMEs. Using numerical simulations, Alvarado-
Gómez et al. (2018) showed that a stronger overlying large-scale dipolar
(poloidal) magnetic field of 75 G could prevent a typical solar CME from
erupting and that only CMEs with magnetic energies $\gtrsim 30$ times larger
than those in typical solar CMEs were able to escape. Those that escaped,
however, were not able to accelerate efficiently due to the strong overlying
magnetic field.

While Aarnio et al. (2011) and Drake et al. (2013) concentrated on flare
energies in the X-ray band, Osten and Wolk (2015) investigated empirical solar
flare–CME relations using the bolometric energy from flares. These authors
provided energy partition calculations that can be used to relate the amount
of radiated flare energy in one bandpass (e.g., in white light or X-rays) to
the total bolometric energy of flares. White-light flares, for example, are
frequently seen in high-cadence observations of missions such as Kepler, K2
and TESS. The long-term monitoring provided by these missions allows us to
build more complete statistical studies of flares from stars of different
spectral types and ages (Maehara et al., 2012, 2015; Davenport et al., 2014),
which can then be used to study stellar CMEs.

An alternative approach to estimate mass-loss rates of CME-dominated winds was
proposed by Cranmer (2017). He used relations between solar magnetic energy
flux and the kinetic energy flux of the solar wind/CME outflows to predict the
evolution of CME mass-loss rates of solar-like stars. His models predict that
both the wind and CME mass-loss rates are larger at younger ages, with CMEs
dominating the mass loss process in the first 0.3 Gyr of the lifetime of a
solar-like star. At these younger ages, Cranmer (2017) estimated that CME
mass-loss rates are a factor of 10 to 100 higher than the quiescent
(overlying) stellar wind. At ages $\gtrsim 1$ Gyr, the mass loss in the
quiescent wind dominates that in CMEs.

Although these models qualitatively agree that CMEs can contribute
significantly to the total mass-loss rates at younger ages, observationally
confirming the presence and amount of stellar CMEs is challenging and their
detection has remained elusive (see, e.g., Leitzinger et al., 2020, for an
updated census). Given that type II radio bursts are related to CMEs in the
Sun (not all CMEs produce type II bursts though), one possible way to detect
stellar CMEs is through observations of radio bursts (see, e.g., Crosley et
al., 2016, and references therein). Solar type II radio bursts are believed to
originate in the shocked material created as the CME propagates outwards at
super-Alfvénic velocities (Gopalswamy et al., 2019). The plasma emission from
this shocked material is seen in the dynamic spectrum as a burst drifting in
time (related to the speed of the shock/CME) and frequency (related to the
density of the corona). Given that the coronal density decreases with height,
type II radio bursts drifts towards lower frequencies as the CME propagates
outwards – starting (maximum) frequencies of $\sim 100$ MHz corresponds to
electron densities of $\sim 10^{8}$ cm-3 (see Equation 5 that will be
discussed below).

Not all CMEs should produce type II bursts though. For example, if stellar
CMEs are not sufficiently accelerated to become super-Alfvénic (see, e.g.,
slowly accelerated CMEs as seen in the models from Alvarado-Gómez et al.
2018), a shock wave would not be formed and thus a type II burst would not be
seen. In spite of this, given that type II bursts are indicative of solar
CMEs, by analogy, detecting stellar type II radio bursts could provide a lower
limit on stellar CMEs (Crosley et al., 2016), which can be used to determine
CME occurrence rates and characterise their properties. Crosley et al. (2016)
relates the shock speed $v_{\rm CME}$ to the frequency drift rate $\dot{f}$
and coronal scale height $H$ as

$v_{\rm CME}=-\frac{2H\dot{f}}{f},$ (2)

where $f$ is the frequency. Although $H$ has to be modelled, both $\dot{f}$
and $f$ are quantities that are directly obtained from type II radio burst
observations. With this, one could derive the CME speed. One further step is
required to convert the CME speed into CME mass. For that, it is assumed that
energy equipartition between the bolometric radiated energy of the flare
($E_{\rm flare,bol}$) and the kinetic energy of its associated CME holds. Thus

$M_{\rm CME}=\frac{2E_{\rm flare,bol}}{v_{\rm CME}^{2}}=\frac{E_{\rm
flare,bol}}{2H^{2}}\left(\frac{f}{\dot{f}}\right)^{2},$ (3)

where Eq. (2) was used in the last equality of Equation (3). $E_{\rm
flare,bol}$ is not a directly observed quantity, but energy partition
calculations can be used to convert from observed flare energies at a given
bandpass to $E_{\rm flare,bol}$ (Osten and Wolk, 2015). Thus, while no type II
burst has yet been observed associated with a stellar flare, if one were
detected, the rate of frequency change would provide a very useful measure of
CME speed (Equation 2) and mass (Eq. (3, cf. Crosley and Osten 2018). With an
observationally derived flare frequency distribution, this could then be used
to estimate the total mass loss in CME-dominated winds.

More recently, the prospects of using coronal dimming have been suggested as a
means to identify stellar CMEs (Harra et al., 2016; Jin et al., 2020). In the
Sun, coronal dimming is seen in certain EUV coronal lines, tracing the post-
CME evacuation of the corona. From the depth of the light curve, one can infer
how much mass has been blown out, and from the slope of the light curve, one
can determine the CME speed (Mason et al., 2016). The suggestion of detecting
the stellar equivalent of solar coronal dimming is still in its first steps
and will be explored with future instrumentations (France et al., 2019). Some
points still need to be elucidated, such as whether dimming could be detected
in the case where CMEs are happening all the time, as could potentially be the
case of young stars, or whether dimming could change the “basal” level of the
stellar EUV emission to such an extent that one would not be able to
disentangle particular CMEs. To answer these and other open questions, more
detailed modelling studies are needed (see, e.g., Jin et al., 2020).

###  2.6 Propagation/suppression of radio emission from a point source
(planet) embedded in a stellar wind

As discussed in Sect. 2.2, a stellar wind, if sufficiently dense, can emit
free-free emission at radio frequencies. We can define a boundary around the
star wherein the bulk of the wind emission comes from. Here, we define this
boundary, also known as the radio photosphere, as the isosurface where the
frequency-dependent optical depth is $\tau_{\nu}$. The value of
$\tau_{\nu}=0.399$, for example, delineates the region within which 50% of the
radiation is absorbed by the stellar wind and 50% of the emission escapes
(Panagia and Felli, 1975). The left panel in Figure 7 illustrates the radio
photosphere at $\nu=30$ MHz (dashed line) and the density profile of the wind
of a sun-like star with a mass-loss rate of $2\times
10^{-12}~{}M_{\odot}~{}{\rm yr}^{-1}$ (Kavanagh and Vidotto, 2020). In this
figure, we are seeing a 2D cut of the radio photosphere, where the observer
sees the system from the negative $x$ axis. In three dimensions, the radio
photosphere resembles a ‘tea cup’ (see Fig. 5 in Kavanagh et al., 2019), which
means that in the plane of the sky, this isocontour would have an
approximately circular shape (the shape is only perfectly circular for a
spherically symmetric wind though).

Figure 7: Left: Number density of the wind of a solar-like star with a mass-
loss rate of $2\times 10^{-12}~{}M_{\odot}~{}{\rm yr}^{-1}$. A planet is
considered to orbit at 0.02 au, with the observer looking towards the system
from the negative $x$-direction. The dashed line shows the radio photosphere
where 50% of a 30-MHz wind emission is produced. The hypothetical 30-MHz radio
emission of this planet is increasingly more attenuated after the planet
ingresses the radio photosphere and the attenuation peaks at orbital phase
$\phi=0.5$. Its emission is least attenuated at $\phi=0$. Given that the
position of the radio photosphere is linked to the stellar wind properties,
monitoring of planetary radio emission could allow one to derive stellar wind
properties. Figure from Kavanagh and Vidotto (2020). Right: The situation on
the left panel only occurs if the plasma emission $f_{p}$ is below the
cyclotron frequency $f_{c}$ of the planetary emission (white area). If the
wind of the host star has a high mass-loss rate and the planet has a weak
magnetic field, such that $f_{p}>f_{c}$, then the planetary radio emission
cannot propagate through the wind of the host star (grey area). Detections of
planetary radio emission can thus place an upper limit on the mass-loss rate
of the star (Equation 8). Note that the wind parameters used to produce this
figure is based on models of the weak-lined T Tauri star V830 Tau. Figure from
Vidotto and Donati (2017).

We can now imagine a situation in which a point-source radio emitter is
embedded in the wind, which is optically thick at radio frequencies. The point
source could be, for example, an exoplanet. Exoplanets, if they are
magnetised, are believed to emit at radio frequencies, analogously to the
magnetised planets in the solar system (Farrell et al., 1999). Their emission
is cyclotronic and thus takes place at cyclotron frequencies

$f_{c}=\frac{eB_{p}}{2\pi m_{e}c}=2.8\left(\frac{B_{p}}{1\rm G}\right){\rm
MHz}$ (4)

where $B_{p}$ is the planet’s magnetic field, $c$ the speed of light, and $e$
and $m_{e}$ are the electron charge and mass. We explore here two different
scenarios. In the first scenario, we investigate whether the emission of the
point source could _propagate_ though the wind of the host star, while, in the
second scenario, we assume the emission can propagate, but it is _attenuated_.

