Solar Angular Momentum Loss Over the Past Several Millennia

TL;DR: In this article, the authors used near-Earth measurements of the solar wind plasma and magnetic field to calculate the torque on the Sun throughout the space age, and reconstructions of solar open magnetic flux were used to estimate the time-varying braking torque during the last nine millennia.

Abstract: The Sun and Sun-like stars lose angular momentum to their magnetized stellar winds. This braking torque is coupled to the stellar magnetic field, such that changes in the strength and/or geometry of the field modifies the efficiency of this process. Since the space age, we have been able to directly measure solar wind properties using in situ spacecraft. Furthermore, indirect proxies such as sunspot number, geomagnetic indices, and cosmogenic radionuclides, constrain the variation of solar wind properties on centennial and millennial timescales. We use near-Earth measurements of the solar wind plasma and magnetic field to calculate the torque on the Sun throughout the space age. Then, reconstructions of the solar open magnetic flux are used to estimate the time-varying braking torque during the last nine millennia. We assume a relationship for the solar mass-loss rate based on observations during the space age which, due to the weak dependence of the torque on mass-loss rate, does not strongly affect our predicted torque. The average torque during the last nine millennia is found to be 2.2 × 1030 erg, which is comparable to the average value from the last two decades. Our data set includes grand minima (such as the Maunder Minimum), and maxima in solar activity, where the torque varies from ~1 to 5 × 1030 erg (averaged on decadal timescales), respectively. We find no evidence for any secular variation of the torque on timescales of less than 9000 yr.

Summary

1. Introduction

• The observed rotation periods of most low-mass stars (M*1.3Me) on the main sequence can be explained by their magnetized stellar winds.
• Such models provide insight on how stellar wind torques evolve on secular timescales (∼Gyr), independently from their understanding of the braking mechanism.
• The authors observe variability in the magnetic field of the Sun on a range of much shorter timescales (DeRosa et al. 2012; Vidotto et al. 2018), which is expected to influence the angular momentum loss rate in the solar wind (Pinto et al.
• By applying a braking law derived from magnetohydrodynamic (MHD) simulations by Finley & Matt (2018), they calculated the time-varying torque on the Sun due to the solar wind.

2. Angular Momentum Loss Formulation

• The formulation of Equation (5) for á ñRA , using fopen, is insensitive to how the coronal magnetic field is structured (i.e., insensitive to the geometry of the magnetic field; Réville et al. 2015a), but the fit constants can be affected by differing wind acceleration profiles (Pantolmos & Matt 2017), and 3D structure in the mass flux.
• By comparing feasible base wind temperatures, Pantolmos & Matt (2017) showed there is at most a factor of ∼2 difference in the prediction of Equation (7) between the coldest and hottest polytropic winds (1.3–4.2MK for the Sun).

3.1. Observed Solar Wind Properties

• Hourly near-Earth solar wind plasma and magnetic field measurements are available from the OMNIWeb service.
• The OMNI data set is compiled from the in situ observations of 4 https://omniweb.gsfc.nasa.gov/.
• The average of this data set is indicated with a gray horizontal line.
• The open magnetic flux roughly declines in time over the past three cycles, with the current sunspot cycle hosting some of the weakest values recorded in the OMNI data set.

3.2. Coronal Mass Ejections

• Equations (8) and (9) do not take into account the effects of coronal mass ejections (CMEs) in the data.
• These appear as impulsive changes (generally increases) in the observed solar wind properties, and clearly violate the assumed isotropy of wind conditions in Equations (8) and (9).
• CMEs occur once every few days at solar minimum, however their occurrence rate tracks solar activity, and at solar maximum they are observed on average five times a day (Webb et al. 2017; Mishra et al. 2019).
• Previous authors have removed these events through the use of CME catalogs (Cane & Richardson 2003) or clipping anomalous spikes (Cohen 2011).
• CMEs carry only a few percent of the total solar mass-loss rate (Cranmer et al. 2017), however, at solar maximum they can provide a significant fraction of the average mass flux in the equatorial solar wind (Webb & Howard 1994).

3.3. Decades of Solar Wind Torque

• The authors use the open magnetic flux and mass-loss rate estimates from Section 3.1 to compute the angular momentum loss rate in The Astrophysical Journal, 883:67 (9pp), 2019 September 20 Finley et al. the solar wind using Equation (7).
• The results from this calculation are shown in the bottom panel of Figure 1.
• The value of á ñRA during the current sunspot cycle ranges from ∼8 to 16Re.
• From the in situ measurements of Pizzo et al. (1983) using the Helios spacecraft, to the recalculation of Li (1999) based on data from the Ulysses spacecraft.

