# Torus-stable zone above starspots

Xudong Sun (孙旭东) ,1 Tibor Török ,2 and Marc L. DeRosa 3
1Institute for Astronomy, University of Hawai‘i at Mānoa, Pukalani, HI 96768,
USA
2Predictive Science Inc., San Diego, CA 92121, USA
3Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA 94304, USA
E-mail<EMAIL_ADDRESS>(XS)

(Accepted 2021 November 5. Received 2021 November 4; in original form 2021
September 14)

###### Abstract

Whilst intense solar flares are almost always accompanied by a coronal mass
ejection (CME), reports on stellar CMEs are rare, despite the frequent
detection of stellar ‘super flares’. The torus instability of magnetic flux
ropes is believed to be one of the main driving mechanisms of solar CMEs.
Suppression of the torus instability, due to a confining background coronal
magnetic field that decreases sufficiently slowly with height, may contribute
to the lack of stellar CME detection. Here we use the solar magnetic field as
a template to estimate the vertical extent of this ‘torus-stable zone’ (TSZ)
above a stellar active region. For an idealised potential field model
comprising the fields of a local bipole (mimicking a pair of starspots) and a
global dipole, we show that the upper bound of the TSZ increases with the
bipole size, the dipole strength, and the source surface radius where the
coronal field becomes radial. The boundaries of the TSZ depend on the
interplay between the spots’ and the dipole’s magnetic fields, which provide
the local- and global-scale confinement, respectively. They range from about
half the bipole size to a significant fraction of the stellar radius. For
smaller spots and an intermediate dipole field, a secondary TSZ arises at a
higher altitude, which may increase the likelihood of ‘failed eruptions’. Our
results suggest that the low apparent CME occurrence rate on cool stars is, at
least partially, due to the presence of extended TSZs.

###### keywords:

Sun: coronal mass ejections (CMEs) – Sun: magnetic field – stars: magnetic
field – starspots – instabilities

††pubyear: 2021††pagerange: Torus-stable zone above starspots–Torus-stable
zone above starspots

##  1 Introduction

###  1.1 Torus instability in solar eruptions

Solar coronal mass ejections (CMEs) are rapid expulsions of magnetized plasma
with velocity up to $3000$ km s-1 and mass up to a few $10^{16}$ g (Webb &
Howard, 2012). The fastest CMEs are often associated with intense flaring
(Vršnak et al., 2005). In these events, the CME kinetic energy and the flare
radiative energy are generally both of order $10^{32}$–$10^{33}$ erg (Emslie
et al., 2012). The interaction between a CME and the Earth’s magnetosphere can
cause severe space weather disturbances (Baker & Lanzerotti, 2016).

Many solar CMEs originate from active regions that harbour kilogauss magnetic
fields and sunspots. Prior to eruption, their coronal fields are thought to
often evolve towards a ‘magnetic flux rope’ configuration, i.e. a current-
carrying magnetic flux tube with coherent twist (Patsourakos et al., 2020).
Evidence of pre-eruption flux ropes in the low corona has been reported
frequently (e.g. Green et al., 2011; Patsourakos et al., 2013; Duan et al.,
2019; Kliem et al., 2021). The twisted structure is clearly visible in a
significant fraction of CMEs (Vourlidas et al., 2013).

The torus instability (Bateman, 1978; Kliem & Török, 2006) is believed to be
one of the main driving mechanisms of solar CMEs. For a toroidal flux rope
(Fig. 1; see also Fig. 1 of Myers et al., 2016), the internal poloidal
magnetic field and toroidal electric current produce an outward-directed
force, known as the ‘hoop force’, that drives its expxansion. The interaction
of the flux rope’s current with an external poloidal field $\bm{B}_{p}$, also
known as the ‘strapping field’, generates an oppositely directed ‘strapping
force’ that provides the equilibrium and can stabilize the system. The torus
instability sets in when $B_{p}=\lVert\bm{B}_{p}\rVert$ decreases fast enough
with respect to the major radius $\mathcal{R}$ of the toroidal flux rope, so
that it can no longer provide sufficient strapping force. For
$B_{p}\propto\mathcal{R}^{-n}$, the ‘decay index’ $n$ should satisfy

$n=-\dfrac{\partial\ln B_{p}}{\partial\ln\mathcal{R}}>n_{c},$ (1)

where $n_{c}$ is a critical decay index, above which the torus instability may
occur. Conversely, the torus instability will be suppressed if $n<n_{c}$. On
the Sun, the coronal flux ropes are akin to partial tori whose legs are
anchored in the photosphere (Chen & Krall, 2003). The height of the flux rope
axis’ apex above the photosphere, $h$, is commonly used in place of
$\mathcal{R}$.

The corona may be divided into ‘torus-stable’ ($n<n_{c}$) and ‘torus-unstable’
zones ($n>n_{c}$), separated by a critical height $h_{c}$ (at which
$n=n_{c}$). The value of $n_{c}$ depends on the details of the system, such as
the aspect ratio of the flux rope, the profile of the toroidal current, etc. A
range of $n_{c}$ between $0.5$ and $2$ has been found for solar cases based on
analytical models (Isenberg & Forbes, 2007; Olmedo & Zhang, 2010),
magnetohydrodynamic (MHD) simulations (Török & Kliem, 2007; Aulanier et al.,
2010; Fan, 2010; Kliem et al., 2014; Zuccarello et al., 2015), and laboratory
plasma experiments (Myers et al., 2015; Alt et al., 2021). In particular, the
value $n_{c}=1.5$, derived for a thin, axisymmetric, full torus (Bateman,
1978), has been reproduced in several aforementioned MHD simulations, and is
widely used in observational studies.

The majority of solar CMEs exhibit a slow expansion followed by an impulsive
acceleration (Zhang et al., 2001; Vršnak, 2001). For the torus instability to
be triggered, the flux rope must first ascend from a torus-stable zone (TSZ)
into an unstable one. This initial slow rise of the ejecta may be due to a
number of mechanisms, such as flux cancellation, shearing or twisting motions
(Green et al., 2018), that facilitate (slow) tether-cutting (Moore et al.,
2001) or breakout (Antiochos et al., 1999) reconnection. Indeed, the values of
$n$ at the eruption onset height based on coronal field models are broadly
consistent with the theoretical $n_{c}$ values (e.g. Filippov & Den, 2001;
McCauley et al., 2015). The occurrence of the torus instability is
particularly supported when the signatures of alternative mechanisms, for
instance flare emission due to magnetic reconnection, lag behind the CME
acceleration or are largely absent (e.g. Song et al., 2013; Cheng et al.,
2020).

Observations show that some upward-moving flux ropes decelerate and come to a
halt with no CME ensuing (Ji et al., 2003; Green et al., 2007; Chen et al.,
2013; Zhou et al., 2019). Such a ‘failed eruption’ may suggest that the
eruption is initiated in a TSZ due to alternative mechanisms, such as the
helical kink instability (Török & Kliem, 2005; Hassanin & Kliem, 2016) or
‘flare reconnection’ (Karpen et al., 2012). In these cases, the early
kinematic evolution resembles that of a successful eruption; the slow-decaying
background field is believed to play an important role in suppressing the
eruption (Huang et al., 2020). To test this hypothesis, many studies have
compared modelled coronal fields prior to flares that occurred with or without
a CME. The models for the latter type of flares, of which many are associated
with failed eruptions, tend to have a stronger background field or a smaller
$n$ at typical eruption onset heights (Wang & Zhang, 2007; Liu, 2008; Cheng et
al., 2011), a smaller $n$ at the flux rope axis’ apex (Jing et al., 2018; Duan
et al., 2019), or a greater $h_{c}$ for a given $n_{c}$ (Wang et al., 2017;
Sarkar & Srivastava, 2018; Baumgartner et al., 2018; Vasantharaju et al.,
2018; Li et al., 2020).

