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The Effect of Magnetic Variability on Stellar Angular Momentum Loss I: The Solar Wind Torque During Sunspot Cycles 23 & 24

TL;DR: In this paper, the angular momentum loss in the solar wind was calculated using MHD simulations from Finley & Matt (2018), with one using the open flux measured in the Sun-like wind, and the other using remotely-observed surface magnetograms.
Abstract: The rotational evolution of cool stars is governed by magnetised stellar winds which slow the stellar rotation during their main sequence lifetimes. Magnetic variability is commonly observed in Sun-like stars, and the changing strength and topology of the global field is expected to affect the torque exerted by the stellar wind. We present three different methods for computing the angular momentum loss in the solar wind. Two are based on MHD simulations from Finley & Matt (2018), with one using the open flux measured in the solar wind, and the other using remotely-observed surface magnetograms. Both methods agree in the variation of the solar torque seen through the solar cycle and show a 30-40% decrease from cycle 23 to 24. The two methods calculate different average values, $2.9\times10^{30}$erg (open flux) and $0.35\times10^{30}$erg (surface field). This discrepancy results from the already well-known difficulty with reconciling the magnetograms with observed open flux, which is currently not understood, leading to an inability to discriminate between these two calculated torques. The third method is based on the observed spin-rates of Sun-like stars, which decrease with age, directly probing the average angular momentum loss. This method gives $6.2\times10^{30}$erg for the solar torque, larger than the other methods. This may be indicative of further variability in the solar torque on timescales much longer than the magnetic cycle. We discuss the implications for applying the formula to other Sun-like stars, where only surface field measurements are available, and where the magnetic variations are ill-constrained.

Summary (2 min read)

1. INTRODUCTION

  • Angular momentum loss through stellar winds explains the rotational evolution of low mass stars (M∗ ≤ 1.3M ) on the main sequence.
  • This allows, for the first time, a more continuous calculation of the angular momentum loss rate.
  • Using the multitude of current observations of the Sun (this work), and multi-epoch studies of other stars from the ZDI community (Paper II), the authors can now evaluate the variation of stellar wind torques over decadal timescales.

2. SEMI-ANALYTIC TORQUE FORMULATIONS

  • FM18 provides semi-analytic prescriptions for the angular momentum loss rate based on over 160 stellar wind simulations using the PLUTO magnetohydrodynamics (MHD) code (Mignone et al. 2007; Mignone 2009).
  • As discussed in Pantolmos & Matt (2017) variations in the chosen wind speed, i.e. a wind comprised of all slow or all fast wind, differ by a factor of ∼ 2 in the predicted torque.
  • For this work, the authors adopt the parameters derived originally in FM18, with a temperature between the extremes (see Pantolmos & Matt 2017), and accept potential discrepancies in the wind acceleration over the solar cycle.
  • 2. Formulation Using Open Magnetic Flux Réville et al. (2015a) show that by parametrising the relationship for the average Alfvén radius in terms of the open magnetic flux, φopen, a scaling behaviour independent of magnetic geometry can be formulated.
  • The simplicity of the semi-analytic derivation for the open flux torque formulation (see Pantolmos & Matt 2017) suggests that this method produces the most reliable torque for a given estimate of the open flux.

3. OBSERVED SOLAR WIND PARAMETERS

  • Information regarding the magnetic properties of the Sun are used here in two forms.
  • Measurements of the solar wind speed and density are also made in-situ by multiple spacecrafts, but here the authors focus on results from Ulysses and ACE.
  • During the 22 years this averages to removing ∼ 1% of the data from each 27-day bin.
  • This process produces complex coefficients αlm, which weight each of the spherical harmonic modes, Br(θ, φ) = l=lmax∑ l=0 m=l∑ m=−l αlmY l m(θ, φ), (7) where θ and φ represent the co-latitude and longitude of the magnetograms respectively.
  • 2. Mass Loss Rates and Magnetic Open Flux Variability From ACE/Ulysses.

4. EVALUATING THE SOLAR WIND ANGULAR MOMENTUM LOSS RATE

  • Here the authors consider three methods for determining the angular momentum loss in the solar wind.
  • The authors aim to characterise any difference between these torque predictions, and attempt to determine the most accurate estimate of the solar wind torque and its variability.
  • The average torque predicted using the open flux method is 2.28 × 1030erg, which is 3.26 times greater than the surface field method in the previous section.
  • Shown in both the observations from ACE and Ulysses.

5. DISCUSSION

  • Using the torque formulations from FM18, the value of the solar wind torque is shown to be lower than the empirical estimate based on the rotation of other Sunlike stars.
  • The authors also find a disagreement between the two predictions from FM18, using either the surface or open flux method for calculating the torque.
  • It is therefore generally accepted that magnetograms require multiplication by an uncertain factor or the inclusion of additional magnetic flux (typically coronal mass ejections or small scale surface fields) in order to bring observations in-line with the extrapolated field strength at 1AU (Wang 1993; Zhao & Hoeksema 1995; Cohen et al.
  • Torques derived using stellar magnetic field observations and equation (3) may be lower than in actuality, due to the FM18 model producing a smaller value of unsigned open flux than measured in the solar wind.

