# Winds and magnetospheres from stars and planets: similarities and
differences

Stan Owocki  Department of Physics & Astronomy, Bartol Research Institute,
University of Delaware, Newark, DE 19716 USA
email<EMAIL_ADDRESS>
(2022)

###### Abstract

Both stars and planets can lose mass through an expansive wind outflow, often
constrained or channeled by magnetic fields that form a surrounding
magnetosphere. The very strong winds of massive stars are understood to be
driven by line-scattering of the star’s radiative momentum, while in the Sun
and even lower-mass stars a much weaker mass loss arises from the thermal
expansion of a mechanically heated corona. In exoplanets around such low-mass
stars, the radiative heating and wind interaction can lead to thermal
expansion or mechanical ablation of their atmospheres. Stellar magnetospheres
result from the internal trapping of the wind outflow, while planetary
magnetospheres are typically shaped by the external impact from the star’s
wind. But in both cases the stressing can drive magnetic reconnection that
results in observable signatures such as X-ray flares and radio outbursts.
This review will aim to give an overview of the underlying physics of these
processes with emphasis on their similarities and distinctions for stars vs.
planets.

###### keywords:

Sun: solar wind; stars: early-type; stars: mass loss; stars: planetary systems

††volume: 370††journal: Winds of Stars and Exoplanets††editors: A. A. Vidotto,
L. Fossati & J. Vink, eds.

##  1 Introduction

Figure 1: Icons to represent analogies for processes inducing the three
different types of steady stellar wind outflow.

To set the stage for this symposium’s exploration of “The Winds from Stars and
Exoplanets”, I have been asked to review the similarities and differences in
the physical processes that drive wind outflows from stars vs. planets,
including the distinct roles that magnetic fields play in trapping and
diverting plasma flows in an associated magnetosphere.

As illlustrated in figure 1, one can identify three broad classes of stellar
wind:

  1. 1\.

the pressure-driven coronal wind of the Sun and other cool stars;

  2. 2\.

radiatively driven winds from OB stars;

  3. 3\.

the slow “overflow” mass loss from highly evolved giant stars.

For the last a key is the greatly reduced gravity, which allows even surface
convection or pulsations to eject outer layers to escape, somewhat like Roche-
lobe overflow in binary systems. The resulting mass loss can be irregular and
difficult to quantify, and since it has little overlap with winds from
planets, we set this aside to focus on the former two.

We first review (§2) the radiative driving of OB winds, which occurs through a
kind of line-driven “suction” effect. We then discuss (§3) how the winds of
the Sun and other cool stars are, in contrast, driven by the gas pressure
associated with hot corona, with mass loss representing now an escape valve
analogous to that of pressure cooker.

This provides a basis for discussion (§4) of planetary outflows, which are
likewise largely driven by gas-pressure expansion, now powered by the UV and
X-ray heating from the chromospheric and coronal emission from the underlying
cool star.

We conclude with a review (§5) of stellar magnetospheres, contrasting their
inside-out, internal filling by the stellar wind with the outside-in, external
stress imposed on planetary magnetospheres by the wind of their host star.

##  2  Radiatively driven winds from OB stars

###  2.1 Radiative acceleration and Eddington parameter

In hot stars with a high luminosity, the outward force from scattering of
stellar radiation can overcome gravity and so drive a stellar wind outflow.
For opacity $\kappa_{\nu}$ at a frequency $\nu$ with radiative flux $F_{\nu}$,
the total radiative acceleration depends on the frequency integral,

$g_{rad}=\int_{0}^{\infty}d\nu\,\frac{\kappa_{\nu}F_{\nu}}{c}\equiv\frac{{\bar{\kappa}}_{F}F}{c}\,$
(1)

where the last equality defines the flux-weighted mean opacity,
${\bar{\kappa}}_{F}$, with $F$ the bolometric flux.

In the idealized case of continuum scattering by free electrons,
${\bar{\kappa}}_{F}$ is just equal to the electron scattering opacity
$\kappa_{e}=(1+X)0.2=0.34\,{\rm cm}^{2}$/g, where the latter value applies to
a fully ionized gas with solar hydrogen mass fraction $X=0.72$. The ratio of
the associated radiative acceleration to gravity defines the classical
Eddington parameter,

$\Gamma_{e}\equiv\frac{\kappa_{e}F/c}{g}=\frac{\kappa_{e}L}{4\pi
GMc}=2.6\times 10^{-5}\frac{L/L_{\odot}}{M/M_{\odot}}\sim
0.26\left(\frac{M}{100\,M_{\odot}}\right)^{2}\,,$ (2)

wherein the inverse-square radial dependence of both the radiative flux
$F=L/4\pi r^{2}$ and gravity $g=GM/r^{2}$ cancels, showing this Eddington
parameter depends only on the ratio $L/M$ of luminosity to mass. The third
equality shows that $\Gamma_{e}$ is very small for stars like the Sun; but if
one applies the standard main-sequence mass-luminosity scaling $L\sim M^{3}$,
the last equality shows that massive stars can have Eddington parameters that
approach unity. This provides a basic rationale for the upper limit to stellar
mass, which is empirically found to be around $200M_{\odot}$, remarkably close
to the mass for which (2) gives $\Gamma_{e}\approx 1$. Stars that approach or
exceed this classical Eddington limit can have strong eruptive mass loss, as
thought to occur in eruptive Luminous Blue Variable stars like $\eta$ Carinae.

But since generally ${\bar{\kappa}}_{F}\gg\kappa_{e}$, even stars with
$\Gamma_{e}\ll 1$ can have a total $\Gamma>1$, and drive a more steady-state
stellar wind mass loss. Ignoring the small gas-pressure acceleration, the
associated steady-state acceleration has the scaling,

$v\frac{dv}{dr}=-\frac{GM}{r^{2}}+\frac{\kappa L}{4\pi
r^{2}}=(\Gamma-1)\frac{GM}{r^{2}}\,,$ (3)

where to simplify the notation, we have set ${\bar{\kappa}}_{F}=\kappa$. For
constant $\Gamma$, this can be trivially integrated to give the variation of
wind velocity with radius $r$,

$v(r)=v_{\infty}\,(1-R/r)^{1/2}~{}~{};~{}~{}v_{\infty}=v_{esc}\sqrt{\Gamma-1}\,,$
(4)

which shows that the wind terminal speed $v_{\infty}$ just scales with the
escape speed $v_{esc}\equiv\sqrt{2GM/R}$ from the wind initiation surface
radius $R$. We thus see that $\Gamma$ represents an effective anti-gravity,
which for the simple case $\Gamma=2$ gives a direct gravitational reversal,
with material flying away from the star, asymptotically reaching the escape
speed.

Moreover, if we multiply (3) by $4\pi\rho r^{2}dr$, and note the standard
definition of mass loss rate ${\dot{M}}\equiv 4\rho vr^{2}$, we find upon
integration a relationship between the wind momentum ${\dot{M}}v_{\infty}$ and
its optical depth $\tau\equiv\int_{R}^{\infty}\kappa\rho dr$,

${\dot{M}}v_{\infty}=\frac{\tau L}{c}\left(\frac{\Gamma-1}{\Gamma}\right)\,.$
(5)

OB stars generally have $\tau<1$ and so fall within the single-scattering
limit, ${\dot{M}}v_{\infty}<L/c$. In contrast, Wolf-Rayet (WR) stars can have
$\tau\approx 1-10$ and so require multi-line scattering to explain their
optically thick winds with ${\dot{M}}v_{\infty}>L/c$.

Figure 2:  Left: Illustration of the resonance nature of line opacity for an
optically thin line, for which the spectral average over the resonance quality
(${\bar{Q}}$) results in an optically thin line acceleration that is of order
a thousand times the gravitationally scaled acceleration from electron
scattering $\Gamma_{e}$. Right: The corresponding acceleration for an
optically thick line, which is reduced by the line optical thickness
$\tau_{line}$, which in an accelerating wind scales with the ratio of local
density to velocity gradient. This line desaturation is a consequence of the
Doppler shift of the line absorption from the increasing wind velocity $v$,
resulting in a net line force that scales as
$\Gamma_{thick}\sim(1/\rho)(dv/dr)$.

###  2.2 The CAK model for line-driven stellar winds

In practice, the enhancement of the flux-weighted mean opacity above the
simple electron scattering value results mainly from the bound-bound resonance
of radiation with electrons bound into heavy ions ranging from CNO to Fe and
Ni. As illustrated in the left panel of figure 2, the resonance nature of such
bound-bound line-scattering greatly enhances the opacity, typically by a
factor ${\bar{Q}}\gtrsim 10^{3}$ (Gayley, 1995). This means any star with
$\Gamma_{e}\gtrsim 1/{\bar{Q}}\approx 10^{-3}$ can have a line-force that
overcomes gravity and so drive a wind outflow.

In practice this maximal line acceleration from optically thin scattering is
reduced by the saturation of the reduced flux within an optically thick line.
But as illustrated in the right panel of figure 2, the Doppler shift
associated with the wind acceleration acts to desaturate this line absorption,
effectively sweeping the absorption through a broad frequency band, extending
out to the frequency associated with the Doppler shift from the wind terminal
speed, $v_{\infty}$. This wind Doppler shift of line resonance concentrates
the interaction of continuum photons into a narrow resonance layer with width
set by the Sobolev length, $\ell\equiv v_{th}/(dv/dr)$ (Sobolev, 1960),
associated with acceleration through the ion thermal speed $v_{th}$ that
broadens the line profile. In the outer wind where $\ell\ll r$, the line
acceleration for optically thick lines is reduced by $1/\tau$, where the
Sobolev optical depth $\tau\equiv{\bar{Q}}\kappa_{e}\rho\ell$, giving then a
line acceleration $\Gamma_{thick}\sim(1/\rho)(dv/dr)$ that itself scales with
the wind acceleration.

Within this Sobolev approximation, Castor et al. (1975, hereafter CAK))
developed a formalism that accounts for the cumulative radiative acceleration
from a power-law ensemble of both optically thick and thin lines,

$\Gamma_{CAK}\approx\frac{{\bar{Q}}\Gamma_{e}}{({\bar{Q}}t)^{\alpha}}\gtrsim
1~{}~{};~{}~{}t\equiv\kappa_{e}c\frac{\rho}{dv/dr}\,,$ (6)

where the CAK power index $\alpha\approx 0.5-0.7$ characterizes the relative
number of strong vs. weak lines. Applying (6) into the equation of motion (3)
and using the fact that the critical solution requires $v(dv/dr)\sim
GM/r^{2}$, we find the maximal CAK mass loss rate has the scaling

${\dot{M}}\sim\frac{L}{c^{2}}\left(\frac{{\bar{Q}}\Gamma_{e}}{1-\Gamma_{e}}\right)^{-1+1/\alpha}\,.$
(7)

For canonical values $\alpha=1/2$, ${\bar{Q}}=2000$, and $L=10^{5}L_{\odot}$,
this CAK scaling gives ${\dot{M}}\approx 10^{-5}M_{\odot}$/yr, which is indeed
a billion times the mass loss rate of the solar wind! Much as in the solar
wind, the terminal wind speeds scale with the surface escape speed
$v_{esc}\equiv\sqrt{2GM/R}$, with values up to $v_{\infty}\approx 2000$ km/s.