A failed eruption may also occur if a TSZ is situated above a torus-unstable
zone. A flux rope may erupt due to the torus instability in the lower layer,
and subsequently get trapped in the higher one. Indeed, some multipolar solar
active regions exhibit a saddle-like $n(h)$ profile that crosses $n=n_{c}$
three times (Guo et al., 2010; Cheng et al., 2011; Wang et al., 2017;
Filippov, 2020). The three critical heights divide the corona into four
layers; a secondary TSZ arises at a higher altitude.

Figure 1: Illustration of the basic magnetic ingredients for the torus
instability. The yellow field lines form a partial toroidal flux rope, whose
toroidal current is clockwise directed. The grey field lines indicate the
strapping field. The red (blue) colour shows positive (negative) photospheric
vertical field. The height of the apex $h$ and the major radius of the torus
$\mathcal{R}$ are marked.

###  1.2 The ‘missing’ stellar CMEs

Stellar flares are common on cool stars, and are thought to be powered by
excess magnetic energy as on the Sun (Benz & Güdel, 2010). The energy-
frequency relation follows a similar power law as the solar flares, but
extends to much higher energy ranges (Shibata et al., 2013). Thousands of
‘super flares’ with $10^{33}$–$10^{36}$ erg bolometric energy have been
detected, many of which occurred on Sun-like stars (e.g. Schaefer et al.,
2000; Maehara et al., 2012; Davenport, 2016; Howard et al., 2019; Günther et
al., 2020).

Yet, whilst stellar flares are observed with regularity, stellar CME
observations are rare. Less than $40$ candidates have been identified using
the signatures of Doppler shift, X-ray absorption, or coronal dimming in
extreme ultraviolet and X-ray (e.g. Moschou et al., 2019; Argiroffi et al.,
2019; Veronig et al., 2021). Surveys of Balmer-line spectra find no signatures
of CMEs in a sample of F–K dwarfs (Leitzinger et al., 2020), and only weak
line asymmetries on M dwarfs where the inferred velocities are mostly below
the escape threshold (Vida et al., 2019; Muheki et al., 2020). Recent radio-
optical observing campaigns report only one case of a type-iv@ radio burst
(Zic et al., 2020) and no type-ii@ radio bursts (Crosley & Osten, 2018a, b;
Villadsen & Hallinan, 2019), the latter being a hallmark for CME-generated
shocks on the Sun.

The lack of CME detection from cool stars is surprising. For the Sun, the
flare-CME association rate $P$ increases with the soft X-ray flux
$F_{\mathrm{SXR}}$ (as measured by the GOES satellite’s $0.1$–$0.8$ nm band),
and reaches unity for the largest events where $F_{\mathrm{SXR}}\geq
10^{-3.5}$ W m-2 (bolometric radiated energy $E$$\,\sim\,$$10^{32}$ erg;
Andrews, 2003; Yashiro et al., 2006). Barring major observational biases, the
stellar CME rate or their typical velocities must be significantly lower than
the corresponding solar values. This will have major implications on the
stellar mass loss and angular momentum loss rates (Drake et al., 2013; Odert
et al., 2017; Wood et al., 2021) as well as on exoplanet habitability
(Khodachenko et al., 2007; Airapetian et al., 2020).

###  1.3 Magnetic confinement on cool stars

NOAA solar active region 12192 in October 2014 hosted the largest sunspot
group since 1990. It was the second most flare-productive region of solar
cycle 24. Surprisingly, none of its major flares were associated with a CME.
Analyses revealed a weaker magnetic twist in the core-region field and a
stronger background field than that of two other flare-CME-rich regions (Sun
et al., 2015).

Active region 12192 serves as a solar analogue to the ‘missing stellar CME
conundrum’, to which failed eruptions may be a solution (Drake et al., 2016;
Osten & Wolk, 2017). Therein, magnetic reconnection driven by the initial
eruption onset could explain the flaring; confinement of the eruption by the
overlying field could explain the lack of CME detection.

Whilst the task of observing stellar flux ropes remains intractable, the
large-scale stellar photospheric fields have been probed for a sizable sample
of cool stars via spectropolarimetry observations (e.g. Marsden et al., 2014;
Petit et al., 2014; Folsom et al., 2016). Inversion techniques such as the
Zeeman Doppler Imaging (Semel, 1989; Donati & Brown, 1997) and the Magnetic
Doppler Imaging (Piskunov & Kochukhov, 2002) have been used to infer low-
resolution stellar magnetic field maps. The field strength and topology may be
very different from those of the Sun, and potentially more effective at
suppressing the CMEs. Here, we posit three relevant factors.

First, many cool stars host large spots over ten degrees in diameter with
kilogauss magnetic fields, compared to typical sunspots of sometimes similar
strength but diameters of just a few degrees (Berdyugina, 2005; Strassmeier,
2009). The unsigned magnetic flux $\lvert\Phi\rvert$ of such starspots will be
much greater than the typical solar value, $10^{22}$ Mx, and can, therefore,
provide strong ‘local confinement’ in the low corona. This is supported by
recent surveys of solar flares, where the flare-CME association rate $P$ is
found to be anti-correlated with $\log\,\lvert\Phi\rvert$ (Li et al., 2020).
For the empirical relation $P=\alpha\log F_{\mathrm{SXR}}+\beta$, the positive
slope $\alpha$ is found to drastically decrease with $\lvert\Phi\rvert$ (Li et
al., 2021). If this trend holds, even superflares with $F_{\mathrm{SXR}}=0.01$
W m-2 ($E$$\,\sim\,$$10^{34}$ erg) are expected to have $P<0.5$ in a large
stellar active region, $\lvert\Phi\rvert=10^{24}$ Mx.

Second, some cool stars can have strong global-scale magnetic fields up to the
kilogauss range, compared to just a few gauss on the Sun (Donati & Landstreet,
2009; Reiners, 2012; Kochukhov, 2021). The efficacy of this ‘global
confinement’ is demonstrated with MHD modelling of stellar coronae based on a
solar template. In Alvarado-Gómez et al. (2018), it is shown that a $75$ G
dipole field is able to fully entrap CMEs with up to $3\times 10^{32}$ erg
kinetic energy. It is interesting to note that these failed eruptions are
still accompanied by intense magnetic reconnection, which will produce flare
radiative energy of the same order.

Third, it has been argued that cool stars with stronger surface magnetic
activity will have a smaller fraction of magnetic flux open to the
asterosphere (Schrijver et al., 2003; Farrish et al., 2019). Consequently, the
larger fraction of closed magnetic flux is expected to exert stronger
confinement on erupting flux ropes. In the widely used potential field source
surface (PFSS) model for solar and stellar coronal fields (Altschuler &
Newkirk, 1969; Schatten et al., 1969), the effect is approximated by higher
‘source surface’ radii where the fields become open and radial due to the
action of stellar winds (Réville et al., 2015, 2016; See et al., 2017, 2018).

###  1.4 Aim and outline

In the study presented here, we estimate the upper bound of the TSZ, i.e. the
critical height $h_{c}$, above bipolar magnetic spots on cool stars. An
extended TSZ is expected to reduce the likelihood of the torus instability
onset and to increase the likelihood of failed eruptions. The suppression of
the torus instability may help to explain the low apparent stellar CME rate.

To this end, we consider an idealised stellar magnetic environment comprising
a local bipole (mimicking a pair of starspots) and a global dipole, which
provide the local- and global-scale confinement, respectively. Henceforth, the
starspot field configuration will be referred to as a (local) ‘bipole’,
whereas the background stellar field configuration will be referred to as a
(global) ‘dipole’. We use the PFSS model to calculate the large-scale coronal
magnetic field that an erupting flux rope would have to encounter, and
investigate how the TSZ above starspots varies with the bipole size, the
dipole strength, and the source surface radius.