The planetary radio emission can only propagate in the stellar wind plasma if
the cyclotron frequency of emission $f_{c}$ is larger than the stellar wind
plasma frequency $f_{p}$ everywhere along the propagation path, where

$f_{p}=\left(\frac{n_{e}e^{2}}{\pi
m_{e}}\right)^{1/2}=9\left(\frac{n_{e}}{1~{}{\rm cm}^{-3}}\right)^{1/2}{\rm
kHz}.$ (5)

The local electron density $n_{e}$ of a fully ionised hydrogen stellar wind is
$n_{e}=n/2$, where $n$ is the total particle density. The condition
$f_{c}>f_{p}$ is met when

${B_{p}}>\left(\frac{n_{e}}{10^{5}~{}\textrm{cm}^{-3}}\right)^{1/2}\textrm{G
~{}~{}~{}~{}or ~{}~{}~{}~{}}n_{e}<10^{5}\left(\frac{1\rm
G}{B_{p}}\right)^{2}~{}\textrm{cm}^{-3}.$ (6)

The density is related to the mass-loss rate of a stellar wind by the mass
continuity equation. Therefore, the condition expressed in Equation (6) can be
translated into

${B_{p}}\gtrsim\left(\frac{\dot{M}}{10^{-12}~{}M_{\odot}~{}{\rm
yr}^{-1}}\right)^{1/2}\left(\frac{400~{}{\rm km/s}}{u(r_{\rm
orb})}\right)^{1/2}\left(\frac{0.05~{}\rm au}{r_{\rm orb}}\right)\textrm{G},$
(7)

or

$\left(\frac{\dot{M}}{10^{-12}~{}M_{\odot}~{}{\rm
yr}^{-1}}\right)\lesssim\left(\frac{B_{p}}{\textrm{G}}\right)^{2}\left(\frac{u(r_{\rm
orb})}{400~{}{\rm km/s}}\right)\left(\frac{r_{\rm orb}}{0.05~{}\rm
au}\right)^{2},$ (8)

where we assumed a mass-loss rate of a steady, spherically symmetric, fully
ionised hydrogen wind: $\dot{M}=4\pi r_{\rm orb}^{2}m_{p}n_{e}(r_{\rm
orb})u(r_{\rm orb})$, with $u(r_{\rm orb})$ being the wind speed at the
orbital distance $r_{\rm orb}$. The right panel in Figure 7 shows the region
of parameter space where radio emission of the exoplanet V830 Tau b could
propagate through the wind of the host star (Vidotto and Donati, 2017). As we
can see, the propagation depends on the combination of planetary and stellar
wind characteristics. As such, if one can detect radio emission from the
planet, the planetary magnetic field can be derived from the frequency of the
emission (Equation 4), and an upper limit of the wind mass-loss rates can be
estimated (Equation 8).

Assuming the planetary radio emission can propagate through the wind of the
host star, there is still the question of by how much the planetary emission
is attenuated by the wind. To answer this question, we turn our attention back
to the left panel in Figure 7 and more specifically to the location of the
planet through its orbit with respect to the radio photosphere (dashed curve).
The planetary radio emission is least attenuated (i.e., least absorbed by the
wind of the host star) when the planet is between the observer and the star at
orbital phase $\phi=0$. As the planet ingresses in the region where the wind
is optically thick, the emission from the planet will get increasingly more
attenuated, due to an increase in optical depth. In the case shown in the left
panel of Figure 7, the reduction in radio emission from the planet starts at
orbital phase $\phi=0.36$, it is maximum when the planet is the furthest from
the observer (at $\phi=0.5$) and decreases back again until the planet emerges
from the radio photosphere (at $\phi=0.64$). Given that the size of the radio
photosphere depends on the physical properties of the stellar wind, if radio
emission is detected from the planet for a certain fraction of the orbit, one
can then use this information to further constrain the stellar wind properties
(Kavanagh and Vidotto, 2020).

###  2.7 Constraining stellar winds (upper limits) though X-ray emission

As I will discuss further on, in the Sun, the bulk of the X-ray emission
originates from active regions, whose magnetic loops can confine hot plasma.
Nevertheless, a hot stellar wind, optically thin in X-rays, can also
contribute to a fraction of the stellar X-ray emission. Assuming that the
total stellar luminosity is $L_{X}=L_{X}^{\rm AR}+L_{X}^{\rm wind}$, with the
superscript ‘AR’ denoting the contribution from active regions, we have that
$L_{X}^{\rm wind}<L_{X}$. Therefore, we can use the observed stellar
luminosity $L_{X}$ to infer the upper limit of the wind contribution
$L_{X}^{\rm wind}$. As I will show next, this can then place a constraint
(upper limit) in the wind mass-loss rate.

The X-ray emission of the wind can be estimated as

$L_{X}^{\rm
wind}=\int_{V}\left(\int_{\nu_{1}}^{\nu_{2}}\epsilon_{\nu}d\nu\right)dV$ (9)

where $\epsilon_{\nu}$ is the X-ray emissivity integrated over the X-ray
frequency range $[\nu_{1},\nu_{2}]$ and $V$ is the volume of the emitting
wind. Considering the wind X-ray emission is caused by free-free radiation,
the emissivity of an optically thin fully ionised hydrogen plasma is given by
(Rybicki and Lightman, 1986)

$\epsilon_{\nu}=6.8\times 10^{-38}\frac{n_{e}n_{i}}{T^{1/2}}g_{\rm
ff}\exp\left(-\frac{h\nu}{k_{B}T}\right)\textrm{erg~{}cm}^{-3}~{}\textrm{s}^{-1}~{}\textrm{Hz}^{-1}$
(10)

where $g_{\rm ff}$ is the Gaunt factor of the order of unit (Karzas and
Latter, 1961), $h$ is the Planck constant and $k_{B}$ is the Boltzmann
constant. Note that, at temperatures from below 1 MK up to several MK, lines
dominate the X-ray spectrum. Therefore, when assuming that the emissivity is
entirely due to free-free radiation, we are providing a lower limit on the
emission, given the probable temperature range of coronal winds. Let us first
estimate the wind emissivity in the X-ray range

$\epsilon_{X}=\int_{\nu_{1}}^{\nu_{2}}\epsilon_{\nu}d\nu=6.8\times
10^{-38}\frac{n_{e}n_{i}}{T^{1/2}}\int_{\nu_{1}}^{\nu_{2}}\exp\left(-\frac{h\nu}{k_{B}T}\right)d\nu$
(11)

where we assumed $g_{\rm ff}\simeq 1$. Solving the integral analytically, one
gets

$\epsilon_{X}=1.4\times
10^{-27}{n_{e}n_{i}T^{1/2}}\left[\exp\left(-\frac{h\nu_{1}}{k_{B}T}\right)-\exp\left(-\frac{h\nu_{2}}{k_{B}T}\right)\right].$
(12)

With this, we get that the wind contribution to the X-ray luminosity is

$L_{X}^{\rm wind}=\int_{V}\epsilon_{X}dV=1.4\times
10^{-27}{T^{1/2}}\left[e^{-{h\nu_{1}}/{k_{B}T}}-e^{-{h\nu_{2}}/{k_{B}T}}\right]\textrm{EM},$
(13)

where we assumed the wind is isothermal and the emission measure is

$\textrm{EM}=\int_{V}n_{e}n_{i}dV=\frac{1}{4}\int_{V}n^{2}dV,$ (14)

where we used the fact that for a fully ionised hydrogen wind, we have
$n_{i}=n_{e}=n/2$.

Equation (13) shows that the X-ray emission coming from the wind depends on
the density profile $n$ of the wind. In general, the relationship between
density and wind mass-loss rate is not straightforward, as $\dot{M}$ depends
also on the velocity structure $u$ of the wind ($\dot{M}\propto ur^{2}n$).
However, for non-magnetised, thermally-driven winds at a given temperature, it
can be shown that the mass-loss rate scales linearly with $n$, as the velocity
structure only depends on the temperature of the wind (see Sect. 5.2.1 and in
particular Equation 42). Therefore, Equation (13) indicates that isothermal
winds with higher densities and thus higher mass-loss rates would have higher
$L_{X}^{\rm wind}$.

To compute $L_{X}^{\rm wind}$, we can use stellar wind models to predict how
the density of the wind varies with $r$ and plug this in Eq. (14). Here,
however, we proceed with a simplified approximation. For an isothermal wind,
we can approximate the density structure below the sonic point by a
hydrostatic density structure

$n=n_{0}\exp\left[-\frac{R_{\star}}{H_{\star}}\left(1-\frac{R_{\star}}{r}\right)\right],$
(15)

where $n_{0}$ is the wind base density, $H_{\star}=k_{B}T/(g_{\star}m_{p}/2)$
is the scale height of a fully ionised hydrogen wind and
$g_{\star}=GM_{\star}/R_{\star}^{2}$ is the stellar gravity at the surface.
Thus

$\displaystyle\textrm{EM}=\frac{1}{4}\int_{V}n^{2}dV=\frac{1}{4}n_{0}^{2}\int_{V}\exp\left[\frac{2R_{\star}}{H_{\star}}\left(\frac{R_{\star}}{r}-1\right)\right]dV$
$\displaystyle\simeq\pi
R_{\star}^{3}n_{0}^{2}\int_{1}^{x_{\max}}\exp\left[\frac{2R_{\star}}{H_{\star}}\left(\frac{1}{x}-1\right)\right]x^{2}dx,$
(16)

where $x=r/R_{\star}$. The limits of the integral above should be from the
stellar surface $x=1$ to the observer $x_{\max}\to\infty$. However, given the
validity of the hydrostatic density structure, the value of $x_{\max}$ should
not be larger than the sonic point. Numerically, we can see that the integrand
decays quite fast with $x$, having its maximum of $1$ at $x=1$. For the sake
of simplicity, we will take $x_{\max}\simeq 2R_{\star}$. For isothermal winds
of solar-like stars, this is actually not a bad assumption, as the bulk of the
emission for winds with temperatures ranging from $1.0$ to $2.5$ MK occurs
within $2R_{\star}$.

For the present-day solar wind, with a temperature of $\simeq 1.5\times
10^{6}$ K and base density of $n_{0}=10^{8}$ cm-3, solving the integral in
Equation (16) numerically gives $\simeq 0.1$ and thus $\textrm{EM}\simeq
10^{48}$ cm-3. Substituting this in Equation (13), for an X-ray range from 0.2
keV to 10 keV, our estimated X-ray emission of the solar wind is quite low, on
the order of $10^{-10}L_{\odot}$. The total X-ray luminosity of the Sun varies
during the solar cycle from about $7.0\times 10^{-8}$ to $1.2\times
10^{-6}L_{\odot}$ (Peres et al., 2000), which is 700 to 12000 times larger
than our estimated X-ray wind emission.

