# Can scallop-shell stars trap dust in their magnetic fields?

H. Sanderson,1 M. Jardine,2 A. Collier Cameron,2 J. Morin,3 J.-F. Donati,4
1Department of Earth Sciences, University of Oxford, South Parks Road, Oxford,
OX1 3AN
2SUPA, School of Physics and Astronomy, North Haugh, St Andrews, Fife KY16
9SS, UK
3LUPM, Université de Montpellier, CNRS, Place Eugène Bataillon, F-34095
Montpellier, France
4Institut de Recherche en Astrophysique et Planétologie, Université de
Toulouse, UPS-OMP, 31400 Toulouse, France E-mail:
<EMAIL_ADDRESS>
(Accepted 2022 November 08. Received 2022 October 11; in original form 2022
August 18.)

###### Abstract

One of the puzzles to have emerged from the Kepler and TESS missions is the
existence of unexplained dips in the lightcurves of a small fraction of
rapidly-rotating M dwarfs in young open clusters and star-forming regions. We
present a theoretical investigation of one possible explanation - that these
are caused by dust clouds trapped in the stellar magnetic fields. The depth
and duration of the observed dips allow us to estimate directly the linear
extent of the dust clouds and their distances from the rotation axis. The dips
are found to be between 0.4-4.8%. We find that their distance is close to the
co-rotation radius: the typical location for stable points where charged
particles can be trapped in a stellar magnetosphere. We estimate the charge
acquired by a dust particle due to collisions with the coronal gas and hence
determine the maximum grain size that can be magnetically supported, the
stopping distance due to gas drag and the timescale on which dust particles
can diffuse out of a stable point. Using the observationally-derived magnetic
field of the active M dwarf V374 Peg, we model the distribution of these dust
clouds and produce synthetic light curves. We find that for 1$\mu$m dust
grains, the light curves have dips of 1% - 3% and can support masses of order
of $10^{12}$ kg. We conclude that magnetically-trapped dust clouds
(potentially from residual disc accretion or tidally-disrupted planetesimal or
cometary bodies) are capable of explaining the periodic dips in the Kepler and
TESS data.

###### keywords:

stars: magnetic fields, stars: coronae, stars: low mass, stars:variable

††pubyear: 2022††pagerange: Can scallop-shell stars trap dust in their
magnetic fields?–6

##  1 Introduction

Over the last few years, observations from K2 (Rebull et al., 2016; Stauffer
et al., 2017, 2018, 2021) and TESS (Zhan et al., 2019) have revealed a class
of rapidly-rotating ($P<1$ d) M dwarfs with unusual asymmetric, rigidly-
periodic light curves. The light curves have broad, quasi-sinusoidal
modulations in amplitude, similar to those expected for starspots but with
sharp smaller features superimposed, whose variability is too rapid to be
explained by starspots alone (Stauffer et al., 2017; Zhan et al., 2019). The
large and broad features in the light curves are stable for years whilst the
smaller and sharper features can survive for a few months and disappear on a
timescale of days (Zhan et al., 2019; Günther et al., 2022). Over the course
of a full year, several distinct manifestations of the scallop phenomenon
appeared, each lasting between 1 and 2 months. The similarities (including
strict periodicity and complex Fourier spectra) between the light curves of
the scallop-shells discovered by Stauffer et al. (2017) and the complex
rotators discovered by Zhan et al. (2019) suggest they are one class of star
(Günther et al., 2022). They will be referred to as scallop-shells for the
rest of this paper. These stars differ from the other classes of variable M
dwarf, dippers and spotted stars with differential rotation, because they have
shorter rotational periods and the variability is more rigidly periodic
(Günther et al., 2022).

The main theories suggested to explain these light curves are: co-rotating
clouds of gas/dust at the Keplerian co-rotation radius
($R_{K}=\left(GM_{\star}/\Omega^{2}\right)^{1/3}$) superposed on an underlying
rotational starspot modulation (Stauffer et al., 2017, 2018; Günther et al.,
2022), starspots with a misaligned dust disc (Zhan et al., 2019), and eclipses
of bound-free emission from gaseous slingshot prominences as they pass behind
the star (Rebull et al., 2016). The lack of an IR excess in the SEDs of these
stars rules out a primordial, accreting dust disc as an explanation (Stauffer
et al., 2017).

A recent study of occurrence rates among the complex rotational variables
implies that, in order to explain these dips, some 30 percent of young
rapidly-rotating M dwarfs would need to have inner dust discs misaligned with
their rotation axes (Günther et al., 2022). We also note that for spot
occultation by a dust disk to produce the observed sharp decreases in
brightness, the occulted spots would have to be bright. Doppler-imaging
studies of non-accreting M dwarfs (e.g. Barnes & Collier Cameron, 2001; Barnes
et al., 2015), however, invariably show significant coverage of dark rather
than bright spots. This argues against models involving starspots and a
misaligned disc.

Previous authors have discounted the co-rotating-clouds hypothesis, because
gas clouds alone could not give the required absorption depth and it was
claimed that the magnetic field at the co-rotation radius is too weak to trap
dust (Günther et al., 2022). However, studies of rapidly-rotating stars show
that local traps exist within their magnetic fields where clouds of cool,
dense gas (known as “slingshot prominences”) may be supported (Ferreira, 2000;
Jardine et al., 2001, 2020). The existence of these mechanically-stable
equilibrium points within stellar magnetic fields merits a closer look at the
confinement of dust in similar locations.

Dust clouds may not be the only source of photometric variation. Spots may
also contribute. Starspot activity is ubiquitous in such young, active stars;
indeed the quasi-sinusoidal modulation of their light curves is the very
phenomenon that allows their rotation periods to be measured photometrically.
In the subset of these stars that exhibit scallop-shell behaviour, the short-
duration absorption dips repeat on the same period as the underlying starspot
modulation. This indicates that the structures causing the dips co-rotate with
the photosphere. Therefore, we propose the scallop-shell behaviour is caused
by a superposition of starspots (the large quasi-sinusoidal modulation) and
co-rotating clouds of dust trapped in magnetic stable points (the narrow
features). Magnetospheric gas (Günther et al., 2022) or dust (Palumbo et al.,
2022) clouds have been briefly explored as an explanation but quantitative
analysis of the dust trapping mechanism and the resulting light curves has not
been done.

We suggest these co-rotating dusty clouds are produced when dust (potentially
from a tidally disrupted body analogous to a sun-grazing comet) becomes
collisionally charged as it enters the corona, and couples to the magnetic
field. The dust moves along magnetic field lines to magnetic stable points,
whilst being decelerated by gas drag from the corona. Once there, the
diffusion and sublimation timescales of dust at the stable points are slow, so
the sharp features produced by the dust are stable over the observed
timescale. Evolution of the stellar magnetic field (due to for example surface
differential rotation or flux emergence) or the onset of thermal collapse
leading to upflow of material into the stable point could lead to the eventual
ejection of the dust cloud and disappearance of the narrow feature (see
Section 4.2).