The remainder of the paper is organized as follows. In Section 2, we describe
the idealised model, and present the results of the model calculations in
Section 3, followed by a discussion of the limitations of the model and the
implications of the results in Section 4.

##  2 Method

###  2.1 PFSS model

We adopt an asterographic coordinate system specified by radius $r$,
colatitude $\theta\in[0,\pi]$, and longitude $\phi\in[-\pi,\pi]$. The PFSS
model solves for the coronal magnetic field $\bm{B}(r,\theta,\phi)$ between
the stellar surface $r=R_{\star}$ and the source surface $r=R_{s}$. The field
is divergence- free (from Maxwell’s equations) and is assumed in the PFSS
model to also be current-free and thus curl-free. As a result, it can be
described by a scalar potential $\Psi(r,\theta,\phi)$ that satisfies

$\begin{split}\nabla\Psi&=-\bm{B},\\\ \nabla^{2}\Psi&=0.\\\ \end{split}$ (2)

The inner boundary condition at $r=R_{\star}$ is commonly set to match the
photospheric radial field (Wang & Sheeley, 1992). Additionally, $\Psi$ must be
constant (usually zero) on the outer boundary at $r=R_{s}$ as the field is
assumed to be radial there. That is,

$\begin{split}\left.\dfrac{\partial\Psi}{\partial
r}\right|_{r=R_{\star}}&=-B_{r}(R_{\star},\theta,\phi),\\\
\left.\Psi\right|_{r=R_{s}}&=0.\\\ \end{split}$ (3)

The solution can be written as (Hoeksema, 1984):

$\begin{split}\Psi=R_{\star}&\sum_{\ell=0}^{\infty}\sum_{m=0}^{\ell}P_{\ell}^{m}(\cos\theta)\,\left(g_{\ell
m}\cos{m\phi}+h_{\ell m}\sin{m\phi}\right)\times\\\
&{\left(\dfrac{R_{\star}}{r}\right)^{\ell+1}\left[1-\left(\dfrac{r}{R_{s}}\right)^{2\ell+1}\right]}\left/{\left[\ell+1+\ell\left(\dfrac{R_{\star}}{R_{s}}\right)^{2\ell+1}\right]}\right.,\\\
\end{split}$ (4)

where $\ell$ and $m$ are respectively the degree and order of the spherical
harmonics expressed in part as the associated Legendre polynomials,
$P_{\ell}^{m}(x)$. The coefficients $g_{\ell m}$ and $h_{\ell m}$ can be
determined by integrating the surface field:

$\genfrac{}{}{0.0pt}{}{g_{\ell m}}{h_{\ell
m}}=c_{\ell}\int\limits^{\pi}_{-\pi}\genfrac{}{}{0.0pt}{}{\cos}{\sin}m\phi\,\mathrm{d}\phi\int\limits^{\pi}_{0}B_{r}(R_{\star},\theta,\phi)\,P^{m}_{\ell}(\cos\theta)\sin\theta\,\mathrm{d}\theta,$
(5)

where the multiplier $c_{\ell}=(2\ell+1)/4\pi$ is a result from the Schmidt
quasi-normalization scheme for the spherical harmonics. An axial dipole field
is represented by the ($\ell=1$, $m=0$) mode, for which $B_{r}$ at the stellar
surface is $g_{10}\cos\theta$, with the amplitude of this mode proportional to
$g_{10}$.

We can evaluate the energy spectrum of the surface $B_{r}$ using the mean
squared field associated with a certain $\ell$ (DeRosa et al., 2012):

$\langle
B^{2}_{r,\ell}\rangle=\dfrac{1}{2\ell+1}\sum_{m=0}^{\ell}\left(g^{2}_{\ell
m}+h^{2}_{\ell m}\right).$ (6)

###  2.2 Bipole model

Solar active regions are often approximated as bipolar magnetic regions. Their
properties, including emergence latitude, spatial extension, tilt angle, and
magnetic flux, have been measured for decades (e.g. Wang & Sheeley, 1989;
Harvey & Zwaan, 1993; Stenflo & Kosovichev, 2012). Bipole models are commonly
used as the source term for surface flux transport models (e.g. van
Ballegooijen et al., 1998; Baumann et al., 2004; Cameron et al., 2010), or as
the boundary condition for the background magnetic field in CME models (e.g.
Amari et al., 1996; Török & Kliem, 2003; Aulanier et al., 2010). They have
also been scaled to estimate the magnetic energy available to stellar flares
(Aulanier et al., 2013).

We adopt a symmetric solar bipole model (Yeates, 2020) that can be scaled to
represent larger starspots. The centroids of the positive and negative
polarities are placed at $(s_{+},\phi_{+})$ and $(s_{-},\phi_{-})$,
respectively. Here $s$ is the sine of latitude $\lambda$, i.e.
$s=\sin\lambda=\cos\theta$. The overall bipole centroid location
$(s_{0},\phi_{0})$, the polarity separation (hereafter ‘bipole size’) $\rho$,
and the tilt angle $\gamma$ with respect to the equator are

$\begin{split}s_{0}&=(s_{+}+s_{-})/2,\\\ \phi_{0}&=(\phi_{+}+\phi_{-})/2,\\\
\rho&=\arccos\left[s_{+}s_{-}+\sqrt{1-s^{2}_{+}}\sqrt{1-s^{2}_{-}}\cos(\phi_{+}-\phi_{-})\right],\\\
\gamma&=\arctan\left[\dfrac{\arcsin(s_{+})-\arcsin(s_{-})}{\sqrt{1-s^{2}_{0}}\,(\phi_{-}-\phi_{+})}\right].\\\
\end{split}$ (7)

The bipole properties are defined by these parameters along with the unsigned
magnetic flux $\lvert\Phi\rvert$.

To obtain the surface magnetic field map, we first rotate the spherical
coordinate $(s,\phi)$ to $(s^{\prime},\phi^{\prime})$ such that the bipole is
centred on the equator $(s^{\prime}_{0},\phi^{\prime}_{0})=(0,0)$ with zero
tilt $\gamma^{\prime}=0$ (Appendix A of Yeates, 2020). This is equivalent to
$s^{\prime}_{+}=s^{\prime}_{-}=0$, and
$\phi^{\prime}_{-}=-\phi^{\prime}_{+}=\rho/2$. With the mapping from
$(\theta,\phi)$ to $(s^{\prime},\phi^{\prime})$, the desired
$B_{r}(R_{\star},\theta,\phi)$ can be evaluated in the new coordinate system
as:

$F(s^{\prime},\phi^{\prime})=-B_{0}\dfrac{\phi^{\prime}}{\rho}\exp\left[-\dfrac{\phi^{\prime
2}+2\arcsin^{2}(s^{\prime})}{(a\rho)^{2}}\right].$ (8)

The constant $a$ controls the axial dipole moment of the bipole, and the
positive $B_{0}$ controls $\lvert\Phi\rvert$.

For active regions in solar cycle 24, $a=0.56$ provides a good match with the
observed dipole moment. The maximum field strength from equation (8) is
$aB_{0}/\sqrt{2e}\approx 0.24B_{0}$.

Figure 2: Illustration of the modelled coronal magnetic topology with
$\rho=20\degr$, $g_{10}=200$ G, and $R_{s}=2.5R_{\star}$. The central sphere
shows the surface $B_{r}$. Open field lines with red (blue) colour have
positive (negative) magnetic polarity. Closed field lines in dark grey have
the same apex height at $r=2.4R_{\star}$ along the equator. Closed field lines
in light grey are rooted within or near the starspots. An interactive version
of the figure is available in the supplementary material and via Zenodo.