4. Solar Wind Torque on Centennial and Millennial Timescales

• Up until now, the authors have examined only direct measurements of the solar wind.
• These indirect measurements are used to estimate longer time variability of the Sun’s open magnetic flux (Lockwood et al.
• Significantly for this work, they do not produce estimates for how the mass-loss rate of the Sun has varied.
• In terms of the open magnetic flux, which is constructed using the range of observed values from Section 3.1.the authors.

4.1. Estimating the Mass-loss Rate, and Wind Torque with the Open Magnetic Flux

• A weak trend of increasing mass-loss rate with increasing open magnetic flux is observed.
• When the authors estimate the mass-loss rate for the historical estimates of the open magnetic flux in Sections 4.3, they will use both Equation (10) and the 2σ bounds.
• This allows us to predict the torque on the Sun due to the solar wind solely from the value of the open magnetic flux.

4.2. Reconstructions of the Solar Open Magnetic Flux

• For the centuries and millennia pre-dating the space age, estimates of the open magnetic flux have been produced using a number of different indirect methods.
• To compare them with indirect methods and over a wide range of timescales, the authors plot the spacecraft data from Figure 1 also in Figure 3, which displays the solar wind parameters versus logarithmic look-back time since 2019.

4.2.1. Centennial Variability

• Geomagnetic disturbances, caused by the interaction of the solar wind and the Earth’s magnetosphere, have been found to The Astrophysical Journal, 883:67 (9pp), 2019 September 20 Finley et al. correlate well with solar activity, and thus the amount of open magnetic flux in the heliosphere (Stamper et al.
• Inspecting the past four centuries, there are also times when the open magnetic flux is shown to weaken for several magnetic cycles (Usoskin et al. 2015).
• The 2σ bounds of Equation (10) roughly encompass the observed variation of the mass-loss rate (as constructed).
• Therefore, provided the 5 However, the additional complications did not statistically improve their Ṁ predictions.

4.2.2. Millennial Variability

• To go back further the open magnetic flux can only be reconstructed using cosmogenic radionuclides.
• This rate is modulated by the geomagnetic field, but also by features in the heliosphere, such as the interplanetary magnetic field and solar wind (Stuiver 1961; Stuiver & Quay 1980).
• Therefore, the concentration of cosmogenic radionuclides can be used as a proxy for solar variability (see review by Beer et al. 2012).
• These smoothed values are plotted with dotted and dashed lines, respectively, in the top panel of Figure 3.
• During the last several millennia, there appear to be times similar to the modern grand maxima, and the grand minima which are observed in the centennial reconstructions.

5.1. Reliability of Open Flux Proxies and Our Predicted Mass-loss Rates

• Indirect reconstructions of the solar open magnetic flux are by no means certain, and require careful examination and calibration.
• Sunspot number records, from which their centennial torque is ultimately generated, often suffer from historical periods that are incomplete or uncertain due to a lack of reliable observers (Vaquero et al.
• This requires knowledge of the physical mechanisms which produce, transport, and deposit each radioisotope (e.g., Reimer et al. 2009; Heikkilä et al. 2013).
• The solar massloss rate is not observed to vary substantially (extremes of 0.7–3.0× 1012 g s−1, see also Cohen 2011), and the torques calculated using Equation (7) are weakly dependent on their choice of mass-loss rate (when compared to the open magnetic flux).

5.2. Impacts of Magnetic Variability on Short Timescales

• Reconstructions of solar activity based on the concentrations of cosmogenic radionuclides incur smoothing effects from the transport and deposition timescales of each radionuclide.
• Therefore, such records struggle to recover short timescales variability, such as the 11 yr sunspot cycle.
• The significance of this effect over the complete nine millennia can be probed in a few ways.
• Each reconstruction is only sensitive to variability on timescales larger than the cadence of the data set.
• For timescales shorter than 27 day, the authors have no measure of how variability affects their average values compared to the true value, but observed variations on shorter timescales may be ever more dominated by spatial variations in the wind, rather than variations in the global, integrated wind properties.

6. Conclusion

• In this paper the authors have investigated the angular momentum loss rate of the Sun on a longer timescale than previously attempted.
• Further exploration of this discrepancy is required, and with Parker Solar Probe making in situ measurements of the solar wind closer to the Sun than previously attempted (Fox et al. 2016), a direct measurement of the angular momentum loss rate would help to validate, or discredit, their calculations.

Figures

Figure 2. Mass-loss rate, Ṁ , vs. open magnetic flux, fopen, derived the in situ observations of the OMNI data set. Values are color-coded by sunspot cycle, 20–24. The black line corresponds to the power-law fit of Equation (10). The dashed red lines indicates the 2σ bounds given by a log-Gaussian centered on the fit line.