An overall point is that in such models the onset of line-driving near the
sonic point represents an effective line-driven suction, which draws up mass
from the underlying hydrostatic equilibrium of the subsonic region. The
reduction in pressure from the outer line-driving induces the underlying
subsonic region to expand upward, much as the suction on a straw draws up
liquid from a glass (see figure 1). This outside-in suction contrasts with the
inside-out thermal expansion of a pressure cooker, and of the analogous gas
pressure-driven solar wind discussed below in §3.

###  2.3 The Conti mechanism

A longstanding question is whether such line-driven winds might, over an
O-star’s main sequence lifetime, lead to loss of the star’s Hydrogen envelope,
representing the so-called “Conti mechanism” for producing the strong
depletion of Hydrogen inferred for Wolf-Rayet (WR) stars. Using the scaling
$L\sim M^{3}$ to estimate the star’s main sequence lifetime $t_{\rm ms}\approx
1\,{\rm Myr}\,(100M_{\odot}/M)^{2}$, we find for the canonical line opacity
factor ${\bar{Q}}=2000$ that the cumulative mass loss fraction follows the
scalings,

$\displaystyle\frac{{\dot{M}}t_{\rm ms}}{M}$ $\displaystyle\approx$
$\displaystyle~{}0.18\left(\frac{M}{100M_{\odot}}\right)^{2}~{};~{}\alpha=1/2$
(8) $\displaystyle\approx$ $\displaystyle
0.016\left(\frac{M}{100M_{\odot}}\right)^{1}~{};~{}\alpha=2/3$

which shows a very sensitive dependence on the CAK power index $\alpha$. When
corrected for wind clumping, empirically inferred mass loss rates agree better
with predictions for higher $\alpha\approx 0.6$, and so seem to disfavor the
Conti mechanism for H-envelope stripping.

As discussed in the review by A. Sander in these proceedings, quantitative
models of OB-wind mass loss endeavor to derive the line-driving opacity self-
consistently from NLTE solution of excitation and ionization of key driving
ions.

Figure 3: Left: Illustration of mechanical energy flux $F_{E}$ deposited over
a damping is balance optically thin cooling $n_{e}n_{i}\Lambda(T)$; the
exponential decrease in densit $\rho$ with height leads to high temperature
and eventual thermal run above the peak in cooling function $\Lambda$. Right:
Schematic to illustrate energy input into the nearly hydrostatic, subsonic
coronal base vs. that into the supersonic wind above the Parker sonic radius
$R_{s}$. For the former, the net input vs. loss by conduction back into the
underlying atmosphere leads to coronal heating that sets the coronal
temperature and location and density at the sonic radius, thus fixing the
associated wind mass loss rate. For the latter, any further energy addition
increases the wind flow speed, with the wind expansion providing the primary
mechanism to carry out the total net amount of coronal heating.

##  3 Pressure-driven coronal winds

###  3.1 Runaway heating of the solar corona and its pressure extension

In the Sun and other cool stars with surface temperatures below $10^{4}$ K,
the recombination of Hydrogen leads to strong subsurface convection and an
associated magnetic turbulence that drives mechanical heating of the upper
atmosphere. As illustrated in the left panel of figure 3, deposition of
mechanical energy flux $F_{E}$ over a damping length $\lambda_{d}$ must be
balanced by radiative cooling. In the upper atmosphere this requires
excitation of ions by electrons, so the associated cooling rate per unit
volume scales with density-squared, $\rho^{2}\sim n_{e}n_{p}$, multiplied by
an optically thin cooling function $\Lambda(T)$ that reflects the excitation
and ionization of the emitting ions.

As the density decreases exponentially with height, balancing the local
heating $F_{E}/\lambda$ occurs at progressively higher temperatures,
reflecting the initially steep increase in $\Lambda(T)$ from more energetic
collisional excitation. However, for temperatures $T\gtrsim 10^{5}$ K, such
collisions begin to ionize away the bound electrons, reducing the efficiency
of radiative cooling, and causing the $\Lambda(T)$ to decline with increasing
$T$. Since radiative cooling can no longer balance the heating, this leads to
a thermal runaway to coronal temperatures $T>$MK, limited now by inward
thermal conduction from the corona to underlying atmosphere, as illustrated in
right panel of figure 3.

Maintaining hydrostatic equilibrium at the coronal base radius $R$ implies a
pressure scale height,

$H\equiv\frac{P}{|dP/dr|}=\frac{kT}{\mu
g}=\frac{2a^{2}}{v_{esc}^{2}}R\equiv\frac{T}{T_{esc}}\,R\,,$ (9)

where $g\equiv GM/R^{2}$ and $v_{esc}\equiv\sqrt{2GM/R}$ are the surface
gravity and escape speed, and for a gas with molecular weight $\mu$, the
isothermal sound scales with temperature as $a\equiv\sqrt{kT/\mu}$. The last
equality defines an “escape temperature” at which $H=R$, which for solar
parameters has a value $T_{esc}\approx 13.8$ MK.

In the solar photosphere the low temperature $T\ll T_{esc}$ implies $H\ll R$
and thus a fixed gravity $g$, with pressure declining exponentially with scale
height $H$. But in the corona, the higher temperature gives
$T/T_{esc}=H/R\lesssim 1$, implying one now has to account for the radial
decline in gravity in the hydrostatic balance.

For a simple isothermal model, the pressure stratification now takes the form,

$P(r)=P(R)e^{(R/H)(R/r-1)}\,.$ (10)

In contrast to the exponential decline in the photosphere, the coronal
pressure at large radii now asymptotically approaches a finite value,
$P_{\infty}$ . Relative to the initial pressure $P_{o}\equiv P(R)$ at the
coronal base, the total drop in pressure for a hydrostatic, isothermal corona
is given by

$\frac{P_{o}}{P_{\infty}}\approx e^{13.8{\rm
MK}/T}~{}~{};~{}~{}\log\frac{P_{o}}{P_{\infty}}\approx\frac{6}{T/{\rm MK}}\,.$
(11)

The latter equality shows the pressure drops by 6 decades for $T=1$ MK, and
only 3 decades for $T=2$ MK.

By comparison, from the solar transition region at the coronal base to the
interstellar medium, the pressure drop is actually much greater,
$\log(P_{tr}/P_{ism})\approx 12$. The upshot is that an extended, hot corona
can not be maintained in hydrostatic equilibrium; instead, as shown by the
outward streamers from the eclipse image in figure 3, it must undergo an
outward, supersonic expansion known as the solar wind.

As illustrated by the right panel of figure 3, this solar wind expansion can
be thought of as analogous to the release valve of a pressure cooker, driven
fundamentally by mechanical heating generated by magnetic turbulence in the
underlying solar atmosphere. Some of this upward energy flux is lost back to
the solar atmosphere through thermal conduction, but the net effect leads to a
thermal runaway that raises the coronal temperature to temperature within a
factor ten of $T_{esc}$.

Figure 4:  Left: Solution topology for Mach number $v/a$ vs. scaled radius
$r/r_{c}$ for an isothermal corona. Solutions with initially low-speed at the
wind base include subsonic “breeze” solutions, which however again have a
large terminal pressure, $P_{\infty}$. The transonic wind solution is the only
with low enough $P_{\infty}$ match the low pressure of the interstellar
medium. Right: Steady-state velocity topology overplotted with snapshots at
$\Delta t=20$ ks intervals of a time-depedent hydrodynamical simulations for
evolution away from an initially hydrostatic corona. As the high pressure
reacts to the lower pressure outer boundary, an expansion develops that
eventually evolves to the supersonic solar wind solution.

###  3.2 Isothermal Solar Wind

The imbalance between outward pressure and inward gravity gives rise to an
outward acceleration, which for a steady-state outflow takes the form

$v\,\frac{dv}{dr}=-\frac{GM}{r^{2}}-\frac{1}{\rho}\frac{dP}{dr}\,.$ (12)

For an isothermal case with $P=\rho a^{2}$, one can use mass conservation of
the spherical outflow, $\rho vr^{2}={\dot{M}}/4\pi$=constant, to eliminate the
density $\rho$ in favor of the radius $r$ and flow speed $v$. This allows one
to split the pressure gradient force into terms that scale with the velocity
and the sphericity, giving

$\left(v-\frac{a^{2}}{v}\right)\frac{dv}{dr}=-\frac{GM}{r^{2}}+\frac{2a^{2}}{r}\,.$
(13)

In the subsonic region $v\ll a$, this reduces to the condition for hydrostatic
equilibrium.

But at a critical (a.k.a. “Parker”) radius,

$r_{c}=\frac{GM}{2a^{2}}=\frac{R^{2}}{2H}=R\left(\frac{T_{esc}}{2T}\right)\,,$
(14)

the spatial component of pressure balances gravity, so making the RHS vanish.
The LHS can likewise vanish if either $dv/dr=0$ or $v=a$.

Figure 4 plots Mach number $M\equiv v/a$ vs. scaled radius $r/r_{c}$ for the
overall topology of solutions. We can eliminate all the bi-valued solutions,
as well as those that start at supersonic speeds near the coronal base. There
is a class of subsonic solutions that have a velocity peak, with thus
$dv/dr=0$, at the critical radius $r=r_{c}$; but, like the hydrostatic case,
all such “breeze” solutions asymptote to a finite pressure at large radii, and
thus again cannot match the required low pressure of the interstellar medium.

The only solution that matches both the requirement of low, subsonic speed at
the coronal base $v(R)\ll a$, and vanishing pressure at large distances,
$P_{\infty}\rightarrow 0$, is the critical transonic solution with
$v(r_{c})=a$.

###  3.3 Energy balance and mass loss rate

While such isothermal models provide a basic rationale for the supersonic
outflow of the solar wind, they effectively ignore the central physics of what
energy mechanisms keep the corona hot against the adiabatic cooling of the
wind expansion. Moreover, because the density scales out of the equation of
motion (13), they do not provide any basis for determining the associated wind
mass loss rate, ${\dot{M}}\equiv 4\pi\rho vr^{2}$. For that one must consider
the overall wind energy balance, which we cast here in terms of integration
from some stellar base radius $R$ to an outer radius $r$,

${\dot{M}}\left[\frac{v^{2}}{2}-\frac{GM}{r}+\frac{5}{2}\frac{P}{\rho}\right]_{R}^{r}=4\pi\int_{R}^{r}r^{\prime
2}q_{net}\,dr^{\prime}+4\pi[R^{2}F_{c*}-r^{2}F_{c}(r)]\,.$ (15)

The terms in parenthesis on the LHS represent the wind kinetic energy,
potential energy, and gas enthalpy, while the RHS include the integral of the
local net heating per unit volume, and the net difference in conductive heat
flux.