As we can see, the present day solar wind indeed contributes to a small
fraction of the total solar X-ray luminosity. Although we know the base
density of the solar wind, this parameter cannot be easily derived for winds
of solar-like stars. We can then ask ourselves, if the Sun were instead a
distance star, what would be the base density and, thus mass-loss rates, we
would derive from its X-ray luminosity? Given that $L_{X}^{\rm wind}\propto
n_{0}^{2}$, we would obtain a density value that is larger by a factor of
$\sqrt{700}$ to $\sqrt{12000}$ the $n_{0}$ value we adopted before. For the
same wind temperature of $\simeq 1.5\times 10^{6}$ K, this implies that our
estimated wind mass-loss rates would be larger by a factor of $\sim 26$ to
$110$ and the derived maximum mass-loss rate of the Sun-as-a-star would have
been $\lesssim 5\times 10^{-13}$ to $2\times 10^{-12}~{}M_{\odot}~{}{\rm
yr}^{-1}$. The Sun-as-a-star experiment illustrates that stellar X-ray
emission can only provide upper limits for wind mass-loss rates.

There has not been many examples in the literature where the technique
presented here has been used to measure mass-loss rates in cool dwarf stars.
Lim and White (1996) used soft X-ray observations of the M dwarf star YZ CMi
to derive an upper limit of its mass-loss rate. Llama et al. (2013) used the
X-ray observations of the K dwarf HD 189733 to constrain the base density
$n_{0}$, which is a free parameter in their models.

###  2.8 Detecting charge-exchange induced X-ray emission

The interaction between an ionised stellar wind with a neutral ISM can lead to
charge-exchange, in which the ISM neutral atom transfers charge to a solar
wind ion during a collision. In this process, a highly-charged ion,
particularly oxygen, is excited to a high excitation state, which is then
followed by a single or a cascade of radiative decays. These decays lead to
emission in the X-ray range. Contrary to the charge-exchange process discussed
in the context of Ly-$\alpha$ observations (Sect. 2.1) that only occurs at the
site of the interaction between wind and ISM, the charge-exchange induced
X-ray emission should take place throughout the stellar wind, as neutral ISM
atoms penetrate in the astrosphere. This gives rise to an X-ray ‘halo’.

This method was first idealised in Wargelin and Drake (2001), using the solar
wind as an example. Considering that low-mass stars have winds that are
embedded in a partially neutral ISM, Wargelin and Drake (2001) further
suggested that stars with mass-loss rates not much greater than solar could
also produce charge-exchange induced X-ray emission. This emission has a
distinct profile with a steep rise closer to the star and then a slow decay
(see Figure 1 of Wargelin and Drake 2001 for a Sun-as-a-star wind emission).
In a follow-up study, the same authors observed Proxima Centauri (Wargelin and
Drake, 2002), an M dwarf star, but unfortunately the signature of charge-
exchange induced X-ray emission was not detected. With their non-detection,
they were able to place an upper limit for the mass-loss rate of Proxima
Centauri of about 14 times that of the present-day Sun.

###  2.9 Accretion onto white dwarfs as probe of the wind of secondary
companion

The last method I would like to present here consists of observing signatures
of mass accretion in close binary systems, in which the primary is a white
dwarf. The assumption is that material from the wind of the secondary is
accreted onto the white dwarf and thus, if one can measure the accretion rate,
the wind mass-loss rate of the secondary can be inferred. To the best of my
knowledge, this method has not yet been used in systems with solar-like stars,
and estimates of mass-loss rates so far exist for some M dwarf stars that are
members of eclipsing binary systems (Debes, 2006; Parsons et al., 2012).

For the interacting stars where only mass-accretion rates onto the white dwarf
companions have been reported (Parsons et al., 2012), these accretion rate
values can be used as lower limits for the mass-loss rates of the M dwarf
stars. These are considered lower limits of wind mass-loss rates, because not
necessarily all the mass lost in the stellar wind of the secondary will be
accreted into the primary white dwarf. Debes (2006) finds that M dwarf mass-
loss rates are about 15 to 100 times larger than white dwarf mass-accretion
rates.

As I mentioned before, this method has been used to model mass-loss rates in M
dwarfs only (see Table 1 in Vidotto and Bourrier 2017 for a compilation of
such values). At the moment, it is unclear if the coronae and winds of M dwarf
stars in these binary systems are similar to those of isolated/non-interacting
M dwarf stars.

From the nine methods presented in Sect. 2, the first four are some of the key
methods of wind detection, with the other proposed methods mostly used as case
studies and applied on individual stars.

##  3 An observed evolutionary sequence for mass-loss rates?

Figure 8 summarises the stellar wind measurements derived using the methods
discussed here. Colour indicates the method used in the derivation: blue for
exoplanets, orange for astrospheres, green for prominences, and the Sun is
indicated in black (values at minimum and maximum of cycle are provided). Note
that for the radio observations, only upper limits have been extracted and
these are represented by the arrows pointing down (a few upper limits obtained
from other methods are also included). The larger symbols are the stars that
are relevant for this review – they are main-sequence, solar-like stars, while
the smaller symbols are either evolved stars or M dwarfs. The solid line is a
fit through the larger circles. Figure 8 shows that, overall, mass-loss rates
increase with X-ray flux $F_{X}$ and, given that young stars have higher
$F_{X}$, this implies that young stars have overall higher mass-loss rates.
The recent stellar wind measurements from Jardine and Collier Cameron (2019,
green circles), when contextualised with other measurements, suggest that
mass-loss rate continues to increase with $F_{X}$ and that a wind dividing
line (cf. Sect. 2.1) is no longer required.

Figure 8: Summary of derived mass-loss rates for low-mass stars combining
results from the different methods discussed in Sect. 2. The $y$-axis is given
in solar values, i.e., $\dot{M}_{\odot}/R_{\odot}^{2}$, with
$\dot{M}_{\odot}=2\times 10^{-14}~{}M_{\odot}~{}{\rm yr}^{-1}$. Colour
indicates the method used in the derivation: blue for exoplanets, orange for
astrospheres, green for prominences, black for the Sun at minimum/maximum of
its sunspot cycle. Grey arrows indicate upper limits, which are mostly derived
from radio observations. The solid line is a power-law fit through the larger
circles. The smaller symbols are either evolved stars or M dwarfs, which were
not included in the fit and neither were the stars for which only upper limits
exist (arrows). The values used in this plot were compiled from the following
works: Drake et al. (1993); Lim et al. (1996); Gaidos et al. (2000); Wood et
al. (2001, 2002, 2005a, 2014); Wood and Linsky (2010); Wood (2018); Wargelin
and Drake (2002); Bourrier et al. (2013); Kislyakova et al. (2014); Fichtinger
et al. (2017); Vidotto and Bourrier (2017); Vidotto and Donati (2017); Jardine
and Collier Cameron (2019); Finley et al. (2019); O Fionnagain et al. (2021).

Quantifying the wind of the same star through the use of different methods is
an ideal approach to verify and validate measurements. Multiple measurements
already exists for some stars in Fig. 8. One example is $\pi^{1}$ UMa – in
Fig. 8, the astrospheric measurement of $\pi^{1}$ UMa can be identified at
$F_{X}=1.7\times 10^{6}\mathrm{\ erg\ cm^{-2}\ s^{-1}}$ and
$\dot{M}/R_{\star}^{2}=0.53~{}\dot{M}_{\odot}/R_{\odot}^{2}$ (Wood, 2004),
while radio observations derived an upper limit of
$\dot{M}/R_{\star}^{2}\lesssim 270~{}\dot{M}_{\odot}/R_{\odot}^{2}$
(Fichtinger et al., 2017). The spin down model of Johnstone et al. (2015a)
predicts a ten times larger than solar mass-loss rate for $\pi^{1}$ UMa, i.e.,
at $\dot{M}/R_{\star}^{2}\sim 11~{}\dot{M}_{\odot}/R_{\odot}^{2}$. Although
the upper limit provided by the radio observation does not contradict the
astrospheric measurement nor the spin down modelling, the astrospheric
measurement for this star is in contradiction with spin-down tracks that are
statistically observed for this type of star. As I discussed in Sect. 2.1, it
is possible that $\pi^{1}$ UMa is in a high-ionisation region, which might
have affected the result obtained in the astrospheric method (see Wood et al.
2014). Another star in Fig. 8 with multiple wind measurements is the M dwarf
Proxima Centauri ($F_{X}=1.4\times 10^{6}\mathrm{\ erg\ cm^{-2}\ s^{-1}}$):
the astrospheric method (Wood et al., 2001), the charge-exchange induced X-ray
emission (Wargelin and Drake, 2002) and the free-free radio emission (Lim et
al., 1996) all produced upper limits of $\lesssim 4\times 10^{-15}$, $\lesssim
2.8\times 10^{-13}$ and $\lesssim 7\times 10^{-12}~{}M_{\odot}~{}{\rm
yr}^{-1}$, respectively. Unfortunately, in this case, the three measurements
are less stringent, as they are all upper limits.

Figure 8 also shows that there is a significant spread in the observations.
Vidotto et al. (2016) noted that the more active stars can show a significant
variation (cyclic or not) of stellar magnetism in timescales of the order of
years. Such variations could increase the spread in stellar wind properties,
that is worsened when the X-ray flux and mass-loss rates are not
contemporaneously derived. Note for example in Fig. 8 the case of the present-
day Sun, shown in black, where there is a variation of about one order of
magnitude in $F_{X}$ during the solar cycle.

Figure 9: The mass-loss rate of the solar wind (solid black line) has a small
variation during the solar cycle. The green line is the estimated open
magnetic flux of the solar wind. Data shows cycles 23 and 24. Figure adapted
from Finley et al. (2019).

