# Centrifugal breakout reconnection as the electron acceleration mechanism
powering the radio magnetospheres of early-type stars

S. P. Owocki1, M. E. Shultz1, A. ud-Doula2, P. Chandra3, B. Das3, P. Leto4
1Department of Physics and Astronomy, University of Delaware, 217 Sharp Lab,
Newark, Delaware, 19716, USA
2Department of Physics, Penn State Scranton, Dunmore, PA 18512, USA
3National Centre for Radio Astrophysics, Tata Institute of Fundamental
Research, Pune University Campus, Pune-411007, India
4INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania,
Italy
E-mail<EMAIL_ADDRESS>
###### Abstract

Magnetic B-stars often exhibit circularly polarized radio emission thought to
arise from gyrosynchrotron emission by energetic electrons trapped in the
circumstellar magnetosphere. Recent empirical analyses show that the onset and
strength of the observed radio emission scale with both the magnetic field
strength and the stellar rotation rate. This challenges the existing paradigm
that the energetic electrons are accelerated in the current sheet between
opposite-polarity field lines in the outer regions of magnetised stellar
winds, which includes no role for stellar rotation. Building on recent success
in explaining a similar rotation-field dependence of H$\alpha$ line emission
in terms of a model in which magnetospheric density is regulated by
centrifugal breakout (CBO), we examine here the potential role of the
associated CBO-driven magnetic reconnection in accelerating the electrons that
emit the observed gyrosynchrotron radio. We show in particular that the
theoretical scalings for energy production by CBO reconnection match well the
empirical trends for observed radio luminosity, with a suitably small, nearly
constant conversion efficiency $\epsilon\approx 10^{-8}$. We summarize the
distinct advantages of our CBO scalings over previous associations with an
electromotive force, and discuss the potential implications of CBO processes
for X-rays and other observed characteristics of rotating magnetic B-stars
with centrifugal magnetospheres.

###### keywords:

stars: magnetic fields – stars: early type – stars: rotation – radio
continuum: stars – magnetic reconnection

††pagerange: Centrifugal breakout reconnection as the electron acceleration
mechanism powering the radio magnetospheres of early-type
stars–References††pubyear: 2002

##  1 Introduction

Hot luminous, massive stars of spectral type O and B have dense, high-speed,
radiatively driven stellar winds (Castor et al., 1975). In the subset
($\sim$10%; Grunhut et al. (2017); Sikora et al. (2019a)) of massive stars
with strong ($>\,100$ G; Aurière et al. (2007); Shultz et al. (2019a)),
globally ordered (often significantly dipolar; Kochukhov et al. (2019))
magnetic fields, the trapping of this wind outflow by closed magnetic loops
leads to the formation of a circumstellar magnetosphere (Petit et al., 2013).
Because of the angular momentum loss associated with their relatively strong,
magnetised wind (ud-Doula et al., 2009), magnetic O-type stars are typically
slow rotators, with trapped wind material falling back on a dynamical
timescale, giving what’s known as a “dynamical magnetosphere” (DM).

But in magnetic B-type stars, the relatively weak stellar winds imply longer
spin-down times, and so a significant fraction that still retain a moderately
rapid rotation; for cases in which the associated Keplerian co-rotation radius
$R_{\rm K}$ lies within the Alfvén radius $R_{\rm A}$ that characterises the
maximum height of closed loops, the rotational support leads to formation of a
“centrifugal magnetosphere” (CM), wherein the much longer confinement time
allows material to build up to a sufficiently high density to give rise to
distinct emission in H$\alpha$ and other hydrogen lines (Landstreet & Borra,
1978). A recent combination of empirical (Shultz et al., 2020) and theoretical
(Owocki et al., 2020) analyses showed that both the onset and strength of such
Balmer-$\alpha$ emission is well explained by a centrifugal breakout (CBO)
model, wherein the density distribution of material within the CM is regulated
to be near the critical level that can be contained by magnetic tension (ud-
Doula et al., 2008). The upshot is that such hydrogen emission arises only in
magnetic stars with both strong magnetic confinement and moderately rapid
stellar rotation.

Another distinctive observational characteristic of many such magnetic B-stars
is their non-thermal, circularly polarized radio emission, thought to arise
from gyrosynchrotron emission by energetic electrons trapped within closed
magnetic loops. An initially favoured model by Trigilio et al. (2004) proposed
that these electrons could be accelerated in the current sheet between field
lines of opposite polarity that have been stretched outward by the stellar
wind, as illustrated in the left panel of Figure 1. But a recent empirical
analysis by Leto et al. (2021) has shown that the observed radio emission has
a clear dependence on stellar rotation, providing strong evidence against this
current-sheet model, which includes no role for rotation. Instead Leto et al.
(2021) noted that their fits to the radio luminosity scale in proportion to a
quantity that has the physical dimension of an electromotive force (EMF),
which they speculated may be suggestive of an underlying mechanism. Indeed,
the EMF is invoked (Hill, 2001) to model auroral emission from the interaction
of high-energy magnetospheric particles with planetary atmospheres. However,
such thermal atmospheric auroral dissipation of EMF-accelerated particles in
the magnetosphere cannot explain the polarized radio emission that likely
arises from gyrosynchrotron processes in the highly conductive magnetosphere
itself.

The alternative theoretical scalings explored here were motivated by a more
recent companion empirical analysis by Shultz et al. (2021, hereafter Paper
I), which confirms the basic results of Leto et al. (2021), but within a
significantly extended sample that allows further exploration of potential
empirical trends and scalings. In particular, we show below (section 3) that
these empirical scalings for nonthermal radio emission can be well fit by
models grounded in the same CBO paradigm that has been so successful for
H$\alpha$ emission. Specifically, it is now the magnetic reconnection
associated with CBO events that provides the nonthermal acceleration of
electrons, which then follow the standard picture of gyrosynchrotron emission
of observed circularly polarized radio.

Figure 1:  Schematic to contrast the previous Trigilio et al. (2004) current-
sheet (CS) model for electron acceleration and radio emission (left) with our
proposed centrifugal breakout (CBO) model for electron acceleration from CBO-
driven magnetic reconnection events (right). Pink-shaded regions indicate
magnetic field lines contributing plasma to the electron acceleration, while
grey shading indicates regions isolated from the locus of electron
acceleration. A key distinction regards the lack of a dynamical role for
rotation in the CS model, in contrast to the inferred empirical dependence on
rotation, which is well matched in the CBO model. Figure adopted from Figure 8
of Paper I, which contains further details on the schematic.

Figure 1 graphically illustrates the key distinctions between this new CBO
paradigm (right) from the previous current-sheet-acceleration model (left)
proposed by Trigilio et al. (2004).

To lay the groundwork for derivation in section 3 of these scalings for radio
emission from CBO-driven reconnection, the next section (2) reviews the basic
CM model and the previous application of the CBO paradigm to H$\alpha$. In
section 4 we contrast our results with the EMF-based picture, and discuss the
potential further application of the CBO paradigm, including for modeling the
stronger X-ray emission of CM vs. DM stars (Nazé et al., 2014). We conclude
(section 5) with a brief summary and outlook for future work.