This paper provides a theoretical basis for this model. We consider the effect
on the dust particles of both the magnetic field and plasma in the solar
corona and demonstrate that these particles can be confined by the magnetic
field in stable points. We use observational data from the complex rotators
presented in Zhan et al. (2019) to confirm the position of the dust clouds
close to the co-rotation radius and determine the cloud sizes. Using these
observational values we derive possible dust-cloud masses and use a magnetic
map of an illustrative rapidly-rotating M dwarf (V374 Peg) to generate
theoretical light curves caused by the subset of these magnetospheric clouds
that transit the stellar disc.

The structure of the paper is as follows: Section 2 outlines the methods for
extracting information from the observations and the theory for the
confinement of dust at magnetic stable points. Section 3 applies the methods
and theory from Section 2 to observed scallop-shell stars and V374 Peg and
generates theoretical light curves for magnetospherically trapped dust.
Section 4 discusses the implications of these results and we conclude in
Section 5.

##  2 Theory and Methods

###  2.1 Observational indicators of dust-cloud locations and sizes

These Stauffer et al. (2017, 2018) M dwarfs all have spectral types M5 or
later, and rotation periods of order half a day or less. Given that the
transients recur on the same period as the starspot modulation, they must co-
rotate with the photosphere. This places them either in Keplerian orbits
synchronous with the stellar rotation, or confined in co-rotating structures.
Given that the transients arise when co-rotating structures transit the
stellar disc, their durations depend on their distances from the stellar
rotation axis. The dip duration is the length of time taken for an isolated
cloud to traverse the stellar diameter at the transverse rotational speed of
its location. Expressing the dip duration $T$ as a fraction of the rotation
period $P$,

$\frac{T}{P}\simeq\frac{2R_{\star}}{2\pi a_{\rm cloud}},$ (1)

where $a_{\rm cloud}$ is the distance of the cloud from the stellar rotation
axis. This assumes that the cloud transits the centre of the disc. Shorter
durations may be seen if the path of the cloud passes well away from the disc
centre.

Zhan et al. (2019) identified 10 scallop stars in their TESS survey from the
existence of high-frequency harmonics of the rotation period in the Fourier
transforms of their light curves. If the highest frequency detectable above
the noise is related to the shortest transient duration by $\nu_{\rm
max}\simeq 1/T_{\rm min}$, we can establish a lower limit on $a_{\rm cloud}$
either from the Fourier transform of the light curve or by measuring the
duration of the shortest distinct dip seen in the light curve. Given an
estimate of the stellar mass and rotation period, we can then compare the
inferred value of $a_{\rm cloud}$ directly with the Keplerian co-rotation
radius. For the results given in Section 3.1 we used the same sample of 10
TESS light curves examined by Zhan et al. (2019) and by Günther et al. (2022).

The cloud radius $r_{\rm cloud}$ can be estimated by considering the observed
fractional depth $k$ of an isolated cloud transient (typically $k=2-4\%$).
Empirically, this tells us the ratio

$k=\frac{A_{\rm cloud}}{A_{\star}}=\left(\frac{r_{\rm
cloud}}{R_{\star}}\right)^{2}.$ (2)

Hence $r_{\rm cloud}=\sqrt{k}R_{\star}$ where $k$ is the fractional flux
deficit at mid-event for a small cloud.

###  2.2 Magnetic field structure and the position of dust clouds

One of the main challenges in understanding this phenomenon is to explain how
these clouds can be confined in discrete locations and forced to co-rotate
with the star. This is possible within a magnetic field if the cloud is
located close to a stable point in the field. These are local gravitational
potential minima as calculated following the direction of the field. These
locations can be determined if the magnetic field structure is known
(Ferreira, 2000; Jardine et al., 2001). They are defined by

$\left(\underline{B}\cdot\underline{\nabla}\right)\left(\underline{g}_{\rm
eff}\cdot\underline{B}\right)<0.$ (3)

Here the effective gravitational acceleration is given by $\underline{g}_{\rm
eff}=(\underline{g}.\underline{B})/|\underline{B}|$ and

$\underline{g}(r,\theta)=\left(-GM_{\star}/r^{2}+\omega_{\star}^{2}r\sin^{2}\theta,\omega_{\star}^{2}r\sin\theta\cos\theta\right),$
(4)

in spherical coordinates (r, $\theta$, $\phi$), with the polar direction
aligned with the rotation axis of the star, where $\omega_{\star}$ is the
stellar angular velocity. For example, for a simple dipolar magnetic field
where the dipole axis is aligned with the rotation axis, the stable points lie
in the rotational equator at radii $r>(2/3)^{1/3}R_{K}$ (Ferreira, 2000). Even
for more complex field structures, the stable points typically cluster close
to, or just beyond, the co-rotation radius (Jardine et al., 2001).

In order to determine the coronal structure of the magnetic field, we can
extrapolate from maps of the surface magnetic field obtained using Zeeman-
Doppler imaging (Donati & Collier Cameron, 1997; Donati et al., 1999; Hussain
et al., 2000; Carroll et al., 2012; Rosén et al., 2015). Here we assume that
the coronal field is potential and therefore supports zero volume currents. We
can then use the well-established Potential Field Source Surface method
(Altschuler & Newkirk, 1969) to perform the extrapolation. Since the field is
potential we may express it as $\underline{B}=-\underline{\nabla}\Psi$, with
the result that $\underline{\nabla}\cdot\underline{B}=0$ then requires
$\nabla^{2}\Psi=0$. We may therefore express $\Psi$ in spherical co-ordinates
$(r,\theta,\phi)$ as

$\Psi=\sum_{l=1}^{N}\sum_{m=-l}^{l}[a_{lm}r^{l}+b_{lm}r^{-(l+1)}]P_{lm}(\theta)e^{im\phi},$
(5)

where all radii are scaled to the stellar radius. The map of the surface
radial magnetic field therefore provides one of the boundary conditions for
Eqn. 5. For the other, we assume that at some radius (know as the source
surface, rss) the field is is forced to become purely radial by the pressure
of the hot coronal gas, and so at $r=r_{ss}$, $B_{\theta}=B_{\Phi}=0$. Hence

$b_{lm}=-a_{lm}r_{ss}^{2l+1}.$ (6)

For further details, the reader is referred to the Appendix.

We show an example of such a field structure and the associated stable points
in Fig. 1. This field extrapolation utilises the surface magnetic field map of
the rapidly-rotating M dwarf V374 Peg (Donati et al., 2006). The stable points
cluster around the co-rotation radius, as would be expected.

Figure 1: Magnetic field structure of V374 Peg with dust-cloud locations shown
as blue clumps. The top panel shows the view as see from the observer’s
viewpoint at inclination 70o (top) while the bottom panel (where the co-
rotation radius is marked as a red circle) shows the view from above the
rotation axis. The radial magnetic field map is shown on the stellar surface
(blue is negative, red is positive polarity).

###  2.3 Dust particle confinement in the magnetic field

Magnetospherically-trapped dust has previously been suggested by Farihi et al.
(2017) to explain the unusual pattern of evolving, short-duration periodic
transits observed by Vanderburg et al. (2015) and Gänsicke et al. (2016) in
the polluted white dwarf WD1145+017. However, a model for trapping charged
dust around M dwarfs differs from that for white dwarfs, because M dwarfs are
larger and have an extensive corona. Collisions with ions in the corona exert
drag on the dust particles and are the main cause of dust-particle ionisation
due to their high number density ($10^{16}-10^{18}\rm m^{-3}$Ness et al.
(2002, 2004)), in contrast to white dwarfs where photoionisation dominates.