The mass-loss rate of the Sun, however, does not seem to vary significantly
during the cycle (Cohen, 2011), with Finley et al. (2019) estimating that the
solar wind mass-loss rate varies in the range $\sim[1.6,3.2]\times
10^{-14}\,M_{\odot}~{}{\rm yr}^{-1}$ as shown by the black line in Fig. 9 (see
also Wang 1998). However, other wind properties, like density and velocity and
their latitudinal distribution vary more significantly during the cycle (Fig.
10). When the Sun is at minimum activity, its large-scale magnetic field is
dominated by a dipolar field, whose axis is roughly aligned with the rotation
axis (Sanderson et al., 2003; DeRosa et al., 2012; Vidotto et al., 2018a). In
this case, the solar wind velocity distribution is organised into a faster
stream ($\sim 700$–$800$ km/s) emerging from high latitude coronal holes and a
slower stream ($\sim 400$ km/s) around the equatorial plane (McComas et al.,
1998). Conversely, at solar maximum, the large-scale magnetic field geometry
is more complex, with a dipolar component vanishing and giving rise to higher-
order fields, in particular, even-mode components become more important, such
as the quadrupole component (DeRosa et al., 2012; Vidotto et al., 2018a). The
solar wind speed in this case has a more complex distribution, as shown in the
right panel of Fig. 10. The solar case illustrates the dependence of wind
properties on stellar magnetism and later on in this review I will come back
to the evolution of stellar magnetism.

Figure 10: Solar wind variations during its magnetic cycle. Left: At cycle
minimum (cycle 22), the solar magnetic field resembles an aligned dipole and
the solar wind shows a bimodal velocity distribution, with faster streams
emerging from high-latitude coronal holes and slower streams remaining in the
equatorial plane. Right: At cycle maximum (cycle 23), the solar magnetic field
geometry is more complicated, which is reflected in the solar wind velocity
distribution. The background image shows a zoom-in of the solar corona
extending out to a few solar radii, while the polar plot shows the solar wind
speed measured by Ulysses at several au from the Sun. Colour indicates the
magnetic field polarity (red for outward, blue for inward). Figures from
McComas et al. (2003).

I would like now to come back to the fits in Figures 4 and 8, which represent
the fits to the observations of mass-loss rates for solar-like stars. Using
measurements derived from the astrosphere method (Fig. 4), Wood et al. (2005a)
found that $\dot{M}/R_{*}^{2}\propto F_{X}^{1.34\pm 0.18}$, with the fit being
valid for $F_{X}\lesssim 10^{6}\mathrm{\ erg\ cm^{-2}\ s^{-1}}$. On the other
hand, the solid line in Fig. 8 fits through all the solar-like star
measurements (larger symbols) derived from several of the methods discussed in
Sect. 2. Note that this fit shows no break within the range of $F_{X}$ shown
in the $x$-axis. For those stars, I find a less steep dependence with $F_{X}$,
with a significant (20%) 1-$\sigma$ uncertainty in the slope of the
fit444Using simple stellar wind models, See et al. (2017) predicted a very
similar relationship: $\dot{M}\propto F_{X}^{0.79}$.

$\frac{\dot{M}}{R_{\star}^{2}}=10^{-2.75\pm 0.68}\left[\frac{F_{X}}{{\rm
erg\,cm}^{-2}{\rm s}^{-1}}\right]^{0.66\pm 0.12}\,\,\frac{M_{\odot}}{{\rm
yr}~{}R_{\odot}^{2}}\,.$ (17)

I can now derive an approximate evolutionary sequence for the solar wind from
this. I am only focusing on stars older than $\sim 600~{}$Myr, after which
their rotational evolution has converged (see Sect. 4.2). These are also stars
that are no longer in the saturated regime, i.e., those for which X-ray
increases with rotation (see Sect. 4.3). There are several relations in the
literature that link X-ray flux or luminosity with age or rotation rate (Ribas
et al., 2005; Güdel, 2007; Reiners et al., 2014). Here, I adopt the relation
from Güdel (2007), who found that $L_{X}\propto t^{-1.5\pm 0.3}$, for solar-
like stars in the non-saturated regime. From this, Eq. (17) becomes

$\dot{M}\propto t^{-0.99}\,.$ (18)

Wood et al. (2005a) did this exercise, albeit using a different X-ray–age
relationship. Using their fit, the line presented in Fig. 4, Wood et al.
(2005a) derived that $\dot{M}\propto t^{-2.33}$, which is steeper than our
derivation. These two power-laws are shown in Fig. 11. Both of these power-
laws indicate that the solar wind had higher mass loss rate in the past. The
dispersion is however quite significant, due to all uncertainties in the
involved power-laws. For example, the uncertainty in the slope led Wood et al.
(2005a) to estimate a solar-wind mass-loss rate in the range $[5,30]\times
10^{-12}\,M_{\odot}~{}{\rm yr}^{-1}$ for when the Sun was 600 Myr-old.

Figure 11: An evolutionary mass loss sequence. Predictions on how the solar
wind mass-loss rate would have been in the past. These power-laws are
discussed in the text and here they are normalised to match the present-day
solar wind mass-loss rates. Note that the power-laws carry large uncertainties
in their slope, which are not illustrated in this figure. The proxy for the
young Sun at 2 Myr, V830 Tau, is shown on the left of the plot. The question
mark here emphasises that, from observational studies, we are not sure how the
solar wind has evolved from the end of the pre-main sequence/beginning of the
main sequence, until today. The mass-loss rate saturation discussed in
Johnstone et al. (2015a) is shown schematically by the horizontal line.

Regardless of which of the two fits we use ($\dot{M}\propto t^{-2.33}$ or
$\dot{M}\propto t^{-0.99}$) for main-sequence stars, there is still a great
unknown on how the solar wind has evolved from early stages in the main
sequence (or final stages of the pre-main sequence) until the age of $\sim
600$ Myr. Can we extrapolate the $t^{-0.99}$ dependence to early times in the
main sequence? It has also been suggested that mass-loss rates increase with
rotation (towards lower ages) but then saturate. This suggestion was discussed
in Johnstone et al. (2015a) and is necessary to avoid the most rapidly
rotating stars to spin down too fast. I will come back to this in Sect. 5.3.
In Fig. 11, I represent the mass-loss rate saturation only schematically.
Figure 11 also shows the mass-loss rate derived for the weak-lined TTauri star
V830 Tau, which is considered to be a 2 Myr-old ‘baby Sun’. With a mass of
about $1M_{\odot}$, this star still has an inflated radius of $2R_{\odot}$.
The upper limit for its mass-loss rate is estimated to be $<3\times
10^{-9}\,M_{\odot}~{}{\rm yr}^{-1}$, with likely values ranging between
$[1\times 10^{-12},{1\times 10^{-10}}]\,M_{\odot}~{}{\rm yr}^{-1}$ (Vidotto
and Donati, 2017).

Summarising the discussion presented in this Section, it seems natural that
the solar wind has evolved from a high mass-loss rate at early ages until
today. However, observations so far do not give us very stringent limits on
how precisely this evolution took place.

##  4 Observed evolution of the main ingredients of the solar wind

The primary ingredient of the solar wind driving is the stellar magnetism,
which can be measured at the surface of solar-like stars via Zeeman broadening
of spectral lines or through polarimetric techniques. Indirectly, surface
magnetism can manifest itself through activity proxies, such as through
brightness variations due to spots and plages as they come in and out of view
in a rotation cycle, emission detected in the core of certain chromospheric
lines (e.g., CaII H&K lines), coronal high energy radiation in X-rays and
ultraviolet, among others. All these magnetic proxies evolve with stellar
rotation, the clock that tracks the passage of time in solar-like stars.

This clock is ultimately regulated by stellar winds, which remove angular
momentum from the star and thus force single solar-like stars to spin down
with time. In this Section, I discuss the observational point of view of the
evolution of stellar rotation, activity and magnetism as they play important
roles in the theory of winds of solar-like stars, which will be discussed in
Sect. 5. For a recent review on the theory of stellar magnetic field
generation and links with stellar rotation, I point the reader to Brun and
Browning (2017).

###  4.1 Evolution of magnetism

Stellar magnetism can be observationally probed with different techniques. A
technique that has been particularly successful in imaging stellar magnetic
fields is the Zeeman Doppler Imaging (ZDI, Donati and Brown 1997; for a review
see Donati and Landstreet 2009). Through a series of circularly polarised
spectra (Stokes V) distributed over one or more rotational cycles, this
technique has allowed the reconstruction of the large-scale surface fields
(intensity and orientation) of more than one hundred cool stars to date (e.g.,
Donati et al., 2006; Morin et al., 2008; Petit et al., 2008; Marsden et al.,
2011; Mengel et al., 2016; Folsom et al., 2016). The top panel in Fig. 12
illustrates an output of this technique (Fares et al., 2009), in which the
three components of the stellar magnetic field are reconstructed: radial,
azimuthal (West-East) and meridional (North-South). For comparison, I show in
the middle panel a synoptic map from the Sun, produced using SOLIS data
(Gosain et al., 2013). The difference between both sets of maps is striking.
Firstly, the solar magnetic field strength is an order of magnitude larger
than the stellar map. Secondly, we see all this salt-and-pepper structure in
solar maps that are not seen in the stellar maps. The reason for these
differences lies ultimately in the resolution of the data. Due to its
proximity, synoptic magnetic maps of the Sun can be reconstructed down to much
smaller scales than stellar maps.

If we decompose these maps using spherical harmonics, a stellar map would
typically reach up to a maximum harmonic order555For fast rotating stars,
higher maximum harmonic orders can be achieved. See for example the case of
the young solar-type star HD 141943 with $\ell_{\max}\sim 30$ (Marsden et al.,
2011)). $\ell_{\max}\sim 10$. For the solar map, on the other hand, the high
resolution allows harmonics of order $\ell_{\max}=192$ to be achieved. One can
estimate how the maximum order is linked to angular resolution as
$\Delta\theta\simeq 180^{\circ}/\ell_{\max}$, which means that the surface of
the Sun can be mapped down to $\sim 1^{\circ}$, while for a star similar to
the one shown in the top panel in Fig. 12, the resolution is on the order of
$\sim 23^{\circ}$. Because of the lower resolution of ZDI maps, magnetic
fields of opposite polarities that fall within an element of resolution cancel
out. Given that the small-scale field (e.g., concentrated in spots and active
regions) have high intensities and that these fields cancel out, the ZDI maps
of older solar-like stars do not reach the large strength fields observed in
the Sun and allow only the large-scale field to be reliably reconstructed
(Johnstone et al., 2010; Arzoumanian et al., 2011; Lang et al., 2014).

