A Solar-cycle Study of Coronal Rotation: Large Variations, Rapid Changes, and Implications for Solar-wind Models

Abstract

Information on the rotation rate of the corona, and its variation over latitude and solar cycle, is valuable for making global connections between the corona and the Sun, for global estimates of reconnection rates and as a basic parameter for solar-wind modeling. Here, we use a time series of tomographical maps gained from coronagraph observations between 2007 and 2020 to directly measure the longitudinal drift of high-density streamers over time. The method reveals abrupt changes in rotation rates, revealing a complex relationship between the coronal rotation and the underlying photosphere. The majority of rates are between −1.°0 to +0.°5 day ⁻¹ relative to the standard Carrington rate of 14.°18 day ⁻¹ , although rates are measured as low as −2.°2 day ⁻¹ and as high as 1.°6 day ⁻¹ . Equatorial rotation rates during the 2008 solar minimum are slightly faster than the Carrington rate, with an abrupt switch to slow rotation in 2009, then a return to faster rates in 2017. Abrupt changes and large variations in rates are seen at all latitudes. Comparison with a magnetic model suggests that periods of equatorial fast rotation are associated with times when a large proportion of the magnetic footpoints of equatorial streamers are near the equator, and we interpret the abrupt changes in terms of the latitudinal distribution of the streamer photospheric footpoints. The coronal rotation rate is a key parameter for solar-wind models, and variations of up to a degree per day or more can lead to large systematic errors over forecasting periods of longer than a few days. The approach described in this paper gives corrected values that can form a part of future forecasting efforts.

Full text

A Solar-cycle Study of Coronal Rotation: Large Variations, Rapid Changes, and
Implications for Solar-wind Models
Liam Edwards , David Kuridze , Thomas Williams , and Huw Morgan
Department of Physics, Aberystwyth University, Ceredigion, Cymru, SY23 3BZ, UK; hmorgan@aber.ac.uk
Received 2021 September 23; revised 2022 February 4; accepted 2022 February 11; published 2022 March 25
Abstract
Information on the rotation rate of the corona, and its variation over latitude and solar cycle, is valuable for making
global connections between the corona and the Sun, for global estimates of reconnection rates and as a basic
parameter for solar-wind modeling. Here, we use a time series of tomographical maps gained from
coronagraph observations between 2007 and 2020 to directly measure the longitudinal drift of high-density
streamers over time. The method reveals abrupt changes in rotation rates, revealing a complex relationship between
the coronal rotation and the underlying photosphere. The majority of rates are between −1°.0 to +0°.5 day
−1
relative to the standard Carrington rate of 14°. 18 day
−1
, although rates are measured as low as −2°. 2 day
−1
and as
high as 1°.6 day
−1
. Equatorial rotation rates during the 2008 solar minimum are slightly faster than the Carrington
rate, with an abrupt switch to slow rotation in 2009, then a return to faster rates in 2017. Abrupt changes and large
variations in rates are seen at all latitudes. Comparison with a magnetic model suggests that periods of equatorial
fast rotation are associated with times when a large proportion of the magnetic footpoints of equatorial streamers
are near the equator, and we interpret the abrupt changes in terms of the latitudinal distribution of the streamer
photospheric footpoints. The coronal rotation rate is a key parameter for solar-wind models, and variations of up to
a degree per day or more can lead to large systematic errors over forecasting periods of longer than a few days. The
approach described in this paper gives corrected values that can form a part of future forecasting efforts.
Unified Astronomy Thesaurus concepts: Solar corona (1483);Solar atmosphere (1477);Solar cycle (1487);Solar
differential rotation (1996);Solar physics (1476);Solar rotation (1524);Solar wind (1534)
1. Introduction
The rotation of the Sun is a complicated subject due to the
variation of results arising from a broad range of observations and
analysis techniques. The variation of results is a symptom of the
complexity of the Sun’s rotation arising from the interdependence
of the solar magnetic dynamo with the convective plasma, leading
to a rotation which is dependent on latitude, depth within the Sun,
and solar-cycle phase; see Howard (1984), Schroeter (1985),
Thompson et al. (2003), and Beck (2000)for reviews. The rotation
of the solar atmospheric layers, in relation to the underlying
photosphere, gives insight into how different atmospheric
structures (e.g., active regions and coronal holes)may be
influenced by subphotospheric motions, and to rates of inter-
change reconnection in the low atmosphere. It is a field made
complicated by many factors including the different dependence
of types of observation on density and geometrical factors (e.g.,
radio, white light, line emission), the temperature dependence of
different observations, uncertainties on a local level as to the
magnetic connectivity between different atmospheric layers, lack
of direct routine observations of the atmospheric magnetic field,
and, above all, the extended line of sight (LOS)through the
optically thin medium.