We propose that the confinement of dust in the stable points occurs as
follows. The dust grains initially move towards the star from an external
source (see Section 4) due to a combination of gravity and Poynting-Robertson
drag. As the dust particle enters the stellar corona it will rapidly become
charged by collisions with the plasma and therefore will experience the
Lorentz force in addition to drag from the coronal plasma. The density of this
plasma ensures that once inside the corona, the radiation force experienced by
the dust particles is negligible in comparison to the collisional drag on the
dust grains. If the Lorentz force exceeds the gravitational force, the
magnetic field may be able to capture the dust grains. Even in this case,
however, dust particles will be free to move along field lines. If they
encounter a stable location they will remain there. The effect of collisional
drag with the coronal gas will be to slow down the motion of the dust grain,
potentially preventing it from reaching a stable point. Typically stable
points are close to the co-rotation radius.

####  2.3.1 Charging of dust grains

In order for the dust particles to be affected by the magnetic field, they
must be charged. The dust grains become charged by collisions with electrons
and protons in the coronal plasma. The dust grains are assumed to acquire a
negative charge, because electrons in the plasma have faster velocities than
electrons due to their lower masses so collide more frequently with the dust
grains. The resulting charge, $q_{d}$ and collision cross section can be
calculated following a method of Ke et al. (2012) to give

$Q=-\frac{30{\pi}{\epsilon}_{0}r_{\rm d}k_{\rm B}T}{e},$ (7)

where $k_{\rm B}$ is the Boltzmann constant, $\epsilon_{0}$ is the
permittivity of free space, $e$ is the charge on one electron, $r_{\rm d}$ is
the radius of the spherical dust grain and T is the plasma temperature.

####  2.3.2 Lorentz force

For a charged dust grain in a dipole field, the magnitude of the Lorentz force
is

$\begin{split}F_{L}&=q_{d}v_{\rm d}B\\\ &=\frac{30\pi\epsilon_{0}r_{\rm
d}k_{\rm B}T}{e}v_{\rm d}B_{\rm
surf}\left(\frac{R_{\star}}{r}\right)^{3}\end{split}$ (8)

where $q_{d}$ is the charge on the dust grain, $v_{\rm d}$ is its velocity
perpendicular to the magnetic field, $B_{\rm surf}$ is the surface magnetic
field strength of the star, $r$ is the distance from the centre of the star
and $R_{\star}$ is the stellar radius. For a dust grain moving towards the
star, its velocity will be a fraction of the escape speed $v_{\rm d}=fv_{\rm
esc}=f\sqrt{\frac{2GM_{\star}}{r}}$ where $0\leq f\leq 1$ is a factor that
accounts for the alignment of the particle’s velocity relative to the magnetic
field and its velocity relative to the escape velocity. For example, a
particle in a circular orbit has $v_{\rm d}=\frac{v_{\rm esc}}{\sqrt{2}}$.
Equation 8 can therefore be rewritten as

$F_{L}=\frac{30\pi\epsilon_{0}r_{\rm d}k_{\rm B}T}{e}f\sqrt{2GM_{\star}}B_{\rm
surf}R_{\star}^{3}r^{-\frac{7}{2}}.$ (9)

####  2.3.3 Gravitational force and maximum grain size

The gravitational force on a dust grain is given by

$F_{G}=\frac{4\pi r_{\rm d}^{3}\rho_{\rm d}}{3}\frac{GM_{\star}}{r^{2}}$ (10)

where $M_{\star}$ is the stellar mass, $\rho_{\rm d}$ is the bulk density of
the dust grains and $G$ is the gravitational constant. For a dust particle to
be supported by the magnetic field against gravity we require $F_{L}>F_{G}$.
The gravitational force is used rather than the effective gravitational force,
because it is a more conservative criterion which takes into account that the
dust is not initially on a circular orbit. In our model, the dust grains are
treated as spherical and we follow Zhan et al. (2019) in using a typical
density of rocky material of 3000 kg $\rm m^{-3}$. The gravitational force on
a particle scales as $r_{\rm d}^{3}$ whereas the charge (and Lorentz force)
scale as $r_{\rm d}$. Therefore grains must be below a critical radius to be
significantly affected by the magnetic field. Combining Equations 8 and 10 and
assuming that the particle is located at a radial distance $r=\beta R_{K}$
gives

$\begin{split}r_{\rm d}<\left(\frac{45\sqrt{2}}{2}\frac{\epsilon_{0}k_{\rm
B}TR_{\star}^{3}fB_{\rm surf}\Omega}{eGM_{\star}\rho_{\rm
d}\beta^{\frac{3}{2}}}\right)^{\frac{1}{2}}.\end{split}$ (11)

####  2.3.4 Drag from coronal plasma

The drag force experienced by the dust grain due to collisions with protons in
the plasma is

$F_{\rm drag}=m_{\rm p}n_{\rm p}\sigma_{\rm col}c_{s}v_{\rm rel}$ (12)

where $c_{s}=\sqrt{\frac{k_{\rm B}T}{\mu m_{\rm p}}}$ is the sound speed in
the plasma and $\mu=0.61$ is the mean molecular weight of the plasma. This was
approximated by primordial solar abundances by mass fraction (Asplund et al.,
2009), which are a reasonable approximation for these young convective stars
which will be well mixed. $\sigma_{\rm col}$ is the collision cross section
between dust and protons in the plasma and $v_{\rm rel}$ is the relative
velocity between the dust and plasma - again taken to be $fv_{\rm esc}$. The
collision cross section between a charged dust grain and protons is enhanced
due to their electrostatic interaction. Following Ke et al. (2012), this
enhancement can be written as $\sigma_{\rm col}=\pi r_{\rm d}^{2}(1+\theta)$
and $\theta=5$ was used. Combining these expressions

$F_{\rm drag}=6\pi r_{\rm d}^{2}n_{\rm p}\sqrt{\frac{2k_{\rm B}Tm_{\rm
p}GM_{\star}}{\mu r}}f.$ (13)

This drag force has a comparable magnitude to the Lorentz force for micrometer
size grains. The drag force decelerates the dust particle, decreasing the
Lorentz force it experiences. The stopping time, $\tau_{\rm s}$ gives an
estimate of how long the particle will be influenced by the magnetic field
before it is brought to rest relative to the gas. The stopping time is given
by

$\begin{split}\tau_{\rm s}&=\frac{m_{\rm d}v_{\rm d}}{F_{\rm drag}}\\\
&=\frac{m_{\rm d}}{n_{\rm p}m_{\rm p}c_{s}\sigma_{\rm col}}\end{split}$ (14)

where $m_{\rm d}=\frac{4}{3}\pi r_{\rm d}^{3}\rho_{\rm d}$ is the mass of a
spherical dust grain. In turn, a stopping distance for a dust grain initially
travelling at some fraction of the escape velocity can be estimated as

$\begin{split}d_{\rm s}&=\frac{v_{\rm d}\tau_{\rm s}}{2}\\\ &=\frac{r_{\rm
d}\rho_{\rm d}f}{9n_{\rm p}}\sqrt{\frac{2GM_{\star}\mu}{m_{\rm p}k_{\rm
B}Tr}}\\\
&=\frac{fr_{d}\rho_{d}(GM_{*}\Omega)^{\frac{1}{3}}}{9n_{p}}\sqrt{\frac{2\mu}{m_{p}k_{B}T\beta}},\end{split}$
(15)

where the third equality comes from substituting $r=\beta\,R_{K}$. This
stopping distance must be large enough that dust grains can travel along field
lines to a stable point once they enter the corona. Larger grains have larger
stopping distances, because although they experience more drag they have
higher inertia.