Figure 12: Top: The reconstructed stellar magnetic field using the ZDI
technique for the F-type star $\tau$ Boo (Fares et al., 2009). Middle:
Synoptic map of the Sun plotted with data from Gosain et al. (2013). The solar
map, due to its increased resolution, shows a lot more structure in the
surface magnetic field. Bottom: To compare the solar map with the stellar map,
I filtered out the small-scale magnetic field structure of the Sun (Vidotto,
2016b). At the top of each panel, I show the maximum harmonic order
$\ell_{\max}$ for each map and its typical spatial resolution.

In the bottom panel of Fig. 12, I show how the Sun would look like if it were
observed as a star. This is done by filtering out the small-scale field of the
solar synoptic map. In practice, I decompose the solar map shown in the middle
panel using spherical harmonics as presented in Vidotto (2016b). This allows
me to calculate the spherical harmonics coefficients for each given harmonic
order $\ell$. Note that the smaller scale structure is described by
increasingly larger values of $\ell$. Thus, to filter out the small-scale
component, I only use the derived coefficients up to a maximum harmonic order
of $\ell_{\max}=5$ for reconstructing the large-scale field. This method is
commonly used in the literature to separate the large-scale field, i.e., the
low harmonic orders (e.g., DeRosa et al., 2012; Petrie, 2013; Vidotto et al.,
2018a; Lehmann et al., 2018, 2019). With this, we can now more easily compare
the large-scale field of the Sun (bottom panel of Fig. 12) to the ZDI map of a
solar-like star (top panel).

A thorough investigation of the limitations of ZDI was recently presented in
Lehmann et al. (2019), a study that incorporated simulated data in the “ZDI
machinery”. The authors then compare what the ZDI method derives against the
controlled input data. This is done for simulated stars with two different
inclination angles of the spin axis with respect to the plane-of-sky (20∘ and
60∘) and three rotational periods (17, 19 and 27 days). Lehmann et al. (2019)
showed that for these low-activity stars, ZDI recovers relatively well the
large-scale field morphologies, but magnetic energies can be underestimated by
up to one order of magnitude (roughly a factor of $\sqrt{10}\sim 3$ in field
strengths). They also showed that the reconstruction of the field geometry is
less reliable if the star is viewed more pole-on (lower inclination angles).
For this reason, over the years, ZDI studies have favoured mapping the
magnetic field of stars that have intermediate inclinations.

One of the major advantages of ZDI maps is that they allow the derivation of
the three components of the stellar magnetic field topology. Even though this
topology is only restricted to the large-scale field, it can still be useful
in stellar wind modelling. As I will show in Sect. 5, surface maps have indeed
been increasingly adopted in 3D simulations of stellar winds (e.g., Vidotto et
al., 2012; Alvarado-Gómez et al., 2016b; Boro Saikia et al., 2020) and the
limited resolution of ZDI magnetograms has been demonstrated not to affect
stellar wind models (Jardine et al., 2017; Boro Saikia et al., 2020). The
reason for this is the following. If you consider a multipolar field, the
dipolar field ($\ell=1$) decays with $r^{-3}$; a quadrupolar field ($\ell=2$)
with $r^{-4}$; an octupolar field ($\ell=3$) with $r^{-5}$; and so on.
Generalising, a field with a harmonic order $\ell$ will decay with
$r^{-(\ell+2)}$. This means that small-scale fields (i.e., with large $\ell$
values) have very short reach. Thus, it is the large-scale field that is
embedded in winds of stars. Stellar winds flow through open magnetic field
lines, which ultimately are formed by the stellar large-scale fields that were
blown open by the wind outflow. The small-scale field can also affect stellar
winds, through, for example, determining the “micro-physics” of wind
acceleration, which is key for understanding the heating of the solar lower
atmosphere (Suzuki and Inutsuka, 2005; Cranmer et al., 2007; Shoda et al.,
2019).

While the ZDI is blind to the small-scale magnetic field, the Zeeman
broadening technique is not. This technique uses Zeeman-induced line
broadening of unpolarized light (Stokes I) to derive the average unsigned
surface magnetic field $\langle|B_{I}|\rangle$ (Johns-Krull, 2007; Shulyak et
al., 2019). When we compare the unsigned average field strength
$\langle|B_{V}|\rangle$ from ZDI maps with $\langle|B_{I}|\rangle$, we see
that $\langle|B_{V}|\rangle$ is only a fraction ($\sim 10\%$) of
$\langle|B_{I}|\rangle$ (Reiners and Basri, 2009; Morin et al., 2010). Note
that here I use the subscripts $I$ and $V$ to differentiate between Stokes I
and Stokes V derived fields. For a review on this technique, see Reiners
(2012).

$\langle|B_{I}|\rangle$ is the product of the intensity-weighted surface
filling factor of active regions $f$ and the mean unsigned field strength in
these regions, which is roughly assumed to be the same as the equipartition
field $B_{\rm eq}$: $\langle|B_{I}|\rangle=fB_{\rm eq}$ (Cranmer and Saar,
2011; See et al., 2019b). The equipartition field is found by balancing the
thermal and magnetic pressure at the photosphere of the star. Recent results
(Cranmer and Saar, 2011; See et al., 2019b; Kochukhov et al., 2020) suggest
that $B_{\rm eq}$ itself does not change significantly from star to star, but
the filling factor $f$ increases for fast rotators (younger stars).
Eventually, the surface of the star becomes covered in active regions and the
filling factor saturates for the very fast rotating stars at $f\sim 1$. This
leads to a saturation in the magnetic flux of fast rotating stars (Fig. 13a,
Reiners, 2012).

Figure 13 shows a compilation of a few trends derived empirically considering
measurements from these two aforementioned techniques: Zeeman broadening (left
panels) and ZDI (right panels). The top panels show how magnetism varies with
Rossby number666The use of Ro instead of $P_{\rm rot}$ is commonly found in
the literature, as it allows comparison across different spectral types and
thus reduces the spread commonly noticed in trends involving $P_{\rm rot}$.
One issue to be aware is that the Rossby number is model dependent, as
$\tau_{c}$ is not an observable. . The Rossby number Ro is defined as the
ratio between rotation period $P_{\rm rot}$ and convective turnover time
$\tau_{c}$

${\rm Ro}=\frac{P_{\rm rot}}{\tau_{c}}$ (19)

Three panels of this Figure show magnetic fluxes, defined as

$\phi_{I}=4\pi R_{\star}^{2}\langle|B_{I}|\rangle~{}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}~{}\phi_{V}=4\pi R_{\star}^{2}\langle|B_{V}|\rangle$ (20)

where the unsigned average field strength from ZDI observations is given by

$\langle|B_{V}|\rangle=\frac{1}{4\pi}\int|B_{\rm
ZDI}(\theta,\varphi)|\sin\theta d\theta d\varphi\,.$ (21)

Here, $\theta$ and $\varphi$ are the surface colatitude and longitude, and
$|B_{\rm ZDI}(\theta,\varphi)|$ is the unsigned surface field strength of the
ZDI map at $(\theta,\varphi)$. Note that in Figure 13b and d, only the radial
component was used to calculate $\phi_{V}$.

Figure 13: Magnetic flux relations for low-mass stars. The columns on the left
show measurements of Zeeman broadening fields and the right columns present
measurements of ZDI fields. The top row shows magnetic field as a function of
Rossby number and the bottom row shows X-ray luminosity as a function of
magnetic flux. (a): Figure from Reiners (2012), where crosses are Sun-like
stars, circles are M-type of spectral class M6 and earlier, red squares are
late M dwarfs. I performed a rough computation of the power-law dependence
with Ro for the unsaturated stars: $\langle|B_{I}|\rangle\propto{\rm
Ro}^{-1.41\pm 0.22}$. (b) and (d): Figures from Vidotto et al. (2014a), where
different symbols correspond to different ZDI surveys (open circles are the
late and mid M dwarfs, not considered in the fits). (c): Figure from Pevtsov
et al. (2003), where dots are for the quiet Sun, squares for X-ray bright
points, diamonds for solar active regions; pluses for solar-disc averages,
crosses for G, K, and M dwarfs, and circles are for T Tauri stars. Solid lines
shown in the last three panels are power-law fits.

The trend we see in Fig. 13a is also seen in other tracers of stellar
activity, such as in X-ray luminosity versus rotation. I will come back to
this later on this Section. This trend consists of two parts: a flat, or
saturated, part that is independent of stellar rotation, and a power-law that
shows magnetism/activity decreasing with the increase of Ro (i.e., towards
slow rotators). Saturation occurs in fast rotating stars and the break occurs
roughly at Ro $\sim 0.1$. Here, we are not focusing on the red squares in
panel a nor on the open circles in panel b, as those represent mid to late-M
dwarfs. Panels a and b in Fig. 13 show that both the large-scale field
$\langle|B_{V}|\rangle$ and the total field $\langle|B_{I}|\rangle$ behave
similarly and are summarised as follows:

  * •

I computed a rough fit for Ro $\gtrsim 0.1$ for panel Fig. 13a and found a
power-law slope of $-1.41\pm 0.22$ (cf, with $-1.2$ from Saar 2001), which is
consistent with the slope found in ZDI studies of $-1.19\pm 0.14$ (panel b).