Figure 1. Several decades of open magnetic flux, fopen, and mass-loss rate, Ṁ , estimated from the OMNI data set (near-Earth measurements), are shown with circles (color-coded by sunspot cycle number, 20–24) in the top two panels. The predicted solar wind torque, τ, using Equation (7) is then shown in the bottom panel. Averages of these three quantities are shown with gray horizontal lines. Over-plotted in each panel are the fopen reconstruction from Owens et al. (2017b), the Ṁ predicted by Equation ((10)), and the τ from Equation (11), with solid black lines. The 2σ bounds for the predicted Ṁ and τ, are indicated with dashed red lines.

Table 1 Average Solar Wind Torques for Historical Activity Maxima and Minima

Figure 3. 9000 yr of solar open flux, fopen, mass-loss rate, Ṁ , and our predicted solar wind torque, τ, vs. inverse logarithmic look-back time from 2019. The results derived from the OMNI data set are plotted as they appeared in Figure 1. The fopen reconstructed by Owens et al. (2017b; group sunspot number) and Lockwood et al. (2014a; geomagnetic, aa-index) are plotted in the top panel with purple and magenta lines, respectively. We calibrate the long-time fopen reconstruction from Wu et al. (2018b; cosmogenic radionuclides), plotted in the top panel in gray, by first averaging the Owens et al. (2017b) and Lockwood et al. (2014a) reconstructions on the same decadal timescale, shown with dashed and dotted black lines, respectively, then we shifted the Wu et al. (2018b) fopen to match by adding a constant value. This reconstruction is shown with a solid black line, in good agreement with the smoothed values in the overlapping time period of ∼1600–1900. Using the fopen from Owens et al. (2017b) and Wu et al. (2018b), the Ṁ predicted using Equation (10) is plotted in the middle panel with solid purple and black lines, respectively. The τ predicted by Equation (11), for each reconstruction is then plotted with solid purple and black lines in the bottom panel. For both predicted Ṁ and τ, the 2σ bound is indicated with dashed red lines. Maxima and minima in solar activity are shaded with color.