The rotation of the optically thin corona can be estimated
using several different types of observations. For heights above
the very lowest corona (or where the atmospheric features
cannot be observed against the disk, necessitating the use of
off-limb observations), most rotation estimates are based on
long time series, subject to an extended LOS. As different
coronal structures rotate through the field of view, the signal’s
modulation gives an estimate of the dominant rotation rate.
This is commonly called flux modulation, and it must use long
time series to determine the rotation rate (several months to
years), thus shorter timescales are inaccessible. We refer the
reader to the introduction of Morgan (2011a)for a more
detailed overview of solar atmospheric rotation.
There have been several works studying coronal rotation
since Morgan (2011a). Vats & Chandra (2011)used long-term
flux modulation of both radio and X-ray imagery to show a
small yet clear differential rotation of the corona, with a north–
south asymmetry in rotation rates, with asymmetry most
pronounced during solar maximum. A similar asymmetry has
been shown using extreme-UV (EUV)observations by Sharma
et al. (2020b). Li et al. (2012)used a very long time series of
solar radio observations to show a small decreasing trend in the
mean coronal rotation rate between years 1947 and 2009, but
no significant link to the Schwabe cycle. Xie et al. (2017)also
found a decreasing trend over the same period, and found
significant periodicities in the temporal variation of rotation
rates ranging from 2 to 10 yr. Conversely, the study of the
coronal green line by Deng et al. (2020)showed an increasing
rotation rate over a similar multidecadal period, finding similar
significant periodicities ranging from 3 to 11 yr, and thus a
possible link between rotation rates, the 11 yr solar cycle, and
the quasi-biennial oscillation. Using radio observations, Bhatt
et al. (2017)found a decreasing rotation rate with greater
altitude, in disagreement with both radio measurements by Vats
et al. (2001), and EUV measurements by Sharma et al. (2020a),
who found an increasing rate with altitude. Obridko &
Badalyan (2020)used potential field source surface magnetic
The Astrophysical Journal, 928:42 (14pp), 2022 March 20 https://doi.org/10.3847/1538-4357/ac54ba
© 2022. The Author(s). Published by the American Astronomical Society.
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of the Creative Commons Attribution 4.0 licence. Any further
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1

models of the corona, based on photospheric magnetic field
estimates, to study differential rotation as a function of latitude
and height, and interpreted their results in terms of links to
subphotospheric rotations. Their interpretation is interesting,
but does not tie in the large body of results gained from decades
of observation. Mechanisms must be invoked and proven (e.g.,
interchange reconnection)to properly link the coronal rotation
and large-scale coronal structural changes to photospheric and
subphotospheric rotation and bulk motions. The large body of
observational results relating to coronal rotation and structure,
as outlined here, in Morgan (2011a), and elsewhere, contain
results that are often contradictory, and consolidation is
required for further understanding. One missing aspect is the
use of long-term simulations to gauge the response of the
coronal rotation to the changing distribution of the photo-
spheric field; global magnetofrictional models may be ideally
suited for this.
Morgan (2011a)first used tomography maps of the corona to
estimate coronal rotation rates at a height of 4 R
e
over the
period 1996–2010, and found large variations of rotation rates
compared to other studies, up to ±3°day
−1
relative to the
Carrington rate. The study found surprising variations in
rotation between latitudes, and rapid changes in rotation rate at
a given latitude. A related work also used a time series of
tomography maps to study the longitudinal drift of density
structures in the equatorial streamer belt during the 1996 solar
minimum (Morgan 2011b). The approach of Morgan (2011a)
differs considerably from a flux modulation approach, since a
reconstruction of the coronal density structure is made using 2
weeks of input data, and the rotation is estimated from the
changing configuration of the reconstructed densities. So while
flux modulation gives the dominant mean rotation rate from a
time series spanning at least a few full rotations (several
months), a time series of tomography maps give a more direct
measurement of rotation with finer time resolution. Projection
effects, and the related limitations of flux modulation, are
examined by Mancuso & Giordano (2013)in the context of
ultraviolet emission-line measurements. In particular, they
show that projection effects can lead to a bias toward finding
a rigid rotation at higher latitudes. The same arguments apply
equally to any study of rotation based on flux modulation, thus
the combination of projection effects and long time-series
averaging may lead to a misleading picture of rigid coronal
rotation.