Since the change in gas enthalpy is generally relatively small, we can write
the energy balance over the full wind by

${\dot{M}}\left(\frac{v_{\infty}^{2}}{2}+\frac{v_{esc}^{2}}{2}\right)\approx
Q_{heat}-Q_{rad}-Q_{cond}\,,$ (16)

where $Q_{rad}$ and $Q_{cond}$ are the energy losses from radiation and
conduction, and the total heating $Q_{heat}$ is associated with MHD waves
and/or magnetic reconnection. Since the wind terminal speed is generally
comparable to the escape speed, $v_{\infty}\approx v_{esc}$, we see that the
mass loss rate scales directly with the net heating vs. cooling,

${\dot{M}}\approx\frac{Q_{net}}{v_{esc}^{2}}\,.$ (17)

Figure 5:  Left: White light eclipse picture of the solar corona, showing how
coronal magnetic field confinement in loops extends into outward steamers by
the wind expansion, while open field coronal holes allow direct wind outflow.
Right: Corresponding X-ray corona from emission by the MK plasma. Image
credits: NASA.

Measurements by interplanetary spacecraft give a typical solar wind speed of
$v_{\infty}\approx 400$ km/s, ranging up to $v_{\infty}\approx 700$ km/s in
high-speed streams thought to originate from open-field coronal holes; these
values thus do indeed straddle the solar escape speed, $v_{esc}\approx 618$
km/s.

The observed mass flux implies to a quite small global mass loss rate,
${\dot{M}}\approx 10^{-14}M_{\odot}$/yr, which is even less than the mass loss
associated with the Sun’s radiative luminosity, ${\dot{M}}_{rad}\equiv
L_{\odot}/c^{2}\approx 5\times 10^{-14}M_{\odot}$/yr. Over the Sun’s entire
$\sim 10$ Gyr lifetime, it will thus lose only $0.01\%$ of its mass via the
solar wind. The wind kinetic luminosity is ${\dot{M}}v_{\infty}^{2}/2\approx
10^{-7}L_{\odot}$, which turns out quite comparable the observed total coronal
emission in X-ray and XUV, $L_{xuv}$.

As discussed in section 4, such coronal emission from cool stars with
exoplanets can play an important role in heating planetary atmospheres and
inducing their own wind outflows.

###  3.4 Angular momentum loss and spindown: the Skumanich law

While the mass loss from coronal winds from the Sun and other cool stars is
negligible, the coronal magnetic field provides an extended moment arm that
can make the associated loss of angular momentum in the magnetized wind have a
significant effect in causing an evolutionary spindown in the star’s rotation.
This is indeed thought to be the key cause of the relatively slow rotation
period of the present-day Sun, $P_{rot}\approx 27$ d.

Following pioneering analysis of Weber and Davis (1967), for a star of
rotation frequency $\Omega$, the loss of stellar angular momentum ${\dot{J}}$
from a wind with mass loss rate ${\dot{M}}$ is given by

${\dot{J}}=(2/3){\dot{M}}\Omega R_{A}^{2}\,,$ (18)

where the effective moment arm is set by the Alfvén radius $R_{A}$, defined to
be where the wind outflow speed equals the Alfvén speed, $V(R_{A})\equiv
V_{A}\equiv B/\sqrt{4\pi\rho}$. For the present-day solar wind and magnetic
field, one finds $R_{A}\approx 20R_{\odot}$, which when applied in (18) gives
a spindown time,

$t_{s}\equiv\frac{J}{{\dot{J}}}\approx 10\,{\rm Gyr}\,.$ (19)

The fact that this comparable to the Sun’s main sequence lifetime is
consistent with the notion that Sun’s present-day slow rotation period of 27 d
is the result of wind spindown.

More quantitatively, observations of a large population of solar-type stars
shows their rotation follow a simple relation, known as the “Skumanich law”
(Skumanich, 2019). From an initial period $P_{o}$, the period $P$ increases
with the square root of the age $t$,

$P(t)\approx P_{o}\left(\frac{2t}{t_{so}}+1\right)^{1/2}\,.$ (20)

If one assumes that the magnetic field scales with $B\sim\Omega\sim 1/P$, then
application of the Weber and Davis (1967) analysis gives a simple scaling for
the initial spindown time with the initial field strength $B_{o}$,

$t_{so}\approx 0.09\,\frac{MV_{w}}{R^{2}B_{o}^{2}}=0.38\frac{\rho
RV_{w}}{B_{o}^{2}}\approx 11.7{\rm
Gyr}\left(\frac{B_{o}}{2\,G}\right)^{-2}\,,$ (21)

where $\rho$ and $R$ are the star’s mean density and radius, and $V_{w}$ is
its wind speed. The last equality provides a numerical evaluation for solar
parameters. Remarkably, the dependence on wind mass loss rate scales out,
through cancellation of the ${\dot{M}}$ in (18) with the inverse dependence in
the Alfvén radius, $R_{A}^{2}\sim 1/{\dot{M}}$.

For example, taking the Sun’s initial rotation period to be $P_{o}\approx 1$
d, achieving the present $P\approx 27$ d for the Sun’s current age $t\approx
4.6$ Gyr requires an initial field $B_{o}\approx 61$ G and initial spindown
time of just $t_{s}\approx 12.6$ Myr. This gives for the present global field
$B_{now}=61/27=2.3$ G.

The upshot is that the young Sun was likely much more rapidly rotating, with
stronger magnetic activity, a stronger wind, and greater coronal EUV emission.
These likely played a role in stripping the initially much denser atmosphere
of Mars, implying that younger, more active stars might have similar effects
on the atmospheres of the exoplanets, as we next discuss.

Figure 6:  Left: Illustration of dust and ion tails from comets. The former is
driven away slowly by solar radiation, thus trailing behind the comet’s
orbital motion. The latter is driven by the much faster solar wind, and so is
oriented nearly in an anti-solar direction. Right: illustration of mass loss
from a hot-Jupiter exoplanet can be similarly driven away in a comet tail by
interaction with the star’s light and wind. Image credits: NASA.

##  4 Planetary winds and mass loss

The above discussion of solar and stellar wind mass loss from expansion of a
hot corona provides a good basis for exploring planetary winds and the
associated depletion of their atmospheres. One key difference is that solar
and stellar coronae form from inside-out heating from the star, in contrast to
the mainly external heating and ablation of a planetary atmosphere from the
parent star and its wind. As illustrated in figure 6, this can be expected to
cause any planetary outflow to take the form of a cometary tail, much as has
been observed extensively for comets orbiting the Sun.

Unlike the more spherical, solar coronal expansion, which must become
supersonic to match the outer boundary condition of a low interstellar
pressure, the cometary tail mass loss from planets could, in principal, be
quite slow and subsonic, even representing the outer layer loss of an
otherwise nearly hydrostatically stratified atmosphere. But even if the
kinetic energy of the outflow thus remains small and negligible, there remains
a key energy requirement for some source of heating to power any mass loss
against the gravitational binding from the planet.

###  4.1 Planetary escape temperatures

In this context, it helpful to consider the general scaling and values of the
escape temperatures of planetary bodies as a function of their mass $M$ and
radius $R$, and the molecular weight $\mu$ (in units of proton mass $m_{p}$)
of their atmospheres,

$T_{esc}\equiv\frac{GM/R}{k/\mu m_{p}}=131\,{\rm
kK}\,\frac{M/M_{J}}{R/R_{J}}=131\,{\rm
kK}\,\frac{\rho}{\rho_{J}}\,\left(\frac{R}{R_{J}}\right)^{2}\,.$ (22)

Here the last two equalities give scalings relative to Jupiter, assuming
$\mu=0.6$ that applies for a fully ionized mix of H and He at solar
abundances. For gas giants, the mean density $\rho$ is (like the Sun)
typically of order unity CGS (e.g., $\rho_{J}\approx 1.3$ g/cm3), and so the
last equality shows that the escape temperature mainly scales quadratically
with the planetary radius.

For smaller, rocky, terrestrial planets like the Earth, the density is a bit
higher ($\rho_{E}\approx 5.5$ g/cm${}^{3}\approx 4.1\rho_{J}$,) but the radius
is much smaller ($R_{E}\approx R_{J}/10$), giving the Earth an escape
temperature of about $4500$ K.

By comparison, if we approximate planetary absorption and emission as a simple
blackbody, then for a planet at a distance $d$ from a star with effective
temperature $T_{\ast}$ and radius $R_{\ast}$, the equilibrium temperature is
given by

$T_{eq}=290\,{\rm
K}\,\frac{T_{\ast}}{T_{\odot}}\sqrt{\frac{R_{\ast}/R_{\odot}}{d/{\rm
au}}}=4250\,{\rm K}\,\frac{T_{\ast}}{T_{\odot}}\sqrt{\frac{R_{\ast}}{d}}\,.$
(23)

The first equality is cast in terms of interplanetary distances like that of
the Earth at 1 au; it actually matches quite well the case of Earth’s actual
mean temperature, but this is due to a rather fortuitous cancelation between
the effect of the high albedo of clouds in reducing the absorption of solar
radiation, and the greenhouse effect in trapping Earth’s own cooling
radiation.

The latter equality frames this in terms of very close-in planets, like the
so-called “hot Jupiters”. A key point here is that even for close-in planets,
the direct heating from the star’s thermal radiation is almost never
sufficient to bring a planet anywhere near the typical escape temperature
associated with the thermal wind expansion from the planet.

Instead, it is the much harder, XUV radiation from stellar coronae that can
lead such escape-level heating, as we next discuss.

###  4.2 Planetary mass loss from heating by XUV coronal radiation

For photons ($\gamma$) of a given wavelength $\lambda$ and thus energy
$E=h\nu=hc/\lambda$, we can cast this energy in terms of an associated
temperature,

$T_{\gamma}\equiv\frac{h\nu}{k}=11.6\,{\rm kK}\,\frac{h\nu}{eV}=14\,{\rm
kK}\,\frac{1\mu m}{\lambda}\,.$ (24)

For photons above the Hydrogen ionization limit with $h\nu>13.6$ eV and
$\lambda<91.2$ nm, the associated escape temperature is $T_{esc}>156$ kK.
Coronal emission at EUV energies can thus readily ionize H with a sufficient
excess energy to heat the gas to temperature well above typical escape
temperature.