Full text

Solar angular momentum loss over the
past several millennia
Article
Published Version
Finley, A. J., Deshmukh, S., Matt, S. P., Owens, M. and Wu,
C.-J. (2019) Solar angular momentum loss over the past
several millennia. The Astrophysical Journal, 883 (1). 67. ISSN
0004-637X doi: https://doi.org/10.3847/1538-4357/ab3729
Available at https://centaur.reading.ac.uk/87799/
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Solar Angular Momentum Loss over the Past Several Millennia
Adam J. Finley
1
, Siddhant Deshmukh
1
, Sean P. Matt
1
, Mathew Owens
2
, and Chi-Ju Wu
3
1
University of Exeter, Exeter, Devon, EX4 4QL, UK; af472@exeter.ac.uk
2
University of Reading, Reading, Berkshire, RG6 6BB, UK
3
Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, Göttingen, Germany
Received 2019 April 8; revised 2019 July 22; accepted 2019 July 29; published 2019 September 20
Abstract
The Sun and Sun-like stars lose angular momentum to their magnetized stellar winds. This braking torque is
coupled to the stellar magnetic field, such that changes in the strength and/or geometry of the field modifies the
efficiency of this process. Since the space age, we have been able to directly measure solar wind properties using
in situ spacecraft. Furthermore, indirect proxies such as sunspot number, geomagnetic indices, and cosmogenic
radionuclides, constrain the variation of solar wind properties on centennial and millennial timescales. We use
near-Earth measurements of the solar wind plasma and magnetic field to calculate the torque on the Sun throughout
the space age. Then, reconstructions of the solar open magnetic flux are used to estimate the time-varying braking
torque during the last nine millennia. We assume a relationship for the solar mass-loss rate based on observations
during the space age which, due to the weak dependence of the torque on mass-loss rate, does not strongly affect
our predicted torque. The average torque during the last nine millennia is found to be 2.2×10
30
erg, which is
comparable to the average value from the last two decades. Our data set includes grand minima (such as the
Maunder Minimum), and maxima in solar activity, where the torque varies from ∼1to5×10
30
erg (averaged on
decadal timescales), respectively. We find no evidence for any secular variation of the torque on timescales of less
than 9000 yr.
Key words: magnetohydrodynamics (MHD) – solar wind – Sun: evolution – Sun: rotation
1. Introduction
The observed rotation periods of most low-mass stars
(M
*
1.3M
e
) on the main sequence can be explained by
their magnetized stellar winds. These winds efficiently remove
angular momentum causing stars to spin-down with age
(Skumanich 1972; Soderblom 1983; Barnes 2003, 2010;
Delorme et al. 2011; Van Saders & Pinsonneault 2013; Bouvier
et al. 2014). Throughout this process, their magnetic field
generation (due to the dynamo mechanism) is strongly linked
with rotation (Brun & Browning 2017), and the strength of the
magnetic field is found to influence the efficiency of angular
momentum transfer through the stellar wind (Weber &
Davis 1967; Mestel 1968; Kawaler 1988; Matt et al. 2012;
Garraffo et al. 2015). The resulting strong dependence of
torque on rotation rate leads to a convergence of rotation
periods with age, as initially fast rotating stars generate strong
magnetic fields and experience a larger braking torque than the
initially slowly rotating stars. This spin-down is also observed
to be a function of stellar mass (Agüeros et al. 2011; McQuillan
et al. 2013; Núñez et al. 2015; Covey et al. 2016; Rebull et al.
2016; Agüeros 2017; Douglas et al. 2017).
Many models now exist to study the rotation period
evolution of low-mass stars (Gallet & Bouvier 2013;
Brown 2014; Gallet & Bouvier 2015; Johnstone et al. 2015;
Matt et al. 2015; Amard et al. 2016
; Blackman & Owen 2016;
Sadeghi Ardestani et al. 2017; Garraffo et al. 2018; See et al.
2018). Such models provide insight on how stellar wind
torques evolve on secular timescales ( ∼Gyr), independently
from our understanding of the braking mechanism. For Sun-
like stars, the torques prescribed by these models are averaged
over fractions of the braking timescale (∼10–100 Myr).
However, we observe variability in the magnetic field of the
Sun on a range of much shorter timescales (DeRosa et al. 2012;
Vidotto et al. 2018), which is expected to influence the angular
momentum loss rate in the solar wind (Pinto et al. 2011; Réville
& Brun 2017; Finley et al. 2018; Perri et al. 2018).
In Finley et al. (2018), the short timescale variability (from
∼27 days up to a few decades) of the solar wind was examined
using in situ observations of the solar wind plasma and
magnetic field. By applying a braking law derived from
magnetohydrodynamic (MHD) simulations by Finley & Matt
(2018), they calculated the time-varying torque on the Sun due
to the solar wind. When averaged over the last ∼20 yr they
found a solar wind torque of 2.3×10
30
erg. This value is in
agreement with previous in situ and data driven calculations
(Pizzo et al. 1983;Li1999), and also recent simulation results
(Alvarado-Gómez et al. 2016; Réville & Brun 2017;Ó
Fionnagáin et al. 2018; Usmanov et al. 2018).
When compared to the torques required by rotation–
evolution models (e.g., Matt et al. 2015), current estimates of
the solar wind torque are smaller by a factor of ∼3 (this
discrepancy was noted already by Soderblom 1983). One
possible explanation for the discrepancy is that the solar wind
torque is variable, and that the torque is currently in a “low
state,” or that the torque has recently, but permanently
weakened (e.g., as suggested by van Saders et al. 2016;
Garraffo et al. 2018; Ó Fionnagáin & Vidotto 2018). For this to
be true, the variations in the torque must have happened on
timescales much longer than the space age (decades), but
shorter than the timescales on which the rotation–evolution
models are sensitive to (∼10
8
yr, for solar-aged stars).
In this work, we employ reconstructions of solar wind
properties from the literature, in order to estimate the solar
wind torque further back in time than has been probed so far
(more than two orders of magnitude). Although we still cannot
probe the timescales of rotational evolution, this helps to
elucidate the types of variability that may occur in the solar
wind torque. We first describe the Finley & Matt (2018)
The Astrophysical Journal, 883:67 (9pp), 2019 September 20 https://doi.org/10.3847/1538-4357/ab3729
© 2019. The American Astronomical Society. All rights reserved.
1