##  2 Background

###  2.1 Dynamical vs. Centrifugal magnetospheres

For a magnetic hot-star with stellar wind mass loss rate ${\dot{M}}$ and
terminal wind speed $v_{\infty}$, MHD simulations (ud-Doula & Owocki, 2002)
show that the channeling and trapping of the stellar wind can be characterised
by a dimensionless wind-magnetic-confinement parameter,

$\eta_{\ast}\equiv\frac{B_{\rm eq}^{2}R_{\ast}^{2}}{\dot{M}v_{\infty}}\,,$ (1)

where $B_{\rm eq}$ is the surface field strength at the magnetic equator and
$R_{\ast}$ is the stellar radius. This characterises the ratio of magnetic
energy to wind kinetic energy. The radial extent of closed magnetic loops can
be characterised by the Alfvén radius, which for an initially dipolar field
with strong-confinement scales as

$\frac{R_{\rm A}}{R_{\ast}}\approx\eta_{\ast}^{1/4}~{}~{};~{}~{}\eta_{\ast}\gg
1\,,$ (2)

Simulations of cases with rotation-aligned dipoles (ud-Doula et al., 2008)
showed further that the dynamical effect of rotation can be similarly
characterised by a dimensionless parameter, now given by the ratio of the
star’s equatorial rotation speed to the near-star orbital speed,

$W=\frac{v_{\rm rot}}{v_{\rm orb}}=\frac{2\pi R_{\ast}}{P_{\rm
rot}}\left(\frac{GM_{\ast}}{R_{\ast}}\right)^{-1/2}\,,$ (3)

with $M_{\ast}$ and $P_{\rm rot}$ the stellar mass and rotation period, and
$G$ the gravitation constant. For magnetically trapped material that is forced
to co-rotate with the underlying star, centrifugal forces balance gravity in
the common equator at the Kepler co-rotation radius, given by

$\frac{R_{\rm K}}{R_{\ast}}=W^{-2/3}\,,$ (4)

For slowly rotating stars with $R_{\rm K}>R_{\rm A}$, rotation has little
dynamical effect, and so wind material trapped in closed magnetic loops below
$R_{\rm A}$ simply falls back to the star on a dynamical timescale, giving
then a dynamical magnetosphere (DM).

In contrast, for stars with both moderately rapid rotation ($W\lesssim 1$) and
strong confinement ($\eta_{\ast}\gg 1$), one finds $R_{\rm K}<R_{\rm A}$. In
the region $R_{\rm K}<r<R_{\rm A}$ magnetic tension still confines material
while the centrifugal force prevents gravitational fallback, thus allowing
material build-up into a much denser centrifugal magnetosphere (CM).

###  2.2 Centrifugal breakout and H$\alpha$ emission

As first analyzed in the appendices of Townsend & Owocki (2005), this CM mass
buildup is limited to a critical surface density for which the finite magnetic
tension can still confine the material against the outward centrifugal
acceleration. For mass buildup beyond this critical density, the magnetic
field lines become stretched outward by the centrifugal force, leading
eventually to centrifugal breakout (CBO) events. Through analysis of 2D MHD
simulations by ud-Doula et al. (2008), Owocki et al. (2020) showed that the
resulting global surface density scales as

$\sigma(r)\approx\sigma_{\rm K}\,\left(\frac{r}{R_{\rm
K}}\right)^{-6}~{}~{};~{}~{}r>R_{\rm K}\,,$ (5)

where the characteristic surface density at the Kepler radius scales with the
magnetic field strength and gravitational acceleration there,

$\sigma_{\rm K}\approx 0.3\,\frac{B_{\rm K}^{2}}{4\pi g_{\rm K}}\,.$ (6)

A key feature of this CBO-regulated density is that it is entirely independent
of the stellar wind mass loss rate ${\dot{M}}$ that controls the CM mass
buildup. This helps explain the initially unexpected empirical finding by
Shultz et al. (2020) that the onset and strength of observed H$\alpha$
emission from magnetic B-stars is largely independent of the stellar
luminosity, which plays a key role in setting the mass loss rate of the
radiatively driven stellar wind.

Motivated by this key result, Owocki et al. (2020) examined the theoretical
implications of this CBO-limited density scaling for such H$\alpha$ emission,
showing that it can simultaneously explain the onset of emission, the increase
of emission strength with increasing magnetic field strength and decreasing
rotation period, and the morphology of emission line profiles (Shultz et al.,
2020; Owocki et al., 2020). As initially suggested by Townsend & Owocki
(2005), the breakout density at $R_{\rm K}$ is set by $B_{\rm K}$, and is
independent of $\dot{M}$; precisely this dependence on $B_{\rm K}$, and lack
of sensitivity to $\dot{M}$, was found by Shultz et al. (2020) for both
emission onset and emission strength scaling. Owocki et al. (2020) found an
expression for the strength $B_{\rm K1}$ necessary for the density at $R_{\rm
K}$ to produce an optical depth of unity in the H$\alpha$ line, and showed
that the threshold $B_{\rm K}/B_{\rm K1}$ neatly divides stars with and
without H$\alpha$ emission. Two-dimensional MHD simulations of CBO by ud-Doula
et al. (2006, 2008) yielded a radial density gradient associated with the CBO
mechanism, which in conjunction with the density at $R_{\rm K}$ set by $B_{\rm
K}$ can be used to predict the optically thick area and, hence, the scaling of
emission strength (Owocki et al., 2020). Finally, a characteristic emission
line profile morphology, common across all H$\alpha$-bright CM host stars, was
reported by Shultz et al. (2020) and shown by Owocki et al. (2020) to be a
straightforward consequence of a co-rotating optically thick inner disk
transitioning to optically transluscent in the outermost region.

A crucial subtlety that deserves emphasis is that, in contrast to expectations
from 2D MHD simulations that CBO should manifest as catastrophic ejection
events accompanied by large-scale reorganization of the magnetosphere (ud-
Doula et al., 2006, 2008), which has indeed never been observed in the densest
inner regions (Townsend et al., 2013; Shultz et al., 2020), the H$\alpha$
analysis performed by Shultz et al. (2020) instead indicates that the
magnetosphere must be continuously maintained at breakout density, with CBO
occuring more or less continuously on small spatial scales. However, it is
worth noting that the ‘giant electron-cyclotron maser (ECM) pulse’ observed by
Das & Chandra (2021) may have been the signature of a large-scale breakout
occuring in magnetospheric regions in which the density is too low to be
probed by H$\alpha$ or photometry.

H$\alpha$ emission and gyrosynchrotron emission occur in the same part of the
rotation-magnetic confinement diagram (see Fig. 3 in Paper I), and H$\alpha$
emission EW and radio luminosity are closely correlated (see Fig. 6 in Paper
I). Since H$\alpha$ emission is regulated by CBO, this suggests that the same
may be true of gyrosynchrotron emission. In the following we develop a
theoretical basis for this connection, which we then compare to the empirical
regression analyses and measured radio luminosities.