6. CONCLUSION

  • In this work the authors have utilised the wealth of current solar observations and the semi-analytic results from FM18 to produce an estimate of the current solar wind torque.
  • The authors thank the ACE MAG and SWEPAM instrument teams and the ACE Science Center for providing the ACE data.

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ORE Open Research Exeter
TITLE
The effect of magnetic variability on stellar angular momentum loss. I. the solar wind torque during
sunspot cycles 23 and 24
AUTHORS
Finley, AJ; Matt, SP; See, V
JOURNAL
Astrophysical Journal
DEPOSITED IN ORE
08 March 2019
This version available at
http://hdl.handle.net/10871/36356
COPYRIGHT AND REUSE
Open Research Exeter makes this work available in accordance with publisher policies.
A NOTE ON VERSIONS
The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of
publication

Accepted for publication in The Astrophysical Journal: July 31, 2018
Preprint typeset using L
A
T
E
X style emulateapj v. 12/16/11
THE EFFECT OF MAGNETIC VARIABILITY ON STELLAR ANGULAR MOMENTUM LOSS I:
THE SOLAR WIND TORQUE DURING SUNSPOT CYCLES 23 & 24
Adam J. Finley*, Sean P. Matt & Victor See
University of Exeter, Devon, Exeter, EX4 4QL, UK
Accepted for publication in The Astrophysical Journal: July 31, 2018
ABSTRACT
The rotational evolution of cool stars is governed by magnetised stellar winds which slow the stellar
rotation during their main sequence lifetimes. Magnetic variability is commonly observed in Sun-like
stars, and the changing strength and topology of the global field is expected to affect the torque exerted
by the stellar wind. We present three different methods for computing the angular momentum loss in
the solar wind. Two are based on MHD simulations from Finley & Matt (2018), with one using the
open flux measured in the solar wind, and the other using remotely-observed surface magnetograms.
Both methods agree in the variation of the solar torque seen through the solar cycle and show a
30 40% decrease from cycle 23 to 24. The two methods calculate different average values, 2.9 × 10
30
erg (open flux) and 0.35 × 10
30
erg (surface field). This discrepancy results from the already well-
known difficulty with reconciling the magnetograms with observed open flux, which is currently not
understood, leading to an inability to discriminate between these two calculated torques. The third
method is based on the observed spin-rates of Sun-like stars, which decrease with age, directly probing
the average angular momentum loss. This method gives 6.2 × 10
30
erg for the solar torque, larger
than the other methods. This may be indicative of further variability in the solar torque on timescales
much longer than the magnetic cycle. We discuss the implications for applying the formula to other
Sun-like stars, where only surface field measurements are available, and where the magnetic variations
are ill-constrained.
Keywords: magnetohydrodynamics (MHD) - stars: low-mass - stars: stellar winds, outflows - stars:
magnetic field- stars: rotation, evolution
1. INTRODUCTION
Angular momentum loss through stellar winds explains
the rotational evolution of low mass stars (M
1.3M
)
on the main sequence. These stars are shown to have
outer convection zones (Marcy 1984; Donati et al. 2006;
Morin et al. 2008; Donati et al. 2008; Petit et al. 2008;
Morgenthaler et al. 2011; Gregory et al. 2012; Reiners
2012; Folsom et al. 2016, 2017), which are able to sup-
port magnetic fields through the interplay of rotation and
convection, forming a dynamo (Brun & Browning 2017).
The magnetic field generation of such dynamos is linked
with rotation (Browning 2008; Reiners et al. 2009; Rein-
ers & Basri 2010; Vidotto et al. 2014; See et al. 2015;
Shulyak et al. 2017), such that a faster rotator will, in
general, produce a larger field strength. Stellar winds are
found to be more effective at slowing rotation in the pres-
ence of these large scale magnetic field (Weber & Davis
1967; Mestel 1968; Keppens & Goedbloed 2000; Matt
et al. 2012; Garraffo et al. 2015; eville et al. 2015a).
Therefore, the relation of stellar rotation, magnetism and
angular momentum loss leads to the convergence of ro-
tation periods at late ages (Skumanich 1972; Soderblom
1983; Barnes 2003, 2010; Delorme et al. 2011; Van Saders
& Pinsonneault 2013; Bouvier et al. 2014).
Observations of the rotation rates of stars at different
ages, and our knowledge of stellar structure, also give
us direct constraints on the total external torque on the
star. This value is independent from any knowledge of
the physical mechanism for that angular momentum loss,
but it probes only a long-time average torque (i.e., only
*af472@exeter.ac.uk
on timescales smaller than the spin-down time, which
can be in the range of tens to hundreds of Myr for main
sequence stars). With the increasing number of accu-
rate rotation period measurements available to compare
with model results (e.g. Ag¨ueros et al. 2011; McQuil-
lan et al. 2013; N´u˜nez et al. 2015; Rebull et al. 2016;
Covey et al. 2016; Douglas et al. 2017; Ag¨ueros 2017),
we are able to examine the physical mechanisms of stel-
lar wind braking in greater detail (Irwin & Bouvier 2009;
Bouvier et al. 2014). A variety of spin evolution models
have been developed to date (e.g. Gallet & Bouvier 2013;
Van Saders & Pinsonneault 2013; Gallet & Bouvier 2015;
Johnstone et al. 2015; Matt et al. 2015; Amard et al.
2016; Sadeghi Ardestani et al. 2017; See et al. 2018),
which relate basic stellar properties; mass, radius, ro-
tation period, field strength and mass loss rate, with re-
sults from analytic or numerical models for the spin down
torque applied to the star, and the subsequent redistri-
bution of internal angular momentum.
Stellar mass and radius remain essentially constant
throughout the main sequence. However, in addition to
the long-time secular changes of the magnetic field due to
rotation, magnetic activity is also observed to vary sig-
nificantly over timescales of years to decades (Baliunas
et al. 1995; Azizi & Mirtorabi 2017). This is routinely
observed for the Sun which is known to have a mag-
netic activity cycle (Babcock 1961; Wilcox & Scherrer
1972; Willson & Hudson 1991; Guedel et al. 1997; G¨udel
2007; Schrijver & Liu 2008), moving from an activity
maximum through minimum and back to maximum in
roughly 11 years. The Sun’s cyclic behaviour is appar-
ent in changes to the large scale magnetic field (DeRosa
arXiv:1808.00063v3 [astro-ph.SR] 23 Aug 2018