Figure 7:  Left: Ly$\alpha$ transit light for hot Jupiter HD1897333b,
comparing data points with values models with various assumed mass loss rates
and EUV fluxes. Figure from Lecavelier Des Etangs et al. (2010). Right:
Ly$\alpha$ transit spectrum for warm Neptune GJ436, overplotted for 4 phases
in which the planet was pre, post, in and out of transit. Note, in particular,
that the post-transit spectrum shows evidence for a comet tail, with a speed
that likely reflects either radiative acceleration by stellar Ly$\alpha$ or
entrainment in the outflowing stellar wind, rather than acceleration by
thermal expansion from the planet. Adapted from Ehrenreich et al. (2015).

In analogy with the scaling in eqn. (17) that was derived for mass loss from
stellar coronae, the associated planetary mass loss depends on the net XUV
heating of its outer atmosphere. For a planet of mass $M_{p}$ and radius
$R_{p}$ at a distance $d$ from a star with coronal XUV luminosity that is a
fraction of $f_{xuv}\equiv L_{xuv}/L_{\ast}$ of the stellar luminosity, the
energy balance to drive a planetary mass loss ${\dot{M}}$ is

${\dot{M}}\frac{GM_{p}}{R_{p}}\approx\epsilon\,f_{xuv}\,\frac{L_{\ast}}{4\pi
d^{2}}\,\pi R_{p}^{2}\,,$ (25)

where $\epsilon$ is an order-unity efficiency factor for this XUV heating.
Solving for the mass loss shows that it scales with the ratio of XUV flux to
the planet’s mean density,

${\dot{M}}=1.5\epsilon f_{xuv}\,\frac{L_{\ast}/4\pi d^{2}}{G\rho_{p}}\,.$ (26)

For example, assuming $\epsilon\approx 1$ and a solar value $f_{xuv}\approx
10^{-7}$ for XUV fraction, we find a hot Jupiter at distance $d=2R_{\odot}$
from a solar-type star should have mass loss rate ${\dot{M}}\approx 10^{10}$
g/s. The associated mass loss time is $t_{atm}\equiv M/(dM/dt)\approx 4\times
10^{12}$ yr. A central conclusion is thus that even for such an extreme case,
the total mass loss from such gas giants is small even over the evolutionary
lifetime of the system.

In contrast, for terrestrial planets like Earth, the atmosphere is a tiny
fraction $f_{atm}\approx 10^{-6}$ of the planet’s mass. Even at the greater
distance $d=1$ au, the associated time for Earth to loss a Hydrogen atmosphere
is $t_{atm}\approx 670$ Myr, and even shorter if one accounts for the likely
stronger coronal emission from the early Sun.

By contrast, the present-day atmosphere of Nitrogen and Oxygen has been mostly
retained, because of its higher molecular weight, although there is evidence
of a weak “polar wind” mass loss from polar regions of open magnetic fiield.

As illustrated in figure 7, mass loss from exoplanets can be observationally
diagnosed through transit light curves and spectra, often centered as here on
the UV Lyman-$\alpha$ transition of Hydrogen at $\lambda\approx 91.2$ nm. The
results here are for two hot giants, with the light curve in the left panel
providing constraints on the planets mass loss and star EUV flux. The overplot
of the Ly$\alpha$ spectral line at four phases of the transit provides
information on the outflow speed and its orientation in a comet tail extending
away from the star.

##  5 Magnetospheres of stars and planets

Mass loss from both stars and planets can be strongly affected by magnetic
fields. Figure 5 illustrates vividly the key role of fields in structuring the
solar coronal X-ray emission and the solar wind outflow. The white-light
coronal image on the left shows how gas trapped by closed magnetic loops near
the Sun gives way to a radially pointed streamer structures. As illustrated in
left panel of figure 8, the basic mechanism was first demonstrated in a
pioneering analysis by Pneuman & Kopp (1971), who examined how an initially
closed magnetic dipole in the solar corona is forced open by gas-pressure
expansion. The opening of opposite polarity field lines induces a current-
sheet in the outer, expanding wind, where they are a central feature of in
situ spacecraft measurements. Analogous magnetic effects likely occur other
cool stars with hot coronal winds, but the complexity of these dynamo-
generated fields leads to significant cancellation in disk-integrated light,
making them difficult to diagnose remotely through spectro-polaritmetry.

Figure 8:  Left: Iterative solution from Pneuman & Kopp (1971) for how an
initial dipole field (dashed lines) in the solar corona is by altered by gas
pressure and solar wind expansion into a stretched and eventually open form,
with lines of opposite polarity (denoted by arrows) separated by a current
sheet. Middle: Snapshot of MHD simulation of analogous distortion of dipole
field from a magnetic hot star, this time by its strong, radiatively driven
wind, with the Alfvén radius $R_{A}$ marking the transition from inner closed
loops to radially stretched field lines in the outer wind. Right: Analogous
MHD snapshot for a hot star with rapid rotation, giving a Kepler co-rotation
radius $R_{K}\ll R_{A}$, forming a wind-fed CM in the region between,
punctuated by episodic CBO events, with associated magnetic reconnection.

###  5.1 Massive-star magnetospheres

In contrast, although massive, luminous, hot stars lack the hydrogen
recombination convection zone that induces the magnetic dynamo cycle of
cooler, solar-type stars, modern spectropolarimetry has nonetheless revealed
that about 10% of O, B and A-type stars harbor large-scale, organized (often
predominantly dipolar) magnetic fields ranging in dipolar strength from a few
hundred to tens of thousand Gauss. These fields, which are likely fossils of
an earlier epoch, channel and trap the strong, radiatively driven winds of
such stars, feeding a circumstellar magnetosphere.

The inside-out building of these wind-fed magnetospheres is in some way
complementary to the outside-in nature of the planetary magnetospheres
impacted the star’s wind. But there are also some interesting similarities in
the role of the characteristic magnetospheric and corotation radii.

####  5.1.1  Alfvén radius and Kepler co-rotation radius

MHD simulation studies (e.g., ud-Doula & Owocki, 2002; Ud-Doula, Owocki, &
Townsend, 2008) show that the overall net effect of a large-scale, dipole
magnetic field in diverting such a hot-star wind can be well characterized by
a single wind magnetic confinement parameter and its associated Alfvén radius,

$\eta_{\ast}\equiv\frac{B_{eq}^{2}\,R_{\ast}^{2}}{\dot{M}\,v_{\infty}}~{}~{};~{}~{}\frac{R_{\mathrm{A}}}{R_{\ast}}\approx
0.3+\left(\eta_{\ast}+0.25\right)^{1/4}\,,$ (27)

where $B_{\mathrm{eq}}=B_{\mathrm{p}}/2$ is the field strength at the magnetic
equatorial surface radius $R_{\ast}$, and $\dot{M}$ and $v_{\infty}$ are the
fiducial mass-loss rate and terminal speed that the star would have in the
absence of any magnetic field. This confinement parameter sets the scaling for
the ratio of the magnetic to wind kinetic energy density. For a dipole field,
the $r^{-6}$ radial decline of magnetic energy density is much steeper than
the $r^{-2}$ decline of the wind’s mass and energy density; this means the
wind always dominates beyond the Alfvén radius, which scales as
$R_{\mathrm{A}}\sim\eta_{\ast}^{1/4}$ in the limit $\eta_{\ast}\gg 1$ of
strong confinement.

Figure 9:  Schematic of stellar magnetospheres, contrasting those from non-
rotating (or slowly rotating) stars (left) from rapidly rotating stars
(right). On the left $R_{A}$ denotes the Alfvén radius, separating the region
of closed magnetic loops from the magnetized outflowing wind, which stretches
the field into open lines of opposite polarity, separated by a current sheet
(CS). On the right $R_{K}$ denotes the Kepler co-rotation radius, above which
the centrifugal force exceeds gravity, trapping wind material in a centrifugal
magnetosphere (CM), until the mass build-up overwhelms the magnetic field
confinement, leading to centrifugal breakout (CBO) events.

For a model with $\eta_{\ast}=100$, the middle panel of figure 8 shows
magnetic loops extending above $R_{\mathrm{A}}\approx 3.2R_{\ast}$ are drawn
open by the wind, while those with an apex below $R_{\mathrm{A}}$ remain
closed, trapping wind upflow from opposite footpoints of closed magnetic
loops. Once this material cools back to near the stellar effective
temperature, it falls back onto the star over a dynamical timescale.

The dynamical effects of rotation can be analogously parameterized (Ud-Doula,
Owocki, & Townsend, 2008) in terms of the orbital rotation fraction, and its
associated Kepler corotation radius,

$W\equiv\frac{V_{\rm rot}}{{V_{\rm orb}}}=\frac{V_{\rm
rot}}{\sqrt{GM_{\ast}/R_{\ast}}}~{}~{};~{}~{}R_{\rm K}=W^{-2/3}\,R_{\ast}$
(28)

which depends on the ratio of the star’s equatorial rotation speed to the
speed to reach orbit near the equatorial surface radius $R_{\ast}$. Insofar as
the field within the Alfvén radius is strong enough to maintain rigid-body
rotation, the Kepler corotation radius $R_{\rm K}$ identifies where the
centrifugal force for rigid-body rotation exactly balances the gravity in the
equatorial plane. Figure 9 contrasts the roles of magnetic confinement and
centrifugal support in stellar magnetospheres.

Figure 10:  Left: Sketch of the regimes for a dynamical vs. centrifugal
magnetosphere (DM vs. CM). The lower panel illustrates the case of a slowly
rotating star with Kepler radius beyond the Alfvén radius
($R_{\mathrm{K}}>R_{\mathrm{A}}$); the lack of centrifugal support means that
trapped material falls back to the star on a dynamical timescale, forming a
DM, with shading illustrating the time-averaged distribution of density. The
upper panel is for more rapid rotation with $R_{\mathrm{K}}<R_{\mathrm{A}}$,
leading then to a region between these radii where a net outward centrifugal
force against gravity is balanced by the magnetic tension of closed loops;
this allows material to build up to the much higher density of CM. Right,
Upper: Contour plot for density at arbitrary snapshot of an isothermal 2D MHD
simulation with magnetic confinement parameter $\eta_{\ast}=100$ and critical
rotation factor $W=1/2$. The overlay illustrates the definition of radial mass
distribution, $\Delta m/\Delta r$, within $10^{\circ}$ of the equator. Right,
Lower: Density plots for log of $\Delta m/\Delta r$, plotted versus radius
(1-5 $R_{\ast}$) and time (0-3 Msec), for a mosaic of 2D MHD models with a
wide range of magnetic confinement parameters $\eta_{\ast}$, and 3 orbital
rotation fractions $W$. The horizontal solid lines indicate the Alfvén radius
$R_{\rm A}$ (solid) and the horizontal dashed lines show Kepler radius $R_{\rm
K}$ (dashed).