##  3 CBO-driven Magnetic Reconnection

###  3.1 Rotational spindown

The rotational energy of a star with moment of inertia $I$ and rotational
frequency $\Omega$ is given by

$E_{{\rm rot}}=\frac{1}{2}I\Omega^{2}\,.$ (7)

If we assume a fixed moment of inertia, the release of rotational energy
associated with a spindown $-d\Omega/dt\equiv-{\dot{\Omega}}$ is

$L_{{\rm rot}}=I\Omega{\dot{\Omega}}\,.$ (8)

For a magnetized star with a wind of mass loss rate ${\dot{M}}$, Weber & Davis
(1967) argued that the loss of the star’s angular momentum $J=I\Omega$ scales
as

${\dot{J}}=I{\dot{\Omega}}={\dot{M}}\Omega R_{\rm A}^{2}\,,$ (9)

which gives the associated release of rotational luminosity the scaling,

$L_{{\rm rot}}={\dot{M}}\Omega^{2}R_{\rm A}^{2}\,.$ (10)

For a star with an equatorial field of strength $B_{\rm eq}$ at the stellar
surface radius $R_{\ast}$, ud-Doula & Owocki (2002) and ud-Doula et al. (2008)
showed that the Alfvén radius depends on the dimensionless wind magnetic
confinement parameter $\eta_{\ast}$ (Eqn. 1). Specifically, for a magnetic
multipole of order $p$ (=1, 2 for monopole, dipole, etc.), with radial scaling
as $B\sim r^{-(p+1)}$, $R_{\rm A}$ scales as

$\frac{R_{\rm A}}{R_{\ast}}=\eta_{\ast}^{1/2p}\,,$ (11)

which for the standard dipole case ($p=2$), reduces to the scaling given in
Eqn. (2).

The Weber & Davis (1967) analysis treated the simple case of a pure radial
field from a spilt monopole, with $p=1$. But ud-Doula et al. (2008) showed a
base dipole field leads to a spindown that follows the Weber & Davis (1967)
scaling (Eqn. 9), where $R_{\rm A}$ is given by Eqn. 11 with a multipole index
set to the $p=2$ value for a dipole.

###  3.2 Breakout from centrifugal magnetospheres

The above wind-confinement scalings work well for wind-magnetic braking, which
operates through wind stress on open field lines, wherein the associated
Poynting flux carries away most of the angular momentum.

But for rapid rotators with a strong field, the magnetic trapping of the wind
into a centrifugal magnetosphere leads to some important differences for
scalings of the associated luminosity.

First, as discussed in the appendices of Townsend & Owocki (2005, see their
equation A7), for trapping and breakout from a CM, the wind speed $v_{\infty}$
in the usual wind confinement parameter $\eta_{\ast}$ is replaced with a
characteristic dynamical speed of the stellar gravity, which we take here to
be the surface orbital speed $v_{{\rm orb}}\equiv\sqrt{GM_{\ast}/R_{\ast}}$
(since this is used in the definition of $W$ and thus $R_{\rm K}$), giving the
centrifugal magnetic confinement parameter

$\eta_{\rm c}\equiv\frac{B_{\rm d}^{2}R_{\ast}^{2}}{{\dot{M}}v_{{\rm
orb}}}\,.$ (12)

A second difference stems from the fact that, even for an initially dipolar
field, the rotational stress of material trapped in the CM has the effect of
stretching the field outwards, thus weakening its radial drop off, and so
reducing the effective multipole index to $p<2$.

Finally, this stretching ultimately leads to centrifugal breakout (CBO)
events, with associated release of energy via magnetic reconnection. In
general the overall total luminosity available from CBO events should follow a
general scaling analogous to that for $L_{{\rm rot}}$,

$\boxed{L_{\rm CBO}\approx{\dot{M}}\Omega^{2}R_{\ast}^{2}\eta_{\rm
c}^{1/p}}\,.$ (13)

####  3.2.1 Split monopole case

As a first example, consider the limit in which field lines are completely
opened by the wind ram pressure into a split monopole, with $p=1$, which gives

$\displaystyle L_{\rm CBO}(p=1)$ $\displaystyle\equiv$
$\displaystyle{\dot{M}}\Omega^{2}R_{\ast}^{2}\eta_{\rm c}$ (14)
$\displaystyle=$ $\displaystyle\frac{\Omega^{2}R_{\ast}^{4}B_{\rm
d}^{2}}{v_{\rm orb}}$ $\displaystyle=$ $\displaystyle W\Omega
R_{\ast}^{3}B_{\rm d}^{2}\,,$

where $W$ is the critical rotation fraction (Eqn. 3). Note that the second
equality recovers the empirical scaling $L_{\rm rad}\propto
B^{2}R_{\ast}^{4}/P_{\rm rot}^{2}$ found by Leto et al. (2021) and verified in
Paper I.

Remarkably, note also that in this monopole field case the dependence on wind
feeding rate ${\dot{M}}$ has canceled. Dimensionally, the scaling now is as if
the total magnetic energy over a volume set by $R_{\ast}^{3}$ is being tapped
on a rotational timescale.

An alternative physical interpretation is that the field acts more like a
conduit, trapping mass in a CM, with total rotational energy tapped on a
breakout timescale, set in this monopole case by the orbital timescale.

####  3.2.2 Dipole case

More generally, this breakout luminosity depends on the wind feeding rate.

In particular, for the pure dipole scaling with $p=2$, we find

$\displaystyle L_{\rm CBO}(p=2)$ $\displaystyle=$
$\displaystyle{\dot{M}}\Omega^{2}R_{\ast}^{2}\eta_{\rm c}^{1/2}$ (15)
$\displaystyle=$ $\displaystyle\frac{L_{\rm CBO}(p=1)}{\sqrt{\eta_{\rm
c}}}\,.$

This has $L_{\rm CBO}\sim\sqrt{\dot{M}}$, with a weaker, linear scaling with
$B_{\rm d}$.

In general, empirical evaluation of $L_{\rm CBO}$ thus requires evaluation of
the wind feeding rate ${\dot{M}}$, where we have used the same CAK mass-loss
rates as adopted in Paper I.

In the applications below, we consider multipole indices $1<p<2$, intermediate
between these monopole and dipole limits.

###  3.3 Application to radio emission

Figure 2: Radio luminosity $L_{\rm rad}$ as a function of breakout luminosity
$L_{\rm CBO}$ for the split monopole case. The thick black line indicates
$L_{\rm rad}=L_{\rm CBO}$; the solid red line shows the same line shifted by
the mean difference. The dashed green line shows the regression. Red circles
indicate stars with detected emission; blue squares, stars without emission.
Green circles indicate HD 171247 and HD 64740.

Let us next consider how well such breakout scalings for rotational luminosity
correlate with observed radio luminosities, $L_{\rm rad}$. Noting that the
dimensional scaling of breakout luminosity is set by the $p=1$ case, it is
convenient to cast the general scaling in the form

$L_{\rm CBO}(p)=L_{\rm CBO}(p=1)\eta_{\rm c}^{-1+1/p}=L_{\rm
CBO}(p=1)\eta_{\rm c}^{q}\,,$ (16)

by which we see an inferred empirical exponent $q$ in $\eta_{\rm c}$ implies
an effective multipole exponent $p=1/(1+q)$.