###  2.3 A model for the stellar coronal magnetic field

We now consider a cool star for which the photospheric magnetic field
comprises a pair of spots embedded in a global, axial dipole (Fig. 2). The
orientations of the bipole and the dipole are aligned such that the coronal
field strength is maximized. The surface $B_{r}$ of the bipole is prescribed
by equation (8), whilst the dipole contribution is $g_{10}\cos\theta$. We
further assume the following:

  * •

The bipole straddles the equator, and its dipole moment is aligned with the
north-south direction. The corresponding coronal field thus points largely
southward. The bipole is specified by $s_{+}=-s_{-}=\sin(\rho/2)$,
$\phi_{+}=\phi_{-}=0$, and $\gamma=\pi/2$.

  * •

The bipole magnetic field resembles that of sunspots. We specifically adopt
$a=0.56$ and $B_{0}=1.25\times 10^{4}$ G. The latter translates to a maximum
field strength of $3$ kG, typical for sunspot umbrae (Solanki et al., 2006).

  * •

The axial dipole moment is directed northward, i.e. $g_{10}>0$. All other
$g_{\ell m}$ and $h_{\ell m}$ coefficients are $0$. The corresponding coronal
field points southward.

The lower boundary condition $B_{r}(R_{\star},\theta,\phi)$ for the coronal
PFSS extrapolation is calculated on a grid with $1\degr$ spatial resolution,
where $B_{r}(R_{\star},\theta,\phi)$ is the sum of the starspot bipole and
global dipole fields. We truncate the series at a maximum $\ell=120$ (Tóth et
al., 2011). The modelled coronal field has a grid size of $0.001R_{\star}$ in
$r$.

The modelled coronal field is largely axisymmetric away from the starspots
(Fig. 2). The closed field lines lie within a same meridional plane, and the
open field lines originate from the polar regions and expand super-radially.
Near the starspots, however, the closed field lines zonally expand away from
the centre; the open-closed field boundaries extend toward the equator.

We acknowledge that our model configuration is highly idealised. In reality,
active regions are believed to emerge preferentially within the ‘active
latitudes’, which lie below $30\degr$ for the Sun but likely higher for rapid
rotators according to dynamo models (Schuessler & Solanki, 1992; Işık et al.,
2018). Their tilt angle with respect to the equator follows Joy’s law on the
Sun, with a median of less than $10\degr$ (Hathaway, 2015). Our estimates of
the TSZ heights presented in Section 3 therefore likely represent upper
limits.

In this set-up, we assume that a possibly eruptive flux rope (not present in
the model) resides above the starspots (see Fig. 1). Its axis is expected to
be nearly parallel to the magnetic polarity inversion line along the equator.
The apex of the flux rope axis is located at $h=r-R_{\star}$, $\theta=\pi/2$,
and $\phi=0$. The external poloidal component of the coronal field is north-
south oriented above the polarity inversion line. Hence, we adopt $B_{p}\equiv
B_{\theta}(r,\pi/2,0)$ as the strapping field. The decay index $n$ in equation
(1) is then

$n(r)=\left.-\dfrac{r-R_{\star}}{B_{\theta}}\dfrac{\partial
B_{\theta}}{\partial r}\right|_{\theta=\pi/2,\,\phi=0}.$ (9)

In what follows, we assume that the critical index $n_{c}$ occurs at a value
of 1.5. Opting for a larger (smaller) $n_{c}$ will generally increase
(decrease) the values of $h_{c}$; it is not expected to change the conclusions
qualitatively.

Hereafter, the length variables will be normalised to the stellar radius
$R_{\star}$; normalised variables will be designated by a tilde. For example,
the normalised source surface radius and critical height are
$\tilde{R}_{s}=R_{s}/R_{\star}$ and $\tilde{h}_{c}=h_{c}/R_{\star}$,
respectively.

Figure 3: Surface properties of the modelled starspots, with the circles
showing the results for various bipole sizes $\rho$. The curves show either a
quadratic or a linear fit. The panels illustrate (a) the single spot area
$A_{\mathrm{s}}$ as a function of $\rho$, (b) the effective spot diameter $d$
as a function of $\rho$, (c) the unsigned magnetic flux of the bipole
$\lvert\Phi\rvert$ (upper branch), and within the spot pair
$\lvert\Phi_{\mathrm{s}}\rvert$ (lower branch) as functions of the area of the
spot pair $A_{\mathrm{s}}$. The solar radius $R_{\odot}=6.96\times 10^{10}$ cm
is used in place of $R_{\star}$. The slopes are $c=1.23\times 10^{20}$ and
$c_{\mathrm{s}}=0.98\times 10^{20}\,\mathrm{Mx}\,\mu\mathrm{Hem}^{-1}$. The
colours in the circles are based on $\rho$ in all three panels.

Figure 4: Height profiles of the strapping field $B_{p}$, for bipoles of
various $\rho$ and dipoles of various $g_{10}$. The nominal
$\tilde{R}_{s}=2.5$ is used.

Figure 5: Energy spectra $\langle B^{2}_{r,\ell}\rangle$ of the surface
$B_{r}$, as derived from equation (6), for bipoles of various $\rho$. Vertical
dotted lines indicate the peaks.

###  2.4 Free parameters

Our coronal field model includes three free parameters: the bipole size
($\rho$), the axial dipole strength ($g_{10}$), and the source surface radius
($R_{s}$). Their values can vary significantly amongst cool stars.

Bipole size.— Known sunspots cover at most $0.1\%$ of the solar surface area,
and their sizes follow a log-normal distribution. The typical (extreme) area
of a single spot is about $60$ ($3000$) microhemispheres ($\mu$Hem), or
$1.3\degr$ ($8.9\degr$) in diameter (Baumann & Solanki, 2005). The average
polarity separation $\rho$ of solar active regions is about $5\degr$ (Wang &
Sheeley, 1989). In contrast, the spot coverage of cool stars typically ranges
from $1\%$ to $10\%$ based on Doppler imaging, and tends to be larger based on
molecular band modelling (Berdyugina, 2005; Strassmeier, 2009). Recent studies
using light curve modulation or occultation mapping (with transiting
exoplanet) suggests that the diameter of a single spot (cluster) can reach
$20\degr$ (Davenport et al., 2015; Morris et al., 2017).

Axial dipole strength.— For the Sun, the axial dipole amplitude $g_{10}$ has
varied from near zero (maximum phase) to about $5$ G (minimum phase) for the
past several activity cycles (DeRosa et al., 2012), and remained below $10$ G
for the past century (Muñoz-Jaramillo et al., 2012). Surveys show that slow-
rotating, solar-mass stars and low-mass M dwarfs both tend to have poloidal,
axisymmetric field, though the field amplitudes can be very different (Donati
& Landstreet, 2009). The estimated dipole coefficient $g_{10}$ ranges from
$\log g_{10}\approx 0.3$ G (solar-like stars) to $\log g_{10}\approx 3.2$ G (M
dwarfs; See et al., 2019, and references therein). The mean field $\langle
B\rangle$ of inactive (active) M dwarfs are in the hectogauss (kilogauss)
range (Kochukhov, 2021). The corresponding $g_{10}\approx 2\langle B\rangle$
(assuming a pure dipolar field) is $1$–$3$ orders of magnitude stronger than
the Sun.