####  2.3.5 Sublimation and diffusion timescales

The features observed in the light curves are stable over an 81 day K2
campaign (Stauffer et al., 2017) and 1-2 sectors (27-54+ days) of TESS
observations (Zhan et al., 2019). Therefore, once at the stable point, dust
grains must have diffusion and sublimation timescales longer than a few months
to remain at the stable point, assuming there is no resupply of dust. For
sublimation timescales of dust grains close to M dwarfs we refer to the recent
paper by Zhan et al. (2019), who considered five different compositions and
three grain sizes (0.1$\mu$m, 1$\mu$m, 10$\mu$m) at distances of
1-7$R_{\star}$. For V374 Peg the co-rotation radius is at
$R_{K}=4.72R_{\star}$, where three (corundum, forsterite and enstatite) of the
five compositions considered by Zhan et al. (2019) for all sizes and 10$\mu$m
iron grains have sublimation timescales greater than a year and one (fayalite)
has a sublimation timescale of one month. These timescales are sufficiently
long that dust grains can survive long enough for the features observed and
sublimation should not be the dominant contributor to the variability.

Random thermal motion of particles within the plasma may of course lead to
diffusion of dust out of the stable point. The diffusion coefficient can be
estimated as $D=\frac{1}{2}\lambda v_{\rm p}$ where $\lambda=\frac{1}{n_{\rm
p}\sigma_{\rm col}}$ is the mean free path for collisions between dust grains
and protons in the plasma (since protons have greater momentum than the
electrons in the plasma) and $v_{p}=\sqrt{\frac{3k_{B}T}{m_{p}}}$ by
equipartition theorem. Assuming that at the stable point the dust is diffusing
out of a spherical cloud of radius $r_{\rm cloud}$ and surface area $A=4\pi
r_{\rm cloud}^{2}$, the diffusion timescale, $\tau_{\rm diff}=A/D$, can be
written as

$\begin{split}\tau_{\rm diff}=\frac{8\pi r_{\rm cloud}^{2}n_{\rm p}\sigma_{\rm
col}}{v_{\rm p}}.\end{split}$ (16)

###  2.4 Dust masses for the absorption and light curves

The mass of dust required to produce the observed absorption must be
physically realistic. To produce the observed absorption, the optical depth,
$\tau$ of the dust cloud must be at least one. The optical depth is given by

$\tau=n_{\rm d}\sigma_{\rm ext}s$ (17)

where $n_{\rm d}$ is the number density of the absorbing species, $\sigma_{\rm
ext}$ is the extinction cross section and $s$ is the path length. For
micrometer size grains in the Kepler passband (400-900 nm), light undergoes
Mie scattering and the extinction cross section can be written as $\sigma_{\rm
ext}=Q_{\rm ext}\pi r_{\rm d}^{2}$ (assuming spherical dust grains). $Q_{\rm
ext}$ is a numerical prefactor that depends on the complex refractive index of
the dust grains and the dimensionless parameter $X=\frac{2\pi r_{\rm
d}}{\lambda}$, which characterises the size of the particle compared to the
wavelength of the light, $\lambda$. Croll et al. (2014) calculated $Q_{\rm
ext}$ as a function of $X$ for a variety of refractive indices, including
those for the same minerals considered by Zhan et al. (2019) which are likely
to be in the dust clouds. For values of $X$ greater than five, Figure 13 in
Croll et al. (2014) indicates there is little variation in $Q_{\rm ext}$ with
increasing $X$ and composition has little effect. Therefore, for the optical
depth calculation, the light was taken to be monochromatic at 650nm as it did
not affect the extinction cross section. The path length for the optical depth
calculation was taken to be twice the radius of the spherical dust cloud, such
that $s=2r_{\rm cloud}$. This maximises the path length for a given cloud. The
number density of the dust was taken to be constant within the cloud, because
the dust clouds are small in comparison to the pressure scale height of the
corona. This allows us to write

$n_{\rm d}=\frac{\tau}{2Q_{\rm ext}\pi r_{\rm d}^{2}r_{\rm cloud}}.$ (18)

The minimum dust number density can be obtained by setting $\tau=1$.

The dust-cloud locations must be estimated to calculate model light curves. As
discussed in Section 2.3, dust clouds can be located at stable points (see
Section 2.2) in the magnetic field. At any given time, not all stable points
will support dust clouds, so we randomly select half of the stable points to
support dust clouds and calculate multiple realisations of the resulting dust-
cloud locations. We show one example in Fig. 1. Equations 2 and 18 can then be
used to calculate the cloud radii and number densities and can be combined
with these dust-cloud locations to generate model light curves (see Section
3.4).

Quantity | Value
---|---
$T$ | 5.7 MK
$M_{\star}$ | $0.28M_{\odot}$
$R_{\star}$ | $0.34M_{\odot}$
$R_{K}$ | $4.72R_{\star}$
$B_{\rm surf}$ | 500G
$\Omega_{*}$ | $1.63\times 10^{-4}\,\rm rad\,\rm s^{-1}$
$\rho_{\rm d}$ | 3000 kg$\,\rm m^{-3}$
$n_{\rm p}$ | $10^{16}\,\rm m^{-3}$
$r_{\rm cloud}$ | $0.1\,R_{\star}$
Table 1: Typical values for quantities in scaling relationships in this paper.
Stellar parameters except temperature are values for V374 Peg from Donati et
al. (2006), with $R_{\star}$ assuming sin$i=70$ °. The coronal temperature of
V374 Peg comes from Jardine & Collier Cameron (2019). The dust particle
density follows Zhan et al. (2019) and the proton number density in the corona
comes are order of magnitude estimates based on Ness et al. (2002, 2004). The
cloud radius was obtained from the amplitude of the scallop-shell features
(see Section 3.3).

##  3 Results

###  3.1 Dust-cloud locations and sizes from observations

In Table 2 we list the lower limits on the radial distances $a/R_{\star}$ of
the co-rotating clouds from the rotation axes of the 10 stars studied by Zhan
et al. (2019) via the two different methods described in Section 2.1 and the
radii of the dust clouds. The dips are between 0.4-4.8% corresponding to cloud
radii of 0.06-0.22 $R_{\star}$. We identified the narrowest dip in each of the
published light curves and measured its fractional depth $k$ and full duration
$T$ at half maximum depth directly. We also computed the Fourier power
spectrum of the TESS data for each of the Zhan et al. (2019) targets, and
determined the timescale corresponding to the highest-frequency peak in the
regularly-spaced pattern clearly visible above the noise. Both methods yielded
similar estimates for $T$, and hence for $r/R_{\star}=P/\pi T$. The distances
$r$ of the clouds from the rotation axis are plotted and compared with the
Keplerian co-rotation radius in Fig. 2.