####  5.1.2 Dynamical vs. Centrifugal Magnetosphere

If $R_{\rm A}<R_{\rm K}$, then material trapped in closed loops will again
eventually fall back to the surface on a dynamical timescale, thus forming
what’s known as a dynamical magnetosphere (DM).

But if $R_{\rm A}>R_{\rm K}$, then wind material located between $R_{\rm K}$
and $R_{\rm A}$ can remain in static equilibrium, forming a centrifugal
magnetosphere (CM) that is supported against gravity by the magnetically
enforced co-rotation. As illustrated in the upper left schematic in figure 10,
the much longer confinement time allows material in this CM region to build up
to a much higher density than in a DM region.

Eventually the centrifugal force of this material overwhelms the confining
effect of magnetic tension, leading to centrifugal breakout (CBO) events, as
illustrated by the right panel of figure 8 for the case $\eta_{\ast}=100$ and
$W=0.5$. (See also the right side of figure 9.)

For general 2D MHD simulations in the axisymmetric case of a rotation-axis
aligned dipole, the mosaic of greyscale plots in figure 10 shows the time vs.
height variation of the equatorial mass distribution $\Delta m/\Delta r$ for
various combinations of rotation fraction $W$ and wind confinement
$\eta_{\ast}$ that respectively increase upward and to the right. This
illustrates vividly the DM infall for material trapped below $R_{\rm K}$ and
$R_{\rm A}$, vs. the dense accumulation of a CM from confined material near
and above $R_{\rm K}$, but below $R_{\rm A}$. For such CM stars, note also the
episodic CBO events.

Figure 11: Classification of 64 observationally confirmed magnetic massive
stars in terms of magnetic confinement vs. rotation fraction, characterized
here by a log-log plot of Kepler radius $R_{\rm K}$ increasing downward vs.
Alfvén radius $R_{\rm A}$ increasing to the right. The labeled ID numbers are
sorted in order of decreasing effective temperature $T_{\mathrm{eff}}$, with
stellar identities given in Table 1 of Petit et al. (2013). Stars to the left
of the vertical solid line have only weakly magnetized winds (with
$\eta_{\ast}<1$), Star below and left of the diagonal solid line have
dynamical magnetospheres (DM) with $R_{\rm A}<R_{\rm K}$, while those above
and right of this line have centrifugal magnetospheres (CM) with $R_{\rm
A}>R_{\rm K}$.

####  5.1.3 Comparison with Observations of Confirmed Magnetic Hot-stars

For observationally confirmed magnetic hot-stars with $T_{\mathrm{eff}}\gtrsim
16$ kK, figure 11 plots positions in a log-log plane of $R_{\mathrm{K}}$ vs.
$R_{\mathrm{A}}$ (Petit et al., 2013). The vertical solid line representing
$\eta_{\ast}=1$ separates the domain of non-magnetized or weakly magnetized
winds to left, from the domain of stellar magnetospheres to the right. The
diagonal line representing $R_{\mathrm{K}}=R_{\mathrm{A}}$ divides the domain
of centrifugal magnetospheres (CM) to the upper right from that for dynamical
magnetospheres (DM) to the lower left.

Figure 12:  Greyscale renditions of dynamic spectrum for variation of doppler-
shifted emission and absorption with rotation phase. The right panel shows
results from an RRM model assuming rotation and parameters quoted in the text
for the strong CM star $\sigma~{}$Ori E. This is in good overall agreement
with the left panel showing observed variations in H$\alpha$ from this star.
Adopted from Townsend, Owocki, & Groote (2005).

For strong CM stars with $R_{\mathrm{K}}\gg R_{\mathrm{A}}$ one can apply a
semi-analytic, rigidly rotating magnetosphere (RRM) model developed by
Townsend & Owocki (2005), which assumes the wind-fed mass accumulates along
surfaces of minimum combined gravitational-centrifugal potential. The strong
confinement leads to a high enough density to make the CM have a net emission
in the Hydrogen Balmer line, H$\alpha$. In the common case that the magnetic
dipole axis has a non-zero titl angle $\beta$ with the rotation axis, the
highest density and strongest emission comes from circumstellar clouds near
the common rotational and magnetic equator. As rotation brings these clouds
from in front of the star to off the limb, the H$\alpha$ line profile shows
corresponding variations from absorption near line center to emission in the
blue and red wings. As shown in figure 12, the overall line-profile variation
can be characterized by a dynamic spectrum that renders on a grayscale the
rotational phase variations of doppler-shifted emission and absorption.

For example, the strong CM star $\sigma~{}$Ori E has a nearly dipole field of
polar strength $B_{p}>10^{4}$ G and strong tilt $\beta>60^{o}$, and rotation
fraction $W\approx 0.25-0.5$. The corresponding confinement parameter
$\eta_{\ast}>10^{6}$ implies an Alfvén radius $R_{\mathrm{A}}\approx
30R_{\ast}$ well beyond the Kepler radius $R_{\mathrm{K}}\approx 2R_{\ast}$.
Figure 12 compares dynamical spectra from a RRM model (left) with that
obtained from actual H$\alpha$ observations of $\sigma~{}$Ori E (right). The
good overall agreement provides strong evidence for the overall CM paradigm,
as well as the basic RRM analysis method.

Figure 13: Left: Observed magnetic B-stars plotted in the $\log B_{K}/B_{K1}$
vs. $\log L/L_{\odot}$ plane, with stars showing H$\alpha$ in filled circles,
and those without in open circles (Shultz et al., 2020). Here $B_{K}$ is the
observationally inferred field strength at the Kepler co-rotation radius
$R_{\mathrm{K}}$, while is $B_{K1}$ is the strength the CBO model predicts is
needed to make the CM have unit optical thickness in H$\alpha$ at
$R_{\mathrm{K}}$. Above a threshold luminosity, the onset of H$\alpha$
emission occurs right at this CBO value, independent of luminosity; this
indicates the associated wind feeding of the CM overwhelms any leakage, to
build the density to its breakout values. Right: Occurrence of radio emission
for magnetic B stars in a log-log plane of the ratio
$R_{\mathrm{A}}/R_{\mathrm{K}}$ vs. luminosity (now increasing from right to
left). Filled circles with detected radio lie the upper diagram, indicating
rapid rotation again plays a key role in radio emission; the lack of clear
dependence on luminosity again indicates the wind feeding rate is relatively
unimportant, as expected for emission tied to CBO events and their associated
magnetic reconnection.

Figure 14:  Left: Comparison of observed non-thermal radio luminosity with the
luminosity scaling for CBO model, showing a strong correlation with an
efficiency factor $\epsilon\approx 10^{-8}$ for conversion to radio emission
(Owocki et al., 2022). Right: Empirical scaling law for radio emission,
comparing early-type stars with ultra-cool dwarfs, and even extending to radio
emission from Jupiter (Leto et al., 2021).

####  5.1.4 Role of centrifugal breakout in H$\alpha$ and radio emission

Despite this success of the RRM model in characterizing the relative density
of wind-fed material along potential minima, the lack of a clear mechanism for
emptying this build-up had precluded any prediction of the absolute density
and associated emission. But recent theoretical analyses and numerical MHD
simulations (Owocki et al., 2020) have provided strong evidence that this
wind-fed density build-up in CM’s proceeds up to a level wherein the outward
centrifugal force exceeds the inward confinement of magnetic, whereupon
material escapes in centrifugal breakout (CBO) events. This CBO analysis gives
a specific, threshold value of the magnetic field strength at the Kepler
radius, $B_{K1}$, that is needed to give a sufficient density for H$\alpha$ to
become optically thick, and thus produce a net emission within the H$\alpha$
line profile. The left panel of figure 13 shows a log-log plot of inferred
ratio of $B_{K}/B_{K1}$ vs. stellar luminosity. Remarkably, the onset of
H$\alpha$ emission, shown by the filled circles, occurs right at the line for
unit ratio, with no apparent luminosity trend for the onset of emission. The
provides strong evidence that CBO is the mechanism for controlling density and
mass balance in the CM.

Moreover, the right panel of figure 13 shows a similar strong dependence on
rotation for the observed incoherent radio emission from such magnetic B-stars
(Shultz et al., 2022). As illustrated in the left side of figure 9, the
previously favored model by Trigilio et al. (2004) proposed that electrons
accelerated at the base of the wind-induced current sheet could be spiral
along closed magnetic loops, producing the radio by gyro-synchrotron emission.
Since this includes no role for stellar rotation, this model is now strongly
disfavored.

A promising alternative, illustrated by the right side of figure 9, proposes
instead that electrons are accelerated near sites magnetic reconnection
associated with CBO events. An analysis by Owocki et al. (2022) show that,
under the assumption that the Alfvén radius follows the split-monopole scaling
$R_{A}\sim\sqrt{\eta_{\ast}}$, the luminosity available from such CBO events
scales as

$L_{CBO}\approx W\Omega B_{\ast}^{2}R_{\ast}^{3}\,.$ (29)

While this seems to suggest the full volume energy of the magnetic field is
dissipated on a rotation timescale, in practice the field and its breakout
effectively just act as a conduit for the large energy reservoir associated
with the stellar rotation.

The left panel of figure 14 compares the observed radio luminosity $L_{rad}$
against this predicted $L_{CBO}$ scaling (Owocki et al., 2022). The results
show a nearly linear correlation, with however a small efficiency factor $\sim
10^{-8}$ for conversion of available CBO energy into radio. Moreover, when
extended downward by several decades, the right panel shows that this
empirical scaling for radio luminosity from early-type stars also matches well
that observed from ultra-cool dwarfs (Leto et al., 2021). Such UCDs are also
inferred to have both strong magnetic fields and rapid rotation, and thus are
also strong candidates for CBO events and associated reconnection.

Even more remarkably, further downward extrapolation shows that the radio
emission from Jupiter follows this same scaling relation, thus suggesting a
potential link in physical process producing radio emission in stellar and
planetary magnetospheres.

Figure 15:  Schematics sketches for the magnetospheres of Earth (left) and
Jupiter (right). Both are externally stressed by the solar wind into elongated
tail, leading to episodes of magnetotail reconnection that drive particle
acceleration manifest as auroral substorms. But Jupiter is also stressed
internally from volcanic ejection of material from the active moon Io, which
upon ionization by solar UV becomes trapped in Jupiters rotating
magnetosphere. Jupiter’s decametric radio emission is thought link to this Io
plasma torus, possibly through CBO-driven reconnection and gyro-emission from
associated energized electrons. Image credits: NASA

###  5.2 Outside vs. inside stressing of planetary magnetospheres

In contrast to the inside-out stressing of stellar magnetospheres, figure 15
illustrates how the impact of an external solar or stellar wind exerts an
outside-in stress on planetary magnetospheres, stretching them into a long
magnetotail away from the star. As seen for the Earth’s magnetosphere on the
left, thermal expansion now forms a “polar wind” outflow, which is then
channeled along field lines toward the tail. The closed-field region near the
equator traps material into a plasma sheet that extends beyond the Earth’s co-
rotation radius ($R_{K}\approx 6R_{E}$), where the combination of centrifugal
and external stresses from the solar wind drive X-line magnetic reconnection
and associated plasma ejection along the magnetotail, a process quite
analogous to CBO events from stellar magnetospheres.