  * •

The saturation in panel a occurs at about $\langle|B_{I}|\rangle_{\rm sat}\sim
2400$ G, or, assuming an average stellar radius of $1\,R_{\odot}$ for the
solar-like stars, a magnetic flux of $\phi_{I,{\rm sat}}\sim 1.4\times
10^{26}$ Mx. Saturation for ZDI measurements is currently not clear – the
indicative grey dashed line in panel b is at $\phi_{V,{\rm sat}}\sim 1.8\times
10^{24}$ Mx (see discussion in Vidotto et al. 2014a)

Another very interesting relation is between stellar X-ray luminosities
$L_{X}$ and magnetic fluxes, as shown in the bottom panels of Fig. 13.
Combining solar data and Zeeman broadening measurements, Pevtsov et al. (2003)
reported a large dynamical range spanning about 12 orders of magnitude in
$L_{X}$ and $\phi_{I}$. If we focus only on dwarf stars (crosses in Fig. 13c),
the dynamical range is more modest and spans about 2 orders of magnitude,
which is also similar to the dynamical range in $\phi_{V}$ (Fig. 13d).
Comparing panels c and d, we see that in both cases, the large-scale field and
the total field increase with X-ray luminosity. This is an indication that
magnetic fields power coronal activity, in the form of X-rays. Considering
only the dwarf stars, Pevtsov et al. (2003) found a power-law slope of
$0.98\pm 0.19$ for Zeeman broadening measurements, while ZDI measurements show
a slope of $1.80\pm 0.20$ (Vidotto et al., 2014a). Technically, these slopes
are similar within $3\sigma$, but the differences in slopes could also be
real. If this is the case, this could indicate different ‘efficiencies’ in
producing large- and small-scale fields (Donati et al., 2008; Morin et al.,
2008). One needs to keep in mind that the X-ray values and magnetic field
values shown in each panel are not contemporaneous. For example, similar to
the Sun, stars have cycles, which affect both field strengths and X-ray
luminosities. Thus, non-contemporaneous X-ray and magnetic measurements would
lead to increased scatter in the relations shown in Fig. 13. The scatter is
seen in all four panels for stellar measurements (it is less evident in panel
c, due to the large dynamical range of the plot).

Figure 13 shows that overall, $\langle|B_{I}|\rangle$ and
$\langle|B_{V}|\rangle$ follow activity trends that are similar to those
reported in chromospheric and coronal activity proxies (e.g., Skumanich, 1972;
Wright et al., 2011). An interesting question still remains: how does the
magnetic field _evolve_ with age itself? Figure 14 shows that the average
magnetic field intensity decays with age-0.65±0.04 (Vidotto et al., 2014a).
This trend is valid over three orders of magnitude in field intensity (from G
to kG fields) and four orders of magnitude in ages (from Myr to several Gyr).
Although the trend is clear, it also presents a large scatter. Part of this
scatter is due to short-term evolution of magnetism, including magnetic
cycles. Additionally, in the case of stars younger than $\sim 600$ Myr, part
of the scatter is also caused by stars of similar ages and masses, but
different rotation rates, showing different levels of magnetisation (see
Figure 7a in Folsom et al., 2016, where we see a factor of 10 in
$\langle|B_{V}|\rangle$ for solar-mass stars at 120 Myr). It is fascinating,
nevertheless, to see that the relation in Figure 14 is in accordance to the
observed decrease of $\Omega_{\star}$ with approximately the square-root of
age. I will discuss age–rotation relations in Sect. 4.2.

Figure 14: The average unsigned large-scale magnetic field decays with age as
$t^{-0.655}$. This is a similar trend as seen in $\Omega_{\star}(t)$ (Equation
22). Different symbols correspond to different ZDI surveys. Note that the Sun
is represented at minimum and maximum phases of its magnetic cycle. Typical
error bars are indicated on the bottom left. Figure from Vidotto et al.
(2014a).

###  4.2 Evolution of stellar rotation

Stellar clusters are laboratories for stellar evolution models. In particular,
observations of rotation rates of stars in clusters at different ages allow us
to draw an evolutionary history for stellar rotation. Figure 15 shows
age–rotation diagrams from Gallet and Bouvier (2015). The observations of
rotation rates of stars with masses $\sim 1~{}M_{\odot}$ are shown as pluses,
the Sun is depicted as an open circle and rotational velocity dispersion of
old field stars in the Galactic disc is represented by the rectangle labeled
‘OD’ (old disc). The right panel shows, in addition, model tracks as solid
(describing the rotation of the envelope) and dotted (same, but for the core)
lines. The observations shown in these diagrams are from stars in open
clusters, which have good age estimates. The variety of stellar rotation rates
found in each cluster results in the vertical alignment of the pluses. For
each cluster, the blue, green and red diamonds represent the 25th, 50th, and
90th rotational percentiles, respectively, and the associated models are shown
in curves with the same colours.

Figure 15: Rotational evolution from solar-mass stars. Pluses are observations
of rotation rates of open cluster stars, circle is the present-day Sun and the
rectangle shows the velocity dispersion of old disc field stars. The grey
shaded bars in panel (a) highlight the velocity dispersion at each age. The
red, green, and blue diamonds represent the 25th, 50th, and 90th rotational
percentiles, respectively. In panel (b), models are over-plotted to these
observations. Image adapted and reproduced with permission from Gallet and
Bouvier (2015), copyright by ESO.

Two clear behaviours are seen in Fig. 15. Firstly, it is noted that stars,
even those with different rotation histories at the early stages of their
lives, converge to a unique rotation-age relation approximately at the age of
the Hyades open cluster ($\simeq 625$ Myr, Delorme et al. 2011). From then on,
observations show that their rotation rates evolve as

$\Omega_{\star}\propto t^{-b},$ (22)

where $b$ is known as the magnetic braking index. The equation above forms the
basis of the gyrochronology dating method (Barnes, 2003), whereby a
measurement of rotation rate allows one to infer the age of the star.
Skumanich (1972) derived a braking index of $b\simeq 0.5$, but more modern
calibrations have suggested small variations around the square-root. For
example, Collier Cameron et al. (2009) and Delorme et al. (2011) obtained
$b=0.56$, Mamajek and Hillenbrand (2008) found $b=0.566\pm 0.008$ and dos
Santos et al. (2016) found $b=0.62\pm 0.36$. The gyrochronology relation
(Equation 22) can be very powerful for stars older than $\sim 600$ Myr, as
rotation rates are relatively simple to measure and allows us to derive the
age of stars. The problem of deriving ages using this method really starts for
stars younger than $\sim 600$ Myr.

The second behaviour seen in Fig. 15 is that younger clusters show a wider
dispersion of rotation rates, as opposed to older ones. Given that the
dispersion of rotational velocities is already seen at very young, Myr-old
open clusters, the dispersion is attributed to different ‘initial’ conditions
acquired at star formation. It is believed that a star born as, say, a fast
rotator will remain as a fast rotator during its evolution, i.e., it will
evolve along the upper envelope shown by the blue solid line in Fig. 15b.
Broadly speaking, the rotational evolution models represent three phases.
Starting from the young ages, we find the first phase, when the profile shows
a constant $\Omega_{\star}$. This is the disc-locking phase, in which the
accretion disc regulates stellar rotation. Meanwhile, the star is contracting
onto the main sequence. Once the disc is dispersed, disc locking ceases to
exist and only contraction continues to take place and, mostly due to angular
momentum conservation, the star then spins up. This is the only phase in which
rotation increases and we see the peak in rotation happening at around 20–30
Myr. From there on, rotation decreases with age, which means that single stars
on the main sequence will spin down with time. The main driver for this spin
down, as we already discussed in the introduction, is the magnetised stellar
wind, which carries away angular momentum.

The particular model I present in Fig. 15 (Gallet and Bouvier, 2015) includes
more physical aspects, such as the evolution of the moment of inertia (mass,
radius) and coupling between the rotations of the core and of the envelope.
Other models exist in the literature – they differ, for example, on which
stellar evolution code is used, different treatments for the core-envelope
decoupling, or different prescriptions for the ‘stellar wind braking law’,
among others (e.g., Kawaler, 1988; MacGregor and Brenner, 1991; Keppens et
al., 1995; Spada et al., 2011; Reiners and Mohanty, 2012; Epstein and
Pinsonneault, 2014; Johnstone et al., 2015a; Matt et al., 2015; Amard et al.,
2019). The braking law describes how angular momentum loss rate $\dot{J}$
depends on the stellar properties, such as mass, radius, stellar magnetism,
rotation, surface metallicity, and stellar wind mass-loss rate. The
constructions of braking laws are based on theoretical models, empirical
scalings, or a combination of both. I refer the reader to Amard et al. (2016)
for a study on how different braking laws can affect rotational evolution
models and thus the goodness of fits to observed rotation rates. Reviews on
the topic include the extensive lecture notes from Palacios (2013) and Bouvier
(2013). I will come back to angular momentum losses in Sect. 5.3.

###  4.3 Evolution of coronal activity

The third wind “ingredient” I would like to discuss is stellar activity. In
particular, I would like to focus on coronal activity. The corona starts above
the chromosphere (and above the transition region) and it is characterised by
a high temperature ($\simeq 10^{6}$K) and low density plasma. The emission
from the corona falls in the X-ray and UV part of the electromagnetic
spectrum. The solar wind is, in a sense, a continuation of the solar corona,
expanding into the interplanetary space. However, in the cool-star community,
it is common to separate the corona as the region of complex, _closed_
magnetic field lines and the wind as belonging to the region of _open_ field
lines. For this reason, the corona is sometimes referred as the ‘closed
corona’ and it is the source of X-ray and UV emission, with the wind-bearing
regions being X-ray dark (see Sect. 2.7). There is still no consensus at how
precisely the corona and the wind are heated to such high temperatures and it
is possible that a combination of different mechanisms heat the plasma in
different regions and at different times, as discussed in the recent review on
solar coronal and wind heating by Cranmer and Winebarger (2019).

Coronal activity is observed in other stars through X-ray and UV observations.
Figure 16 shows how the normalised X-ray luminosity $R_{X}={L_{X}}/{L_{\rm
bol}}$ evolves with Rossby number. Similar to what was seen in Figures 13a and
b, the emission saturates at $R_{X}\simeq 10^{-3}$ for fast rotating stars
with Ro $\lesssim 0.1$ and a decrease in X-ray emission is seen with Ro-2, or
$\sim\Omega_{\star}^{2}$, in the unsaturated part (Pizzolato et al., 2003;
Reiners et al., 2014). As discussed in Sect. 4.2, rotation can be used as a
proxy for age and thus this decay in X-ray can be attributed to age evolution.
Three factors contribute to the large spread seen in Fig. 16: inaccuracies in
X-ray measurements, variability caused by stellar cycles and intrinsic
differences between stars (some being more active than others for same mass,
age and rotation, Johnstone et al. 2020).