Let us first examine how well this basic, dimensional scaling of the monopole
model, with $p=1$ and so $q=0$, fits the observed radio emission. Fig. 2 shows
the observed radio luminosity $L_{\rm rad}$ as a function of $L_{\rm CBO}$ for
the monopole case (Eqn. 14, i.e. with no dependence on $\dot{M}$). The thick
black line shows $L_{\rm rad}=L_{\rm CBO}$, while the thick red line shows the
same relationship shifted by about 8 dex, the mean difference between $L_{\rm
rad}$ and $L_{\rm CBO}$ for radio-bright stars for the monopole scaling. This
line is almost indistinguishable from a regression of $L_{\rm rad}$ vs.
$L_{\rm CBO}$ for radio-bright stars. The relationship yields a correlation
coefficient $r=0.87$, and separates radio-bright from radio-dim stars with a
K-S probability of about $10^{-8}$. Further, there are very few radio-dim
stars with scaled breakout luminosities greater than the upper limits on their
radio luminosities, i.e. to the right of the red line; those radio-dim stars
that are to the right of the line, are very close to it.

Figure 3: Ratio between $L_{\rm rad}$ and $L_{\rm CBO}$ as a function of
bolometric luminosity (left) and $\cos{\beta}$ (right).

Eqn. 14 does not yield quite as high of a correlation coefficient as the
purely empirical scaling in Paper I. To see if there is some dependence on the
mass-loss rate, the left panel of Fig. 3 shows the residual radio luminosity
after subtraction of the monopole $L_{\rm CBO}$ as a function of bolometric
luminosity. There is only a weak dependence on $L_{\rm bol}$, with a
correlation coefficient $r=0.19$ and a slope $b=0.16$.

Another factor that may affect radio luminosity is the obliquity angle $\beta$
of the magnetic dipole axis from the rotational axis. Indeed, in the empirical
regression analysis in Paper I the factor $f_{\beta}=(1+\cos{\beta})/2$ was
found to improve the correlation. The plasma distribution in the CM is a
strong function of $\beta$, since the densest material accumulates at $R_{\rm
K}$ at the intersections of the magnetic and rotational equatorial planes
(Townsend & Owocki, 2005). For the special case of an aligned rotator
($\beta=0^{\circ}$) this will result in plasma being evenly distributed around
$R_{\rm K}$. With increasing $\beta$ the plasma distribution becomes
increasingly concentrated at the two intersection points, leading to a warped
disk that eventually becomes two distinct clouds. Therefore, the mass confined
within the CM will be a maximum for $\beta=0^{\circ}$ and a minimum for
$\beta=90^{\circ}$. If reconnection in the CM is the source of the high-energy
electrons that populate the radio magnetosphere, we would then naturally
expect that radio luminosity should decrease with increasing $\beta$. The
right panel of Fig. 3 shows the residual radio luminosity as a function of
$\cos{\beta}$, and demonstrates that radio luminosity in fact does increase
with decreasing $\beta$; in fact, the relationship is much stronger than for
$\log{L_{\rm bol}}$, with $r=0.49$ and $b=0.64$.

Figure 4: As Fig. 2, with $L_{\rm CBO}$ modified to account for the residual
dependence on bolometric luminosity (i.e. the surface mass-flux from the wind)
and the tilt angle of the magnetic dipole (Fig. 3).

Fig. 4 replicates Fig. 2, with the difference that corrections for $\dot{M}$
and $\beta$ are accounted for. Following Eqn. 16, $\dot{M}$ dependence was
determined by scaling Eqn. 14 with $\eta_{\rm c}^{q}$. A purely empirical
correction for $\beta$ was adopted as $f_{\beta}^{x}=((1+\cos{\beta})/2)^{x}$,
such that $f_{\beta}(\beta=0^{\circ})=1$ and $f_{\beta}(\beta=90^{\circ})\neq
0$. By minimizing the residuals, the best-fit exponents are $q=-0.09$ and
$x=2$. The former exponent corresponds to $p=1.1$, implying only a very slight
departure from the monopole scaling. The latter indicates an increase in
$L_{\rm rad}$ by a factor of 4 as $\beta$ decreases from $90^{\circ}$ to
$0^{\circ}$. As can be seen in Fig. 4, these corrections lead to a tighter
correlation ($r=0.92$) and a somewhat reduced ratio between $L_{\rm CBO}$ and
$L_{\rm rad}$ to around 7 dex.

####  3.3.1 Emission threshold

Figure 5: Ratio between surface magnetic field strength and the threshold
magnetic field strength necessary to achieve a flux density of 1 mJy, as a
function of bolometric luminosity.

In their development of a breakout scaling relationship for H$\alpha$ emission
from CMs, Owocki et al. (2020) defined a threshold magnetic field strength
$B_{\rm K1}$ as the strength of the magnetic field at $R_{\rm K}$ necessary to
confine a sufficient quantity of plasma at $R_{\rm K}$ for the optical depth
to reach unity. They demonstrated that all magnetic early B-type stars with
$B_{\rm K}/B_{\rm K1}>1$ are H$\alpha$-bright, while all stars with $B_{\rm
K}<B_{\rm K1}$ do not display H$\alpha$ emission. By solving Eqn. 14 for
$B_{\rm d}$ we can derive a similar threshold value for the radio luminosity:

$B_{\rm thresh}=\left(\frac{\epsilon L_{\rm CBO}P_{\rm rot}}{2\pi
R_{\ast}^{3}W}\right)^{1/2},$ (17)

where $\epsilon\sim 10^{-8}$ is an efficiency scaling determined from the
empirical ratio between $L_{\rm CBO}$ and $L_{\rm rad}$, and additional
dependence on $\beta$ and $\eta_{\rm c}$ is implic itly ignored. Since what is
actually observed is a flux density $F_{\rm rad}$ rather than a luminosity
$L_{\rm rad}\propto F_{\rm rad}d^{2}$, Eqn. 17 is necessarily a function of
distance $d$. Amongst the radio-bright stars, the median flux density
uncertainty is 0.1 mJy, while the median significance of a detection is around
10$\sigma$ i.e. 1 mJy. We therefore take the flux density detectability
threshold for the sample as 1 mJy and solve Eqn. 17 accordingly to obtain
$B_{1}$ (i.e. the surface magnetic field necessary to generate 1 mJy of flux
density at the star’s distance). The results are shown in Fig. 5.

As expected, all radio-bright stars have $\log{B_{\rm d}/B_{1}}\gtrsim 0$. The
relationship for non-detected stars is not as clean as for the similar plot
for H$\alpha$ shown by Owocki et al. (2020), as there are a large number of
stars with surface magnetic fields above this threshold. However, the radio
observations comprising this sample, having been obtained at a variety of
observatories with different capabilities over a span of over 30 years, are
quite heterogeneous, with a wide range of upper limits, and many of the non-
detected stars in this regime have upper limits comparable to 1 mJy. Further,
generally only a single snapshot at one frequency is available, and it is
possible that they were observed at inopportune rotational phases. These stars
should certainly be reobserved with modern facilities.

The dashed line in Fig. 5 shows the theoretical detection limit for radio
telescopes such as the upcoming Square Kilometre Array able to achieve $\mu$Jy
precision, under the assumption that a 10$\sigma$ detection (i.e. 10 $\mu$Jy)
is necessary for the star’s radio emission to be securely detected. As can be
seen, such facilities can at least double the number of stars with measured
gyrosynchrotron emission. This is especially true when stars that have not yet
been observed in the radio are included: the green triangles in Fig. 5 show
those stars from the samples studied by Aurière et al. (2007), Sikora et al.
(2019a, b), and Shultz et al. (2019b) without radio observations, essentially
all of which are expected to have radio flux densities es above 10 $\mu$Jy and
about half of which should have flux densites above 1 mJy.