With the exception of one outlier, which could be the product of an elongated
cloud complex with an extent greater than the stellar diameter, the narrowest
clearly-identifiable dips in all the stars in the sample of Zhan et al. (2019)
lie between 5 and 10 stellar radii from the rotation axis. These should be
considered as lower limits, since the narrowest events may result from grazing
transits of clouds in an equatorial ring tilted relative to the line of sight.
As with transiting planets, the probability that a randomly-inclined object at
distance $r$ will transit the stellar disc is $\mathcal{P}(\rm
transit)=R_{\star}/r$. This produces a geometrical selection effect resulting
from the random inclinations of the stellar rotation axes, favouring grazing
transits when the co-rotation radius is further from the stellar rotation
axis. We validated this explanation by using the dip widths in the synthetic
light curve of V374 Peg discussed and illustrated in Section 3.4, to estimate
the radial distance in the same way. The derived distance, shown as a green
lower limit in Fig. 2, was about 60 percent of the co-rotation radius, whereas
the model clouds were all close to co-rotation. We conclude that the dips in
these stars arise from co-rotating structures predominantly located close to,
or inside, the co-rotation radius.

Figure 2: Lower limits on the radial distances $r/R_{\star}$ of the co-rotating clouds from the rotation axes of the 10 stars studied by (Zhan et al., 2019), versus Keplerian co-rotation radius. The blue points are measured using the FWHM of the smallest dip feature in each light curve, while the orange points are derived from the highest frequency clearly visible in the Fourier power spectrum of the light curve. The grey line of unit slope allows the cloud locations to be compared with the co-rotation radius. The green point, derived in the same way from the duration of the narrowest dip produced by clouds near the co-rotation radius in the synthetic light curve of V374 Peg (Fig. 6), illustrates the lower-limit nature of these measurements. Table 2: Estimated cloud sizes and radial distances for the smallest dip features identified in TESS light curves of the ten 10 stars from (Zhan et al., 2019). The first 6 columns give the TIC number , stellar mass and radius in solar units, the rotation period in days, the fractional duration $T/P$ of the smallest dip, and its fractional depth $k$. The next three columns list the Keplerian co-rotation radius, minimum cloud distance $r_{\rm FWHM}$ estimated from the FWHM of the smallest dip, and the minimum cloud distance $r_{\rm Fourier}$ estimated from the highest frequency in the Fourier power spectrum of the light curve. The final column gives the cloud size $r_{\rm cloud}$ estimated from the fractional depth of the smallest dip. TIC | $M_{\star}/M_{\odot}$ | $R_{\star}/R_{\odot}$ |  $P_{\rm rot}$ (day) | $T/P$ | $k$ | $R_{\rm K}/R_{\star}$ | $r_{\rm FWHM}/R_{\star}$ | $r_{\rm Fourier}/R_{\star}$ | $r_{\rm cloud}/R_{\star}$
---|---|---|---|---|---|---|---|---|---
206544316 | 0.35 | 0.46 | 0.3217 | 0.100 | 0.048 | 6.1 | 3.2 | 14.8 | 0.22
224283342 | 0.34 | 0.25 | 0.8873 | 0.036 | 0.019 | 22.3 | 8.9 | 9.8 | 0.14
425933644 | 0.38 | 0.51 | 0.4863 | 0.053 | 0.036 | 7.0 | 6.0 | 7.0 | 0.19
425937691 | 0.3 | 0.32 | 0.2007 | 0.050 | 0.010 | 6.7 | 6.4 | 6.9 | 0.10
38820496 | 0.25 | 0.28 | 0.656 | 0.050 | 0.008 | 18.0 | 6.4 | 3.6 | 0.09
201789285 | 0.18 | 0.24 | 0.1516 | 0.033 | 0.018 | 8.8 | 9.6 | 5.0 | 0.13
234295610 | 0.37 | 0.38 | 0.7615 | 0.050 | 0.005 | 12.9 | 6.4 | 8.4 | 0.07
177309964 | 0.49 | 0.49 | 0.4533 | 0.050 | 0.014 | 6.4 | 6.4 | 6.3 | 0.12
332517282 | 0.19 | 0.25 | 0.4023 | 0.042 | 0.010 | 16.0 | 7.6 | 8.3 | 0.10
289840928 | 0.29 | 0.38 | 0.1999 | 0.040 | 0.003 | 5.7 | 8.0 | 5.5 | 0.06

###  3.2 Can the dust be confined by the magnetic field?

Typical values for dust and stellar parameters can be substituted into the
equations from Section 2.3 to determine if dust trapping in magnetic stable
points is possible. The typical values used are summarised in Table 1 and
stellar parameters have been taken for the magnetically active M dwarf V374
Peg, but the results here are presented as scaling relationships so they can
be applied to other stars. Although V374 Peg does not show the lightcurve dips
that are characteristic of the scallop-shell stars, it was used as an
illustrative example of the magnetic field structures that we might expect to
find on rapidly-rotating mid-to-late M dwarfs. It is well observed (Donati et
al., 2006; Morin et al., 2008; Hallinan et al., 2009; Vida et al., 2016) and
magnetic field maps are available for it, which enables the modelling of the
stable points discussed in Section 2.2.

From Equation 11 the critical size below which Lorentz forces dominate over
gravitational ones is

$r_{\rm
d}<12\frac{\left(\frac{f}{1}\right)^{\frac{1}{2}}\left(\frac{T}{5.7\,\rm M\rm
K}\right)^{\frac{1}{2}}\left(\frac{R_{\star}}{0.34R_{\odot}}\right)^{\frac{3}{2}}\left(\frac{B_{\rm
surf}}{500\,\rm G}\right)^{\frac{1}{2}}\left(\frac{\Omega}{1.6\times
10^{-4}\,\rm rad\,\rm
s^{-1}}\right)^{\frac{1}{2}}}{\left(\frac{M_{\star}}{0.28M_{\odot}}\right)^{\frac{1}{2}}\left(\frac{\rho_{\rm
d}}{3000\,\rm kg\,\rm
m^{-3}}\right)^{\frac{1}{2}}\left(\frac{\beta}{1}\right)^{\frac{3}{4}}}\mu\rm
m$ (19)

This indicates dust particles can be supported in the stellar magnetosphere
against gravity. The strength of gravitational force relative to the Lorentz
force as a function of grain size, distance to the star, surface magnetic
field strength and coronal temperature is shown in Figures 3 and 4. The
Lorentz force is larger than the gravitational force except for the largest
dust particles furthest from the star. The strength of the Lorentz force in
comparison to gravity increases at higher coronal temperatures and surface
magnetic field strengths, which increases the maximum grain size that can be
supported in the field against gravity.