2 A. Finley, S. Matt & V. See
et al. 2012), which significantly modifies the solar wind
structure and outflow properties (Smith & Balogh 1995;
McComas et al. 2000; Wang et al. 2000; Tokumaru et al.
2010). Activity cycles on other stars are quantified using
activity proxies such as the long term monitoring of Ca II
HK emission (Baliunas et al. 1995; Egeland et al. 2017),
observed lightcurve modulation due to star spots (Lock-
wood et al. 2007), X-ray activity (Hempelmann et al.
1996) and more recently Zeeman Doppler Imaging, ZDI
(Semel 1989; Donati et al. 1989; Brown et al. 1991; Do-
nati & Brown 1997). The mass loss rate of the Sun is
shown to vary with the magnetic cycle (McComas et al.
2013) and is fundamentally connected with magnetic ac-
tivity (Cranmer et al. 2007). This behaviour is expected
to be similar for other low mass stars.
Previous theoretical studies have shown the variation
in angular momentum loss over magnetic cycles (Pinto
et al. 2011; Garraffo et al. 2015; R´eville et al. 2015b;
Alvarado-G´omez et al. 2016; R´eville & Brun 2017). How-
ever they require costly MHD simulations which attempt
to simultaneously fit the mass loss rate and magnetic field
strengths for single epochs. In contrast, by utilsing stel-
lar wind braking formulations from eville et al. (2015a),
Finley & Matt (2017), Pantolmos & Matt (2017) and
Finley & Matt (2018), hereafter FM18, which can eas-
ily predict the torque for any known mass loss rate and
magnetic field strength/geometry. This allows, for the
first time, a more continuous calculation of the angular
momentum loss rate.
Using the multitude of current observations of the Sun
(this work), and multi-epoch studies of other stars from
the ZDI community (Paper II), we can now evaluate the
variation of stellar wind torques over decadal timescales.
We briefly reiterate the angular momentum loss prescrip-
tions from FM18 in Section 2, collate solar observations
in Section 3, and implement them in Section 4 to pro-
duce the most up-to-date determination of the solar brak-
ing torque, using methods based on the surface magne-
togram data obtained from SOHO/MDI and SDO/HMI,
and evaluating the open magnetic flux from the Ulysses
and the Advanced Composition Explorer (ACE) space-
crafts, along with an estimate based on the rotational
behaviour of other Sun-like stars. Section 5 then dis-
cusses our result and addresses the observed discrepancy
between surface field and open flux methods, along with
the differences between our torque value and the derived
long-time average result.
Previous theoretical studies have shown the variation
in angular momentum loss over magnetic cycles (Pinto
et al. 2011; Garraffo et al. 2015; eville et al. 2015b;
Alvarado-G´omez et al. 2016; R´eville & Brun 2017). How-
ever they require costly MHD simulations which attempt
to simultaneously fit the mass loss rate and magnetic field
strengths for single epochs. By contrast, using the stellar
wind braking formulations from eville et al. (2015a),
Finley & Matt (2017), Pantolmos & Matt (2017) and
Finley & Matt (2018), hereafter FM18, which can eas-
ily predict the torque for any known mass loss rate and
magnetic field strength/geometry, without need for new
simulations. This allows, for the first time, a more con-
tinuous calculation of the angular momentum loss rate.
Using the multitude of current observations of the Sun
(this work), and multi-epoch studies of other stars from
the ZDI community (Paper II), we can now evaluate the
variation of stellar wind torques over decadal timescales.
We briefly reiterate the angular momentum loss prescrip-
tions from FM18 in Section 2, collate solar observations
in Section 3, and implement them in Section 4 to pro-
duce the most up-to-date determination of the solar brak-
ing torque, using methods based on the surface magne-
togram data obtained from SOHO/MDI and SDO/HMI,
and evaluating the open magnetic flux from the Ulysses
and the Advanced Composition Explorer (ACE) space-
crafts, along with an estimate based on the rotational
behaviour of other Sun-like stars. Section 5 then dis-
cusses our result and addresses the observed discrepancy
between surface field and open flux methods, along with
the differences between our torque value and the derived
long-time average result.
2. SEMI-ANALYTIC TORQUE FORMULATIONS
FM18 provides semi-analytic prescriptions for the an-
gular momentum loss rate based on over 160 stellar wind
simulations using the PLUTO magnetohydrodynamics
(MHD) code (Mignone et al. 2007; Mignone 2009). The
simulations in FM18 use a polytropic equation of state,
which equates to a thermally driven wind with a coronal
temperature of 1.7MK for the Sun, and a polytropic in-
dex of γ = 1.05, which is nearly isothermal. The use of
a nearly isothermal wind leads to some discrepancy with
the observed multi-speed solar wind, which is known to
be bimodal in nature (Ebert et al. 2009). Nevertheless,
work by Pantolmos & Matt (2017) has shown changes to
this assumed global wind acceleration can be understood
within these models, and have a well described impact
on our result.
As discussed in Pantolmos & Matt (2017) variations
in the chosen wind speed, i.e. a wind comprised of all
slow or all fast wind, differ by a factor of 2 in the
predicted torque. In reality the solar wind is comprised
of both components, with the relative fraction of slow
and fast wind changing with magnetic activity, which
means the true torque is between these two extremes.
For this work, we adopt the parameters derived originally
in FM18, with a temperature between the extremes (see
Pantolmos & Matt 2017), and accept potential discrep-
ancies in the wind acceleration over the solar cycle.
The simulations of FM18 are axisymmetric, so derived
torques neglect 3D effects as observed in the simulations
of R´eville & Brun (2017). The advantage of these formu-
lations is that calculations can be performed much faster
than MHD simulations. This allows us to use all the
available data to produce the most coherent picture of
solar angular momentum loss over the last 22 years.
The torque, τ, due to the solar wind is then given by,
τ =
˙
M
R
2
hR
A
i
R
2
, (1)
where,
˙
M is the solar wind mass loss rate, the stellar
rotation rate is assumed to be solid body (no differential
rotation) with
=
= 2.6 × 10
6
rad/s, and R
is
the stellar radius for which we adopt R
= R
= 6.96 ×
10
10
cm. As with previous torque formulations, equation
(1) defines the average Alfv´en radius, hR
A
i, to behave
as a lever arm, or efficiency factor for the stellar wind in
braking the stellar rotation (Weber & Davis 1967; Mestel
1968).

