ORE Open Research Exeter
TITLE
Direct detection of solar angular momentum loss with the wind spacecraft
AUTHORS
Finley, AJ; Hewitt, AL; Matt, SP; et al.
JOURNAL
Astrophysical Journal Letters
DEPOSITED IN ORE
25 November 2019
This version available at
http://hdl.handle.net/10871/39781
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publication
Direct Detection of Solar Angular Momentum Loss with the Wind Spacecraft
Adam J. Finley
1
, Amy L. Hewitt
1
, Sean P. Matt
1
, Mathew Owens
2
, Rui F. Pinto
3
, and Victor Réville
3
1
University of Exeter, Exeter, Devon, EX4 4QL, UK; af472@exeter.ac.uk
2
University of Reading, Reading, Berkshire, RG6 6BB, UK
3
IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, Toulouse, France
Received 2019 October 2; revised 2019 October 16; accepted 2019 October 22; published 2019 November 6
Abstract
The rate at which the solar wind extracts angular momentum (AM) from the Sun has been predicted by theoretical
models for many decades, and yet we lack a conclusive measurement from in situ observations. In this Letter we
present a new estimate of the time-varying AM flux in the equatorial solar wind, as observed by the Wind
spacecraft from 1994 to 2019. We separate the AM flux into contributions from the protons, alpha particles, and
magnetic stresses, showing that the mechanical flux in the protons is ∼3 times larger than the magnetic field
stresses. We observe the tendency for the AM flux of fast wind streams to be oppositely signed to the slow wind
streams, as noted by previous authors. From the average total flux, we estimate the global AM loss rate of the Sun
to be 3.3× 10
30
erg, which lies within the range of various magnetohydrodynamic wind models in the literature.
This AM loss rate is a factor of ∼2 weaker than required for a Skumanich-like rotation period evolution (
Wµ
*
stellar age
−1/2
), which should be considered in studies of the rotation period evolution of Sun-like stars.
Unified Astronomy Thesaurus concepts: Solar wind (1534); Solar rotation (1524); Solar evolution (1492); Stellar
evolution (1599); Stellar rotation (1629); Magnetohydrodynamics (1964)
1. Introduction
During the last ∼4 billion years, the Sun’s rotation period is
thought to have changed significantly due to the solar wind
(Gallet & Bouvier 2013, 2015; Brown 2014; Johnstone et al.
2015; Matt et al. 2015; Amard et al. 2016, 2019; Blackman &
Owen 2016; Sadeghi Ardestani et al. 2017; Garraffo et al.
2018; See et al. 2018). This process, broadly referred to as
wind braking, appears to explain the observed rotation periods
of many low-mass (i.e., 1.3M
e
), main-sequence stars
(Skumanich 1972; Soderblom 1983; Barnes 2003, 2010;
Delorme et al. 2011; Van Saders & Pinsonneault 2013; Bouvier
et al. 2014). Due to the interaction of the large-scale magnetic
field on the outflowing plasma, this process is very efficient at
removing angular momentum (AM), despite only a small
fraction of a star’s mass being lost to the stellar wind, during
the main sequence (Weber & Davis 1967; Mestel 1968;
Kawaler 1988).
Generally, the stellar magnetic field is thought of as
providing a lever arm for the wind, which many authors have
attempted to quantify using results from magnetohydrodynamic
(MHD) simulations (Matt et al. 2012 ; Garraffo et al. 2015;
Réville et al. 2015; Finley & Matt 2017, 2018; Pantolmos &
Matt 2017). However, the AM loss rates from these MHD
models have thus far been dif ficult to reconcile with the rates
required by models of rotation period evolution for low-mass
stars (Finley et al. 2018, 2019b; See et al. 2019). Since many
solar quantities are known to high precision (such as mass,
radius, rotation rate, and age), the Sun is often used to calibrate
these rotation period evolution models. However, there are
relatively few works that have attempted to model the current
AM loss rate of the Sun (e.g., Alvarado-Gómez et al. 2016;
Réville & Brun 2017; Finley et al. 2018; Usmanov et al. 2018
;
Ó Fionnagáin et al. 2019) and only a few studies that used
in situ measurements of the solar wind plasma and magnetic
field (Lazarus & Goldstein 1971; Pizzo et al. 1983; Marsch &
Richter 1984a;Li1999). Consequently, the value of the solar
AM loss rate remains uncertain, and the discrepancy between
these two approaches remains in the literature.