In the case (right panel) of the much larger magnetosphere of Jupiter, the
volcanic moon Io provides an additional plasma source, at an orbital radius
well beyond Jupiter’s co-rotation radius, $r_{Io}\approx 2.6R_{K}$. This can
combine with other plasmas to drive CBO reconnection, accelerating electrons
whose gyro-emission in magnetic loops produce the decametric radio emission
thought linked to Io, again a process similar to the CBO-linked radio emission
from stellar magnetospheres.

But a key overriding effect of such planetary magnetospheres is to shield the
planet from the atmospheric ablation that can result from direct impact of
coronal wind ions. Such ablation by the solar wind is thought to be a key
factor in depleting the initially dense atmosphere of Mars, which lacks a
strong global magnetic field. The presence or absence of sufficiently strong,
global magnetic field can thus be a key factor in whether rocky exoplanets
retain a sufficient atmosphere to host liquid water and perhaps life.

##  6 Summary

Let us conclude with a list of key points of this review:

  * •

Hot-star winds are driven by line-scattering of the star’s radiation.

  * •

Gas pressure drives supersonic solar coronal winds.

  * •

XUV heating of planetary atmospheres drives analogous thermal
expansion/escape.

  * •

This and ablation deplete atmospheres of rocky planets, but not gas giants.

  * •

Solar/stellar magnetospheres guide and trap their wind outflow.

  * •

The associated magnetized winds lead to spindown of stellar rotation.

  * •

10% of OBA stars have strong fields with wind-fed magnetospheres…

    * –

characterized by the Alfvén radius $R_{\mathrm{A}}$ and Kepler co-rotation
radius $R_{\mathrm{K}}$.

    * –

If $R_{\mathrm{A}}<R_{\mathrm{K}}$, forms a dyanmical magnetosphere (DM).

    * –

If $R_{\mathrm{K}}<R_{\mathrm{A}}$, forms a centrifugal magnetosphere (CM).

  * •

For CM’s, centrifugal breakout (CBO) controls H$\alpha$ and radio emission.

  * •

Planetary magnetospheres both shield atmospheres from winds, and trap
outflows.

## Acknowledgments

I thank the IAUS370 SOC, and particularly its chair, Aline Vidotto, for the
invitation to give this opening review. I acknowledge numerous helpful
conversations with B. Das, M. Shultz, J. Sundqvist, R. Townsend, A. ud-Doula,
and other members of the Magnetism in Massive Stars (MiMeS) collaboration.
This work was supported in part by NASA ATP grant 80NSSC22K0628.

## References

  * Castor et al. (1975) Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, ApJ, 105, 157.
  * Gayley (1995) Gayley, K. G. 1995, ApJ, 454, 410.
  * Lecavelier Des Etangs et al. (2010) Lecavelier Des Etangs, Ehrenreich, D.. Vidal-Madjar, A., Ballester. G. E., Désert, J.-M., Ferlet, R., Hébrard, G., Sing, D. K., Tchakoumegni, K.-O. and Udry, S. 2010, A&A, 514, A72.
  * Leto et al. (2021) Leto P., Trigilio C., Krtička J., Fossati L., Ignace R., Shultz M. E., Buemi C. S., et al., 2021, MNRAS, 507, 1979.
  * Owocki et al. (2020) Owocki S. P., Shultz M. E., ud-Doula A., Sundqvist J. O., Townsend R. H. D., Cranmer S. R., 2020, MNRAS, 499, 5366.
  * Owocki et al. (2022) Owocki S. P., Shultz M. E., ud-Doula A., Chandra P., Das B., Leto P., 2022, MNRAS, 513, 1449.
  * Petit et al. (2013) Petit V., Owocki S. P., Wade G. A., Cohen D. H., Sundqvist J. O., Gagné M., Maíz Apellániz J., et al., 2013, MNRAS, 429, 398.
  * Pneuman & Kopp (1971) Pneuman G. W., Kopp R. A., 1971, SoPh, 18, 258.
  * Shultz et al. (2020) Shultz M. E., Owocki S., Rivinius T., Wade G. A., Neiner C., Alecian E., Kochukhov O., et al., 2020, MNRAS, 499, 5379.
  * Shultz et al. (2022) Shultz M. E., Owocki S. P., ud-Doula A., Biswas A., Bohlender D., Chandra P., Das B., et al., 2022, MNRAS, 513, 1429.
  * Skumanich (2019) Skumanich, A. 2019, ApJ, 878, 35.
  * Sobolev (1960) Sobolev, V. V. 1960, Cambridge: Harvard University Press, 1960.
  * Townsend & Owocki (2005) Townsend R. H. D., Owocki S. P., 2005, MNRAS, 357, 251.
  * Townsend, Owocki, & Groote (2005) Townsend R. H. D., Owocki S. P., Groote D., 2005, ApJL, 630, L81.
  * Trigilio et al. (2004) Trigilio C., Leto P., Umana G., Leone F., Buemi C. S., 2004, A&A, 418, 593.
  * ud-Doula & Owocki (2002) ud-Doula A., Owocki S. P., 2002, ApJ, 576, 413.
  * Ud-Doula, Owocki, & Townsend (2008) Ud-Doula A., Owocki S. P., Townsend R. H. D., 2008, MNRAS, 385, 97.
  * Weber and Davis (1967) Weber, E. J. and Davis, L. 1967, ApJ, 148, 217.


# Generalized Lomb-Scargle Analysis of 22 years of Super-Kamiokande solar 8B
neutrino data

Vibhavasu Pasumarti<EMAIL_ADDRESS>Shantanu Desai  E-mail:
<EMAIL_ADDRESS>Department of Physics, IIT Hyderabad, Kandi,
Telangana-502284, India

###### Abstract

We apply the generalized Lomb-Scargle periodogram to 22 years data of solar 8B
neutrino fluxes detected by Super-Kamiokande. The primary motivation of this
work was to check if the sinusoidal modulation at a frequency of 9.43/year
(with a period of 38 days), which we had found to be marginally significant
with the first five years of Super-K data, persists, with the accumulated
data. We use four different metrics for the calculation of significance of any
peaks in the Lomb-Scargle periodogram, which could be indicative of
periodicities. We do not find any evidence for periodicity at the
aforementioned frequency or any other frequency with the updated data.
Therefore, the observed peak at 9.43/year with the first five years of Super-
Kamiokande data was only a statistical fluctuation and its significance is
negligible with the updated data.

##  I Introduction

For more than two decades, Sturrock and collaborators have argued that the 8B
solar neutrino flux seen in Super-Kamiokande-I from 1996-2001 Yoo et al.
(2003) contains a sinusoidal modulation at a frequency of 9.43/year, which
corresponds to a period of around 38 days (See Ref. Sturrock (2022) and
references therein). They asserted that this peak is due to the synodic
rotation of the solar core, for which the sidereal rotation rate is around
10.43/year Sturrock (2008); Sturrock et al. (2012). The first independent
analysis of the Super-K data reported $p$-values for this peak ranging from
0.007 to 0.019, depending on the analysis technique used Sturrock et al.
(2005). Another work found $p$-values from 0.018 to 0.12, depending on whether
one incorporates the asymmetric errors or not Ranucci (2006). However, these
results were at odds with the analysis done by the Super-K collaboration,
which did not find any evidence for periodicity at any frequency Yoo et al.
(2003).

In order to investigate this issue, we applied the generalized Lomb-Scargle
periodogram Lomb (1976); Scargle (1982); Zechmeister and Kürster (2009) to
Super-K-I data (binned in intervals of 5-days) Desai and Liu (2016). We then
calculated the $p$-value using the method proposed in Scargle (1982), as well
as using Bayesian information criterion (BIC) Krishak and Desai (2020). We
were able to confirm the peak (previously observed in works by Sturrock and
collaborators) at 9.43/year with a $p-$value of 0.015 (significance of 2.2
$\sigma$), having a BIC value close to 5, pointing to marginal significance
using qualitative “strength of evidence” rules Krishak and Desai (2020). This
peak was also confirmed in other works Pommé et al. (2017, 2018). The
amplitude of this modulation at a period of 38 days is equal to 6.9%. Most
recently, it has been shown that the significance of this peak is due to the
fortuitous alignment of six data points Pommé and Pelczar (2022). If these
data points are excluded, the amplitude gets reduced to 5.3% thereby reducing
the significance of the peak. Therefore, Ref. Pommé and Pelczar (2022) has
asserted that the aforementioned peak is not real and only a statistical
fluctuation. Nevertheless, the only way to resolve this imbroglio is to redo
the same procedure with additional data.

Another reason for possible sinusoidal modulations in the solar neutrino flux
could be due to time variations in the solar magnetic field, if the neutrino
has a non-zero magnetic moment, because of Resonant Spin Flavor precession
Pulido (1994); Akhmedov and Pulido (2003); Abe et al. (2023). Finally, a
periodic variation in the solar core temperature could also induce a
periodicity in the solar neutrino fluxes Raychaudhuri (1971). Therefore
observing such a periodic modulation could enable us to gain insights on a
variety of physical processes in the solar interior and also some of the
fundamental properties of neutrinos.

Twenty years after the first search for periodicities, the Super-K
collaboration has recently carried out another search for periodicities with a
livetime of around 22 years Abe et al. (2023). Therefore, this data would
provide a “smoking gun” test on whether the tantalizing hints for a periodic
signal at 38 days seen in the first five years of data were only a fluctuation
or signatures of a real sinusoidal signal. However, no evidence for
periodicity was found in this analysis. This data has also been made publicly
available.

In this work, we again apply the generalized Lomb-Scargle periodogram (as our
previous work Desai and Liu (2016)) to the aforementioned Super-K data Abe et
al. (2023). We evaluate the statistical significance using four independent
methods, similar to our recent works on searching for periodicities in nuclear
$\beta$-decay rates Tejas and Desai (2018); Dhaygude and Desai (2020);
Gururajan and Desai (2020). This manuscript is structured as follows. We
provided a brief account of the generalized Lomb-Scargle periodogram in Sect.
II. We recap the latest Super-K search for periodicity using more than 20
years of data in Sect. III. Our analysis and results can be found in Sect. IV.
We conclude in Sect. V.

##  II Generalized Lomb-Scargle Periodogram

The Lomb-Scargle (L-S) Lomb (1976); Scargle (1982); VanderPlas (2018);
VanderPlas and Ivezić (2015); Ivezić et al. (2014) periodogram is a widely
used robust technique to look for periodicities in unevenly sampled data. The
main goal of the L-S periodogram is to determine the frequency ($f$) of a
periodic sinusoidal signal in a time-series ($y(t)$) as follows:

$y(t)=a\cos(2\pi ft)+b\sin(2\pi ft).$ (1)

The L-S periodogram calculates the power as a function of frequency, from
which one can assess the statistical significance for any frequency.