Source surface radius.— For the Sun, the value of $\tilde{R}_{s}$ may be
empirically determined by comparing the PFSS modelling results with various
imaging and in situ observations; $\tilde{R}_{s}=2.5$ is widely used
(Hoeksema, 1984). The optimal value likely varies between $1.5$ and $4$
depending on the magnetic activity level, the cycle phase, and the observable
used for calibration (Lee et al., 2011; Arden et al., 2014). For cool stars,
several scalings have been proposed. First, $\tilde{R}_{s}$ is required to
increase with the surface magnetic activity (in the rotation-unsaturated
regime) in order to reproduce the observed spin-down rate. For a star ten
times more active than the Sun, the optimal $\tilde{R}_{s}$ is found to be
about $19$ (Schrijver et al., 2003). Along the same lines, a power-law scaling
$\tilde{R}_{s}\propto P_{\star}^{-0.84}$ was inferred by See et al. (2018),
where $P_{\star}$ is the rotation period. Second, an ‘effective’
$\tilde{R}_{s}$ can be derived by requiring the PFSS open magnetic flux to
match ab initio stellar wind MHD models (Vidotto et al., 2014; Réville et al.,
2015). The range $\tilde{R}_{s}\in[2.7,10.7]$ was found for six solar-mass
stars, which increases with the surface field (Réville et al., 2016). Third,
$\tilde{R}_{s}$ should be greater than the co-rotation radius
$\tilde{R}_{\mathrm{co}}$ so the closed field lines can support co-rotating
‘slingshot prominences’. For the well-studied star AB Dor with
$\tilde{R}_{\mathrm{co}}=2.7$, $\tilde{R}_{s}$ was set to $3.4$ (Jardine et
al., 2002).

For this study, we sample this parameter space by computing a grid of models
with the following parameters: $\rho\in\left[3\degr,25\degr\right]$ sampled
every $1\degr$; $g_{10}\in\left[0,1000\right]$ G sampled every $10$ G;
$\tilde{R}_{s}=2$, $2.5$, $3$, $4$, $5$, $10$, and $20$.

Figure 6: (a) Decay index $n$ as a function of height $\tilde{h}$ for an axial
dipole field, for various source surface radii $\tilde{R}_{s}$. The critical
decay index $n_{c}=1.5$ is shown as a vertical dotted line. The critical
heights $\tilde{h}_{c}$ are indicated by circles and horizontal dotted lines.
(b) Fractional open flux $f$ and $\tilde{h}_{c}$ as functions of
$\tilde{R}_{s}$ for a dipole field. The cases in (a) are marked by coloured
circles and dotted lines.

Figure 7: (a) Decay index $n$ as a function of height $\tilde{h}$ for
starspots’ field, for various bipole sizes $\rho$. The curves diverge into two
branches at $\tilde{h}\approx 0.4$, in which the lower (thicker lines) and
upper (thinner lines) branches represent the cases for $\tilde{R}_{s}=2.5$ and
$20$, respectively. The two cases are almost identical for smaller
$\tilde{h}$. For comparison, the thin black curve shows the pure dipole case
with $\tilde{R}_{s}=2.5$. The critical decay index $n_{c}=1.5$ is shown as a
vertical dotted line. The critical heights $\tilde{h}_{c}$ are indicated by
circles and horizontal dotted lines. (b) Critical height $\tilde{h}_{c}$ as a
function of $\rho$ for $\tilde{R}_{s}=2.5$. The cases in (a) are marked by
circles and dotted lines. The best fit linear function is shown. The
equivalent $\rho$ in degree and $h_{c}$ in Mm (assuming solar radius
$R_{\odot}=696$ Mm) are noted on the top and right side, respectively.

##  3 Results

###  3.1 Model properties

The properties of the modelled starspots are controlled by the bipole size
$\rho$. Using a minimum penumbral $B_{r}$ of $700$ G (Solanki et al., 2006) to
define the boundary of the spots, we derive the following relations (Fig. 3):

  * •

The area of a single spot is $A_{\mathrm{s}}\approx
33\rho^{2}\,\mu\mathrm{Hem}\,\mathrm{deg}^{-2}$, where $\rho$ is measured in
asterographic degrees.

  * •

The effective angular diameter $d$ of a single spot is $d\approx 0.93\rho$,
assuming a circular spot whose area is $2\pi R_{\star}(1-\cos(d/2))$.

  * •

The unsigned flux $\lvert\Phi\rvert$ of the bipole scales with the area of a
single spot as $\lvert\Phi\rvert\approx cA_{\mathrm{s}}$, where
$c=1.23\left(R_{\star}/R_{\odot}\right)^{2}\times 10^{20}$ Mx $\mu$Hem-1.

  * •

The unsigned flux within the starspots is
$\lvert\Phi_{\mathrm{s}}\rvert\approx c_{\mathrm{s}}A_{\mathrm{s}}$, where
$c_{\mathrm{s}}=0.98\left(R_{\star}/R_{\odot}\right)^{2}\times 10^{20}$ Mx
$\mu$Hem-1. This accounts for $79\%$ of the bipole flux.

These empirical scalings allow for direct conversion between the bipole size
and the properties of the starspots. For example, for a bipole with
$\lvert\Phi\rvert=10^{24}$ Mx (Section 1.3), $A_{\mathrm{s}}\approx
8100\times\left(R_{\odot}/R_{\star}\right)^{2}$ $\mu$Hem and $d\approx
14\aas@@fstack{\circ}6\times\left(R_{\odot}/R_{\star}\right)$. For the largest
bipole considered, $\rho=25\degr$, $\lvert\Phi\rvert\approx
2.5\times\left(R_{\star}/R_{\odot}\right)^{2}\times 10^{24}$ Mx.

The decay index $n(\tilde{h})$ can be inferred from the negative slope of the
$B_{p}(\tilde{h})$ curve in a log-log plot (Fig. 4). In our model, $B_{p}$
from the smaller, solar-like bipole ($\rho=5\degr$) decreases with height much
faster than the largest bipole considered ($\rho=25\degr$). Therefore, the
decay index of the small bipole is larger at all heights. Similarly, the decay
index of the bipole’s magnetic field is larger than that of the global dipole
field (see Sec. 3.3).

In essence, $n(\tilde{h})$ depends on the characteristic spatial scale of the
coronal field, which can be characterized by the spherical harmonic degree
$\ell_{\mathrm{max}}$ at the peak of the energy spectrum $\langle
B^{2}_{r,\ell}\rangle$ (Fig. 5). Using equation (6), we find
$\ell_{\mathrm{max}}=38$ for the small bipole with $\rho=5\degr$, in sharp
contrast with $\ell_{\mathrm{max}}=7$ for a larger bipole with $\rho=25\degr$.
Because the magnetic field components associated with larger $\ell$ decrease
faster with height, their confining effect is expected to be more localised.

The critical height $\tilde{h}_{c}(\rho,g_{10},\tilde{R}_{s})$ depends on the
interplay between the spot’s and the dipole’s magnetic fields, which is
further modulated by the source surface radius. Below, we discuss the
behaviours of $n(\tilde{h})$ and $\tilde{h}_{c}$ for an axial dipole (Section
3.2), for a pair of bipolar spots (Section 3.3), and for combinations of the
two (Section 3.4). The effect of the source surface is covered in each
subsection.

###  3.2 TSZ for an axial dipole field

For an axial dipole field, $n$ can be calculated from equations (2) and (4):

$n(r)=3\left(1-\dfrac{R_{\star}}{r}\right)\left[1-\left(\dfrac{r}{R_{s}}\right)^{3}\right]^{-1}.$
(10)

Plots of $n(\tilde{h})$ for various values of $\tilde{R}_{s}$ are shown in
Fig. 5(a). The resultant $\tilde{h}_{c}$ is independent of $g_{10}$. Fig. 6(b)
shows that $\tilde{h}_{c}$ increases approximately linearly with
$\log\tilde{R}_{s}$ for $\tilde{R}_{s}\lesssim 4$, and becomes almost constant
for $\tilde{R}_{s}\gtrsim 10$. For the nominal solar value,
$\tilde{R}_{s}=2.5$, $\tilde{h}_{c}=0.590$. As $\tilde{R}_{s}\to\infty$, which
represents a closed-dipole case with no open field, we find
$\tilde{h}_{c}=n_{c}/(3-n_{c})=1$. This value of $\tilde{h}_{c}$ sets the
upper limit of the TSZ.