The most direct, previous measurement of solar AM loss was
performed using data from the two Helios spacecraft by Pizzo
et al. (1983) and Marsch & Richter (1984a). Despite requiring
significant corrections to account for errors in spacecraft
pointing, and using less than one year’s worth of data, these
authors were able to separate the individual contributions of the
protons, alpha particles, and magnetic field stresses. Interest-
ingly, they showed that the alpha particles in the solar wind had
an oppositely signed AM flux to the proton and magnetic
components. Moreover, fast–slow stream-interactions appeared
to transfer AM away from the fast component of the wind
(causing the fast wind to often carry negative AM flux, like the
alpha particles), which had also been noted by Lazarus &
Goldstein (1971). When compared, the contribution of the
protons (F
AM,p
) and magnetic field stresses (F
AM,B
) were found
on average to be comparable in strength (F
AM,p
/F
AM,B
∼1),
although the AM flux in the protons was one of the most poorly
determined components of the total flux. This result differs from
previous work by Lazarus & Goldstein (1971) using the Mariner
5 spacecraft, who found the AM flux of the protons to dominate
over the magnetic field stresses (F
AM,p
/F
AM,B
∼4.3). Marsch &
Richter (1984a) showed that the ratio of AM flux in the particles
and magnetic field stresses varies considerably with heliocentric
distance and different solar wind conditions.
More recently, Finley et al. (2018) combined observations of
the solar wind (spanning ∼20 yr) with a semi-analytic relation
for the AM loss rate, derived from MHD simulations. Theirs
was a semi-indirect method, requiring in situ measurements of
only the mass flux and magnetic flux. They found a global AM
loss rate that varied in phase with the solar activity cycle, and
had an average value of 2.3×10
30
erg, compatible with the
results from Pizzo et al. (1983) and Li (1999)(∼3×10
30
erg
and 2.1×10
30
erg respectively). By examining proxies of
solar activity which span centuries and millennia into the
Sun’s past, Finley et al. (2019a) showed this value to be
The Astrophysical Journal Letters, 885:L30 (7pp), 2019 November 10 https://doi.org/10.3847/2041-8213/ab4ff4
© 2019. The American Astronomical Society. All rights reserved.
1
representative of the average over the last ∼9000 yr. However,
this value is lower than the AM loss rate of ∼6×10
30
erg used
in models that reproduce the rotational history of the Sun (and
Sun-like stars)(Gallet & Bouvier 2013, 2015; Matt et al. 2015;
Finley et al. 2018; Amard et al. 2019). Deviation from the
rotational evolution value has significant implications for our
understanding of stellar rotation rates (van Saders et al. 2016;
Garraffo et al. 2018), as well as for the technique of
gyrochronology (e.g., Barnes 2003; Metcalfe & Egeland 2019),
in which stellar ages are derived from rotation rates.
In this Letter, we provide a new direct measurement of the
solar AM loss, which follows that of Pizzo et al. (1983) and
Marsch & Richter (1984a) but uses data from the Wind
spacecraft. These data span a period of ∼25 yr and appear not
to require the pointing corrections that were applied to the
Helios data. This Letter is organized as follows: in Section 2 we
describe the data available from the Wind spacecraft and
calculate the time-varying mass flux and AM flux observed in
the equatorial solar wind. Then in Section 3, we estimate the
global AM loss rate and discuss the possible implications for
the rotation period evolution of Sun-like stars.
2. Observed Properties of the Solar Wind
2.1. Spacecraft Selection
The measurements required to accurately constrain the AM
content in the solar wind particles are challenging to make (see
the discussion in Section 3a of Pizzo et al. 1983). Not only are
the fluctuations in the AM flux comparable to the average
value, but from an instrument standpoint, small errors in
determining the wind velocity translate to large errors in the
AM flux (because the radial wind speed is 2–3 orders of
magnitude larger than the typical tangential speed of
1–10 km s
−1
at 1 au). The latter problem appears to be the
main reason why data from most spacecraft have not been used
to measure AM (see Figure 6 of Sauty et al. 2005, which shows
data from the Ulysses spacecraft; there is an approximately 1 yr
periodicty in the observations that is likely due to spacecraft
pointing). The magnetic field direction is generally more
accurately determined because it is not as radial as the flow,
and the instruments used are less sensitive to spacecraft
pointing than the particle detectors (which get different
exposures as the spacecraft pointing changes). Therefore, the
magnetic stress component of the AM flux is typically better
constrained.