This current study extends the work of Morgan (2011a)by
studying the coronal rotation for years 2007 to 2020. Here we
use a much-improved tomography method to gain the coronal
density structure, and the drift of streamers in longitude are
traced manually, thus avoiding uncertainties or ambiguities
associated with automated methods. The observations, tomo-
graphy method, and the manual method for measuring rotation
are summarized in Section 2(an automated correlation-based
method is described in the Appendix)with the results presented
in Section 3. We also provide interpretation, based on magnetic
models, of what may be causing abrupt changes in rotation
rates in Section 4.1, and briefly describe the relevance to solar-
wind models in Section 4.2. Conclusions are given in
Section 5.
2. Method
Morgan (2019, hereafter Paper II)presents a recent
advancement in coronal rotational tomography, which gives
maps of the coronal electron density at heliocentric distances
greater than ≈3R
e
, at all periods of the solar cycle. The
method is based on a spherical harmonic model of the coronal
density, constrained by coronagraph data that are preprocessed
and calibrated as described by Morgan (2015, hereafter
Paper I). Initial results and further method developments are
presented in Morgan & Cook (2020, hereafter Paper III). The
resulting maps clearly show the large-scale distribution of the
streamer belt, albeit showing structure that is smoother than the
true density distribution. The maps are based on a static
reconstruction, or the smooth and positive distribution of
density that best satisfies two weeks of input data.
The COR2 coronagraphs are part of the Sun Earth
Connection Coronal and Heliospheric Investigation (SECCHI;
Howard et al. 2002)suite of instruments aboard the twin Solar
Terrestial Relations Observatory (STEREO A and B; Kai-
ser 2005). Half a solar rotation (or ≈2 weeks)of COR2A
observations are needed to create an electron density map using
the calibration processes of Paper I, the tomographic inversion
of Paper II, and the refinement methods of Paper III. Using the
facilities of SuperComputing Wales, for data spanning 2007
March 17 to 2019 September 5, tomography maps have been
created at ≈2 day increments, resulting in over 2000 sets of
maps over the period. The period includes the 2014–2015 data
gap when STEREO A was traversing behind the Sun. Since
each map at a given date is a static reconstruction created from
±1 week of data from that date, the time series of maps over
several years can be considered as a “sliding window”of
reconstructions. For each date, a set of nine maps are created
for heliocentric distances between 4 and 8 R
e
at 0.5 R
e
increments: this work uses the 4 R
e
maps only.
Figure 1shows an example of density maps for four dates in
early 2011. When a time series of such maps are viewed, it is
common to see a drift of the high-density streamers in
longitude, as well as other changes in structure. For example,
for Figure 1from the earliest map at 2011 February 3 through
to 2011 May 2, the high-density streamer structure drifts
generally to lower longitudes. This simple example shows how
we can extract densities at constant latitudes from the maps in
order to analyze the coronal rotation.
Profiles of density at a given latitude are stacked in time,
giving a time–longitude array of densities. Figure 2shows an
example time–longitude map for the ≈2 yr period beginning on
2012 April, at a latitude of −60°. Density structures that rotate
at the Carrington rate appear as horizontal features in these
plots. This example shows, from the end of 2012 to the end of
2013, one high-density streamer that has a consistent and linear
drift with a negative gradient in Carrington longitude, meaning
the streamer is rotating slower than the Carrington rate. Across
the whole 12 yr of data, and across all latitudes, there are
numerous examples of positive and negative drifts. For each
clear example of a coherent drifting streamer, we manually
trace a straight line, as shown in Figure 2. The gradient is
calculated, which gives the mean rotation rate of the streamer
relative to Carrington rotation. For the example of Figure 2,we
have a gradient of D
D==-

--
y
x1 . 64 day
360
220
1. Manually
identifying and tracing these broad and varying features
involves uncertainty. For example, the streamers only drift
slowly and sometimes contain multiple density peaks at the
same longitude over time (as can be seen at the start of the solid
white line in Figure 2). Furthermore, the path described by the
streamer over time may not follow an exact straight line, which
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is also seen in the example. Quantifying the uncertainty is
difficult. Given approximate time uncertainty of ±30 days and
a longitudinal uncertainty of ±10°, for a streamer that drifts
through 360°over 1 yr, the uncertainty in the resulting rotation
rate is around 15%.
The solar-cycle latitudinal distribution of streamers is shown
in Figure 3, which shows the density averaged over all
longitudes as a function of time and latitude. This distribution
shows that a full 12 yr analysis at all latitudes is not possible.
So at increasing latitudes from the equator, the study becomes
increasingly limited to the years surrounding solar maximum,
when high-density streamers exist at higher latitudes.