braking law, hereafter FM18, in Section 2. Then we estimate
the angular momentum loss rate, due to the solar wind, through
the space age using in situ data in Section 3. Finally, in
Section 4, we use reconstructions of the Sun’s open magnetic
flux (which are based on sunspot number, geomagnetic indices,
and cosmogenic radionuclide records), to estimate the angular
momentum loss rate on centennial and millennial timescales.
2. Angular Momentum Loss Formulation
Generally, the torque on a star due its magnetized wind can
be written as
t =W
áñ
MR
R
R
,1
2
A
2
*
*
*
˙
()
⎛
⎝
⎜
⎞
⎠
⎟
where
M
˙
is the mass-loss rate, Ω
*
is the stellar rotation rate, R
*
is the stellar radius, and
á
ñRR
A
*
can be thought of as an
efficiency factor for the angular momentum loss rate which,
under the assumption of ideal steady-state MHD, scales as the
average Alfvén radius (Weber & Davis 1967; Mestel 1968).
We use a semi-analytic formula for
á
ñR
A
, which depends on
the open magnetic flux, f
open
, and mass-loss rate,
M
˙
, in the
wind (Strugarek et al. 2014; Réville et al. 2015a, 2015b, 2016;
Finley & Matt 2017; Pantolmos & Matt 2017; FM18).We
define the open magnetic flux as the total unsigned flux that
permeates the stellar wind,
f = BAd ,2
open
A
∮
∣· ∣ ()
where
B
is the magnetic field strength in the wind, and A is a
closed surface that is located outside the last closed magnetic
field line. In a steady state, the last closed magnetic field line
resides within the Alfvén radius, R
A
, which is defined as the
location where the wind speed becomes equal to the Alfvén
speed,
pr==vR v B 4
AAA
A
()
, where the subscript A
denotes values taken at R
A
. Considering a steady MHD flow,
along a one-dimensional magnetic flux tube, mass and
magnetic flux are conserved. Therefore, in a steady-state stellar
wind, where the flow is spherically symmetric, the magnetic
field strength at R
A
is specified by flux conservation as
fp=
B
R4
A
open
A
2
()
. The Alfvén speed is then,
fp
pr
=v
R4
4
,3
A
2
open
2
2
A
4
A
()
()
which by rearranging, and then substituting for
M
˙
, produces a
relation for R
A
,
f
ppr
f
p
==R
vvR
vM
44
4
.4
A
2
open
2
2
A
A
A
A
2
open
2
2
A
() ( )
()
˙
()
Since real stellar winds are multi-dimensional in nature, several
authors (e.g., Matt & Pudritz 2008; Pinto et al. 2011; Matt et al.
2012; Cohen & Drake 2014; Réville et al. 2015a, 2015b;
Garraffo et al. 2016; Finley & Matt 2017; Pantolmos &
Matt 2017; FM18) have employed MHD numerical simulations
to derive semi-analytic scalings for the wind torques. A few
of these studies have derived a relationship similar to
Equation (4), which has the form
f
áñ
=
R
R
K
R
Mv
,5
m
A
open
22
esc
*
*
˙
()
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
where
á
ñRR
A
*
is calculated from the simulations by inverting
Equation (1), and K and m are fit constants. In Equation (5),
compared to Equation (4), v
A
has been replaced by the surface
escape speed,
=vGMR2
esc 
, and any dependence v
A
has
on f
open
and
M
˙
is absorbed into the fit constants. These fit
constants also account for the multiplicative factor of (4π)
2
, and
any effects introduced by the flow being multi-dimensional in
nature. The formulation of Equation (5) for
á
ñR
A
, using f
open
,is
insensitive to how the coronal magnetic field is structured (i.e.,
insensitive to the geometry of the magnetic field; Réville et al.
2015a), but the fit constants can be affected by differing wind
acceleration profiles (Pantolmos & Matt 2017), and 3D
structure in the mass flux.
We adopt the fit parameters from FM18. For the Sun,
Equation (5) then reduces to,
f
=
´
´
´
-
-
RR
M
12.9
1.1 10 g s
8.0 10 Mx
,
6
A
12 1
0.37
open
22
0.74
⟨⟩ ( )
˙
()
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟

using values of the solar mass, M
e
=1.99×10
33
g, and
radius, R
e
=6.96×10
10
cm. For the solar wind torque,
Equation (1) becomes,
t
f
=´
´
´
´
-
M
2.3 10 erg
1.1 10 g s
8.0 10 Mx
,
7
30
12 1
0.26
open
22
1.48
(())
˙
()
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
using the solar rotation rate Ω
e
=2.6×10
−6
rad s
−1
. The
torque depends only on f
open
and
M
˙
, given the choice of
polytropic base wind temperature used in FM18. By comparing
feasible base wind temperatures, Pantolmos & Matt (2017)
showed there is at most a factor of ∼2 difference in the
prediction of Equation (7) between the coldest and hottest
polytropic winds (1.3–4.2 MK for the Sun). The simulations
of FM18, from which we derived Equations (6) and (7),
correspond to a base wind temperature of ∼1.7 MK, which sits
at the lower edge of this temperature range (where the torques
are strongest).
3. Solar Wind Torque During the Space Age
3.1. Observed Solar Wind Properties
Hourly near-Earth solar wind plasma and magnetic field
measurements are available from the OMNIWeb service.
4
The
OMNI data set is compiled from the in situ observations of
4
https://omniweb.gsfc.nasa.gov/ (Accessed in 2018 July ).
2
The Astrophysical Journal, 883:67 (9pp), 2019 September 20 Finley et al.