For this analysis, we use the generalized (or floating-mean) L-S periodogram
Zechmeister and Kürster (2009); Bretthorst (2001). The main difference with
respect to the ordinary L-S periodogram is that an arbitrary offset gets added
to the mean values. More details on the differences between the two
implementations can be found in VanderPlas (2018); VanderPlas and Ivezić
(2015). The generalized L-S periodogram has been shown to be more sensitive
than the normal one, for detecting peaks when the data sampling overestimates
the mean VanderPlas (2018); Zechmeister and Kürster (2009); Vanderplas et al.
(2012). To determine the significance of any peak in the L-S periodogram, we
must calculate its false alarm probability or $p$-value. A large number of
metrics have been constructed to estimate the $p$-value of peaks in the L-S
periodogram Scargle (1982); VanderPlas (2018); Pommé (2020); Pommé and De
Hauwere (2020). We now briefly describe these myriad metrics, which we label
based on the command-line options in Python, which are used to calculate these
$p$-values:

  * •

Baluev

This method uses extreme value statistics for stochastic process, to compute
an upper-bound of the $p-$value in case of no aliasing. The analytical
expression for the $p-$value using this method can be found in Baluev (2008);
VanderPlas (2018).

  * •

Bootstrap

This method makes use of non-parametric bootstrap resampling VanderPlas
(2018). It applies L-S periodograms on synthetic data constructed at the same
observation times as the real data. The bootstrap is the most robust estimate
of the $p$-value, as it makes minimal assumptions about the periodogram
distribution, and the observed times also fully account for survey window and
dead-time effects VanderPlas (2018). However, the bootstrap method does not
correctly account for correlated noise in the observations Ivezić et al.
(2014). One also needs a large number of bootstrap resamples to compute the
$p-$value with very good accuracy. We use 1000 bootstrap samples which can
provide $p$-values with an accuracy of about 1%. We also note that for our
analysis, we are using binned data, which consists of observations averaged
over a five-day period, and hence we are not fully incorporating the detector
livetime and dead-time effects.

  * •

Davies

This method is similar to Baluev, but is not accurate at large $p$-values,
where it shows values greater than 1 Davies (2002). Nevertheless, for
completeness we also calculate the $p$-value using this method.

  * •

Naive

This method is based on the assertion that well-separated areas in the
periodogram are independent of each other. The total number of independent
frequencies is dependent on the sampling rate and observation duration
VanderPlas (2018).

We note that all the aformentioned $p$-values are global $p$-values that
evaluate the significance of peaks in the periodogram after accounting for the
“look elsewhere effect”: the trials factors associated with the fact that many
frequencies are being searched at once Gross and Vitells (2010). However, for
two specific frequencies (9.43/year and 1/year), we also calculate the local
$p$-value using the expression in Table 1 of Baluev (2008). This local
$p$-value has been implemented using the single option in the L-S module
provided in astropy. Once the $p-$value is known, one can evaluate the
significance or $Z$-score Ganguly and Desai (2017). The smaller the $p$-value,
the more significant is the peak. A rule of thumb for any peak to be deemed
interesting is that $p$-value should be less than $0.05$. However for a peak
to be statistically significant, its $Z-$score must be greater than 5$\sigma$
Lyons (2013), which corresponds to $p$-value $<10^{-7}$.

##  III Recap of SK23

We briefly describe the latest search for periodicity carried out by the
Super-K collaboration using more than 20 years of 8B solar neutrino data from
1996-2018 Abe et al. (2023). Super-K is a 50 kiloton water Cherenkov detector
located in the Kamioka mine in Japan which detects neutrinos with energies
from MeV range Fukuda et al. (1999) to over a TeV Desai et al. (2008), which
has been taking data since 1996. Super-K has produced physics and astrophysics
results from a wide range of topics from neutrino oscillations Fukuda et al.
(1998) to dark matter Desai et al. (2004). The total livetime used for the
analysis of solar neutrino data until 2018 is equal to 5,804 days combining
four different phases of data taking spanning about 22 years. Super-K detects
about 20 solar neutrino interactions per day. The total dataset was divided
into 1343 time bins, where the average width of each bin is around five days.
A search for perodicities was done using the Maximum likelihood method (by
incorporating the energy information in addition to the fluxes) as well as the
L-S periodogram following the prescription in Press et al. (1992). The L-S
analysis was done by looking for 100,000 frequencies from $10^{-6}$/day to
0.2/day. No significant periodicities apart from the annual modulation due to
the revolution of the Earth around the Sun were found. As a supplement to this
paper, the data for 5-day interval fluxes covering the above observing period
have been made publicly available. The publicly released dataset consists of
mean time, start and end time of the observed data, the measured solar
neutrino flux along with its upper and lower flux errors. However, this data
does not include corrections due to the varying Earth-Sun distance. Unlike the
Super-K-I dataset, these correction factors due to the eccentricity of the
Earth’s orbit around the Sun were not provided along with the dataset at the
time of writing. So we obtained these correction factors by calculating the
average Earth-Sun distance binned in 1-day intervals ($D$) for each 5-day bin.
The distance to the Sun was calculated using the astropy Astropy Collaboration
et al. (2022) module. We then scaled the raw flux and the errors by $D^{2}$
after normalizing the distance by 1 AU. This plot of both the uncorrected and
corrected fluxes as a function of time since the start of Super-K can be found
in Fig. 1.

Figure 1: The Super-K 8B solar neutrino flux as a function of time. The top
panel shows the raw flux and the bottom panel is the corrected flux rescaled
by the square of the distance between the Sun and the Earth averaged over each
5-day bin. Data used for this plot has been obtained from Abe et al. (2023).
For brevity, we have ignored the X-axis bin width in the above plots.

##  IV Analysis and Results

We now apply a generalized L-S periodogram to this dataset. We used the L-S
implementation in astropy.time_series Astropy Collaboration et al. (2022)
module in Python. We provided the mid-point of each time bin, measured flux
and the flux errors as inputs to the L-S periodogram. The flux errors were
obtained from an average of the asymmetric error bars. However, as a sanity
check we also redid the analyses by considering the larger values among the
errors for each data point, but the results are comparable to those obtained
by considering the average of the errors and are not reported here. The
recommended frequency resolution and maximum frequency up to which the
generalized L-S method can robustly detect sinusoidal modulations are given by
the reciprocal of five times the total duration of the dataset and five times
the mean Nyquist equivalent frequency, respectively VanderPlas (2018). For the
Super-K dataset, this default frequency resolution is equal to 0.0091/year,
which is used for our work. Based on the above recommendation, the maximum
frequency up to which the L-S periodogram would be sensitive to any potential
peaks is equal to 152.8/year. However for brevity, we only show the results
for frequencies up to 20/year (similar to our previous work), since previous
searches had shown a statistically significant result only at 9.43/year Desai
and Liu (2016) and the frequencies associated with solar rotation are between
8-14/year. However, we also checked if the updated data exhibited any
statistically significant peaks for frequencies greater than 20/year and up to
70/year (which is equal to the width of each time bin). Since we did not find
any peaks with large statistical significance above a frequency of 20/year, we
do not show any results for the same. We followed the same normalization
convention for the L-S periodogram as in our previous works Tejas and Desai
(2018); Dhaygude and Desai (2020); Gururajan and Desai (2020), where the L-S
power can take values between zero and one. In the appendix, we also show the
results of the same analyses using only the first five years of Super-K data
until July 2001.

The L-S power as a function of frequency for both the uncorrected as well as
the corrected 8B neutrino fluxes are found in Fig. 2. We see a maximum peak
for the uncorrected flux at a frequency of 1/year. However, this peak is not
the strongest, once we rescale the fluxes by the square of the distance. We
calculated the $p$-value for the first four significant peaks. These include
the frequency of 9.39/year, which is close to 9.43/year previously found to be
marginally significant in Desai and Liu (2016). This table containing the L-S
powers and the $p$-values (using all the four methods outlined in Sect. II)
for the top four frequencies with the largest L-S powers can be found in
Tables 1 and 2, respectively. We find that for both the corrected as well as
uncorrected flux, we do not find any statistically significant peaks. All the
observed $p$-values (using any method) are greater than 0.4. For the
uncorrected flux, the maximum power is seen at a frequency of 1.06/year with
the lowest $p$-value of 0.47 using the Naive method. For the flux, scaled by
the square of the distance, the maximum power is seen at a frequency of
9.39/year (which is close to the frequency of 9.43/year, which we had
previously found in Super-K-I data Desai and Liu (2016)). However, its
statistical significance is negligible (with the minimum $p$-value of 0.71).
This is in contrast to the value of 0.015 we had found for Super-K-1.
Therefore, we conclude that with the latest updated data, the marginally
significant peak close to the frequency of 9.43/year using the first five
years of data has disappeared. Since no other significant peaks were found, we
therefore assert that there are no periodic sinusoidal signals at any
frequencies in the 8B solar neutrino flux.  We also applied the generalized
L-S periodogram to all the Super-K data post July 2001 (which was not included
in the earlier analysis) and calculated the local $p-$value for the frequency
of 9.39/year. We find the $p$-values at this frequency to be equal to 0.074
(1.4$\sigma$) and 0.056 (1.6$\sigma$) for the uncorrected and corrected
fluxes, respectively. Therefore, the local $p-$values are not statistically
significant for the frequency of 9.39/year, when analyzing the data after the
phase-I of Super-K. For the frequency of 1/year, the local $p-$value is equal
to 0.001 and 0.54 for the uncorrected and corrected flux, respectively using
the full 22 years of Super-K data. Therefore, the uncorrected flux corresponds
to a local significance of $3.1\sigma$. Once we correct for the Earth’s
eccentricity, the significance becomes negligible.

To summarize, we do not find evidence for periodicities at any frequency using
22 years of Super-K data, in accord with the results found in Abe et al.
(2023).