3. Results
Figure 4shows an example time–longitude density plot for
(a)the solar minimum and ascending phase 6 yr period starting
on 2007 July 1, and (b)the following 6 yr period including
solar maximum and the descending phase. This latter period
also contains a long period of missing tomography data from
≈2014 July to 2016 January caused by telemetry disruption as
STEREO A passed behind the Sun. The figure shows a clear
rotation rate approximately 0°.25 day
−1
faster than the
Carrington rate from 2007 January to 2009 October, followed
by a 2 yr period of rotation considerably slower than the
Carrington rate. During the years surrounding solar maximum,
there are less clear signatures of rotation rates, and the large-
scale structures are short lived compared to solar minimum and
the ascending phase: lasting months rather than years. Despite
this, some of these can be traced over time to calculate rates.
From mid-2016 onwards the rotation returns to a rate faster
than the Carrington rate, similar to the 2008 solar minimum
rate. We attempt to explain the abrupt change in rate during
2009 October by the connection between coronal streamers
and the underlying lowest corona and photosphere in the
discussion.
Figure 5shows the density evolution for latitude −50°for
the period from when streamers first appear at this latitude
(early 2011)through to 2014 January. Slow rotation can be
seen up to around mid-2013. The streamer distribution then
becomes incoherent and uncertain to interpret in terms of
rotation. Figure 6shows the density near to the south pole at
−80°for 2013 April to 2014 June. Although less clear than at
lower latitudes, there is a clear negative gradient drift of the
main streamer.
Table 1lists information on all manually traced streamers in
the south corona throughout the data set, including the latitude
of each structure, the mid-date and duration in days, and the
estimated rotation rate relative to the Carrington rate. Table 2
shows information for northerly and equatorial streamers. The
top plot of Figure 7visualizes this information, showing all
measured rotation rates over the solar cycle as a function of
Figure 1. Maps of the coronal electron density at a distance of 4 R
e
for dates
(a)2011 February 3, (b)2011 March 2, (c)2011 April 2, and (d)2011 May 2.
The longitude and latitude are Carrington spherical coordinates. The density is
as given in the color bars, in units of 10
5
cm
−3
, with all maps sharing a
common color scale. The horizontal (constant latitude)dashed white line is at a
latitude of 40°, and illustrates how a latitudinal slice can be extracted from the
density map in order to analyze the coronal rotation.
Figure 2. An example of density as a function of time and Carrington
longitude, with density indicated by the color bar. One clear example of a
streamer drifting in longitude over time is traced by the solid white line. Note
that the longitude axis is wrapped beyond the ±180°in order to show
continuation, thus the same information is repeated at the bottom and top of the
plot. The ±180°lines are shown by the white dashed lines.
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time and latitude. The majority of values are slower than the
Carrington rotation rate. Faster-than-Carrington rotation is seen
near the equator during solar minimum in 2008, from
equatorial to mid-latitude regions in the north during the
ascending phase to maximum in 2010–2012, and in the polar
north during solar maximum. There are also some examples of
faster-than-Carrington rates in the polar south during solar
Figure 3. Density, averaged over all longitudes, as a function of latitude and time. The vertical black blocks are either data gaps, periods where data is too scarce to
apply tomography, or periods where the tomography has failed to reconstruct.
Figure 4. Density in the equatorial plane at a height of 4 R
e
, as a function of time and Carrington longitude for dates (a)2007 July 1 to 2013 July 1, and (b)2013 July
1 to 2019 July 1. There is a long data gap from 2014 July to 2016 January. Interpolation is used to fill shorter data gaps (the times of these shorter data gaps can be seen
in Figure 3).
Figure 5. Density at the south mid-latitude (−50°)for 2011 January to 2014
January.
Figure 6. Density at high south latitude (−80°)for 2013 April to 2014 June.