Figure 3: Dust grain support as a function of grain size and distance from the
star, for a variety of coronal temperatures and surface field strengths. The
temperature down each column of the subplot is constant and the surface field
strength across each row of the subplot is constant. Figure 4: Dust grain
support as a function of surface magnetic field strength and coronal
temperature, for a variety of grain sizes, $r_{d}$ and distances from the
star, $a$. The distance from the star down each column of the subplot is
constant and the grain size across each row of the subplot is constant.

Using Equation 15, the approximate stopping distance of a dust grain is

$d_{\rm s}=0.08\frac{\left(\frac{f}{1}\right)\left(\frac{r_{\rm d}}{1\mu\rm
m}\right)\left(\frac{\rho_{\rm d}}{3000\,\rm kg\,\rm
m^{-3}}\right)\left(\frac{M_{\star}}{0.28\,M_{\odot}}\right)^{\frac{1}{3}}\left(\frac{\Omega_{*}}{1.63\times
10^{-4}\,\rm s^{-1}}\right)^{\frac{1}{3}}}{\left(\frac{n_{\rm p}}{10^{16}\rm
m^{-3}}\right)\left(\frac{T}{5.7\,\rm M\rm
K}\right)^{\frac{1}{2}}\left(\frac{R_{\star}}{0.34\,R_{\odot}}\right)}R_{\star}.$
(20)

The short distance in comparison to the difference between co-rotation radius
and cloud distance for the large co-rotation radius in Figure 2 may mean that
only a proportion of the dust grains make it to the stable point, but the
distance is not so small that no dust grains could reach the stable points.
Additionally, larger dust grains or smaller coronal number densities or lower
temperatures could lead to larger stopping distances, as shown in Figure 5. A
full simulation of the dust motion on the field lines in the corona is needed
to improve this estimate of stopping distances and constrain the fraction of
dust that settles in a stable point.

Figure 5: Stopping distance of dust grain as a function of grain size based on
Equation 15. In the left panel the number density is fixed at
$n_{p}=10^{16}\,m^{-3}$ and the coronal temperature varies. In the right
panel, the temperature is fixed at $T=1\times 10^{6}\,K$ and the coronal
number density varies. All other constants have the values in Table 1

The diffusion timescale of the dust trapped at a stable point is

$\begin{split}\tau_{\rm diff}&=2.2\times 10^{8}\left(\frac{r_{\rm
cloud}}{0.1\,R_{\star}}\frac{R_{\star}}{0.28\,R_{\odot}}\right)^{2}\left(\frac{r_{\rm
d}}{1\mu\rm m}\right)^{2}\left(\frac{n_{\rm p}}{10^{16}\,\rm m^{-3}}\right)\\\
&\left(\frac{T}{5.7\,\rm MK}\right)^{-\frac{1}{2}}\rm yr.\end{split}$ (21)

This timescale is $10^{9}$ times larger than the length of observations and
indicates once dust is at a stable point it can remain there unless the gas to
which it is coupled moves, the dust sublimates (which has short enough
timescales for certain compositions) or the field structure changes (see
Section 4). Observations of M dwarfs have shown their magnetic field
structures evolve only slowly, changing little over timescales of months (Vida
et al., 2016), so these dust clouds could remain fixed for long enough to
produce the stable signals observed, contrary to suggestions in Günther et al.
(2022).

###  3.3 Dust-cloud masses and number densities

The results in Section 3.2 combined with the sublimation timescales from Zhan
et al. (2019) suggest it is possible to trap dust at stable points in the
magnetosphere for long enough to produce the observed absorption. However, the
dust masses required must also be physically realistic and the model must
reproduce the observed light curves.

A typical dust number density in the cloud can be calculated using Equation
with 1$\mu m$ dust grains which have $Q_{\rm ext}=3$ and dust-cloud radii from
observations in the range $0.06-0.22\,R_{\star}$ (see Table 2). 18,

$n_{\rm d}=2200\left(\frac{\tau}{1}\right)\left(\frac{Q_{\rm
ext}}{3}\right)^{-1}\left(\frac{r_{\rm d}}{1\mu\rm
m}\right)^{-2}\left(\frac{s}{0.1\,R_{\star}}\frac{R_{\star}}{0.34\,R_{\odot}}\right)^{-1}\rm
m^{-3}.$ (22)

Number densities and values of $Q_{\rm ext}$ are given in Table 3. For a given
dust cloud this is a minimum number density as it assumes the longest path
length through the cloud and an optical depth of one. This gives a dust to gas
number ratio of order $10^{-13}$ relative to the corona. This is consistent
with a model of dust particles entrained in coronal plasma and neglecting
collisions between dust particles. The dust to gas mass ratio is of order one,
so the dust mass is a significant fraction of the mass at a stable point. For
a spherical dust cloud with spherical dust grains the cloud mass is

$\begin{split}m_{\rm cloud}&=1.6\times
10^{12}\left(\frac{\tau}{1}\right)\left(\frac{Q_{\rm
ext}}{3}\right)^{-1}\left(\frac{r_{\rm
cloud}}{0.1\,R_{\star}}\frac{R_{\star}}{0.34\,R_{\odot}}\right)^{2}\\\
&\left(\frac{r_{\rm d}}{1\mu\rm m}\right)\left(\frac{\rho_{\rm d}}{3000\,\rm
kg\,\rm m^{-3}}\right)\rm kg.\end{split}$ (23)

This is two orders of magnitude smaller than masses of a small rubble-pile
asteroid (Veras et al., 2014) and many orders of magnitude smaller than
typical Vesta (main-belt, rocky) and Eris (Kuiper-belt, rocky) asteroids in
the Solar System (Malamud & Perets, 2020b). It is also two orders of magnitude
smaller than the mass predicted for cold gas clouds (known as “slingshot
prominences”) trapped at stable points and at least one order of magnitude
less than the observed gas mass in a single gas ejection event from V374 Peg
(Vida et al., 2016; D’Angelo et al., 2018)

$r_{\rm d}/\;\mu\rm m$ | $Q_{\rm ext}$ | $n_{\rm d}/\;\rm m^{-3}$ | $m_{\rm cloud}/\;\rm kg$
---|---|---|---
0.1 | 0.2 | $3.4\times 10^{6}$ | $1.6\times 10^{9}$
0.5 | 2 | $1.3\times 10^{4}$ | $2.0\times 10^{11}$
1 | 3 | 2200 | $1.6\times 10^{12}$
10 | 2 | 34 | $1.6\times 10^{15}$
Table 3: $Q_{\rm ext}$ and number densities of dust grains for an optical
depth of one for a variety of grain sizes assuming an optical depth of one.

###  3.4 Theoretical light curves

Using this number density and corresponding gas to dust ratio, the magnetic
map of V374 Peg was used to simulate the light curves. Since not all potential
stable points will be filled at any one time, a fraction (50$\%$) of all the
stable points was selected at random to support the dust clouds. The locations
of all possible stable points and the field structures that support them are
shown in Figure 1 and three sample realisations of the resulting light curves
for 1$\mu$m dust grains are shown in Fig. 6. For each random selection of
stable points, a similar pattern of dips is found. These show depths of
1-3$\%$, consistent with observations.