several spacecraft, from 1963 to present. We use measurements
of the solar wind to estimate the open magnetic flux using
fp=á ñRB R4,8
R
open
2
1hr 27days
∣()∣ ()
where we average the radial magnetic field B
R
, (taken from a
single observing location) at a distance R from the Sun, over a full
solar rotation (27 days), and assume that the solar wind is roughly
isotropic on our averaging timescale, in order to estimate the open
magnetic flux. Smith & Balogh (1995) were able to show that
R
BR
R
2
∣()∣
is approximately independent of heliographic latitude,
as the solar wind is thought to redistribute significant variations in
magnetic flux due to latitudinal magnetic pressure gradients
caused by non-isotropy (Wang & Sheeley 1995; Lockw ood et al.
2004; Pinto & Roui lla rd 2017). Subsequently, the use of a single
point measurement to infer the global open magnetic flux has
been shown to be a reasonable approximation at distances less
than ∼2 au by Owens et al. (2008).
The open magnetic flux calculated using Equation (8),
during the space age, is plotted in the top panel of Figure 1. The
27 day averages are shown with circles that are colored
according to the different sunspot cycles in our data set. The
average of this data set is indicated with a gray horizontal line.
The open magnetic flux roughly declines in time over the past
three cycles, with the current sunspot cycle hosting some of the
weakest values recorded in the OMNI data set. Due to
kinematic effects that occur between the Alfvén surface and the
measurements taken at 1 au, our estimate of the open magnetic
flux is likely an upper limit (Owens et al. 2017a).
Similarly to Equation (8) for the open magnetic flux, the
solar mass-loss rate is estimated from in situ measurements
using
pr=á ñMRvRR4,9
R
2
27 days
˙
()() ()
which is plotted in the middle panel of Figure 1. Equation (9)
assumes the mass flux evaluated at a single observing location
in the solar wind is representative of all latitudes when
averaged over 27 days. Using data from the fast latitude scans
of the Ulysses spacecraft, Finley et al. (2018) showed that the
calculation of
M
˙
from Equation (9) varies by a few 10ʼsof
percent when the spacecraft was immersed in slow, versus fast,
solar wind streams (see also Phillips et al. 1995). Thus, the
errors due to latitudinal variability are comparable to, but
appear somewhat smaller than, the time variability (see, e.g.,
McComas et al. 2013). The cyclical variations of
M
˙
are less
clear than for the open flux, but they show a similar decreasing
trend over the past three cycles.
3.2. Coronal Mass Ejections
Equations (8) and (9) do not take into account the effects of
coronal mass ejections (CMEs) in the data. These appear as
impulsive changes (generally increases) in the observed solar
wind properties, and clearly violate the assumed isotropy of
wind conditions in Equations (8) and (9). CMEs occur once
every few days at solar minimum, however their occurrence
rate tracks solar activity, and at solar maximum they are
observed on average five times a day (Webb et al. 2017; Mishra
et al. 2019). Previous authors have removed these events
through the use of CME catalogs (Cane & Richardson 2003) or
clipping anomalous spikes (Cohen 2011). CMEs carry only a
few percent of the total solar mass-loss rate (Cranmer et al.
2017), however, at solar maximum they can provide a
significant fraction of the average mass flux in the equatorial
solar wind ( Webb & Howard 1994).
Finley et al. (2018) examined the effect of removing periods
of high wind density (>10 cm
3
) and high magnetic field
strength (>10 nT) , thought to correspond to the CMEs. They
determined that the average open magnetic flux and mass-loss
rate, over their ∼20 yr of data, decreased by ∼4% after these
cuts were applied. As the role of CMEs in removing angular
momentum is still in question (see, e.g., Aarnio et al. 2012),
and their inclusion here is limited to a few percent, we present
our results using the full unclipped data set.
3.3. Decades of Solar Wind Torque
We use the open magnetic flux and mass-loss rate estimates
from Section 3.1 to compute the angular momentum loss rate in
Figure 1. Several decades of open magnetic flux, f
open
, and mass-loss rate,
M
˙
,
estimated from the OMNI data set (near-Earth measurements), are shown with
circles (color-coded by sunspot cycle number, 20–24) in the top two panels.
The predicted solar wind torque, τ, using Equation (7) is then shown in the
bottom panel. Averages of these three quantities are shown with gray
horizontal lines. Over-plotted in each panel are the f
open
reconstruction from
Owens et al. (2017b), the
M
˙
predicted by Equation ((10)), and the τ from
Equation (11), with solid black lines. The 2σ bounds for the predicted
M
˙
and τ,
are indicated with dashed red lines.
3
The Astrophysical Journal, 883:67 (9pp), 2019 September 20 Finley et al.