####  3.3.2 How CBO reconnection can lead to radio emission

Breakout events are accompanied by centrifugally driven reconnection of
magnetic fields that have been stretched outward by rotational stress acting
against the magnetic tension of the initially closed loops. As these loops
reconnect, the associated release of magnetic energy can strongly heat the
ejected plasma. Some fraction of this reconnection energy can accelerate both
ions and electrons to highly super-thermal energies, with some of these
particles becoming trapped into gyration along closed magnetic field loops
near the reconnection site. The associated gyrosynchrotron emission of the
much lighter electrons can then produce the observed radio emission.

The basic scenario of electron acceleration in reconnection events, followed
by gyrosynchrotron emission along magnetic loops, is indeed already a central
component of the model for radio emission (Trigilio et al., 2004). However,
this model is based on wind-driven reconnection, with no inclusion for any
role of stellar rotation. As such, the available reconnection luminosity is
expected to scale with the wind kinetic energy $L_{\rm
wind}={\dot{M}}v_{\infty}^{2}/2$. By comparison, the rotational luminosity for
a multipole exponent $p$ is larger by a factor

$\displaystyle\frac{L_{\rm CBO}(p)}{L_{\rm wind}}$ $\displaystyle=$
$\displaystyle 2\eta_{\rm c}^{1/p}\,\left(\frac{v_{\rm
rot}}{v_{\infty}}\right)^{2}$ (18) $\displaystyle=$ $\displaystyle\eta_{\rm
c}^{1/p}\,W^{2}\left(\frac{v_{\rm esc}}{v_{\infty}}\right)^{2}$
$\displaystyle\approx$ $\displaystyle\frac{\eta_{\rm c}^{1/p}\,W^{2}}{9}\,,$

where the last equality stems from the standard result that the stellar wind
speed scales with the escape speed as $v_{\infty}\approx 3v_{\rm esc}$. For
typical values for B-star magnetospheres with $\eta_{\rm c}\approx 10^{6}$,
$W\approx 1/2$ (Petit et al., 2013) and $p=4/3$, we find $L_{\rm
CBO}(p=4/3)/L_{\rm wind}\approx 880$ (using the empirically derived value of
$p=1.1$ yields an even greater ratio of almost 8000). For these $W$ and $p$
values, the ratio is greater than unity for even moderate confinement values
$\eta_{\rm c}>120$.

Regardless of the relative values, a central empirical result here is the
finding that $L_{\rm rad}$ has a clear scaling with rotation frequency as
$\Omega^{2}$ and with surface field as $B_{\rm d}^{2/p}$, dependences which
are entirely missing from $L_{\rm wind}$. Indeed, we find that $\log(L_{\rm
rad})$ shows only weak correlation with $L_{\rm wind}$, with $r=0.3$. This
strongly disfavors the wind-driven reconnection model proposed by Trigilio et
al. (2004); but it is consistent with the scenario proposed here that
centrifugal-breakout reconnection provides the underlying energy that leads to
the radio luminosity through gyrosynchrotron emission.

While we have cast available energy in terms of loss of the star’s rotational
energy, one has to be careful not to take this too literally. Most (70+%) of
the angular momentum loss in spindown is through magnetic field Poynting
stresses. But the CBO material that leads to reconnection should share the
same basic Weber-Davis scaling with $R_{\rm A}$, and it is that component that
this scenario associates with the reconnection and the resulting electron
acceleration and radio emission.

##  4 Discussion

###  4.1 Comparison with alternative theoretical interpretations

The empirical scaling relationship discovered by Leto et al. (2021) and
confirmed in paper I, $L_{\rm rad}\propto B_{\rm d}^{2}R_{\ast}^{4}/P_{\rm
rot}^{2}=(\Phi/P_{\rm rot})^{2}$, is explained above as a consequence of
electron acceleration via centrifugal breakout. However, Leto et al. pointed
out that $\Phi/P_{\rm rot}$ has the physical dimension of an electromotive
force $\mathcal{E}$ (EMF), which they speculated may be suggestive of an
underlying theoretical mechanism. In this section we examine gyrosynchrotron
emission from this standpoint. We perform a theoretical analysis to test if
the physical conditions able to sustain large scale electric currents within
the stellar magnetosphere can be verified.

This empirical association by Leto et al. (2021) of radio emission with the
voltage of an EMF also stands in contrast with the previous theoretical model
by Trigilio et al. (2004), which associates the acceleration of radio-emitting
non-thermal electrons with the wind-induced current sheet that forms in the
middle magnetosphere. However, Leto et al. (2021) conclusively demonstrated
that the wind does not provide sufficient power to the middle magnetosphere to
drive the observed levels of radio emission.

The highly ionized plasma in these magnetospheres implies a very high
conductivity, and so currents can form even with a vanishingly small EMF.
Instead, the current density $J$ is set by Ampere’s law as a result of a curl
induced in a stressed field,

$J=\frac{c}{4\pi}\,\nabla\times B\,.$ (19)

Even in a non-rotating wind-fed magnetosphere, large-scale stressing of the
magnetic field by the wind ram pressure forces outlying closed loops to open,
with a Y-type neutral point at the top of the last closed loop; above this
there develops a split monopole, with a current sheet separating field lines
of opposite polarity. But unless there are instabilities or induced
variability, this current sheet does not by itself lead to energy dissipation
that can heat the plasma or accelerate electrons. This, together with the lack
of observed radio emission from stars with slow rotation, thus strongly
disfavors the Trigilio et al. (2004) model based on the wind-induced current
sheet.

While the Leto et al. (2021) empirical association of observed radio emission
with an EMF is interesting and insightful, there are some challenges to using
this as a basis for a self-consistent theoretical model. The general
principles regarding current vs. EMF scenarios can be well illustrated by a
simple circuit model, using an Ohm’s law $I=\mathcal{E}/\mathcal{R}$ to
related current $I$ and EMF $\mathcal{E}$ through a resistance $\mathcal{R}$.
The associated dissipated power, or luminosity, scales as

$L_{\rm
emf}=I\mathcal{E}=I^{2}{\mathcal{R}}=\frac{\mathcal{E}^{2}}{\mathcal{R}}\,.$
(20)

If one fixes the current $I$ (as induced by the globally imposed magnetic
curl), then the second equality shows that in the limit of vanishing
resistivity $\mathcal{R}\rightarrow 0$, the luminosity also vanishes; this
underlies one fundamental issue with the current-sheet model advocated by
Trigilio et al. (2004).