Figure 6: Model light curves for V374Peg created by filling a fraction of the
stable points with dust particles of size 1$\mu$m. Three different random
samples of stable points are shown. The narrowest features have a width of
about 0.05 in phase. The corresponding radial acceleration under-estimates the
distance of the cloud from the rotation axis by about 40 percent, as
illustrated by the green arrow in Fig.2. Figure 7: Model light curves for
V374Peg created by filling a fraction of the stable points with dust particles
of size 1$\mu$m. Four different inclinations of the rotation axis to the line
of sight are shown (including $i=70$, the value used in Fig.6).

The inclination of the stellar rotation axis to the line of sight will of
course affect which dust clouds transit in front of the star. For a star with
a mainly aligned dipolar field, the stable points will lie within a torus that
is located around and beyond the co-rotation radius, as shown in Fig. 1.
Reducing the inclination to zero ensures that no dust-clouds will transit
(unless they lie over the rotation pole). We therefore expect that the
scallop-shell phenomenon will predominantly be found in stars whose rotation
axes have large inclinations. We illustrate this in Fig 7 which shows light-
curves calculated for a range of inclinations, including the value of
$i=70^{\circ}$ (Donati et al., 2006) used in Fig. 6. For inclinations below
$40^{\circ}$ no dips can be found.111Kraus et al. (2014) published projected
equatorial rotation speeds for five of the stars in the sample of Zhan et al.
(2019) which belong to the Tuc-Hor moving group. Comparison of these $v\sin i$
values with the stellar radii and rotation periods in Table 2 suggests that
all five have rotation axes inclined at less than 40 degrees to the line of
sight. One of them, TIC 206544316, was observed independently by Zhan et al.
(2019) with spectral resolving power R=24000 on the ANU 2.4-m telescope,
yielding vsini=77 km s-1. This is very different from the 46.8 km s-1 reported
by Kraus et al. (2014) for the same star. The single measurement by Zhan et
al. (2019) indicates an inclination close to 90 degrees for TIC 206544316. The
reason for the discrepancy is unclear, but we note that the probability of a
randomly-chosen sample of 5 stars all having inclinations less than 40 degrees
is $7.0\times 10^{-4}$. It would be interesting to re-determine $v\sin i$ for
the entire sample.

##  4 Discussion

###  4.1 Wider implications

Section 3 demonstrates that dust can be trapped at the co-rotation radius and
the resulting dust clouds can produce light curve dips similar to those
observed for the scallop-shell stars. The scallop-shell phenomenon is only
observed in a small subset of all rapidly-rotating stars with spectral types
later than mid-M. This suggests that either the scallop-shell phenomenon is
sporadic, or that it is seen only in those stars with favourable viewing
geometries.

Many such stars generate strong dipole-dominated fields, which would trap dust
close to the equatorial plane. Clouds confined near the stellar equatorial
plane will only transit the disc if the inclination of the stellar rotation
axis is close to 90 degrees. At low inclinations the clouds may be present but
unobservable.

The probability of a dust cloud at the co-rotation radius crossing the line of
sight and causing a dip is

$P_{\rm transit}=\frac{R_{\star}+r_{\rm cloud}}{R_{K}}.$ (24)

For the 10 scallop-shells discovered by Zhan et al. (2019), typically $r_{\rm
cloud}\simeq 0.1R_{\star}$ and $5R_{\star}\lesssim R_{K}\lesssim R_{\star}$
Hence $P_{\rm transit}$ ranges from 0.22 for $R_{K}=5R_{\star}$ to 0.055 for
$R_{K}=20R_{\star}$ Therefore, if there was no limit on dust supply, within
the subset of young ($<50\,\rm Myr$) and rapidly rotating ($P<1\,\rm day$) M
dwarfs, approximately 10% would be expected to be scallop-shells. Indeed,
Stauffer et al. (2018) and Günther et al. (2022) found 9% and 6% of young,
rapid rotators were scallop-shells respectively. Nonetheless, Günther et al.
(2022) report that even in stars found to exhibit scallop-shell behaviour, the
phenomenon is not always present. This suggests the injection of dust is
sporadic, and that the clouds have a finite lifetime.

Some very low-mass M dwarfs have, however, been shown to generate weaker and
multipolar magnetic fields (Morin et al., 2010). This may also affect the
spatial distribution of stable points where dust can be trapped and hence the
transit probability.

These light curves have only been observed in young stars: 5-10 Myr for those
observed by Stauffer et al. (2018) and 45 Myr for those observed by Zhan et
al. (2019) and Palumbo et al. (2022). The appearance of this phenomenon in
young stars is likely to be because there is more dust and scattering of
planetesimals in young stellar systems (Williams & Cieza, 2011). These stars
are also rapid rotators with periods less than one day. Rapid rotation is
required to ensure that the co-rotation radius (where most stable points are
located) lies within the star’s corona. Rapid rotation also ensures that the
field is strong enough for the Lorentz force to oppose gravity.

One possible constraint on dust availability is if dust delivery is sporadic.
The lack of IR excess in these systems (Stauffer et al., 2018) indicates the
absence of a dusty accretion disc around these stars and so in our model dust
moving towards the star is likely to originate from tidally disrupted
planetesimals. This is consistent with the dust masses obtained in Section
3.3, which are all orders of magnitude lower than asteroid types observed in
the Solar System. This supports a model where a small fraction of the
planetesimal material reaches the stable point due to the size distribution of
planetesimal fragments and the differing effects of gravity and drag on dust
grains of different sizes (see Section 3.2). If the rate of tidal disruption
of planetesimals around the star is slower than the rate at which dust clouds
are destroyed (see Section 4.2) then stars which have not recently had a
tidally disrupted planetesimal will not exhibit this phenomenon. Tidal
disruption of planetesimals around white dwarfs has been modelled extensively
to explain white dwarf pollution e.g. Jura (2003, 2008); Veras et al. (2014);
Malamud & Perets (2020a). These models could be applied to M dwarfs to
investigate origins of the dust and determine if tidal disruption is a
plausible origin. Resulting theoretical predictions could be tested using
observations of scallop-shell light curves. The frequency of occurrence of
scallop shell features could constrain the frequency of planetesimal
disruption within the system whilst the timescale for appearance of these
features in the light curve (e.g. between Sectors 10i and 10ii in Figure 9 of
Günther et al. (2022)) could be used to test predicted timescales between
disruption and dust accretion.

Dust trapped in magnetic stable points in these stars provides a new way to
map the large-scale loop structures in these stars’ coronae. Rather than
needing to observe in H$\alpha$ which is costly and can be done for limited
time, dust can be observed in the optical with instruments, such as TESS for a
long period of time (for example TIC 177309964 in Günther et al. (2022) was
observed for one year). This could enable high cadence observations of the
evolution of M dwarf magnetic fields with these dust clouds acting as a
tracer. Magnetic field structures deduced from optical observations could then
be compared with those from Zeeman-Doppler imaging.