By integrating $\lvert B_{r}\rvert$ on both the stellar photospheric surface
and the source surface, one can determine both the unsigned magnetic flux
$\lvert\Phi_{\star}\rvert$ and the open magnetic flux $\lvert\Phi_{s}\rvert$,
respectively (See et al., 2018). The fractional open flux $f$ is

$f(\tilde{R}_{s})=\dfrac{\left|\Phi_{s}\right|}{\left|\Phi_{\star}\right|}=\dfrac{3\tilde{R}_{s}^{2}}{1+2\tilde{R}_{s}^{3}},$
(11)

which decreases with $\tilde{R}_{s}$ and is anti-correlated with
$\tilde{h}_{c}$ (Fig. 6(b)). In other words, the TSZ expands as the a larger
fraction of the magnetic flux becomes more closed. For $\tilde{R}_{s}=2.5$,
$f=0.581$.

Figure 8: Decay index $n$ as a function of height $\tilde{h}$ for models with
both starspots and dipole, for $\tilde{R}_{s}=2.5$. Panels (a) and (b) are for
fixed bipole sizes $\rho=25\degr$ and $5\degr$, respectively. Coloured curves
indicate various dipole strengths $g_{10}$. The thin black curve shows the
dipole-only case. The critical decay index $n_{c}=1.5$ is shown as a vertical
dotted line. The critical heights $\tilde{h}_{c}$ are indicated by circles and
horizontal dotted lines. In (b), the $g_{10}=100$ G curve intersects $n=n_{c}$
three times, with the shaded region highlighting the height range of the
secondary TSZ.

Figure 9: Extent of the TSZ for models with both starspots and a dipole, for
$\tilde{R}_{s}=2.5$. (a) Critical height $\tilde{h}_{c}$ shown as a curved
surface in the ($\rho$, $g_{10}$, $\tilde{h}$) space. Two surface fragments
marked by white arrows outline additional values of $\tilde{h}_{c}$ that
bracket secondary TSZs. The apparent jagged edges are due to the coarse
sampling of the $\rho$-$g_{10}$ space. (b) Critical height
$\tilde{h}_{c}(\rho,g_{10})$, shown as the background colour. The diagonal
white contour indicates where the spot and dipole strapping fields $B_{p}$ are
equal at $\tilde{h}_{c}$; it roughly separates the dipole- and the spot-
dominated regimes. The red contour on the lower left outlines the regime where
a secondary TSZ exists; the colour therein shows the lowest $\tilde{h}_{c}$.
The typical solar value is marked by a yellow ‘$\odot$’ symbol. The various
regimes discussed in Section 3.4.2 are marked on the top (A and B) and on the
right-hand side (1–4). (c) $\tilde{h}_{c}(g_{10})$, for
$\rho\in\left[5\degr,25\degr\right]$ sampled every $5\degr$ (vertical lines in
(b)). The lower and upper dashed curves show the two values of $\tilde{h}_{c}$
that bracket the secondary TSZs, for $\rho=5\degr$ and $10\degr$. (d)
Parameter $K_{g}$ (equation (12)) for the data points shown in (c). (e)
$\tilde{h}_{c}(\rho)$, for $g_{10}\in\left[0,1000\right]$ G sampled every
$100$ G (horizontal lines in (b)). The dashed curves show the secondary TSZs
for $g_{10}=100$ and $200$ G. (f) Parameter $K_{g}$ for the data points shown
in (e). An interactive version of panel (a) is available in the supplementary
material and via Zenodo.

###  3.3 TSZ for a bipole field

The $n(\tilde{h})$ profile above a bipole (as a pair of starspots) increases
more rapidly with $\tilde{h}$ compared to the dipole. The TSZ is much smaller
($\tilde{h}_{c}\lesssim 0.2$) and is confined to the low corona (Fig. 7(a)).
Empirically, the extent of the TSZ linearly scales with the bipole size,
$\tilde{h}_{c}\approx 0.445\rho+0.005$ (Fig. 7(b)).

The value of $\tilde{h}_{c}$ is largely independent of $\tilde{R}_{s}$. For
$\tilde{R}_{s}=2.5$ and $20$, the relative difference is only $0.8\%$ for the
largest spots considered, $\rho=25\degr$. The effect of $\tilde{R}_{s}$ on $n$
manifests only higher up in the corona ($\tilde{h}\gtrsim 0.4$; Fig. 7(a)).

###  3.4 TSZ for starspots & dipole

For configurations including both starspots and a global dipole, the extension
of the TSZ depends on the relative strengths of the two fields. When the
bipolar starspot component predominates, this leads to a $\tilde{h}_{c}$ lower
in the corona. In contrast, when the large-scale dipole is dominant,
$\tilde{h}_{c}$ will be located at a greater height. We also find that the
effect of the source surface to be more pronounced for dipole-dominated cases.
In the following discussion, we first consider cases where
$\tilde{R}_{s}=2.5$, and then examine the dependence on $\tilde{R}_{s}$ at the
end of the section.

We define two non-dimensional parameters to quantify the impact of the dipole
and the source surface,

$\begin{split}K_{g}&={\tilde{h}_{c}(\rho,g_{10},\tilde{R}_{s})}/{\tilde{h}_{c}(\rho,0,\tilde{R}_{s})},\\\
K_{R}&={\tilde{h}_{c}(\rho,g_{10},\tilde{R}_{s})}/{\tilde{h}_{c}(\rho,g_{10},2.5)}.\\\
\end{split}$ (12)

The first parameter $K_{g}$ relates $\tilde{h}_{c}$ to the zero-dipole
($g_{10}=0$) case, and the second parameter $K_{R}$ relates $\tilde{h}_{c}$ to
the fiducial source surface radius ($\tilde{R}_{s}=2.5$) case.

####  3.4.1 Two contrasting examples

For the largest-spot cases with $\rho=25\degr$, $n$ monotonically increases
with $\tilde{h}$ regardless of the value of $g_{10}$ (Fig. 8(a)). For the
cases with $g_{10}=10$ and $100$ G, the dipole contribution is relatively
small (Fig. 4). The $n(\tilde{h})$ profiles therefore resemble those of the
pure spots (Fig. 7), yielding similarly low TSZs: $\tilde{h}_{c}\leq 0.210$,
$K_{g}\leq 1.07$. For $g_{10}=1000$ G, however, the influence of the dipole
field increases significantly, and the extent of the TSZ almost doubles:
$\tilde{h}_{c}=0.378$, $K_{g}=1.92$.

The situation is drastically different for the smaller-spot cases with
$\rho=5\degr$ owing to their fast decaying field. As $\tilde{h}$ increases,
the large-scale dipole contribution quickly exceeds that of the spots, even
for relatively weak dipole fields (Fig. 4). The $n(\tilde{h})$ profile is
distorted into an S shape (Fig. 8(b)). For $g_{10}=10$ G, the distortion is
relatively small, so the TSZ is similar to that of the spots:
$\tilde{h}_{c}=0.043$, $K_{g}=1.01$. For $g_{10}=1000$ G, however, the
distortion is so large that the TSZ approaches the dipole upper bound:
$\tilde{h}_{c}=0.586$, $K_{g}=13.83$ (cf. Fig. 4).

An interesting case arises for the intermediate dipole $g_{10}=100$ G. The
$n(\tilde{h})$ profile crosses $n=n_{c}$ three times, yielding a secondary,
extended TSZ at higher altitude, $\tilde{h}\in\left[0.167,0.542\right]$. This
resembles the ‘saddle-like’ decay index profiles occasionally found on the Sun
(Section 1.1). The primary TSZ is similar to the TSZ of the spots,
$\tilde{h}_{c}=0.049$, $K_{g}=1.17$. Its extent, $\delta\tilde{h}=0.049$, is
much narrower than that of the secondary TSZ, $\delta\tilde{h}=0.375$.