Solar Torque Variability 3
2.1. Formulation Using Surface Magnetic Field
In equation (1) the torque depends on the average
Alfv´en radius. Simulations of FM18 showed that hR
A
i
can be predicted using the wind magnetisation parame-
ter,
Υ =
B
2
R
2
˙
Mv
esc
, (2)
where the total axisymmetric field strength is evaluated
using the polar field strengths from the lowest order
modes B
= |B
l=1
| + |B
l=2
| + |B
l=3
| ( l is the magnetic
order, for which 1, 2 and 3 correspond to the dipole,
quadrupole and octupole modes respectively), and the
escape velocity is given by v
esc
=
p
2GM
/R
, for which
we adopt M
= M
= 1.99 × 10
33
g.
For mixed geometry axisymmetric fields, the average
simulated Alfv´en radius is found to behave as a broken
power law of the form,
hR
A
i
R
= max
(
K
dip
[R
2
dip
Υ]
m
dip
,
K
quad
[(|R
dip
| + |R
quad
|)
2
Υ]
m
quad
,
K
oct
[(|R
dip
| + |R
quad
| + |R
oct
|)
2
Υ]
m
oct
,
(3)
which approximates the stellar wind solutions from
FM18, using K
dip
= 1.53, K
quad
= 1.70, K
oct
= 1.80,
m
dip
= 0.229, m
quad
= 0.134, and m
oct
= 0.087. The
variables describing the field geometry, R
dip
, R
quad
, and
R
oct
are defined as the ratios of the polar strengths of
each mode over the total; i.e., R
dip
= B
l=1
/B
, etc.
The scaling of equation (3) is such that for most field
strengths, in the solar case, we find that the dipole only
term dominates the angular momentum loss (i.e. the
dipole-only formulation of Matt et al. 2012 holds).
2.2. Formulation Using Open Magnetic Flux
R´eville et al. (2015a) show that by parametrising the
relationship for the average Alfv´en radius in terms of the
open magnetic flux, φ
open
, a scaling behaviour indepen-
dent of magnetic geometry can be formulated. Such a
general formula for the torque is very useful. However,
the open magnetic flux cannot be observed for other stars
than the Sun. We define the unsigned open flux as,
φ
open
=
I
A
|B · dA|, (4)
where A is a closed spherical surface located outside of
the last closed field loop, i.e. in the magnetically open
wind. The wind can then be parametrised with the open
flux wind magnetisation,
Υ
open
=
φ
2
open
/R
2
˙
Mv
esc
, (5)
and the average Alfv´en radius given by,
hR
A
i
R
= K
o
open
]
m
o
, (6)
where, from FM18, K
o
= 0.33 and m
o
= 0.371. Here
we assume the dipolar coefficients as the dipolar fraction
of the total field R
dip
remains significant throughout the
solar cycle, with few exceptions.
The simplicity of the semi-analytic derivation for the
open flux torque formulation (see Pantolmos & Matt
2017) suggests that this method produces the most reli-
able torque for a given estimate of the open flux. This
method is insensitive to surface geometry and any details
of how the field is opened. The only factors that cause
the angular momentum to deviate from this formulation
is the wind acceleration and the 3D structure of the mass
flux.
3. OBSERVED SOLAR WIND PARAMETERS
Information regarding the magnetic properties of the
Sun are used here in two forms. Firstly, synoptic magne-
tograms of the surface magnetic field produced by both
the Michelson Doppler Imager on-board the Solar and
Heliospheric Observatory (SOHO/MDI) and the Helio-
seismic and Magnetic Imager on-board the Solar Dy-
namic Observatory (SDO/HMI), from which we calcu-
late time-varying magnetic field strengths for the dipole,
quadrupole and octupole field components. Secondly,
measurements of the interplanetary magnetic field (IMF)
strength are taken in-situ by the Ulysses and ACE space-
crafts, which we use to produce an estimate of the time-
varying solar open flux. Measurements of the solar wind
speed and density are also made in-situ by multiple
spacecrafts, but here we focus on results from Ulysses
and ACE.
During the calculation of our solar wind quantities,
we perform 27-day averages to remove any longitudinal
variation and produce more representative values for the
global wind. In doing this we have removed information
of any temporal or spatial variation on smaller scales
than this, which has been shown by previous authors
(e.g. DeForest et al. 2014).
An additional complication arises from Coronal Mass
Ejections (CMEs), or Interplanetary CMEs (ICMEs) as
they arrive at the spacecraft detectors. ICMEs are ob-
served in the data as impulsive increases in the in-situ
solar wind properties. CMEs occur on average up to 5
times a day at solar maximum and 1 every 2-3 days at so-
lar minimum (Webb et al. 2017). Some authors have re-
moved these events from their datasets (e.g. Cohen 2011)
using CME catalogues (e.g. Cane & Richardson 2003)
and by identifying anomalous spikes. CMEs carry only
a few percent of the total mass loss rate which is mainly
located near the maximum of activity, and due to the
distribution of their ejection trajectories into the Helio-
sphere, a reduced fraction of these events are recorded at
the in-situ detectors.
In order to gauge the impact of the enhanced mag-
netic field strengths and densities carried by ICMEs, we
re-ran the analysis, removing periods when the wind den-
sity and field strength are greater than 10cm
3
and 10nT
respectively from the hourly spacecraft data (as done for
Ulysses by Cohen 2011). This results in 3% of the
hourly data being cut in each 27-day average at solar
maximum, and 0% at the minimum. During the 22
years this averages to removing 1% of the data from
each 27-day bin. We find by removing the ICMEs the av-
erage open flux and mass loss rate we derive are reduced
by 4%. However, as CMEs should have a contribution
to the total torque we prefer to include these events in our
derived mass loss rate and open flux, even though there
is not yet a model to show how their angular momentum
loss per mass loss rate may be different than that of a
steady global wind (see, e.g. Aarnio et al. 2012). As such