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Table 1
Measured Rotation Rates (in degrees per day, Relative to the Carrington Rate), for Streamers in the South
Latitude Mid-date Duration Rot. Rate Latitude Mid-date Duration Rot. Rate
−85 2013/06/07 70 −1.36 −85 2013/10/09 160 −1.84
−85 2013/11/09 100 1.60 −85 2014/04/28 120 −0.75
−80 2012/11/18 140 0.29 −80 2013/04/07 140 −1.50
−80 2014/01/02 160 −0.72 −75 2012/10/05 190 −0.63
−75 2013/04/13 150 −1.37 −75 2014/04/03 160 0.28
−70 2013/02/11 240 −1.35 −70 2013/08/15 170 −1.00
−70 2014/03/28 160 0.28 −65 2013/02/05 230 −1.46
−60 2012/01/07 150 −1.70 −60 2013/01/11 150 −1.43
−60 2013/03/22 150 −1.33 −60 2013/09/23 140 −0.86
−55 2012/01/06 150 −1.73 −55 2013/01/05 360 −1.13
−55 2013/01/20 210 −0.98 −50 2011/12/14 170 −1.97
−50 2013/01/17 210 −1.17 −50 2013/02/06 250 −0.88
−45 2011/10/01 270 −0.67 −45 2012/08/21 180 −0.83
−45 2013/02/22 150 −1.00 −45 2013/02/17 240 −0.98
−40 2011/09/14 330 −0.50 −40 2012/10/28 350 −0.60
−40 2013/03/12 200 −0.93 −35 2010/05/27 170 −1.00
−35 2010/09/04 170 −0.91 −35 2013/01/21 190 −0.76
−35 2013/03/22 170 −0.97 −35 2016/06/19 420 0.18
−30 2010/04/12 240 −1.04 −30 2010/09/19 160 −0.97
−30 2011/06/06 400 −0.46 −30 2012/06/05 250 −1.10
−30 2013/01/31 190 −1.00 −30 2013/05/26 260 −1.21
−30 2016/10/12 190 −0.42 −30 2017/09/07 370 −0.04
−25 2010/04/15 260 −1.02 −25 2011/05/30 380 −0.59
−25 2012/05/29 230 −1.33 −25 2013/06/13 270 −1.19
−25 2016/09/20 200 −0.35 −25 2017/02/02 130 −0.77
−25 2017/08/31 410 −0.18 −20 2007/11/29 170 0.41
−20 2009/12/08 150 −1.73 −20 2010/12/03 130 0.35
−20 2011/05/02 430 −0.44 −20 2013/08/14 160 −1.16
−20 2014/03/02 200 0.28 −20 2017/08/13 440 −0.09
−15 2008/03/13 340 0.09 −15 2008/06/16 370 0.16
−15 2012/08/24 130 0.77 −15 2013/07/30 210 −0.90
−15 2017/03/06 80 −1.25 −10 2008/03/13 440 0.15
−10 2009/10/24 120 −1.38 −10 2013/08/14 180 −0.78
−10 2017/03/01 90 −1.33 −10 2018/06/24 110 0.95
−5 2007/11/14 320 −0.27 −5 2007/11/04 180 0.42
−5 2008/04/22 720 0.17 −5 2013/08/14 180 −0.72
−5 2017/03/11 130 −1.08 −5 2018/10/17 240 −0.13
Note. The duration is given in days.
Table 2
Measured Rotation Rates for Streamers at the Equator and in the North
Latitude Mid-date Duration Rot. Rate Latitude Mid-date Duration Rot. Rate
0 2008/09/14 670 0.22 0 2012/06/05 410 −0.05
0 2013/08/24 200 −0.50 0 2017/03/16 440 0.24
5 2008/11/08 380 0.17 5 2009/12/23 120 −0.83
5 2010/10/29 140 0.61 5 2017/01/30 330 0.27
5 2018/03/01 200 −0.18 10 2008/05/27 670 0.23
10 2009/10/24 300 −0.25 10 2010/10/29 160 0.69
10 2012/06/05 390 −0.14 10 2019/04/05 240 −0.92
15 2008/03/03 700 0.22 15 2009/08/15 140 −0.61
15 2010/11/18 100 1.30 15 2012/04/01 240 −0.33
15 2018/03/11 200 0.68 20 2008/04/12 580 0.16
20 2009/04/12 90 −1.00 20 2010/11/28 240 0.69
20 2014/01/16 150 0.33 20 2019/02/09 130 −1.19
25 2009/03/23 130 −0.85 25 2011/01/12 150 0.90
25 2017/02/24 160 0.59 30 2011/01/18 120 0.92
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maximum. After the recovery of STEREO during the descending
phase to minimum (2017 and 2018), there is a mixture of slow
and fast rotation. In general over the solar cycle, faster rotations
occur more frequently in the north.
The bottom plot of Figure 7shows this same information but
relative to the underlying photospheric rotation rate at the same
latitude, as given by Howard (1984). This plot shows that, in
general, the corona rotates faster then the photosphere at the
same latitude, up to 4°day
−1
for some periods. The generally
accepted view of coronal rotation is of a rigid rotator compared
to the photosphere, with less differential rotation. Based on this
view, we would expect the corona to rotate faster than the
photosphere at high latitudes. Our results shows that this is true
at the majority of times, but is an oversimplistic model of
coronal rotation. At certain latitudes and times the corona can
rotate slower than the photosphere at the same latitude, and at
some times even low-latitude streamers can rotate considerably
faster than the equatorial photosphere. The discussion gives
some further insight into this complicated pattern of rotation in
terms of how the streamer belt is magnetically connected to the
low atmosphere. Nevertheless, the bottom plot of Figure 7
shows that the corona is more likely to be rotating faster than
the underlying photosphere at all latitudes, which lends support
for the connection between large-scale coronal structure and
subphotospheric motions.