To understand the ramifications of the last equality of Eqns. 20 within the
context of the empirical association of the gyrosynchrotron scaling law with
an EMF, let us return to consideration of plasma conditions in such
magnetospheres. In the notation of the present paper, the EMF can be written
as $\mathcal{E}=B_{\ast}R_{\ast}^{2}\Omega/c$, where the speed of light $c$
comes in from the CGS form for the induction equation. In terms of a plasma
resistivity $\rho$ (with units of time), the circuit resistance scales with
resistivity times a length over area, which in this context gives
$\mathcal{R}\approx\rho R_{\ast}/R_{\ast}^{2}=\rho/R_{\ast}$. Thus Eqn. 20
becomes

$\displaystyle L_{\rm emf}$ $\displaystyle=$
$\displaystyle\frac{B_{\ast}^{2}R_{\ast}^{5}\Omega^{2}}{\rho c^{2}}$ (21)
$\displaystyle=$
$\displaystyle(B_{\ast}^{2}R_{\ast}^{3}\Omega)\,\left[\frac{v_{\rm
rot}R_{\ast}}{\rho c^{2}}\right]\,,$ (22)

where in the latter equality, $v_{\rm rot}=\Omega R_{\ast}$ is the surface
rotation speed at the stellar equator. Here the term in parenthesis separates
out the dimensional luminosity, while the dimensionless ratio in square
brackets can be identified as a magneto-rotational Reynold’s number,

$Re_{\rm mr}\equiv\frac{v_{\rm rot}R_{\ast}}{\rho c^{2}}\,.$ (23)

Rather remarkably, Eqn. 22 has a form very similar to that derived above (cf.
Eqn. 14) for the monopole ($p=1$) CBO model. However an important, indeed
crucial difference is that the CBO rotation scaling is relative to the near-
surface orbital speed, $v_{\rm orb}=\sqrt{GM_{\ast}/R_{\ast}}$. As such in
this CBO model the associated dimensional rotation parameter $W=v_{\rm
rot}/v_{\rm orb}$ is always less than unity.

Indeed, eqn. (21) is very similar to the scaling invoked by Hill (2001, see
their Eqn. 4) to model auroral emission from Jupiter. In this case, the
magnetospheric EMF accelerates ions and electrons to high-energy, which upon
penetrating into the underlying Jovian atmosphere is dissipated through the
low atmospheric conductance $\Sigma_{\rm J}$ (corresponding to high
resistivity $\rho$), resulting in heating and associated thermal bremstrahlung
to give auroral emission.

In contrast, because of the typically very low resistivity of the ionized
plasma in hot-star magnetospheres, the associated dimensionless Reynolds
number is expected to be very large. The associated dissipation luminosity
(Eqn. 22) in this EMF scenario would thus be enormous, leading in effect to a
“short circuit” that would quickly draw down the available pool of magnetic
energy.

In principle, a theoretical model grounded in the EMF could invoke a stronger
resistivity in some local dissipation layer, which would enter directly into
the predicted scalings for the generated luminosity. But it is unclear how
this small-scale dissipation could be reconciled with the large-scale EMF that
is taken to scale with the stellar radius, and how such a dissipation could
remain fixed over the range of stellar and magnetospheric parameters, in order
to preserve the inferred empirical scaling of the observed radio luminosity
with the global EMF.

These difficulties with an association of gyrosynchrotron scaling with EMF,
and with the current sheet model, stand in contrast to some key advantages of
the CBO mechanism proposed here.

First, this CBO model specifies a more modest magnetic dissipation rate, set
by the base dimensional rate $B_{\ast}^{2}R_{\ast}^{3}\Omega$ reduced by the
rotation factor $W<1$, instead of the enormous $Re_{\rm rm}\sim 10^{12}$
enhancement of an EMF mechanism. This CBO dissipation can be quite readily
replenished over time by the centrifugal stretching of closed magnetic field
lines by the constant addition of mass from the stellar wind. As such, the
ultimate source of energy thus comes not from the field – which acts merely as
a conduit – but from the star’s rotational energy.

Second, these eventual centrifugal breakout events lead naturally and
inevitably to magnetic reconnection. This thus preserves the longstanding
notion (Trigilio et al., 2004) that such reconnection provides the basic
mechanism to accelerate electrons to high energies, whereupon the gyration
along the remaining field lines connecting back to the star results in the
gyrosynchrotron emission of the observed radio luminosity.

Third, and perhaps most significantly, instead of the previous notion
(Trigilio et al., 2004) that this reconnection is driven by the stellar wind –
with no consideration of any role for stellar rotation – our model for CBO-
driven reconnection puts rotation at the heart of the process, and so yields a
scaling for luminosity that matches the strong dependence on rotation rate, as
well as on magnetic field energy. Indeed, while the wind can certainly open
the magnetic field and lead to the formation of a current sheet, this does not
itself provide a power source, but merely results in a slower radial decline
of the magnetic field strength as compared to that of a dipole. By contrast,
CBO provides a clear power source for the acceleration of electrons to high
energies.

Thus, although only a small fraction of the breakout luminosity $L_{\rm CBO}$
ends up as radio luminosity, with an inferred effective efficiency
$\epsilon\approx 10^{-8}$, the strong correlation between observed and
predicted scalings provides strong empirical support for such a CBO model.

###  4.2 Energy source - magnetic or rotational?

The energy term in Eqn. 14 is $B^{2}R^{3}$, and it would therefore be natural
to assume that the magnetic field is the energy source powering radio
emission. However, as suggested in § 3.2.1, this is probably not the case. The
mean magnetic energy $E_{\rm mag}$ amongst the radio-bright stars is about
$10^{42}$ erg, whereas the mean rotational kinetic energy $E_{\rm rot}$ in the
same sub-sample is about $10^{47}$ erg, i.e. the star’s rotation is a vastly
greater energy reservoir. Indeed $E_{\rm mag}>E_{\rm rot}$ for only 3 stars
(HD 46328, HD 165474, and HD 187474), all of which have $P_{\rm rot}\sim$
years (and none of which are, of course, detected at radio frequencies).

An additional consideration is that, if the magnetic field were the energy
source, radio emission should over time draw down the magnetic energy of the
star. The peak radio luminosity is around $10^{29}~{}{\rm erg~{}s^{-1}}$,
implying that the magnetic energy of the most radio-luminous stars would be
consumed in about $10^{13}~{}{\rm s}\sim 0.3~{}{\rm kyr}$. To the contrary,
fossil magnetic fields are stable throughout a star’s main sequence lifetime.
For Ap/Bp stars below about 4 M⊙, the decline in surface magnetic field
strength is entirely consistent with flux conservation in an expanding stellar
atmosphere (e.g. Kochukhov & Bagnulo, 2006; Landstreet et al., 2007; Sikora et
al., 2019b), while for more massive stars there is an additional, gradual
decay of flux (e.g. Landstreet et al., 2007, 2008; Fossati et al., 2016;
Shultz et al., 2019b) that is however, much longer than the abrupt field decay
timescale that would be implied if the breakout luminosity was powered by the
magnetic field. Furthermore, the most plausible mechanism for flux decay is
found in small-scale convective dynamos formed in the opacity-bump He and Fe
convection zones inside the radiative envelope (e.g. MacDonald & Petit, 2019;
Jermyn & Cantiello, 2020), which naturally explains why flux does not decay in
A-type stars (which lack these convection zones), and why flux apparently
decays more slowly for the strongest magnetic fields (Shultz et al., 2019b)
since strong fields inhibit convection (Sundqvist et al., 2013; MacDonald &
Petit, 2019).