4 A. Finley, S. Matt & V. See
the results presented in the remainder of this work use
the full, unclipped data set.
3.1. Surface Magnetic Variability From SOHO/MDI
and SDO/HMI
Using synoptic magnetograms taken from MDI and
HMI
1
, the complete radial surface magnetic field
strength, B
r
, is recorded over the past 22 years for each
Carrington-rotation (CR)
2
from July 1996 (CR 1910) to
present. Both instrument teams provide polar field cor-
rected data sets (Sun et al. 2011; Sun 2018), accounting
for projection effects on the line of sight magnetic field
measurements which result in a large amount of noise
at the poles, along with other affects such as the Sun’s
tilt angle which periodically hides these areas from view.
The two instruments observed the Sun over different time
periods with an overlap from the beginning of HMI in
May 2010 (CR 2097) until the end of MDI in December
2010 (CR 2104). Therfore the datasets have been cali-
brated to produce consistent results. For this work, we
apply a multiplicative factor to the HMI field strengths
of 1.2, as suggested by Liu et al. (2012).
We use a total of 282 synoptic magnetograms which
cover the entirety of sunspot cycle 23 (August 1996 -
December 2008, CR 1913 - CR 2078), and cycle 24 up
to January 2018 (CR 2199). These magnetograms are
decomposed into their spherical harmonic components
using the pySHTOOLS code (Wieczorek 2011). The
magnetograms require remapping from the sine-latitude
format of the observations onto an equal sampled grid,
which the code can use. Each map is then decomposed
into a set of spherical harmonic modes Y
l
m
which have
order l = 1, 2, 3, ..., l
max
(a truncation limit placed at
l
max
= 150) and degree l m l. This process pro-
duces complex coefficients α
l
m
, which weight each of the
spherical harmonic modes,
B
r
(θ, φ) =
l=l
max
X
l=0
m=l
X
m=l
α
l
m
Y
l
m
(θ, φ), (7)
where θ and φ represent the co-latitude and longitude of
the magnetograms respectively.
This method was performed by DeRosa et al. (2012)
on 36 years of observations from the Wilcox Solar Ob-
servatory and, similarly to this work, the MDI data set.
The results from our decomposition agree strongly with
the results presented in DeRosa et al. (2012) for the MDI
observations. Appendix A contains a full breakdown of
the dipole, quadrupole and octupole components we cal-
culate.
For the calculation of the solar wind torque based
on the surface field, we require the dipole (l = 1),
quadrupole (l = 2) and octupole (l = 3) component
strengths. The pySHTOOLS code produces a strength
for the axisymmetric component (m = 0) and the sub-
sequent non-axisymmetric components (0 < |m| l)
for each harmonic order l. The formulation from FM18
is produced using axisymmetric simulations only, here
we produce a combined field strength including all m,
1
http://mdi.stanford.edu/data/synoptic.html
2
There are some Carrington-rotations within the SOHO/MDI
sample that have missing data and as such, they are excluded from
our analysis.
rather than neglecting the non-axisymmetric components
(m > 0). We adopt the quadrature addition of field com-
ponents,
B
l
=
v
u
u
t
l
X
m=l
(B
l
m
)
2
, (8)
where B
l
m
= α
l
m
max(|Y
l
m
(θ, φ|), characterises the polar
field strength of each mode. This results in B
l=1
repre-
senting a combined dipole strength using all the spherical
harmonic components with l = 1 and, m = {−1, 0, 1}.
Similarly this is done for the quadrupole (B
l=2
) and oc-
tupole (B
l=3
) modes. The left 3 panels of Figure 1 show
how these combined dipole, quadrupole and octupole
field strengths (solid lines) vary as a function of time
over 22 years of magnetogram observations.
3.2. Mass Loss Rates and Magnetic Open Flux
Variability From ACE/Ulysses
Along with the magnetic properties of the Sun, the
mass loss rate is required to calculate the loss of angu-
lar momentum in the solar wind. The ACE spacecraft
3
,
has been performing in-situ monitoring of the fundamen-
tal solar wind properties since its arrival at the L
1
La-
grangian point (on the Sun-Earth line, approximately 1.5
million km from Earth) in December 1996. A global mass
loss rate is constructed from the 27-day average
4
over the
spacecraft data, assuming the observed solar wind flux
to be characteristic of the total wind (i.e the wind is
isotropically the value observed by ACE),
˙
M = 4πhR
2
v
r
(R)ρ(R)i
27-day
, (9)
where
˙
M is the observed mass loss rate, R is the radial
distance from the Sun of a given observation, v
r
is the
radial wind speed, and ρ is the mass density of the wind.
The mass loss rate produced from ACE data is shown
in the top right panel of Figure 1 using a solid black line.
The estimated mass loss rate varies between 0.432.72×
10
12
g/s, with an average value of 1.14 × 10
12
g/s, which
is consistent with previous works (Wang 1998; Cranmer
2008; Cranmer et al. 2017).
The same calculation is performed on the data avail-
able from the Ulysses spacecraft
5
, shown in light grey.
Ulysses again made in-situ observations of the solar wind.
However it took a polar orbit around the Sun with per-
ihelion at 1.35AU (Astronomical Unit) and aphelion
at 5.4AU. The spacecraft was launched in late 1990
and received a gravity assist from Jupiter in 1992 which
modified the inclination of the orbit to around 80
. No-
tably, the Ulysses spacecraft made three fast latitude
scans of the solar wind, each passing from the north pole
to the south pole in approximately a year. These passes
occurred between, August 1994 - July 1995, November
2000 - September 2001, and February 2007 - January
2008, which corresponds to periods of minimum, max-
imum and minimum solar activity respectively. These
time periods are highlighted in Figure 1 in magenta.
3
http://srl.caltech.edu/ACE/ASC/level2/
4
An averaging period of 27-days is chosen to match the average
synodic period of a Carrington rotation.
5
http://ufa.esac.esa.int/ufa/