Figure 8shows rotation rates averaged over all time as a
function of latitude, with variances over time shown by the shaded
areas. That is, over the whole period of study, the mean and
standard deviation of all rotation rates for a given latitude give the
mean and standard deviation shown in the plot. These averaged
values are fitted to several equations, a fourth-order polynomial
function of latitude (Equation (1)) and its simplified, lower-order
variants:
()w ffff=++++AB C D Esin sin sin sin . 1
234
Fitting this polynomial to the data gives values for the
coefficients as follows: A=14.152, B=−0.736, C=−2.362,
D=0.584, and E=1.839, which is shown by the solid red
lines in Figure 8. The first simplified function used is
()wf=+ABsin , 2
2
where fis the latitude shown as the dotted red line in Figure 8.
Fitting our results to this function gives coefficients A=13.929
and B=−0.551. The second simplified function is
()wff=+ +AB Csin sin , 3
24
as shown with the dashed red line in Figure 8. Fitting our
results to this function gives coefficients A=14.152, B=
−2.362, and C=1.839. For comparison, several estimates of
the photospheric and coronal differential rotation by studies
over the past century are also shown (Newton & Nunn 1951;
Fisher & Sime 1984; Howard et al. 1984; Wöhl et al. 2010;
PoljančićBeljan et al. 2017; Dorotovičet al. 2018; Jha et al.
2021). This plot emphasizes that when rotation rates are aver-
aged over long periods, the resulting means tend to show rigid
rotation over latitude. A similar result was shown using
tomography by Morgan (2011a), and by many other studies
based on flux modulation. However, this hides the considerable
variance in rotation rates over time (as shown by the shaded
areas which show this variance). The corona can, at times,
possess considerable latitudinal differential rotation.
Another significant result from Figure 8is the asymmetry
shown between north and south in the bottom panel. The mid-
latitudes in the north dip to slow average rotation rates, well below
1°day
−1
slower than Carrington, which is not seen in the south.
Table 2
(Continued)
Latitude Mid-date Duration Rot. Rate Latitude Mid-date Duration Rot. Rate
30 2012/07/21 100 −0.75 30 2017/02/20 110 0.50
30 2017/08/09 190 −0.42 35 2010/06/03 270 −1.26
35 2011/03/15 160 −1.16 35 2012/07/17 100 −0.75
35 2017/11/23 130 0.69 40 2010/07/27 190 −1.29
40 2011/02/22 130 −1.19 40 2012/07/26 110 −0.68
40 2012/11/03 70 −0.86 45 2010/07/02 245 −1.08
45 2011/04/05 150 −0.50 45 2013/02/28 340 −0.44
45 2013/10/31 170 0.50 45 2013/11/20 110 0.64
50 2010/04/05 60 −2.42 50 2012/02/04 100 −1.05
50 2013/04/04 190 −0.37 50 2013/10/26 140 0.57
50 2013/11/15 120 0.58 55 2011/10/18 320 −1.11
55 2013/04/30 240 −0.27 55 2013/11/11 130 0.50
55 2013/11/16 100 0.65 60 2011/07/22 190 −1.03
60 2012/07/21 180 0.44 60 2013/02/11 270 −0.11
60 2013/07/06 140 −0.64 60 2014/01/27 110 −0.77
65 2011/09/06 220 −1.25 65 2012/10/10 580 −0.20
65 2013/07/02 130 −0.73 65 2013/12/09 130 −1.04
70 2011/07/27 60 −1.00 70 2013/02/16 160 −0.31
70 2013/07/16 160 −0.50 70 2014/03/23 120 0.50
75 2012/04/19 240 −1.13 75 2013/01/19 250 −0.50
75 2013/07/23 140 −0.18 75 2014/01/19 120 −0.71
80 2012/07/27 140 −1.50 80 2013/08/31 140 0.82
80 2013/09/25 110 0.77 85 2012/03/10 190 −2.03
85 2013/03/10 120 −1.17 85 2013/08/27 120 0.83
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Figure 10(a)of Morgan (2011a)shows latitudinal profiles of
coronal rotation from the previous solar cycle (1996–2009),and
did not find significant asymmetry, although some asymmetry
was present when different phases of the cycle were isolated
(Figures 10(b)–(d)). Several previous studies of coronal rotation
have shown a north–south asymmetry (Wöhl et al. 2010;
Bagashvili et al. 2017; Badalyan & Obridko 2018; Dorotovič
et al. 2018;Hrazdíraetal.2021).Sharmaetal.(2020b)found a
high asymmetry during the 2011–2014 solar maximum, with less
asymmetry in the ascending and descending phases. These results,
as with ours, are contrary to Mancuso et al. (2020), who suggest
that the asymmetry is less pronounced than previously inferred.