Figure 16: The X-ray-rotation activity relation shows how the normalised X-ray
luminosity $R_{X}={L_{X}}/{L_{\rm bol}}$ varies as a function of Rossby number
(fast rotators on the left, slow rotators on the right of the $x$ axis).
Figure from Reiners et al. (2014).

Figure 17 shows the evolution of high-energy radiation of solar-type stars.
The left panel divides the high-energy flux into 6 bands: the [1, 20]Å band
corresponds to hard X-rays, [20, 100]Å to the ROSAT band, [100, 360]Å to the
EUVE band and [920, 1180]Å corresponds to FUSE (Far Ultraviolet Spectroscopic
Explorer) bands. There is a gap between 360 and 920Å for solar-like stars, so
the fit for this band shown in the figure relies on interpolations from
neighbour wavelength bands. The slopes in the power-law fits are shown in
Table 5 of Ribas et al. (2005) – they range from the steepest value of $-1.92$
for the hard X-rays to $-0.85$ for the less energetic band. The fluxes are
normalised to the present-day Sun values. We see that the contribution of the
X-ray part of the spectrum is quite strong at young ages, compared to present-
day solar values.

The EUV part of the spectrum (100–920Å) is relevant in studies of atmospheric
escape in planets and exoplanets. Sanz-Forcada et al. (2011) estimated that
the EUV luminosity $L_{\rm EUV}$ is related to the X-ray luminosity (5–100Å)
as

$L_{\rm EUV}=10^{4.8}L_{X}^{0.86},$ (23)

with luminosities given in erg s-1. Using the X-ray-rotation activity relation
(similar to shown in Fig. 16), the equation above and rotational evolution
models (similar to shown in Fig. 15), Tu et al. (2015) derived an evolutionary
model to describe the evolution of $L_{X}$ and $L_{\rm EUV}$ that is shown in
the right panel of Figure 17. The spread in rotation rates observed for young
stars implies a spread in XUV (X-ray plus UV) luminosities, such that slow
rotators at a given age will have lower XUV luminosities than a fast-rotating
star at the same age (Tu et al., 2015).

Figure 17: Evolution of high-energy irradiation of solar type stars. Left: The
observed evolution of high-energy flux at a given orbital distance at
different bands. Right: The X-ray (left axis) and extreme ultraviolet (right
axis) luminosities derived from rotational evolution tracks (e.g., Fig. 15) as
a function of stellar age. Images reproduced with permission from [left] Ribas
et al. (2005), copyright by AAS; and [right] Tu et al. (2015), copyright by
ESO.

More recently, Johnstone et al. (2020) derived a relationship between the
surface fluxes in X-ray and in the EUV bands (100–360Å) and (360–920Å):

$\log(F_{\rm EUV}^{[100-360]})=2.04+0.681\log(F_{X})\,,$ (24)

with

$\log(F_{\rm EUV}^{[360-920]})=-0.341+0.920\log(F_{\rm EUV}^{[100-360]})\,,$
(25)

with fluxes given in $\mathrm{erg\ cm^{-2}\ s^{-1}}$. The latter equation is
constrained using the Sun only, as there is essentially no measurements in the
band (360–920Å) for stars other than the Sun. To compare to the ‘total’ EUV
band (100–920Å) used in Eq. (23), one should then sum $F_{\rm
EUV}^{[100-360]}$ and $F_{\rm EUV}^{[360-920]}$, obtaining

$F_{\rm EUV}=10^{2.04}\left(F_{X}^{0.681}+0.31F_{X}^{0.626}\right).$ (26)

Comparing Eqs. (23) and (26), we notice a smaller exponent is the latter. This
is because one relation uses flux and the other luminosity. Johnstone et al.
(2020) demonstrate that using surface fluxes reduce spread in empirical
relations between X-ray, Ly-$\alpha$ and overall EUV emissions.

In the Sun, X/EUV flux is very well correlated with sunspot number and the
solar surface magnetic flux (Figs. 1 and 8 in Hazra et al., 2020). Figure 18
shows how the solar XUV flux varies with surface magnetic flux. The XUV flux
is calculated over the range 5 to 912 Å, using data from the Solar Extreme
Ultraviolet Experiment (SEE) instrument on the NASA Thermosphere Ionosphere
Mesosphere Energetics Dynamics (TIMED) spacecraft (Woods et al., 2005). The
solar synoptic maps used to compute the magnetic fluxes are from the
Helioseismic Magnetic Imager (HMI) on board the Solar Dynamics Observatory
(Scherrer et al., 2012). The empty symbols consider the HMI data in its full
resolution, while the filled symbols only considers magnetic field components
with spherical harmonics orders below $\ell_{\max}=10$. This represents the
Sun-as-a-star case.

Figure 18 shows the relatively tight correlation between surface XUV flux and
magnetic fluxes considering both the full resolution maps as well as the low
resolution maps (large-scale fields). This tight correlation led Hazra et al.
(2020) to suggest that the large-scale fields observed in other stars (e.g.,
Fig. 12) could be used to infer the stellar X/EUV fluxes of stars through the
relation

$F_{\rm XUV}\propto\phi_{V}^{1.04\pm 0.026},$ (27)

with $\phi_{V}$ defined in Eq. (20). Note, however, that if one assumes a
lower harmonics order (e.g., $\ell_{\max}\sim 5$) for the ‘Sun as a star’
case, the scatter in the relation increases substantially. The reason why XUV
emission is better related with higher harmonics order is that the solar XUV
emission mostly comes from compact active regions, which are characterised by
smaller-scale fields, i.e., with higher harmonics orders (see discussion at
the beginning of Section 4.3).

Figure 18: Solar XUV flux as a function of the surface magnetic flux. Open
symbols represent the surface magnetic flux calculated directly from HMI
synoptic maps. Filled symbols represent the Sun-as-a-star, where only
harmonics up to order $\ell_{\max}=10$ were used. Colours show temporal
evolution with darkest colour corresponding to year 2010.5 and lightest colour
to year 2019.5 (solar cycle 24). The solid lines show power-law fits to the
data. Further discussion is shown in Hazra et al. (2020).

##  5 Where are we in terms of modelling?

The three observed ingredients discussed in Sect. 4 are often used to inform
stellar wind models, which developed from models of the present-day solar
wind. Here, I do not go into details of the present-day solar wind models. I
refer the reader to some recent reviews in the area, such as Gombosi et al.
(2018) for a review using fluid-based description and Marsch (2006); Echim et
al. (2011) for reviews adopting kinetic theory. Other comprehensive reviews in
the area, discussing the heating and acceleration of the present-day solar
wind, include Cranmer (2009); Cranmer and Winebarger (2019); Richardson
(2018).

This section is focused on models of winds of solar-type stars and, in
particular, on creating an evolutionary sequence that considers and reproduces
the stellar observations available to us, namely rotation, magnetism and
activity (Sect. 4). In Sect. 5.1, I provide a brief overview of the main types
of models used to describe winds of solar-type stars. One important physical
process that is not frequently dealt with in solar wind models is angular
momentum loss. This is mostly because solar wind studies have focused on the
properties of the Sun _today_ : its open magnetic flux, mass flux (densities
and velocities) impinging on the solar system objects, and its mass-loss rate.
The solar wind is usually treated as a snapshot in evolutionary timescales,
and rotation, although considered in solar wind models, has its feedback
neglected on the Sun’s long-term evolution. For stellar astronomers, rotation
is the key for the evolution of the Sun and its wind properties. Thus, in
Sect. 5.3, I focus on modelling angular momentum losses, after presenting a
brief overview of magnetohydrodynamics (MHD) wind models in Sect. 5.2.

###  5.1 Overview on the different treatments used in stellar wind models

Studies of the solar wind have provided guidance for models of winds of low-
mass stars. The detailed physical mechanism(s) that provides the heating and
acceleration of solar-like winds is not known, similarly to the solar wind
itself (e.g., Cranmer, 2009). These winds are believed to be magnetically-
driven, in which coupling between stellar magnetism and convection transports
free magnetic energy, which in turn is converted into thermal energy in the
upper atmosphere of stars (Matsumoto and Suzuki, 2014), giving rise to a hot
corona and a hot wind ($\gtrsim 10^{6}$K). The X-ray emitting stellar corona,
set by energy input, varies with the properties of the star (Jardine, 2004;
Güdel, 2004), as do the stellar wind properties (Ó Fionnagáin et al., 2019).

To the best of my knowledge, the currently-available stellar wind models are
based on fluid descriptions (magnetohydrodynamics, MHD from now on). Although
used to study the solar wind itself, I am not aware of models that have
adopted kinetic theory to study the evolution (or even a snapshot) of winds of
solar-type stars. In terms of the fluid description, two modelling approaches
are used in the study of the hot coronal winds of low-mass stars. The first
one explicitly considers atmospheric heating that is deposited at the
photospheric (or sometimes the chromospheric) level, while for the second one,
a million Kelvin-temperature wind is assumed (i.e., heating is assumed a
priori and thus is “implicit”). I discuss these different treatments next.

##### (i) Explicitly including heating from the photosphere:

A possible scenario to convert magnetic into thermal energy in the atmosphere
of stars involves the dissipation of waves and turbulence, based on scenarios
developed for the solar wind (e.g., Holzer et al., 1983; Cranmer, 2008;
Cranmer and Saar, 2011; Suzuki et al., 2013; Matsumoto and Suzuki, 2014). In
addition to depositing energy, the gradient of wave pressure provides a
volumetric force that accelerates the wind, similar to thermal pressure
gradients. The modelling approach that explicitly considers heating from the
photosphere involves a more rigorous computation of the wave energy and
momentum transfer, i.e., the computations are done from “first principles”
(e.g., Hollweg, 1973; Holzer et al., 1983; Vidotto and Jatenco-Pereira, 2006;
Cranmer, 2008; Suzuki et al., 2013; Shoda et al., 2019). In these models, the
increase in temperature from the colder photosphere to the hotter corona
arises naturally in the solution of the equations as does the wind
acceleration. Most of the models that treat the stellar wind acceleration
starting from the photosphere have focused on the wind dynamics along a single
open magnetic flux tube, as, depending on the level of details of the physics
involved in the wind acceleration/heating mechanisms, models can become
computationally intensive. In particular, a challenging numerical aspect is
the large contrast between the dense and cold photosphere and the hot and
rarefied corona (e.g., Matsumoto and Suzuki, 2012).