Citations
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Journal ArticleDOI
12 Dec 2019-Nature
TL;DR: Observations of solar-wind plasma at heliocentric distances of about 35 solar radii reveal an increasing rotational component to the flow velocity of the solar wind around the Sun, peaking at 35 to 50 kilometres per second—considerably above the amplitude of the waves.
Abstract: The prediction of a supersonic solar wind1 was first confirmed by spacecraft near Earth2,3 and later by spacecraft at heliocentric distances as small as 62 solar radii4. These missions showed that plasma accelerates as it emerges from the corona, aided by unidentified processes that transport energy outwards from the Sun before depositing it in the wind. Alfvenic fluctuations are a promising candidate for such a process because they are seen in the corona and solar wind and contain considerable energy5–7. Magnetic tension forces the corona to co-rotate with the Sun, but any residual rotation far from the Sun reported until now has been much smaller than the amplitude of waves and deflections from interacting wind streams8. Here we report observations of solar-wind plasma at heliocentric distances of about 35 solar radii9–11, well within the distance at which stream interactions become important. We find that Alfven waves organize into structured velocity spikes with duration of up to minutes, which are associated with propagating S-like bends in the magnetic-field lines. We detect an increasing rotational component to the flow velocity of the solar wind around the Sun, peaking at 35 to 50 kilometres per second—considerably above the amplitude of the waves. These flows exceed classical velocity predictions of a few kilometres per second, challenging models of circulation in the corona and calling into question our understanding of how stars lose angular momentum and spin down as they age12–14. Data collected by the Parker Solar Probe in the solar corona are used to determine the organization of Alfven waves, revealing an increasing flow velocity peaking at 35–50 km s−1.