Our results, which are not based on a long time-series analysis and
instead are based on resolved coronal structure through
tomography, show clearly that there is a north–south asymmetry
during the recent solar cycle, with a significant slower rotation at
northern latitudes of 50°–65°.
Figure 9shows a detailed comparison between our results
and those of Bagashvili et al. (2017), derived from detailed
measurements of coronal holes. The agreement is excellent in
the north. In the south, our estimates tend to vary considerably
compared to the smooth curve of Bagashvili et al. (2017), and
are significantly higher polewards of −40°. The rotation of
coronal streamers must be strongly linked to that of coronal
holes, yet the comparison in the south shows significant
differences. One major reason may be that high-latitude coronal
holes, in the lowest corona, may be measured at extended
periods over the solar cycle by Bagashvili et al. (2017),
whereas we can only measure the rotation of high-latitude
streamers for a year or two surrounding solar maximum.
Another reason may be the highly nonradial structure of the
corona, where the measurements of Bagashvili et al. (2017)are
made in the lowest corona and ours are made at extended
distances.
Figure 10 shows, for all measured streamers, the percentage of
time at each rotation rate. The distribution tends toward negative,
or slower, rotation rates, with the most probable rate around
−1°day
−1
relative to Carrington. The bulk of rates are between
−1and0°.5day
−1
, although significant time is spent at slower
(down to −2°.2 day
−1
)and faster (up to 1°.6day
−1
)rates.
Figure 7. Top: rotation rates for coronal streamers as a function of time and latitude. Red (blue)colors represent faster (slower)rotation rates relative to Carrington
rotation, as indicated in the color bar. The horizontal span of each line represents the period over which the rotation rate was estimated. Bottom: rotation rates for
coronal streamers relative to the underlying photospheric rotation rate at the same latitude, with photospheric rotation rates as given by Howard (1984).
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4. Discussion
4.1. Rotation Rates and the Latitudes of Streamer Footpoints
A central issue for understanding coronal rotation rates is
how the coronal streamers at 4 R
e
are connected to the low
solar atmosphere and photosphere. As a simple illustration of
this issue, Figure 11(a)shows an image of the 2009 solar
minimum corona composed from an EUV observation of the
disk and lowest corona by the EUV Imaging Telescope (EIT;
Delaboudiniere et al. 1995)on board the Solar and Helio-
spheric Observatory (SOHO; Domingo et al. 1995), a Mauna
Loa Solar Observatory (MLSO; Fisher et al. 1981)MK4
coronameter observation of the low corona, and a Large Angle
and Spectrometric Coronagraph/SOHO (LASCO/SOHO;
Brueckner et al. 1995)C2 observation. If we measure the
coronal rotation rate for the most prominent equatorial
streamers during this time, the composite image suggests that
the footpoints of these streamers encompass a large latitudinal
range bridging the equator. The apparent footpoint therefore
encompasses a large range of different photospheric rotation
rates. For the 2010 ascending phase shown in Figure 11(b), the
prominent streamer between the equator and mid-latitudes in
the southeast corona has an apparent narrower footpoint that
ranges from south mid-latitudes to just above the equator.
While these composite images cannot give us the detailed
three-dimensional information needed to link coronal streamers
to the low atmosphere, they do serve to illustrate the argument
that the connection between streamers in the extended corona
and the lower atmosphere is not straightforward, and that we
would expect a model of rotation rates based on these
connections and photospheric rotation rates to lead to abrupt
changes over the solar cycle as the global coronal magnetic
field evolves.
Figure 12 explores photospheric–coronal connections using
a Potential Field Source Surface (PFSS)magnetic extrapolation
model (Newkirk & Altschuler 1969; Schatten et al. 1969),
using the Solarsoft PFSS package based on Helioseismic
Magnetic Imager (Scherrer et al. 2012; Schou et al. 2012)/
Solar Dynamics Observatory data. The left column of
Figure 12 shows plots from solar minimum (2009 January)
when we measure a positive (a little faster than Carrington)
coronal rotation rate near the equator. The right column show
plots from a year later (2010 January)when we measure a
strong negative rotation rate. The change from positive to
negative rate occurs abruptly at the end of 2009, as can be seen
in Figures 4and 7. Figures 12(a)and (b)show the longitude–
latitude tomography density maps for 2009 January and 2010
January, respectively. There is no obvious large structural
difference between these two periods, only small changes. For
example, in 2010 there is a northerly high-density streamer that
extends to above 30°north, whereas the 2009 streamer belt is
restricted to within ±30°. Therefore, from inspection of the
tomography maps, there does not seem to be an obvious
structural change that may be connected to the change in
rotation rate.