Models validated for the solar wind have been applied to stellar wind studies.
Suzuki et al. (2013) presented a parametric study of Alfvén-wave driven winds
applied to solar-like stars.777Alfvén waves are MHD waves that propagate along
magnetic field lines and have been suggested as a possible heating mechanism
for the solar corona (Alfvén, 1947). The authors simulated a wide range of
input wave fluxes, starting from the photosphere, by adopting different values
of magnetic field strengths and turbulent velocities, representative of
stellar values. The left panel of Fig. 19 shows a summary of their simulation
results, where the output kinetic luminosity of the stellar wind
($\dot{M}u_{r}^{2}/2$, $y$-axis) is plotted against the the input wave
luminosity ($x$-axis). Different values of average magnetic field strengths
are shown by the different symbols. Here, $B_{r,0}$ represents the magnetic
field intensity at the bottom of a magnetic flux tube and $f_{0}$ represents
the fraction of the star that is covered in open flux tubes. Note the vertical
scatter, indicating that similar input wave energies give rise to different
wind kinetic energies. The right panel of Fig. 19 shows a similar plot, but
instead of the input wave luminosity, the $x$-axis now shows the wave
luminosity as measured at the transition region (top of the chromosphere).
Note that these values are smaller than the input wave luminosity at the
photosphere, because a fraction of the waves is lost as they reflect back
downward, being mostly radiated away. Suzuki et al. (2013) found that a
fraction between 1 and 30% of the original input wave energy is able to get
into the chromosphere. Of this survival wave energy, part of the energy is
lost in the form of radiation and part is used to do work against the
gravitational force – thus, only a small amount of the input wave energy at
the photosphere is actually used to drive the stellar winds. This study shows
that waves cannot easily penetrate in the corona/wind. Overall, for a given
magnetic field, an increase in the wave input generates a larger output
kinetic energy, until saturation occurs. The consequence of this is that, for
a given value of $B_{r,0}f_{0}$, the mass-loss rate of the stellar wind
increases with input wave energy up to a saturation value $\dot{M}$sat. Beyond
this point, the increase in input wave energy seems to no longer increase
mass-loss rate. The saturation is also seen to occur at higher values for
larger average fields: $\dot{M}$${}_{\rm sat}=7.86\times
10^{-12}(B_{r,0}f_{0}/[1\rm{G}])^{1.62}\,M_{\odot}~{}{\rm yr}^{-1}$. This
result indicates that stars more magnetically active would have higher
‘maximum’ mass-loss rates. It is interesting to note that this model links not
only the magnetic field of an open flux tube, but the product between the
magnetic field of a flux tube and the amount of open flux tubes in the
surface.

Figure 19: Left: The output kinetic luminosity ($\dot{M}u_{r}^{2}/2$,
$y$-axis) of stellar winds increases with the input wave luminosity ($x$-axis)
at the photosphere (subscript ‘0’). Different values of magnetic field
strengths are shown by the different symbols. Curves of constant ratios
between $y$-axis and $x$-axis (labeled $c_{E}$) are shown by the solid lines.
Right: The same as the left panel, but instead of the input wave luminosity,
the $x$-axis now shows the wave luminosity as measured at the transition
region (top of the chromosphere, subscript ‘tc’). Images reproduced with
permission from Suzuki et al. (2013), copyright by ASJ.

In Sect. 4.1, I discussed surface filling factor of active regions, $f$,
mentioning that recent results suggest that the magnetic field in active
regions does not change significantly from star to star, but the filling
factor $f$ increases for fast rotators/younger stars (Cranmer and Saar, 2011;
See et al., 2019b; Kochukhov et al., 2020). The filling factor in active
regions is likely linked to the filling factor of open flux tubes – if one
assumes that the field lines in active regions will remain closed, at the very
least we could imagine a scenario where $f\simeq 1-f_{0}$, i.e., the star has
only $f_{0}$ open and $f$ closed field regions (there are no unmagnetised
regions in the star in my speculative scenario). Then, the question is: if $f$
saturates and eventually $f\to 1$ for high rotation, does it mean that the
star will have no open flux tube and thus no stellar wind if it is very
active? This is unlikely.

Unfortunately, to determine the amount of open magnetic flux, one needs
models, such as potential field extrapolations (e.g., Jardine et al., 2002) or
stellar wind models (e.g. Vidotto et al., 2014b; Réville et al., 2015).
Nevertheless, even without models, there is an observationally-derived proxy
that could tell us a bit more about how open magnetic flux, i.e., the one
embedded in the wind, varies with stellar activity. A large fraction ($\sim
90$%, Reiners and Basri 2009; Morin et al. 2010) of the unsigned field derived
in Zeeman broadening measurements can be missed in ZDI measurements due to
flux of different polarities cancelling out within an element of resolution
(Lang et al., 2014). The bulk of the “missed” field is thus in small-scale
fields (in regions that are likely smaller than active-region sizes), which
are mostly connected to each other forming closed-field structures. The
smaller the loop size, and thus the larger its harmonic order $\ell$, the
faster is its decay with radial distance, given that the magnetic field of
order $\ell$ decays with $r^{-(\ell+2)}$. These small loops, thus, decay very
fast with distance and they do not thread the wind, which represents a more
global-scale quantity. The magnetic field that will thread the wind is, on the
contrary, a large-scale field, with See et al. (2018) showing that the open
flux is largely dominated by the dipolar field alone. ZDI measures large-scale
fields, a fraction of which will open up and contribute to the open magnetic
flux. Of course, one cannot tell which fraction of the ZDI field will open up
and contribute to the wind without a model, but in light of the two available
observational quantities, namely the total field (small+large scales) derived
in Zeeman broadening measurements and the large-scale field derived in ZDI
measurements, the latter represents a better proxy for the wind-bearing
magnetic field lines. Thorough discussions on the fraction of open field in
the present-day Sun and on other cool dwarfs can be found in Cranmer (2017)
and See et al. (2019b). Given that the magnetic flux in ZDI fields increases
with X-ray luminosities (see Fig. 13d), this probably means that the magnetic
flux in open field lines increase for more active stars. Thus, even if the
closed-field region increases in coverage for fast rotators, the remaining
area of open field lines, even occupying a smaller portion of the star,
harbours a larger amount of magnetic flux, which ultimately pertains to the
stellar wind.

Theory and numerical simulations predict that wind quantities such as mass-
and angular momentum-loss rates depend on the amount of open magnetic flux
(e.g., Washimi and Shibata, 1993; Vidotto et al., 2014b; Réville et al., 2015;
See et al., 2019a), thus estimating the fraction $f_{0}$ of open field lines
is important for stellar wind models. This is also seen in the work of Suzuki
et al. (2013) – Figure 19 (left), for example, shows that the output kinetic
luminosity of the wind depends on the fraction $f_{0}$ of the photosphere
covered in open flux tubes, along which Alfvén waves with luminosity $L_{A,0}$
can propagate. (The wave luminosity itself depends on the $\sim$kG-level
magnetic field intensity at the bottom of a magnetic flux tube:
$L_{A,0}\propto v_{A,0}\propto B_{r,0}$.)

##### (ii) Implicitly assuming heating in wind models:

The models that explicitly solve for the energy input in the upper atmosphere
of Sun-like stars, such as from Suzuki et al. (2013), reproduce the rapid
increase in temperature from the photosphere to the corona. One of their
limitations is that, because they are computationally expensive, they are
computed along a flux tube, and usually do not consider rotation (however, see
Shoda et al. (2020) for first efforts in including rotation in these types of
model). These two limitations can be overcome in the second approach for
modelling winds of solar-like stars that I discuss now (albeit other
limitations appear). This approach adopts a simplified energy equation,
usually assuming that the thermal variables are connected by a polytropic
relation. Typically, a power-law between the thermal pressure $P$ and the
density $\rho$ or temperature $T$ are used: $P\propto\rho^{\Gamma}$, or
$T\propto\rho^{\Gamma-1}$ where $\Gamma$ is known as the polytropic index. A
value of $\Gamma=1$, for example, indicates that the temperature is
independent of the density, i.e., mimicking an isothermal relation. In the
solar wind, the measured effective polytropic index in the corona is around
$1.1$ (Van Doorsselaere et al., 2011), increasing to $1.46$ near Earth (Totten
et al., 1995) and decreasing with distance in the outer heliosphere (Elliott
et al., 2019). Usually, polytropic wind models assume constant values of
$\Gamma$ ranging from $1$ to $1.15$, although a few models treating a
distance-dependent $\Gamma$ exist for solar wind models (Roussev et al., 2003;
Cohen et al., 2007) and stellar wind models (Vidotto, 2009; Johnstone et al.,
2015b, a).

In polytropic wind models (which can be magnetised or not), the computation
often starts at the point where the temperature has already reached coronal
values $\sim 10^{6}$ K (e.g., Pneuman and Kopp, 1971; Washimi and Shibata,
1993; Keppens and Goedbloed, 2000; Matt et al., 2012; Vidotto et al., 2009b;
Réville et al., 2015). This approach ignores the physical reason of what led
temperatures to increase from photospheric to coronal values. Technically, it
is as if the wind base occurred at the corona – if we take the height of the
solar corona to be at about 2000 km (e.g., Gary, 2001; Yang et al., 2009),
this means that the base of the corona, and of these wind models, occurs at
$\sim 1.003\,R_{\odot}$. In practice, models usually start at $1\,R_{\odot}$
and this small difference in neglected in simulations.

By assuming a polytropic wind, the energy equation is simplified and no