336 citations

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TL;DR: In this paper, an extended grid of state-of-the-art stellar models for low-mass stars including updated physics (nuclear reaction rates, surface boundary condition, mass-loss rate, angular momentum transport, torque and rotation-induced mixing prescriptions) is presented.
Abstract: We present an extended grid of state-of-the art stellar models for low-mass stars including updated physics (nuclear reaction rates, surface boundary condition, mass-loss rate, angular momentum transport, torque and rotation-induced mixing prescriptions). We aim at evaluating the impact of wind braking, realistic atmospheric treatment, rotation and rotation-induced mixing on the structural and rotational evolution from the pre-main sequence to the turn-off. Using the STAREVOL code, we provide an updated PMS grid. We compute stellar models for 7 different metallicities, from [Fe/H] = -1 dex to [Fe/H] = +0.3 dex with a solar composition corresponding to $Z=0.0134$. The initial stellar mass ranges from 0.2 to 1.5\Ms with extra grid refinement around one solar mass. We also provide rotating models for three different initial rotation rates (slow, median and fast) with prescriptions for the wind braking and disc-coupling timescale calibrated on observed properties of young open clusters. The rotational mixing includes an up-to-date description of the turbulence anisotropy in stably stratified regions. The overall behaviour of our models at solar metallicity -- and its constitutive physics -- is validated through a detailed comparison with a variety of distributed evolutionary tracks. The main differences arise from the choice of surface boundary conditions and initial solar composition. The models including rotation with our prescription for angular momentum extraction and self-consistent formalism for angular momentum transport are able to reproduce the rotation period distribution observed in young open clusters over a broad mass-range. These models are publicly available and may be used to analyse data coming from present and forthcoming asteroseismic and spectroscopic surveys such as Gaia, TESS and PLATO.

86 citations

Journal ArticleDOI
TL;DR: In this article, an empirical model was developed to estimate mass-loss rates via coronal mass ejections (CMEs) for solar-type pre-main-sequence (PMS) stars.
Abstract: We develop an empirical model to estimate mass-loss rates via coronal mass ejections (CMEs) for solar-type pre-main-sequence (PMS) stars. Our method estimates the CME mass-loss rate from the observed energies of PMS X-ray flares, using our empirically determined relationship between solar X-ray flare energy and CME mass: log(M_CME [g]) = 0.63 x log(E_flare [erg]) - 2.57. Using masses determined for the largest flaring magnetic structures observed on PMS stars, we suggest that this solar-calibrated relationship may hold over 10 orders of magnitude in flare energy and 7 orders of magnitude in CME mass. The total CME mass-loss rate we calculate for typical solar-type PMS stars is in the range 1e-12 to 1e-9 M_sun/yr. We then use these CME mass-loss rate estimates to infer the attendant angular momentum loss leading up to the main sequence. Assuming the CME outflow rate for a typical ~1 M_sun T Tauri star is 1e-10 M_sun/yr, as permitted by our calculations, the CME spin-down torque may influence the stellar spin evolution after an age of a few Myr.

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References
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Journal ArticleDOI
TL;DR: Matplotlib is a 2D graphics package used for Python for application development, interactive scripting, and publication-quality image generation across user interfaces and operating systems.
Abstract: Matplotlib is a 2D graphics package used for Python for application development, interactive scripting,and publication-quality image generation across user interfaces and operating systems

23,312 citations


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  • ...Figures within this work are produced using the python package matplotlib (Hunter 2007)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors presented new models for low-mass stars down to the hydrogen-burning limit that consistently couple atmosphere and interior structures, thereby superseding the widely used BCAH98 models.
Abstract: We present new models for low-mass stars down to the hydrogen-burning limit that consistently couple atmosphere and interior structures, thereby superseding the widely used BCAH98 models. The new models include updated molecular linelists and solar abundances, as well as atmospheric convection parameters calibrated on 2D/3D radiative hydrodynamics simulations. Comparison of these models with observations in various colour-magnitude diagrams for various ages shows significant improvement over previous generations of models. The new models can solve flaws that are present in the previous ones, such as the prediction of optical colours that are too blue compared to M dwarf observations. They can also reproduce the four components of the young quadruple system LkCa 3 in a colour‐magnitude diagram with one single isochrone, in contrast to any presently existing model. In this paper we also highlight the need for consistency when comparing models and observations, with the necessity of using evolutionary models and colours based on the same atmospheric structures.