Figure 10: Response of the critical height $\tilde{h}_{c}$ to the source
surface radius $\tilde{R}_{s}$. The lower surface is for $\tilde{R}_{s}=2.5$
(same as Fig. 9(a)). The upper surface is for $\tilde{R}_{s}=20$, the colour
of which illustrates the parameter $K_{R}$ (equation (12)). For regions with
multiple critical heights, only the lowest is shown. An interactive version of
the figure is available in the supplementary material and via Zenodo.

####  3.4.2 The $\rho$–$g_{10}$ space

The function $\tilde{h}_{c}(\rho,g_{10})$ is quite non-linear (Fig. 9(a)).
Large gradients appear at the transition between the dipole- and spot-
dominated regimes (upper-left vs lower-right regions in the figure; see also
panel (b)), especially for the smaller spots where the influence of the dipole
is stronger (larger $K_{g}$).

The secondary TSZ exclusively appears in the outlined region in the lower-left
corner of Fig. 9(b), for smaller spots $\rho\in\left[3\degr,10\degr\right]$
and weak to intermediate dipoles $g_{10}\in\left[20,290\right]$ G. For the
secondary TSZ to appear, the dipole needs to be strong enough to distort the
$n(\tilde{h})$ profile, but not too strong such that $\tilde{h}_{c}$ is
located too high up in the corona (see Fig. 8(b)).

We use vertical cuts in the $(\rho,g_{10})$ space, $\tilde{h}_{c}(g_{10})$, to
evaluate how the TSZ above starspots of a certain size varies for different
dipole strengths (Fig. 9(c)). Two regimes (A and B) emerge (cf. Fig. 9(b)).

  1. A.

$\rho\in\left[3\degr,10\degr\right]$: a single $\tilde{h}_{c}$ initially
increases with $g_{10}$ in the low corona. As $g_{10}$ continues to increase,
an inflection point will be reached where three instances of $\tilde{h}_{c}$
arise; the higher two define a secondary TSZ that expands in size. Finally, as
$g_{10}$ increases further, another inflection point is reached, above which
only a single $\tilde{h}_{c}$ remains, located higher up in the corona.

  2. B.

$\rho\in\left[11\degr,25\degr\right]$: $\tilde{h}_{c}$ increases with $g_{10}$
with no secondary TSZ. For smaller $\rho$, $\tilde{h}_{c}$ increases faster
and reaches a greater maximum.

The behaviour of $\tilde{h}_{c}(g_{10})$ in regime A can be explained in the
context of Fig. 8(b). Therein, $\tilde{h}_{c}$ is visualised by the intercepts
between the $n(\tilde{h})$ curve and $n=n_{c}$. As $g_{10}$ increases, the
number of intercepts increases from one to three, then reduces back to one.
During the process, the higher two intercepts move apart. The lower two
approach each other and finally disappear altogether.

Similarly, we use horizontal cuts $\tilde{h}_{c}(\rho)$ to evaluate how the
TSZ varies for different spot sizes on a star with a certain dipole strength
(Fig. 9(e)). Four regimes ($1$–$4$) emerge (cf. Fig. 9(b)).

  1. 1.

$g_{10}\in\left[0,10\right]$ G: $\tilde{h}_{c}$ linearly increases with $\rho$
in the low corona.

  2. 2.

$g_{10}\in\left[20,160\right]$ G: a secondary TSZ appears; its extent shrinks
with $\rho$. The lowest $\tilde{h}_{c}$ increases linearly with $\rho$.

  3. 3.

$g_{10}\in\left[170,290\right]$ G: the profile is similar to regime 2, except
only one high-lying $\tilde{h}_{c}$ exists for the smallest values of $\rho$.

  4. 4.

$g_{10}\in\left[300,1000\right]$ G: the secondary TSZ completely disappears.
The value of $\tilde{h}_{c}$ monotonically decreases with $\rho$.

These behaviours can be broadly understood as a transition from the spot-
dominated to the dipole-dominated regime. The Sun lies in regime 1, where the
spots determine the extent of the TSZ. For regime-4 stars with strong dipoles,
large spots act to shrink the TSZ: the inclusion of the spot field changes the
slope of $B_{p}(\tilde{h})$. At $g_{10}=1000$ G, $K_{g}$ decreases from the
maximum $21.81$ for $\rho=3\degr$ to $1.92$ for $\rho=25\degr$ (Fig. 9(f)).

####  3.4.3 Effect of $\tilde{R}_{s}$

The behaviour of $\tilde{h}_{c}(\rho,g_{10})$ is qualitatively similar for all
considered values of $\tilde{R}_{s}$. The dipole (spot) component controls the
upper left (lower right) portion of the parameter space, with a sharp
transition in between (analogous to the large gradient evident in Fig. 10).
The effect of $\tilde{R}_{s}$ is more pronounced in the dipole-dominated
regime, in which larger values of $\tilde{R}_{s}$ yield higher values of
$\tilde{h}_{c}$, which approach the dipole upper bound. The spot-dominated
regime, in contrast, remains largely unaffected by $\tilde{R}_{s}$. This is
reflected in the $K_{R}$ values for $\tilde{R}_{s}=20$, which are
approximately $1.7$ and $1$ in the dipole- and spot-dominated regime,
respectively. The former number is simply the ratio of the two dipole critical
heights, i.e. $0.998$ (for $\tilde{R}_{s}=20$) vs $0.590$ (for
$\tilde{R}_{s}=2.5$; see Fig. 6).

The secondary TSZ is present for all instances of $\tilde{R}_{s}$, similarly
confined to an approximately triangular region in the lower-left corner of the
$(\rho,g_{10})$ space (see Fig. 9(b)). The upper-right tip of the triangle
shifts to the right as $\tilde{R}_{s}$ increases, reaching $\rho=15\degr$ for
$\tilde{R}_{s}=20$. As a result, the jump to higher $\tilde{h}_{c}$ occurs at
smaller $g_{10}$ for the same $\rho$. $K_{R}$ becomes large at the transition
region (Fig. 10).

##  4 Summary & Discussion

Using an idealised coronal magnetic field model, we estimate the extent of the
TSZ above a bipolar stellar active region (that is, a pair of starspots)
embedded in a global dipole field. We find that the upper bound of the TSZ,
defined by the torus instability critical height, $\tilde{h}_{c}$, increases
with the spot size $\rho$, the dipole strength $g_{10}$, and the source
surface radius $\tilde{R}_{s}$. The upper bound of the TSZ, and the presence
or not of a secondary TSZ, depend on the interplay between the spots’ and the
dipole’s magnetic fields. Our main findings are as follows.

  * •

For a purely dipolar field, the upper bound of the TSZ is located at a
significant fraction of the stellar radius, and it is independent of the
strength of the dipole.

  * •

Increasing the source-surface radius will move this upper bound farther out,
as the amount of closed flux increases. The upper bound of the TSZ ranges from
$0.59R_{\star}$ for a solar-like case with ${R}_{s}=2.5R_{\star}$ to
$R_{\star}$ for a fully closed dipole.

  * •

The field associated with a pair of bipolar starspots alone yields TSZ upper
bounds of about half the corresponding bipole size, typically below
$0.2R_{\star}$, i.e. much lower than the upper bounds of a pure dipole field.

  * •

When both a global dipole and a pair of starspots are present, the TSZ
extension is determined by the relative strength of their magnetic fields. The
TSZ’s upper bound can change drastically at the transition between the dipole-
dominated and the spot-dominated regimes, especially for smaller spots.

  * •

A secondary TSZ forms at a higher altitude for small spots (a few degrees) and
intermediate dipoles (deca- to low hectogauss).