In contrast to the magnetic field, which decays slowly or not at all, magnetic
braking is quite abrupt (Shultz et al., 2019b; Keszthelyi et al., 2020),
making the larger rotational energy reservoirs of rapidly rotating stars a
more far more plausible power source. Quantitatively, for the most radio-
luminous stars in the sample it would take about 30 Myr for the energy
radiated by gyrosynchrotron emission to remove the total rotational energy of
the star. For stars with masses above 5 M⊙ (the mass range of the brightest
radio emitters), this is comparable to or greater than the main sequence
lifetime.

It therefore seems that the magnetic field cannot serve as the energy source,
but rather acts as a conduit for the extraction of rotational energy and its
conversion into gyrosynchrotron emission. The magnetic energy lost in breakout
events is immediately replenished as mass is injected into the CM by the wind,
with the ion-loaded magnetic field then stretching under the centrifugal
stress acting on the co-rotating plasma.

###  4.3 The case of Jupiter

Leto et al. (2021) showed that the scaling relationship for the non-thermal
radio emission from dipole-like rotating magnetospheres also fits the radio
luminosity of Jupiter111This radio emission arises within Jovian
magnetosphere, and so is distinct from the optical auroral emission discussed
above, which arises from interactions in the upper Jovian atmosphere.,
suggesting an underlying similarity in the physics driving gyrosynchrotron
emission from giant planets and magnetic hot stars. Adopting the same
parameters as used by Leto et al. ($B_{\rm eq}=4$ G, $P_{\rm rot}=0.41$ d,
$M_{\rm J}=1.9\times 10^{27}$ kg, and $R_{\rm J}=7.1\times 10^{5}$ km) gives
$W=0.3$. The breakout luminosity is then $\log{L_{\rm
CBO}/L_{\odot}}\sim-15.9$ or, at 1 cm, $L_{\nu}\sim 10^{7}{\rm
erg~{}s^{-1}~{}Hz^{-1}}$, translating to an expected flux density of around 40
Jy at a distance of 4 AU. This is about an order magnitude higher than the
observed radio luminosity of Jupiter (de Pater & Dunn, 2003; de Pater et al.,
2003). However, it is worth noting that in the extrapolation shown by Leto et
al., Jupiter’s EMF of 376 MV is near the lower envelope of the range of
uncertainty inferred from hot stars, i.e. Jupiter is somewhat less luminous
than predicted by a direct extrapolation of the hot star scaling relationship.
Furthermore, 1 dex is at the upper range of the scatter about the $L_{\rm
CBO}$ relationship (see Figs. 2 and 4).

One possible explanation for Jupiter being less luminous than predicted is
that Jupiter’s primary ion source, the volcanic moon Io, is effectively a
point source offset from the centre of the Jovian magnetosphere. This is in
contrast to stellar winds, which feed the magnetosphere isotropically and
continuously from the centre. The result is that hot star magnetospheres are
relatively more populated, and there is therefore more material available for
the generation of gyrosynchrotron emission. Another potential issue is that in
the Jovian magnetosphere reconnection takes place in the magnetotail due to
stretching by the solar wind; its azimuthal extent will therefore be limited,
in analogy to the obliquity dependence found in stellar magnetospheres.
Exploring whether the approximate consistency between the Jovian and stellar
radio luminosities is indeed due to a similarity in the underlying physics, or
is merely coincidental, will require a detailed analysis that is outside the
scope of this paper.

###  4.4 A solution to the low-luminosity problem?

It is notable that magnetospheres are detectable in radio frequencies in stars
with CMs that are too small to be detectable in H$\alpha$. In addition to
being a more sensitive magnetospheric diagnostic, this may also suggest an
answer to the low-luminosity problem identified by Shultz et al. (2020) and
Owocki et al. (2020). While CBO matches all of the characteristics of
H$\alpha$ emission from CM host stars, emission disappears entirely for stars
with luminosities below about $\log{L_{\rm bol}/L_{\odot}}\sim 2.8$. This
could be either a consequence of a ‘leakage’ mechanism, operating in
conjunction with CBO to remove plasma via diffusion and/or drift across
magnetic field lines (Owocki & Cranmer, 2018), or due to the winds of low-
luminosity stars switching into a runaway metallic wind regime (Springmann &
Pauldrach, 1992; Babel, 1995; Owocki & Puls, 2002). In the former case the
leakage mechanism only becomes significant when $\dot{M}$ is low. In the
latter case, H$\alpha$ emission is not produced for the simple reason that the
wind does not contain H ions. Notably, the peculiar surface abundances of
magnetic stars may lead to enhanced mass-loss rates as compared to non-
magnetic, chemically normal stars (Krtička, 2014).

Since CBO apparently governs gyrosynchrotron emission, and is seen in stars
down to $\log{L_{\rm bol}/L_{\odot}}\sim 1.5$, the leakage scenario seems to
be ruled out as an explanation for the absence of H$\alpha$ emission. This
therefore points instead to runaway metallic winds. One possible complication
is that, as is apparent from the direct comparison of H$\alpha$ emission
equivalent widths to radio luminosities (see Fig. 6 in Paper I), stars without
H$\alpha$ so far are also relatively dim in the radio (at least for those
stars for which H$\alpha$ measurements have been obtained). These stars have
systematically lower values of $B_{\rm K}$ than have been found in more
luminous H$\alpha$-bright stars (Fig. 3 in Paper I). Thus, a crucial test will
be examination of both H$\alpha$ and radio for a star with a luminosity well
below 2.8, but $B_{\rm K}\sim 3$, i.e. it must be cool, very rapidly rotating
($P_{\rm rot}\sim 0.5$ d), and strongly magnetic $(B_{\rm d}\sim 10~{}{\rm
kG})$. So far no such stars are apparently known.

A further complication to the runaway wind hypothesis is provided by 36 Lyn, a
relatively cool ($T_{\rm eff}\sim 13$ kK), radio-bright star which, while it
does not show H$\alpha$ emission, does display eclipses in H$\alpha$ (Smith et
al., 2006) and therefore must have H inside its magnetosphere which,
presumably, originated in the stellar wind. Why no other star in 36 Lyn’s
$T_{\rm eff}$ range should show evidence of a similar phenomenon is not
currently understood, although its peculiar magnetosphere may be related to
the remarkably high toroidal component of its magnetic field in comparison to
other magnetic stars, in which the toroidal component is generally quite weak
(Oksala et al., 2018; Kochukhov et al., 2019). Alternatively, this may be due
to a simple selection effect: 36 Lyn’s eclipses are only detectable for about
10% of its rotational cycle, and eclipse absorption would be broader and
shallower for more rapidly rotating stars.

###  4.5 X-rays from CBO?

Figure 6: For the standard case of half-critical rotation ($W=0.5$),
comparison of the X-ray luminosities from the XADM model (dashed) of ud-Doula
et al. (2014) with CBO models of various indices $p$ (solid), plotted vs.
associated magnetic confinement parameter, with all luminosities normalized by
the kinetic energy luminosity of the stellar wind, $L_{\rm
wind}={\dot{M}}v_{\infty}^{2}/2$. The horizontal dotted line represents the
asymptotic luminosity for XADM in the strong-confinement limit. Note that, for
this $W=0.5$ case, the CBO luminosities are significantly enhanced over that
from the XADM model. Results for other rotations can be readily determined by
scaling the CBO models with $W^{2}$.