While the Advanced Composition Explorer spacecraft’ s
nonradial solar wind speed measurements show the expected
behaviors during periods of high variability (Owens &
Cargill 2004), they appear to suffer from the same spacecraft-
pointing-related issues as Ulysses over longer time averages, in
this case showing a strong ∼6 month periodicity. The Wind and
Interplanetary Monitoring Platform 8 (IMP8) spacecraft do not
obviously show such features. Furthermore, during the period
of overlap between Wind and IMP8, there is good agreement in
tangential wind speed, both in terms of the distributions and
time series (linear regression of r=0.81 at the hourly
timescale), suggesting limited instrumental effects.
In this work we focus on the high time cadence Wind
observations. Wind was designed to be a comprehensive solar
wind laboratory for long-term solar wind measurements, and
has certainly stood the test of time; currently approaching its
25th yr since launch (1994 November 1st). During its mission
lifetime the Wind spacecraft completed multiple orbits of the
Earth–Moon system, before relocating to a halo orbit about the
L
1
Lagrangian point (on the Sun–Earth line) in 2004 May. All
the while collecting plasma and magnetic field measurements
of the solar wind and Earth’s magnetosphere with the Solar
Wind Experiment (Ogilvie et al. 1995; Kasper et al. 2006) and
Magnetic Field Investigation instruments (Lepping et al. 1995).
2.2. In Situ Measurements from the Wind Spacecraft
We analyze data recorded by the Wind spacecraft
4
from 1994
November to 2019 June. Using data taken when the spacecraft
was immersed in the solar wind, i.e., outside the Earth’s
magnetosphere. Additionally, we remove times when the
spacecraft encountered interplanetary coronal mass ejections
(ICMEs) using the catalogs
5
of Cane & Richardson (2003) and
Richardson & Cane (2010) because ICMEs can produce large,
nonradial, local flows that are not likely representative of global
AM loss (Owens & Cargill 2004). For times not covered by the
ICME catalog (1994 November–1996 June), we remove data
with properties that are indicative of ICMEs, specifically data
with a proton density greater than 70 cm
−3
or field strengths
greater than 30nT (a similar method was used by Cohen 2011
on Ulysses data).
Measurements of the solar wind magnetic field vector,
proton density, and velocity are available throughout the entire
Wind mission at ∼2 minute cadence. These parameters have a
small number of entries flagged by the instrument team as
containing unusable data, which we simply remove. Similarly,
measurements of the alpha particle density and velocity are
available; however, the number of unusable data entries (where
the proton and alpha particle populations cannot be decon-
volved by the detector) is far greater. Therefore, when the alpha
particles are flagged as unusable, we assume that the alpha
particle density is 4% of the proton density (a representative
value taken from Borrini et al. 1983) and that the alphas’
velocities are identical to the protons’. We transform the vector
quantities of velocity and magnetic field from GSE coordinates
to RTN coordinates, where R points from the Sun to the
spacecraft, T points perpendicular to the Sun’s rotation axis in
the direction of rotation, and N completes the right-handed
triad (further details are available in Fränz & Harper 2002).
For each quantity derived using Wind data in this work, we
calculate values at the smallest available cadence (∼2 minutes)
and then average them over each Carrington rotation (CR,
∼27 days) in our data set. This helps to remove longitudinal
variability caused by the rotation of features on the solar
surface and smooths local fluctuations that occur on a range of
shorter timescales. Finally, we require that each CR-average
has more than 50% of the data from that time period (after
our cuts have been made). Otherwise, that CR is removed. In
the top panel of Figure 1, we plot the tangential wind speed
of the protons and alpha particles as observed by Wind. For the
tangential speeds shown in Figure 1, we have weighted the CR
averages by density, in order to reduce the obscuring effect of
wind stream-interactions (see the discussion in Section 3.3).
Figure
1 shows typical tangential flow speeds of a few km s
−1
,
with variability that appears genuine and not to suffer from the
4
https://wind.nasa.gov/data.php—Data accessed in 2019 June.
5
http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm—
Data accessed in 2019 September.
2
The Astrophysical Journal Letters, 885:L30 (7pp), 2019 November 10 Finley et al.
errors present in data from other spacecraft (as discussed in
Section 2.1).