the solar wind using Equation (7). The results from this
calculation are shown in the bottom panel of Figure 1.We
calculate the average torque on the Sun during the space age
to be 2.97×10
30
erg, which is larger than the value obtained
by Finley et al. (2018) of 2.3×10
30
erg, due to the fact
that Finley et al. (2018) only examined the past ∼20 yr.
Averaging over each individual sunspot cycle, we find values
of 2.67×10
30
erg, 3.66×10
30
erg, 3.70×10
30
erg, 2.69×
10
30
erg, and 2.06×10
30
erg, for cycles 20–24, respectively.
Using Equation (6),
á
ñR
A
is calculated to have its largest value
in cycle 21 of 20.4R
e
, and minimum value of 7.7R
e
in cycle
22. The value of
á
ñR
A
during the current sunspot cycle ranges
from ∼8to16R
e
.
The time-varying torque computed here is in agreement with
previous calculations of the solar wind torque. From the in situ
measurements of Pizzo et al. (1983) using the Helios space-
craft, to the recalculation of Li ( 1999) based on data from the
Ulysses spacecraft. Both of these estimates agree within the
scatter of the 27 day averages computed in this work.
4. Solar Wind Torque on Centennial and Millennial
Timescales
Up until now, we have examined only direct measurements
of the solar wind. These observations have been facilitated by
the exploration of near-Earth space, which began a few decades
ago. For the centuries and millennia before this, only indirect
measurements are available, such as sunspot observations
(Clette et al. 2014), measurements of geomagnetic activity
(Echer et al. 2004), and studies of cosmogenic radionuclides
found in tree rings or polar ice cores (Usoskin 2017). These
indirect measurements are used to estimate longer time
variability of the Sun’s open magnetic flux (Lockwood et al.
2004; Vieira & Solanki 2010; Owens et al. 2011; Wu et al.
2018b). However, these indirect measurements have limita-
tions. Significantly for this work, they do not produce estimates
for how the mass-loss rate of the Sun has varied.
In this section we produce a relation for the mass-loss rate of
the Sun, in terms of the open magnetic flux, which is
constructed using the range of observed values from
Section 3.1. We then use this prescription for the mass-loss
rate, and Equation (7), to evaluate the torque on the Sun due to
the solar wind based on indirect reconstructions of the open
magnetic flux.
4.1. Estimating the Mass-loss Rate, and Wind Torque with the
Open Magnetic Flux
Predicting the mass-loss rates for low-mass stars, such as the
Sun, is a difficult challenge, which has been attempted by
previous authors to varying success (Reimers 1975, 1977;
Mullan 1978; Schröder & Cuntz 2005; Cranmer & Saar 2011;
Cranmer et al. 2017). The mass-loss rates from Section 3.1 are
plotted against their respective open magnetic flux values in the
top panel of Figure 2, colored by sunspot cycle. A weak trend
of increasing mass-loss rate with increasing open magnetic flux
is observed. We fit a power-law relation for the mass-loss rate
in terms of the open magnetic flux,
f
=´
´
-
M 1.26 10 g s
8.0 10 Mx
,10
fit
12 1
open
22
0.44
˙
()
()
()
⎛
⎝
⎜
⎞
⎠
⎟
which is plotted as a solid black line.
There is a large scatter around the fit of Equation (10), which
we wish to propagate through our calculation. We show the 2σ
limits of a log-Gaussian function, centered on the fit, with red
dashed lines. These lines are given by
=
-
MM0.64
fit
fit
˙˙
, and
=
+
MM1.57
fit
fit
˙˙
. When we estimate the mass-loss rate for the
historical estimates of the open magnetic flux in Sections 4.3,
we will use both Equation ( 10) and the 2 σ bounds.
With the mass-loss rate prescribed in terms of the open
magnetic flux, we simplify Equation (7) further to
t
f
=´
´
2.4 10 erg
8.0 10 Mx
,11
30
open
22
1.59
() ()
⎛
⎝
⎜
⎞
⎠
⎟
where the solar wind torque is now given solely as a function of
open magnetic flux. Similarly, the 2σ bound of Equation (10) is
propagated through Equation (7) to give, τ
−
=0.89τ(f
open
),
and τ
+
=1.12τ(f
open
). This allows us to predict the torque on
the Sun due to the solar wind solely from the value of the open
magnetic flux. Note that large (∼50%) uncertainties in
M
˙
translates to only a ∼10% uncertainty in torque, due to the
weak dependence of τ on
M
˙
in Equation (7).
4.2. Reconstructions of the Solar Open Magnetic Flux
For the centuries and millennia pre-dating the space age,
estimates of the open magnetic flux have been produced using a
number of different indirect methods. To compare them with
indirect methods and over a wide range of timescales, we plot
the spacecraft data from Figure 1 also in Figure 3, which
displays the solar wind parameters versus (inverse) logarithmic
look-back time since 2019.
4.2.1. Centennial Variability
Geomagnetic disturbances, caused by the interaction of the
solar wind and the Earth’s magnetosphere, have been found to
Figure 2. Mass-loss rate,
M
˙
, vs. open magnetic flux, f
open
, derived the in situ
observations of the OMNI data set. Values are color-coded by sunspot cycle,
20–24. The black line corresponds to the power-law fit of Equation (10). The
dashed red lines indicates the 2σ bounds given by a log-Gaussian centered on
the fit line.
4
The Astrophysical Journal, 883:67 (9pp), 2019 September 20 Finley et al.