The reconnection energy from CBO events might also be an important for the
X-rays observed from magnetic stars.

For slow rotators with only a dynamical magnetosphere, and no centrifugal
magnetosphere component, the observed X-rays follow quite closely the scaling
predicted by the “X-rays from Analytic Dynamical Magnetosphere” (XADM) model
developed by ud-Doula et al. (2014), as shown in Figure 6 of Nazé et al.
(2014). With a 10% scaling adjustment to account for the X-ray emission duty
cycle seen in MHD simulations, the overall agreement between observed and
predicted X-ray luminosities is quite remarkable for such DM stars (denoted by
open circles and triangles), spanning more than four orders of magnitude in
X-ray luminosity!

However there are several stars with observed X-ray luminosities well above
(by 1-2 orders of magnitude) the 10%-XADM scaling; all are CM stars. These are
the very stars that the analysis here predicts to have CBO reconnection events
that could power extra X-ray emission, and so supplant the X-rays from wind
confinement shocks predicted in the XADM analysis.

To lay a basis to examine whether CBO reconnection X-rays might explain this
observed X-ray excess for CM stars, Figure 6 compares the CBO vs. XADM
predicted scalings for X-ray luminosity, both normalized by the wind kinetic
energy luminosity $L_{\rm wind}={\dot{M}}V_{\infty}^{2}/2$, and plotted versus
their associated confinement parameter $\eta_{c}$ or $\eta_{\ast}$. The XADM
plot is based on eqn. (42) of ud-Doula et al. (2014), with velocity exponent
$\beta=1$. As seen from eqn. (18) here, the CBO scalings also depend on the
multipole exponent $p$, so the various curves show results for various $p$, as
given by the legend. Also, the values shown are for a fixed, fiducial value
for the critical rotation fraction $W=1/2$, but the results for other
rotations can be readily determined by scaling with $W^{2}$.

Note that the CBO scalings increase as $\eta_{c}^{1/p}$, while the XADM
scaling saturates at large $\eta_{\ast}$. For the chosen default rotation
$W=0.5$, the CBO scalings are generally above those for the XADM, but this
will change for lower $W$.

To test this possibility that CBO plays a role in augmenting X-ray emission, a
next step should be to test whether the observed X-rays from CM stars follow
the CBO scaling with $\eta^{1/p}W^{2}$, as given by eqn. (18) when scaled to
$L_{\rm wind}$, or more generally by eqns. (12) and (13).

##  5 Summary and Future Outlook

The radio luminosities of the early-type magnetic stars were empirically found
to be related to the stellar magnetic flux rate (Leto et al., 2021, Paper I).
Leto et al. (2021) did not provide a definitive physical explanation regarding
the origin of the non-thermal electrons. To provide the theoretical support
for explaining how non-thermal electrons originate, in this paper we have
extended the centrifugal breakout model that successfully predicts the
H$\alpha$ emission properties of stars with centrifugal magnetospheres (Shultz
et al., 2020; Owocki et al., 2020), deriving a breakout luminosity $L_{\rm
CBO}\propto(B^{2}R_{\ast}^{3}/P_{\rm rot})W$, where the first term in brackets
has natural units of luminosity, and the dimensionless critical rotation
parameter $W$ is an order-unity correction that includes the additional
$R_{\ast}$ and $P_{\rm rot}$ dependence. The radio luminosity is then $L_{\rm
rad}=\epsilon L_{\rm CBO}$, where $\epsilon\sim 10^{-8}$ is an efficiency
factor. Crucially, there is a nearly 1:1 correspondence between $L_{\rm rad}$
and $\epsilon L_{\rm CBO}$.

The basic scaling relationship is appropriate for a split monopole.
Generalization to higher-order multipoles is accomplished with a correction
$\eta_{\rm c}^{1/p}$, where $\eta_{c}$ is the centrifugal magnetic confinement
parameter and $p$ is the multipolar order (1 for a monopole, 2 for a dipole,
etc.). The small residual dependence of radio luminosity on bolometric
luminosity is removed by adopting $p\sim 1.1$, i.e. a nearly monopolar field.
The minimal residual dependence on $L_{\rm bol}$ (which in line-driven wind
theory sets the mass loss rate through a scaling ${\dot{M}}\sim L_{\rm
bol}^{1.6}$) confirms that the radio magnetosphere is nearly independent of
the mass-loss rate. However, we find that there is a stronger dependence of
the residuals on the obliquity $\beta$ of the magnetic axis with respect to
the rotation axis, with $L_{\rm rad}$ increasing by about a factor of 4 from
$\beta=90^{\circ}$ to $0^{\circ}$. This is consistent with expectations from
the rigidly rotating magnetosphere model that the amount of plasma trapped in
a centrifugal magnetosphere is a strong function of $\beta$ (Townsend &
Owocki, 2005), since with less plasma in the CM, there will be fewer electrons
available to populate the radio magnetosphere.

While radio emission and H$\alpha$ emission are explained by a unifying
mechanism, they probe different parts of the magnetosphere as well as
different parts of the centrifugal breakout process. H$\alpha$ emission probes
the cool plasma trapped in the CM, which has not yet been removed by breakout.
During a breakout event, some of the energy released by magnetic reconnection
accelerates electrons to relativistic velocities, which then return to the
star, emitting gyrosynchrotron radiation as they spiral around magnetic field
lines. Following the result reported in this paper, we explain the radiation
belt model proposed by Leto et al. (2021) to be the magnetic shell connected
to the centrifugal breakout region close to the magnetic equator. This largely
preserves the Trigilio et al. (2004) model, with the primary difference being
the mechanism of electron acceleration.

Overall, the results here provide a revised foundation on which to build a
detailed theoretical model for how centrifugal-breakout reconnection leads to
acceleration of electrons and the associated radio gyrosynchrotron emission.
In particular, we might be able to quantify the level of reconnection heating
through MHD simulations, and how it scales with $W$, $\eta_{\ast}$, etc., as
has been done for other scalings like spindown.

Future theoretical work should focus on the details of the acceleration of the
electrons through reconnection, and their subsequent gyrosynchrotron emission
of polarized radio emission (and perhaps other observable spectral bands like
X-rays), with the specific aim to understand, and quantitatively reproduce,
the inferred emission efficiencies $\epsilon$. This work should also extend to
explore the connection with electron cyclotron maser (ECM) radio emission that
has been detected in many of the same stars showing gyrosynchrotron emission
(Das et al., 2021).

## Acknowledgments

MES acknowledges the financial support provided by the Annie Jump Cannon
Fellowship, supported by the University of Delaware and endowed by the Mount
Cuba Astronomical Observatory. AuD acknowledges support by NASA through
Chandra Award 26 number TM1-22001B issued by the Chandra X-ray Observatory 27
Center, which is operated by the Smithsonian Astrophysical Observatory for and
on behalf of NASA under contract NAS8-03060. PC and BD acknowledge support of
the Department of Atomic Energy, Government of India, under project no.
12-R&D-TFR-5.02-0700.

## Data Availability Statement

No new data were generated or analysed in support of this research. All data
referenced were part of the associated Paper I, Shultz et al. (2021).

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