2.3. Proton and Alpha Particle Properties
The solar wind removes AM from the Sun at a rate
proportional to the mass flux (ρv
r
) multiplied by the specific
AM per unit mass (Λ). Using data from the Wind spacecraft, we
plot the mass flux in the protons, alpha particles, and their total
in the middle panel of Figure 1. We multiply each by 4πr
2
for
an estimate of the global mass-loss rate,
˙
() ()pr r»á + ñ
a
a
Mrv v4,1
p
rp r
2
,,CR
where the spacecraft’s radial distance from the Sun is r, the
radial wind speed is v
r
, the solar wind density is ρ, the
subscripts p and α denote the proton and alpha particle
components, and
á
ñ
CR
denotes an average over a ( ∼27 day) CR.
The total mass flux is dominated by the proton component of
the wind and varies in a way that does not precisely correlate
with the Sun’s activity cycle (see also Phillips et al. 1995;
McComas et al. 2000; Finley et al. 2018; Mishra et al. 2019).
By contrast, the alpha particle mass flux appears to be more
strongly correlated with solar activity throughout the Wind data
set (which is not surprising as the relative abundance of helium
in the equatorial solar wind is strongly correlated with solar
activity, see Kasper et al. 2007).
We define the specific AM as the AM flux divided by the
proton mass flux (i.e., the specific AM per proton in the solar
wind), which is given by,
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
()q
r
rpr
L= + -
a
a
a
rvv
v
v
BB
v
sin
4
,2
tp t
r
p
rp
tr
p
rp
,,
,
,,
CR
where θ is the heliographic latitude of the spacecraft, v
t
is the
tangential wind velocity, B
r
is the radial magnetic field strength
and B
t
is the tangential magnetic field strength. The first term in
Equation (2) is the mechanical AM carried by the protons, the
second term relates to the relative contribution of the alpha
particles, and the final term describes the AM content of the
Figure 1. Top: CR averages of the density-weighted, tangential speed of the protons and alpha particles in the solar wind vs. time, plotted in orange and blue
respectively. Middle: CR averages of mass flux in the protons, alpha particles, and their total (orange, blue, and black lines), each multiplied by 4πr
2
, vs. time. The
prediction of Equation (5) for the open magnetic flux during the same time period is overplotted using a green line, y-axis on the right (see Section 3.1). Bottom: CR
averages of specificAM(defined as the AM flux per proton mass flux; density-weighted velocities are used here, see Section 3.3 ) in the protons, alpha particles, and
magnetic field stresses (orange, blue, and green lines) vs. time. The total specific AM is plotted with a black line.
3
The Astrophysical Journal Letters, 885:L30 (7pp), 2019 November 10 Finley et al.
magnetic field stresses. Equation (2) does not include the
correction factor for the magnetic stresses which accounts for
thermal pressure anisotropies, as it is expected to be negligible
(see Marsch & Richter 1984b). In the bottom panel of Figure 1,
we plot the total specific AM along with the individual proton,
alpha particle, and magnetic field components. We use density-
weighted tangential velocities, as in the top panel of Figure 1,
to reduce the effect of wind stream-interactions (see the
discussion in Section 3.3). Figure 1 shows the protons to
dominate the speci fic AM of the solar wind, with the magnetic
field stresses and alpha particles carrying much less specific
AM (per proton).
2.4. AM Flux Detection
The total AM flux in the protons, alpha particles, and
magnetic field stresses is given by multiplying the specificAM
by the proton mass flux,
⎟
⎞
⎠
(
()
rqr
r
p
=á Lñ =
+-
a
aa
Fv r vv
vv
BB
sin
4
.3
p
rp
p
rp tp
rt
tr
AM , CR , ,
,,
CR
We plot the AM fluxes (multiplied by radial distance squared) in
the protons, alphas, magnetic field,andtheirtotalinthetoppanel
of Figure 2. There is a large scatter/variability in the AM flux,
despite averaging over whole CRs. The variability is mainly due
to the varying specificAM(i.e., in the tangential wind speed),
rather than changes in the mass flux (see Figure 1), and which is
likely affected by local fluctuations in the solar wind, caused by
transients (Roberts et al. 1987; Tokumaru et al. 2012). The solid
black line in Figure 2 shows a 13-CR (i.e., ∼1yr) moving
average on the total AM flux, which more clearly describes the
longer-term variability of the AM flux. Our data set contains
sunspot cycles 23 and 24 (left and right halves of the figures,
respectively), which have notable differences in their AM fluxes.