Active cool stars are expected to have a strong dipole field and a large
source surface radius (Section 2.4), both of which can significantly extend
the TSZ compared to solar conditions. Assuming that CMEs on cool stars
originate from torus-unstable flux ropes in the same way as on the Sun, the
presence of an extended TSZ should reduce the CME occurrence above starspots.
The processes facilitating the onset of the torus instability (see Section
1.1) would have to operate longer, or more efficiently, to lift a flux rope
out of the TSZ. Even if an eruption were driven predominantly by (very
efficient) reconnection above and below a flux rope, it appears that
overcoming the confining effect of a strong dipole field would remain
difficult (DeVore & Antiochos, 2008).

The calculations presented here suggest that eruptions from small spots will
be preferentially impacted. First, the $K_{g}$ value is large under strong
dipoles (regimes A4, Fig. 9(b)). The critical height $\tilde{h}\approx
K_{g}\times 0.445\rho$ can be an order of magnitude larger than that of the
spots alone. A pre-eruption magnetic flux rope would need to be stretched into
an extremely prolate shape before the torus instability can set in. The
required work against the magnetic tension force makes it energetically
unfavourable. Second, the presence of the secondary TSZ may lead to failed
eruptions under intermediate dipoles (regimes A2 and A3). The lower, torus-
unstable zone now allows for the onset of the torus instability, while the
extended, higher TSZ provides a persistently strong strapping force to
potentially entrap the ejecta.

The Sun’s dipole field is generally too weak to produce a secondary TSZ
(regime A1). In certain multipolar active regions, however, several sets of
closed field lines can co-exist with very different connectivities and apex
heights. The higher-lying field lines can mimic the effect the global dipole
field, yielding a similar S-shaped decay index profile (Section 1.1).

The modelled spots have a maximum surface field $B_{r}=3$ kG. This value is
typical for sunspot umbrae, and is consistent with the $2$–$4$ kG estimate for
active dwarf spots using the Zeeman broadening technique (Berdyugina, 2005). A
set of ab initio MHD simulations of spots on cool stars finds the umbral field
to be in the range of $3$–$4.5$ kG, which is largely determined by the fluid
pressure at the unity optical depth (Panja et al., 2020).

Our model yields an empirical relation between the bipole magnetic flux and
the spot area. To compare with the literature, we normalise the flux as
$\lvert\Phi_{21}\rvert=\lvert\Phi\rvert/(10^{21}\,\mathrm{Mx})$, and the area
of the spot pair as $A_{\mathrm{s}18}=(2A_{\mathrm{s}}\times 10^{-6}\times
2\pi R^{2}_{\star})/(10^{18}\,\mathrm{cm}^{2}\,\mu\mathrm{Hem})$. This yields
$\lvert\Phi_{21}\rvert=2.0A_{\mathrm{s}18}$ for $R_{\star}=R_{\odot}$, similar
to the value found for the Sun, $\lvert\Phi_{21}\rvert=2.3A_{\mathrm{s}18}$
(Wang & Sheeley, 1989). The empirical power-law
$\lvert\Phi\rvert\propto\rho^{2}$ is somewhat steeper than that of the Sun,
$\rho^{1.3}$, possibly because the latter accounts for extended plage regions
with weaker, hectogauss fields.

The empirical spot-generated TSZ, $\tilde{h}_{c}\approx 0.445\rho$, is similar
to the $0.5\rho$ estimate from solar coronal field extrapolation models (Wang
et al., 2017) and idealised MHD simulations (Török & Kliem, 2007). However, a
different series of MHD simulations show that $\tilde{h}_{c}$ can be
comparable to $\rho$ (Aulanier et al., 2010; Zuccarello et al., 2015). The
scaling may depend on the detailed surface flux distribution, e.g. the
‘compactness’ of the spots, or the prescribed flux rope properties.

The modelled spots are centred on the equator and aligned with the global
dipole (Fig. 2). A different, more realistic bipole centroid or orientation
(Section 2.3) would reduce the dipole contribution to the strapping field in
our model. Hence, the height estimates of the TSZ shown in this study should
be considered as upper limits.

If a flux rope in an active region remains stable (or is even entirely
absent), reconnection between sheared magnetic fields can still lead to flares
without a CME (Li et al., 2019), a scenario that has been proposed to explain
the observed behaviour of NOAA active region 12192 (Jiang et al., 2016). This
scenario may serve as another resolution to the missing stellar CME conundrum,
but it remains to be seen how frequently such flares actually occur. It
appears that very strong fields would be needed to produce sufficient flaring
from reconnection processes that are not driven by an actual eruption.

Our model does not consider elongated quiescent filament channels, which
constitute a main source for solar CMEs, especially during phases of low solar
activity. Due to the weak photospheric field strength (several gauss),
eruptions stemming from such channels tend to have little flare emission. Such
‘flare-less’ CMEs may have been overlooked on other cool stars.

The torus instability theory is applicable in a low plasma-$\beta$ environment
where the magnetic fields determine the plasma dynamics. The plasma-$\beta$
can vary significantly in the solar atmosphere, due to the changing field
strength, plasma temperature, and density (Gary, 2001; Bourdin, 2017).
Nevertheless, the $\beta\ll 1$ criterion is expected to be satisfied in the
low corona above sunspots, at least up to a couple tenths of the solar radius
where most active region CMEs erupt. Active stars with stronger magnetic
fields are expected to have a more extended low-$\beta$ zone, so the results
presented here remain relevant.

Our idealised model is intended to provide a first-order description of the
TSZ in the $(\rho,g_{10},\tilde{R}_{s})$ parameter space, applicable to cool
stars with a single active region. The coronal magnetic field configurations
of magnetically more active stars are likely to be more complex. Data-
constrained numerical models (e.g. Alvarado-Gómez et al., 2018; Lynch et al.,
2019; Jin et al., 2020) will be needed to explore their effects on eruptions.
In particular:

  * •

The model parameters are expected to be positively correlated with one
another. For instance, surface flux transport simulations tailored to
represent active stars yield stronger dipole fields when more large bipoles
are included (Schrijver & Title, 2001; Lehmann et al., 2018).

  * •

Many K and early M dwarfs have non-axisymmetric or predominantly toroidal
global fields (e.g. Donati & Landstreet, 2009). The effect of such fields on
the TSZ extension is unclear.

  * •

Large sunspot groups are typically fragmented (see Aulanier et al., 2013, for
a discussion in the context of stellar flares). Episodes of emergence, decay,
and coalescence frequently occur. Forward modelling suggests the same
(clustering and nesting) for starspots (Schrijver, 2020; Işık et al., 2020).
If the spots are no longer monolithic, the presence of multiple shorter
magnetic flux ropes becomes more likely than the presence a single, long rope.

  * •

Large polar spots with a ring of azimuthal fields are sometimes observed on
rapid rotators (see Berdyugina, 2005; Strassmeier, 2009, and references
therein). Surface flux transport models suggest that they form due to the
accumulation of remnant active-region flux, with one polarity at the center
and the opposite polarity at the periphery (Schrijver & Title, 2001; Mackay et
al., 2004). It is unclear whether the high-latitude flares observed on rapid
rotators (Maggio et al., 2000; Ilin et al., 2021) originate from such long-
lived polar spots, or from newly emerged bipolar active regions.

## Acknowledgements

We thank Bernhard Kliem and Jamie Tayar for helpful discussions. X. Sun is
supported by NSF award 1854760. T. Török is supported by NSF award 1854790 and
NASA award 80NSSC20K1274. M. L. DeRosa is supported by NASA award
80NSSC18K0029 to Lockheed Martin. Visualization softwares include the Plotly
Python package.

## Data availability

The interactive version of Fig. 2, Fig. 9(a), and Fig. 10, and the data sets
used to generate Figs. 3–10 are available at
https://doi.org/10.5281/zenodo.5498419.

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