Frequently asked questions

Q1. What have the authors contributed in "Solar angular momentum loss over the past several millennia" ?

In this paper, the authors used near-Earth measurements of the solar wind plasma and magnetic field to calculate the torque on the Sun throughout the space age. 

Q2. Why is the torque in the solar wind so variable?

Due to the nonlinear dependence of Equation (11) on the open magnetic flux in the solar wind, short-term variability in the open magnetic flux, even around a fixed average value, will increase the long-term average torques. 

Q3. What is the effect of the cosmogenic radionuclides on the solar flux?

Reconstructions of solar activity based on the concentrations of cosmogenic radionuclides incur smoothing effects from the transport and deposition timescales of each radionuclide. 

Q4. What is the effect of CMEs on the solar wind?

CMEs carry only afew percent of the total solar mass-loss rate (Cranmer et al. 2017), however, at solar maximum they can provide a significant fraction of the average mass flux in the equatorial solar wind (Webb & Howard 1994). 

Q5. How did they calculate the torque on the Sun due to the solar wind?

By applying a braking law derived from magnetohydrodynamic (MHD) simulations by Finley & Matt (2018), they calculated the time-varying torque on the Sun due to the solar wind. 

Q6. Why is the open magnetic flux an upper limit?

Due to kinematic effects that occur between the Alfvén surface and the measurements taken at 1 au, their estimate of the open magnetic flux is likely an upper limit (Owens et al. 2017a). 

Q7. What is the effect of the magnetic field on the rotation of low-mass stars?

The resulting strong dependence of torque on rotation rate leads to a convergence of rotation periods with age, as initially fast rotating stars generate strong magnetic fields and experience a larger braking torque than the initially slowly rotating stars. 

Q8. What are the fit constants for the wind torques?

These fit constants also account for the multiplicative factor of (4π)2, and any effects introduced by the flow being multi-dimensional in nature. 

Q9. How do the authors estimate the open magnetic flux?

The authors use measurements of the solar wind to estimate the open magnetic flux usingf p= á ñR B R4 , 8Ropen 2 1 hr 27 days∣ ( )∣ ( )where the authors average the radial magnetic field BR, (taken from a single observing location) at a distance R from the Sun, over a full solar rotation (27 days), and assume that the solar wind is roughly isotropic on their averaging timescale, in order to estimate the open magnetic flux. 

Q10. How do the authors calculate the braking torque of the Sun?

The authors then utilize reconstructions of the solar open magnetic flux, based on geomagnetic indices (Lockwood et al. 2014a), sunspot number records (Owens & Lockwood 2012), and concentrations of cosmogenic radionuclides (Wu et al. 2018b), to estimate the braking torque over the last four centuries, and then the last nine millennia. 

Q11. What is the simplest way to estimate the mass loss rate for the open magnetic flux?

When the authors estimate the mass-loss rate for the historical estimates of the open magnetic flux in Sections 4.3, the authors will use both Equation (10) and the 2σ bounds. 

Q12. What is the reason why the original data sets have a larger torque than their smoothed?

By comparing the average torques from the smoothed reconstructions of Lockwood et al. (2014a) and Owens et al. (2017b) to their original data sets, the authors find the original data sets have alarger torque by ∼4% than their smoothed counterparts; a result of the nonlinearity of the torque on open magnetic flux. 

Q13. How have previous authors removed these events?

Previous authors have removed these events through the use of CME catalogs (Cane & Richardson 2003) or clipping anomalous spikes (Cohen 2011). 

Q14. How did Finley and Matt determine the solar wind torque?

In Finley et al. (2018), the short timescale variability (from ∼27 days up to a few decades) of the solar wind was examined using in situ observations of the solar wind plasma and magnetic field.