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Journal ArticleDOI
TL;DR: In this paper, the authors presented new models for low-mass stars down to the hydrogen-burning limit that consistently couple atmosphere and interior structures, thereby superseding the widely used BCAH98 models.
Abstract: We present new models for low-mass stars down to the hydrogen-burning limit that consistently couple atmosphere and interior structures, thereby superseding the widely used BCAH98 models. The new models include updated molecular linelists and solar abundances, as well as atmospheric convection parameters calibrated on 2D/3D radiative hydrodynamics simulations. Comparison of these models with observations in various colour-magnitude diagrams for various ages shows significant improvement over previous generations of models. The new models can solve flaws that are present in the previous ones, such as the prediction of optical colours that are too blue compared to M dwarf observations. They can also reproduce the four components of the young quadruple system LkCa 3 in a colour-magnitude diagram with one single isochrone, in contrast to any presently existing model. In this paper we also highlight the need for consistency when comparing models and observations, with the necessity of using evolutionary models and colours based on the same atmospheric structures.

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"The Effect of Magnetic Variability ..." refers background in this paper

  • ...…= 6.2× 1030 erg ( I 6.90× 1053 g cm2 ) × ( Ω 2.6× 10−6 rad/s )( 4.55 Gyr t )( 2 p ) , (13) where we have input fiducial values for the solar moment of inertia (Baraffe et al. 2015), representative rotation rate (Snodgrass & Ulrich 1990), age (Guenther 1989), and p. Appendix B discusses the…...

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  • ...The assumption of constant moment of inertia is correct to better than 2%, for solar-mass stars in the age range from 600 Myr to that of the Sun (Baraffe et al. 2015)....

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"The Effect of Magnetic Variability ..." refers background in this paper

  • ...This is routinely observed for the Sun which is known to have a magnetic activity cycle (Babcock 1961; Wilcox & Scherrer 1972; Willson & Hudson 1991; Guedel et al. 1997; Güdel 2007; Schrijver & Liu 2008), moving from an activity maximum through minimum and back to maximum in roughly 11 years....

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Frequently Asked Questions (15)
Q1. What contributions have the authors mentioned in the paper "The effect of magnetic variability on stellar angular momentum loss i: the solar wind torque during sunspot cycles 23 & 24" ?

The authors present three different methods for computing the angular momentum loss in the solar wind. The authors discuss the implications for applying the formula to other Sun-like stars, where only surface field measurements are available, and where the magnetic variations are ill-constrained. This may be indicative of further variability in the solar torque on timescales much longer than the magnetic cycle. 

The authors need additional information to discriminate between these possibilities. The required variability of ( a ) suggests the authors should observe stars like the Sun that are on average significantly more active ( i. e., that they have larger torques ) such that the average is correct. From the dynamical models, ( b ), uncertainties remain in the wind acceleration and effects of non-axisymmetric field components which both require further study to disentangle. Using the FM18 formula, predictions of the angular momentum loss rates for these stars based on their surface measurements may be smaller than in reality. 

As the mass loss rate is typically evolved self consistently in these models, differences in the modelled torque value is often due to discrepant mass loss rates when compared to observations (as this is a challenging problem). 

In order to gain information about the mass loss rate and wind properties of these distant stars, the authors rely on proxies such as the strength of Lyman-α absorption at their astropauses (Wood 2004) and more recently the observed erosion of exoplanet atmospheres (Vidotto et al. 2011; Vidotto & Bourrier 2017). 

In general, variability in thewind temperature over the cycle will affect both torque formulas from FM18 and so represents an uncertainty on their results, i.e. for a fixed Ṁ , a faster wind will open more flux with a weaker resulting torque. 

Magnetic variability can lead to estimates of the angular momentum loss which are, in the solar case, up to a factor of ∼ 10 different from one observation to another. 

The PLUTO code is used to construct 3D wind solutions for each WSO magnetogram, this produces global values for the mass loss rate and open magnetic flux (Réville, private communication), which are used to generate a torque in the bottom panel of Figure 1. 

3. Long-time variability may also play a role, and with the difficultly ascertaining the true magnetic behaviour of other Sun-stars, i.e. if they are cyclic or stochastic, the corresponding estimate of their angular momentum loss rate may be discrepant from rotation evolution model predictions. 

In their calculation of the solar torque, based on surface magnetogram observations, the authors include the strength of the non-axisymmetric components through equation (8)which adds the components in quadrature to produce a combined strength for each mode l. 

In order to assess the impact of including the nonaxisymmetric components with equation (8), the authors performed the torque analysis using both, only the axisymmetric components, and the combined strength approach of equation (8). 

The averages of the Alfvén radii predicted from the open flux method are nearly constant between cycles, but as cycle 24 is currently moving into a minimum the average is expected to move lower as it becomes complete. 

Differences in the dynamical torque estimates for the current Sun and the long-time-average value may then be due to magnetic variation on longer timescales than the 22 year magnetic cycle. 

The torque calculated in Section 4.1 is controlled largely by the combined dipole field strength, which appears to be out of phase with solar activity, displayed in the top left panel of Figure 1 (note the use of absolute magnitude field strengths). 

Differences in the average present-day torques to the spin-evolution torques, could be due to, (a) variability on a longer timescale than probed by the presentday variability presented here (but less than a spin-down time), (b) errors in using the dynamical models inferring present-day torque, or (c) that stars spin-down significantly different than Skumanich at ages of a few to several Gyr. 

Observations of other Sunlike stars will therefore suffer from considerable uncertainty in their derived angular momentum loss rates based on a single or small number of observations.