###  4.2 Disappearance of features

A mechanism for removal of dust clouds is required to explain the lower than
expected occurrence rates (Section 4.1) and the sudden disappearance of the
narrow features in the light curve - as demonstrated between Sector 11ii and
Sector 12i in Figure 9 of Günther et al. (2022). The sublimation timescales of
dust grains are too long to explain this (see 2.3.5). The magnetically
confined, co-rotating dust clouds hypothesis provides a potential process. If
the dust is trapped in the magnetic field, the disappearance of the dust
signature implies a change in the magnetic field structure. The surface
magnetic fields of M dwarfs have been observed to evolve only slowly over
timescales of months (Morin et al., 2008) so this change is unlikely to be due
to a sudden change in coronal field configuration and disappearance of a
stable point. However, this could be explained by the formation (and
subsequent ejection) of a “slingshot prominence” (a condensation of coronal
plasma) at a magnetic stable point. These condensations are observed as
travelling absorption features in H$\alpha$ (Collier Cameron & Robinson,
1989). The radial acceleration of these features shows that they typically
originate from material at or beyond the co-rotation radius. At these radial
distances, they must be supported against centrifugal ejection, most plausibly
by the magnetic field.

The long diffusion timescale in Equation 16 indicates that the dust would be
entrained in this condensation. These condensations have observed lifetimes of
only around one day. This finite lifetime is a simple consequence of the
nature of the gas upflow that forms the prominence (Jardine & Collier Cameron,
2019). This upflow is sufficiently hot that it is supersonic by the time it
reaches the stable point. As a result, the mass of gas at the stable point
will increase until it can no longer be supported by the magnetic field. At
this point, any gas (and any entrained dust) will be ejected from the support
site. Because the upflow is supersonic, the surface will nonetheless continue
to supply mass leading to another ejection. This limit cycle of ejections
would ensure the removal of any dust that might have been trapped at the
stable point. A similar explanation - centrifugal breakout - has been
suggested for disappearance of features in the dipper star TIC 234284556 by
Palumbo et al. (2022).

###  4.3 Dust particle size

The size of the dust particles requires further investigation. The upper
panels in Figure 10 of Günther et al. (2022) show weak colour dependence,
which could be predominantly due to underlying starspots rather than dust.
This dependence and the broadband K2 data suggest the dust grain radii could
be anywhere in the visible range or slightly larger. 1$\mu m$ dust grains were
suitable for reproducing the signal, as shown in Figure 6. Equation 11
indicates an upper limit on particle size of around 5$\mu m$ and Equation 15
demonstrates that larger particles have larger stopping distances and more
likely to reach a stable point. Therefore, longer wavelength IR observations
are required to confirm the particle size distribution within the range
0.5-5$\mu m$ using methods similar to Croll et al. (2014).

###  4.4 Further Work

The possible coexistence of dust clouds and prominences could be investigated
through time-resolved H$\alpha$ spectroscopy of these stars, with simultaneous
photometry. Such a coexistence would depend on rate of prominence formation
and dust delivery. Stellar coronae have many stable points, so for frequent
dust delivery both scallop-shell light curves and prominences should be
observed. This would tell us to what extent the spatial distribution of dust
mirrors that of the neutral hydrogen condensations. In addition, if the
photometric dips were found always to occur in rotational anti-phase to the
H$\alpha$ transients, this would indicate that dips can also arise from bound-
free emission being eclipsed as gaseous prominences pass behind the star, cf.
Rebull et al. (2016).

As discussed in Section 4.1, theoretical work on the delivery of dust to the
star is required. As well as models of planetesimal delivery and tidal
disruption, it would be instructive to create a magnetohydrodynamic simulation
of dust as it enters the corona, including coupling to the field and coronal
drag as it travels to a stable point. This would investigate what proportion
of dust from a planetesimal will end up in stable points and what the size
distribution of this material is.

##  5 Conclusions

This work has demonstrated that magnetospherically-trapped dust clouds provide
a feasible explanation for the high-frequency variability of the scallop-shell
stars, seen superposed on the underlying photospheric starspot modulation.
Analysis of observations of ten scallop-shells observed by TESS (Zhan et al.,
2019) has shown that the narrow features appear with the same periodicity as
the quasi-sinusoidal starspot modulations and hence are co-rotating. Their
transit times are consistent with their being located at or inside the co-
rotation radius. We have developed a model for coupling of electrostatically-
charged dust particles to the magnetic field, and shown that the existence of
magnetic stable points enables trapping of dust around the co-rotation radius.
The long sublimation and diffusion timescales of dust at this location agrees
with observations of the stability of this signal. Using the magnetic map of
V374 Peg, we have generated synthetic realisations of the light curves that
would be produced by these co-rotating dust clouds transiting the stellar
disc. These have similar features (depth, duration and number of dips) to the
observed light curves.

All of the above confirms that co-rotating dust clouds are a possible
explanation for scallop-shell stars. For the phenomenon to be observed in a
given star requires both a dust delivery mechanism and a favourable viewing
geometry. Observations of these stars could provide a test for theoretical
models of tidal disruption and planetary dynamics of these systems. The
presence of these dust clouds also provides a new method for observing the
magnetic stable points of these stars, which could be compared with H$\alpha$
observations and results from Zeeman-Doppler imaging. Further multi-wavelength
observations of these stars as well as the development of models of dust
delivery processes are important to confirm the validity of this hypothesis
and to better understand these stars.

## Acknowledgements

The authors would like to thank the Royal Society of Edinburgh for the RSE
Cormack Vacation Scholarship which initially enabled this work to be carried
out. The authors thank the reviewer for their helpful comments which improved
the quality of the manuscript. They would also like to thank Rose Waugh, Doug
Lin and Amy Bonsor for helpful discussions. MMJ and ACC acknowledge support
from STFC consolidated grant number ST/R000824/1. HS acknowledges funding on
NERC studentship NE/S007474/1 and a graduate scholarship from Exeter College,
University of Oxford. JFD acknowledges funding from the European Research
Council (ERC) under the H2020 research and innovation programme (grant
agreement #740651 NewWorlds). For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

## Data Availability

Archival data underpinning the plots is available at polarbase
(http://polarbase.irap.omp.eu).

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##  6 Appendix

The solution to Eqn.5 may be written as

$B_{r}=\sum^{N}_{l=1}\sum^{l}_{m=-l}B_{lm}P_{lm}(\theta)f_{l}(r,r_{ss})r^{-(l+2)}e^{im\phi}$
(25)
$B_{\theta}=-\sum^{N}_{l=1}\sum^{l}_{m=-l}B_{lm}\frac{dP_{lm}(\theta)}{d\theta}g_{l}(r,r_{ss})r^{-(l+2)}e^{im\phi}$
(26)
$B_{\phi}=-\sum^{N}_{l=1}\sum^{l}_{m=-l}B_{lm}\frac{P_{lm}(\theta)}{\sin\theta}img_{l}(r,r_{ss})r^{-(l+2)}e^{im\phi}$
(27)

where the functions $f_{l}(r,r_{ss})$ and $g_{l}(r,r_{ss})$ which describe the
modification of the field structure by the outflowing hot gas are given by

$f_{l}(r,r_{ss})=\left[\frac{l+1+l(r/r_{ss})^{2l+1}}{l+1+l(1/r_{ss})^{2l+1}}\right]$
(28)
$g_{l}(r,r_{ss})=\left[\frac{1-(r/r_{ss})^{2l+1}}{l+1+l(1/r_{ss})^{2l+1}}\right].$
(29)