Frequency [year-1] | L-S Power | Bootstrap | Naive | Baluev | Davies
---|---|---|---|---|---
1.06 | 0.013 | 0.92 | 0.47 | 0.84 |  $>$ 1
9.39 | 0.011 | 0.99 | 0.86 | 0.99 |  $>$ 1
15.05 | 0.010 | 1.000 | 0.98 | 1.000 |  $>$ 1
18.76 | 0.008 | 1.000 | 1.000 | 1.000 |  $>$ 1
Table 1: L-S power and $p$-value (last four columns) for the uncorrected 8B Super-K flux using four different methods for the four frequencies showing the largest L-S powers in descending order. All the $p$-values are greater than 0.5, implying that none of the peaks are statistically significant, and there is no evidence for sinusoidal modulations in the data. Frequency [year-1] | L-S Power | Bootstrap | Naive | Baluev | Davies
---|---|---|---|---|---
9.39 | 0.012 | 0.98 | 0.71 | 0.971 |  $>$ 1
15.05 | 0.010 | 1.000 | 0.96 | 1.000 |  $>$ 1
0.28 | 0.008 | 1.000 | 1.000 | 1.000 |  $>$ 1
18.76 | 0.008 | 1.000 | 1.000 | 1.000 |  $>$ 1
Table 2: L-S power and $p$-value (last four columns) for the rescaled 8B
Super-K flux after correcting for the eccentricity of the Earth’s orbit, using
four different methods for the four frequencies showing the largest L-S powers
in descending order. Similar to the uncorrected flux, all the $p$-values are
greater than 0.5, implying that none of the peaks are statistically
significant, and there is no evidence for sinusoidal modulations in the data.
Figure 2: L-S power as a function of frequency using the generalized L-S
periodogram for the raw flux (top panel) and the corrected flux (bottom
panel).

##  V Conclusions

Multiple groups have found evidence for sinusoidal modulations in the 8B solar
neutrino flux observed using the first five years of Super-K data at a
frequency of around 9.43/year, corresponding to a period of 38 days. These
peaks were asserted to be due to the synodic rotation of the solar core. Our
own analysis (in 2016) of this data using the generalized L-S periodogram
found a $p$-value of around 0.015. In November 2023, the Super-K collaboration
updated its results for periodicity searches using 22 years of data using two
independent methods, one of which includes the (normal) L-S analysis. This
work did not find evidence for any statistically significant peak (apart from
the variation due to Earth’s orbit around the Sun) Abe et al. (2023). The
dataset used for this analysis was also made publicly available.

We carried out an independent search for periodicity with the same data using
the generalized L-S periodogram to ascertain if the updated data again
contains a modulation at or near 9.43/year, and if its detection significance
is enhanced with five times more data. We analyzed both the raw solar neutrino
fluxes and also the fluxes rescaled by the square of the distance between the
Earth and the Sun. We estimated the $p$-values using four independent methods.
Our plots for the generalized L-S power as a function of frequency can be
found in Fig. 2. The $p$-values we found for the top four frequencies in
descending order of their L-S powers can be found in Table 1 and Table 2,
respectively. We no longer find any statistically significant peak at a
frequency close to 9.43/year. For the corrected flux, the minimum $p$-value we
found was equal to 0.71 (at a frequency of 9.39/year), which is not
significant. Therefore, we conclude that the entire 22 years of Super-K data
no longer contains any signatures of the sinusoidal modulations close to a
frequency of 9.43/year (period of 38 days) or any other frequency. Our results
are also in accord with the corresponding analyses carried out by the Super-K
Collaboration.

In the spirit of open science, we have made our analysis codes for this work
publicly available, which can be found at
https://github.com/DarkWake9/Project-LP.

## Acknowledgements

We are grateful to the Super-K collaboration for making their solar neutrino
dataset publicly available, to Jonghee Yoo for clarifications about the
released dataset and to the anonymous referees for very useful feedback on the
manuscript.

## References

  * Yoo et al. (2003) J. Yoo et al. (Super-Kamiokande), Phys. Rev. D 68, 092002 (2003), eprint hep-ex/0307070.
  * Sturrock (2022) P. A. Sturrock, Space Science Reviews 218, 23 (2022), eprint 2203.05069.
  * Sturrock (2008) P. A. Sturrock, Solar Physics 252, 221 (2008), eprint 0802.3399.
  * Sturrock et al. (2012) P. A. Sturrock, A. G. Parkhomov, E. Fischbach, and J. H. Jenkins, Astroparticle Physics 35, 755 (2012), eprint 1203.3107.
  * Sturrock et al. (2005) P. A. Sturrock, D. O. Caldwell, J. D. Scargle, and M. S. Wheatland, Phys. Rev. D 72, 113004 (2005), eprint hep-ph/0501205.
  * Ranucci (2006) G. Ranucci, Phys. Rev. D 73, 103003 (2006), eprint hep-ph/0511026.
  * Lomb (1976) N. R. Lomb, Astrophysics and Space Sciences 39, 447 (1976).
  * Scargle (1982) J. D. Scargle, Astrophys. J. 263, 835 (1982).
  * Zechmeister and Kürster (2009) M. Zechmeister and M. Kürster, Astron. & Astrophys. 496, 577 (2009), eprint 0901.2573.
  * Desai and Liu (2016) S. Desai and D. W. Liu, Astroparticle Physics 82, 86 (2016), eprint 1604.06758.
  * Krishak and Desai (2020) A. Krishak and S. Desai, JCAP 2020, 006 (2020), eprint 2003.10127.
  * Pommé et al. (2017) S. Pommé, K. Kossert, and O. Nähle, Solar Physics 292, 162 (2017).
  * Pommé et al. (2018) S. Pommé, H. Stroh, T. Altzitzoglou, J. Paepen, R. Van Ammel, K. Kossert, O. Nähle, J. D. Keightley, K. M. Ferreira, L. Verheyen, et al., Applied Radiation and Isotopes 134, 6 (2018).
  * Pommé and Pelczar (2022) S. Pommé and K. Pelczar, Space Science Reviews 218, 64 (2022).
  * Pulido (1994) J. Pulido, Physics Letters B 323, 36 (1994).
  * Akhmedov and Pulido (2003) E. K. Akhmedov and J. Pulido, Physics Letters B 553, 7 (2003), eprint hep-ph/0209192.
  * Abe et al. (2023) K. Abe et al. (Super-Kamiokande) (2023), eprint 2311.01159.
  * Raychaudhuri (1971) P. Raychaudhuri, Astrophysics and Space Sciences 13, 231 (1971).
  * Tejas and Desai (2018) P. Tejas and S. Desai, European Physical Journal C 78, 554 (2018), eprint 1801.03236.
  * Dhaygude and Desai (2020) A. Dhaygude and S. Desai, European Physical Journal C 80, 96 (2020), eprint 1912.06970.
  * Gururajan and Desai (2020) G. Gururajan and S. Desai, European Physical Journal C 80, 1071 (2020), eprint 2009.12906.
  * VanderPlas (2018) J. T. VanderPlas, Astrophys. J. Suppl. Ser. 236, 16 (2018), eprint 1703.09824.
  * VanderPlas and Ivezić (2015) J. T. VanderPlas and Ž. Ivezić, Astrophys. J. 812, 18 (2015), eprint 1502.01344.
  * Ivezić et al. (2014) Ž. Ivezić, A. Connolly, J. Vanderplas, and A. Gray, _Statistics, Data Mining and Machine Learning in Astronomy_ (Princeton University Press, 2014).
  * Bretthorst (2001) G. L. Bretthorst, in _Bayesian Inference and Maximum Entropy Methods in Science and Engineering_ , edited by A. Mohammad-Djafari (2001), vol. 568 of _American Institute of Physics Conference Series_ , pp. 246–251.
  * Vanderplas et al. (2012) J. Vanderplas, A. Connolly, Ž. Ivezić, and A. Gray, in _Conference on Intelligent Data Understanding (CIDU)_ (2012), pp. 47 –54.
  * Pommé (2020) S. Pommé, Nuclear Instruments and Methods in Physics Research A 968, 163933 (2020).
  * Pommé and De Hauwere (2020) S. Pommé and T. De Hauwere, Nuclear Instruments and Methods in Physics Research A 956, 163377 (2020).
  * Baluev (2008) R. V. Baluev, MNRAS 385, 1279 (2008), eprint 0711.0330.
  * Davies (2002) R. B. Davies, Biometrika pp. 484–489 (2002).
  * Gross and Vitells (2010) E. Gross and O. Vitells, European Physical Journal C 70, 525 (2010), eprint 1005.1891.
  * Ganguly and Desai (2017) S. Ganguly and S. Desai, Astroparticle Physics C94, 17 (2017), eprint 1706.01202.
  * Lyons (2013) L. Lyons, arXiv e-prints arXiv:1310.1284 (2013), eprint 1310.1284.
  * Fukuda et al. (1999) Y. Fukuda et al. (Super-Kamiokande), Phys. Rev. Lett. 82, 2430 (1999), eprint hep-ex/9812011.
  * Desai et al. (2008) S. Desai et al. (Super-Kamiokande), Astropart. Phys. 29, 42 (2008), eprint 0711.0053.
  * Fukuda et al. (1998) Y. Fukuda et al. (Super-Kamiokande), Phys. Rev. Lett. 81, 1562 (1998), eprint hep-ex/9807003.
  * Desai et al. (2004) S. Desai et al. (Super-Kamiokande), Phys. Rev. D 70, 083523 (2004), eprint hep-ex/0404025.
  * Press et al. (1992) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, _Numerical recipes in FORTRAN. The art of scientific computing_ (1992).
  * Astropy Collaboration et al. (2022) Astropy Collaboration, A. M. Price-Whelan, P. L. Lim, N. Earl, N. Starkman, L. Bradley, D. L. Shupe, A. A. Patil, L. Corrales, C. E. Brasseur, et al., Astrophys. J. 935, 167 (2022), eprint 2206.14220.

## APPENDIX

In order to provide a direct comparison with the results using data only from
the phase-I of Super-K, we redo the same analysis as done in the main paper
using data only up to July 2001. For this purpose, we use the same data
provided in Abe et al. (2023), but culled all the data after July 2001, and
used the same correction factors to correct for Earth’s orbit around the Sun,
which we had calculated. Our results for the same can be found in Fig 3, both
with and without the correction factors. Note that we had analyzed the same
data (based on the data and correcton factors provided in Yoo et al. (2003))
in Desai and Liu (2016) using the generalized LS implemenation in the astroML
library.) Once again, the highest peak is seen at a frequency of 9.43/year.
The $p-$values using all the four methods are shown in Table 3. The minimum
$p-$ value is found for the Naive method with a $p$-value of 0.028 (after
correcting for the distance). This corresponds to a $Z$-score of 1.9$\sigma$.
This is close to (but slightly larger than) the value of 0.015 we had found in
Desai and Liu (2016).

Figure 3: L-S Periodogram of Super-K data from April 1996 to July 2001 | Frequency [year-1] | L-S Power | Bootstrap | Naive | Baluev | Davies
---|---|---|---|---|---|---
Not Corrected Flux | 9.43 | 0.056 | 0.17 | 0.045 | 0.14 | 0.15
Corrected Flux | 9.43 | 0.058 | 0.107 | 0.028 | 0.091 | 0.096
Table 3: L-S power and $p$-value (last four columns) for 8B Super-K flux from
April 1996 to July 2001 using four different methods for the four frequencies
showing the largest L-S powers.