Generally, during times of increased solar activity the specific
AM of the protons and magnetic field stresses increase together,
such that F
AM,p
/F
AM,B
does not vary with solar activity. We find
cycle 24, which is currently in its declining phase, has a much
lower average AM flux than cycle 23 (∼40% of cycle 23).
The average value for the AM flux, and that of each
constituent, is listed and compared to previous estimates in
Table 1. The Wind total is primarily composed of the proton
and magnetic field components, with the alpha particles
contributing a small and mostly negative AM flux contribution.
In comparison with the work of Pizzo et al. (1983) and Marsch
& Richter (1984a), the Wind data show a much stronger AM
flux in the protons and a large reduction (in amplitude) to the
AM flux carried by the alpha particles. These differences could
be related to long-term change in the solar wind. For example,
the solar wind appears denser in the last decade compared to
the Helios era (see McComas et al. 2013). Or alternatively, due
to the exchange of momentum between protons and alphas as
the wind propagates into the heliosphere (for which there is
some evidence in Sanchez-Diaz et al. 2016).
The AM flux in the magnetic field stress in the Wind data
is similar to that determined by Pizzo et al. (1983) and Marsch
& Richter (1984a) but is smaller than that determined by
Lazarus & Goldstein (1971). Interestingly, the dominant
contribution to the Wind -measured AM flux comes from the
protons, with the magnetic field of secondary importance. In
simplified MHD simulations of the solar wind (such as those of
Finley & Matt 2017)
, the ratio F
AM,p
/F
AM,B
depends on
parameters such that the larger the Alfvén radius (R
A
) the larger
the contribution of the magnetic field. The average ratio
measured by Wind is F
AM,p
/F
AM,B
=2.6, which is signifi-
cantly different from the ratio of ∼1 found by Pizzo et al.
(1983). Marsch & Richter ( 1984a) showed that Helios data
from smaller heliocentric distances gives larger ratios, which
might account for the difference. The proton-dominated regime
shown by the Wind data is consistent with MHD simulations
that have cylindrically averaged R
A
smaller than 15R
e
.
3. Discussion
Using data from the Wind spacecraft, we have evaluated the
flux of AM in the equatorial solar wind. In this section, we
estimate the global AM loss rate of the Sun and compare with
an MHD model and rotational evolution models. Additionally,
we discuss the effect of ICMEs and interacting wind streams on
our data set.
3.1. Comparison to Theory
To show our result in the context of current theoretical
predictions, we compare to the AM loss rate of Finley et al.
(2018), which was derived using MHD simulations. In their
work, the AM loss rate is given by,
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
˙
([])
˙
[]
[]
()
f
=´
´
´
´
-
J
M
2.3 10 erg
1.1 10 g s
8.0 10 Mx
,4
FM18
30
12 1
0.26
open
22
1.48
where the AM loss rate of the Sun is parameterized in terms of
the mass-loss rate,
˙
M
, and the open magnetic flux,
f
open
. The
open magnetic flux in the solar wind is estimated by,
∣∣ ()fp=á ñrB4, 5
rhr
open
2
1CR
where the average value of the radial magnetic field is assumed
to be representative of the global open magnetic flux in the
solar wind. This assumption has been discussed by many
previous authors (Wang & Sheeley 1995; Lockwood et al.
2004; Pinto & Rouillard 2017) and has observational support
(Smith & Balogh 1995; Owens et al. 2008). Using Equation (5)
we plot the open magnetic flux using data from the Wind
spacecraft in the middle panel of Figure 1 with a solid
green line.
Using Equation (4) we calculate the predicted AM loss rate
of the solar wind, where the mass-loss rate and open magnetic
flux (Equations (1) and (5)) are calculated using data from the
Wind spacecraft. We then relate the AM loss rate and AM flux
using,
∮
˙
·()()
òò
qqf==
pp
FAJdF rddsin , 6
A
AM AM, eq
0
2
0
23
where A represents a closed surface in the heliosphere (we
adopt a sphere of radius r), f is heliographic longitude, and
F
AM,eq
is the AM flux in the solar equatorial plane, assumed to
4
The Astrophysical Journal Letters, 885:L30 (7pp), 2019 November 10 Finley et al.