Figure 12 panels (c)and (d)show information extracted from
the PFSS models for these dates. The green areas are regions of
high convergence at the source surface (placed at 2.5 R
e
). This
is a value that quantifies how widely spaced the footpoints of
open magnetic field lines are at the photosphere: if neighboring
field lines at the source surface arise from widely separated
photospheric regions, the convergence is high. This value,
readily derived from PFSS models through field-line tracing,
was used by Morgan (2010), and is equivalent to the squashing
factor Qdeveloped in an advanced generalized field-mapping
method by Titov (2007). High-density coronal streamers (or
pseudostreamers)are expected to exist at regions of high
convergence, or Q. The gray and red lines show field lines that
are open at the source surface equator. The red lines are field
lines that are close (within 2°in latitude)to a region of high
convergence at the source surface (so likely associated with a
high-density streamer). We record the location of the photo-
spheric footpoints of these red lines, thus creating a record of
the footpoint location of the segments of coronal streamers that
are near the equator. The latitudes of these photospheric
footpoints are shown in Figure 12 panels (e)and (f).
Figure 12(e), for 2009 January, shows a latitudinal range of
approximately −65°to 65°, with a high count of field lines at
the equator, a high count at the extreme south near −60°, and a
Figure 8. Rotation rates averaged over all time as a function of latitude (black
circles), with the standard deviation over time shown by the shaded areas
(darker gray and lighter gray for one and two standard deviations, respectively).
The dotted red line in the top two panels is a fit to the function of latitude given
by Equation (2), and the dashed red line is a fit given by Equation (3). Colored
lines, as indicated in the legend, show photospheric and coronal differential
rotation estimates by Newton & Nunn (1951), Howard et al. (1984), Fisher &
Sime (1984), Wöhl et al. (2010), PoljančićBeljan et al. (2017), Dorotovičet al.
(2018)and Jha et al. (2021). The bottom panel highlights the asymmetry of the
rotation by fitting to Equation (1)and comparing the fit with results from Vats
& Chandra (2011).
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more even distribution between the equator and 50°in the
north. Figure 12(f), for 2010 January, shows a slightly wider
distribution between −70°and 65°. There are two peaks in the
distribution near the high-latitude extremities, a peak at −30°
and a lesser peak at 15°. The major difference between the two
dates is that 2010 January has no connections to the equator,
whereas 2009 January has a strong connection. This gives a
possible explanation of the faster rotation in 2009 since the
photospheric rotation rate is fastest at the equator. In 2010, the
equatorial streamers are linked to higher latitudes where the
rotation is slower.
Figure 13 explores this idea further by applying a similar
analysis to the period from mid-2007 to 2020. Every 4 months
over this period, we calculate a PFSS model and associated
convergence map at the source surface. We extract the field
lines that are associated with the equatorial streamers and
calculate the median absolute latitude of their photospheric
footpoints. These are plotted as a function of time in
Figure 13(a), with the error bars giving the standard deviation
of absolute latitudes from the median. The median latitudes are
approximately 30°from the equator at solar minimum
(2007–2009), dropping to approximately 20°in 2010. There
is a gradual decrease to 10°from 2010 to 2016, then a large
increase back to 40°between 2016 and 2020. Thus, over the
cycle, the segments of coronal streamers near the equator have
footpoints spread over a wider latitudinal range, and extending
to higher latitudes, at solar minimum. At solar maximum, this
range becomes more limited and moves closer to the equator.
Based on the photospheric rotation rates of Howard (1984),
Figure 13(b)shows the rotation rates at the latitudes of
Figure 13(a). The trend is, of course, the opposite of the
latitudes, with faster (slower)rates at lower (higher)latitudes.
Figure 9. Rotation rates for south (top)and north (bottom). Our mean rotation rates and variance over time are shown as the data points with shaded areas (darker gray
and lighter gray for one and two standard deviations, respectively). The blue colored lines show estimates of coronal hole rotation from Bagashvili et al. (2017). The
red lines show latitudinal fits to our estimated rates according to Equations (1)(solid red lines)and (3)(dashed red lines).
9
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These rates are slower than the Carrington rate at solar
minimum, increasing to the Carrington rate and slightly above
during maximum, then decreasing to very slow rates in the
descending phase to the next solar minimum. This is not in
agreement with the rotation rates measured by the tomography
(see Figure 7(a)), where we see small positive rates (relative to
Carrington)at solar minimum, and large negative rates after
2010, and a return to positive in 2017. To highlight this
disagreement, Figure 13(c)shows equatorial rotation rates
gained from a correlation analysis of the tomographical maps
(see the Appendix for details of this procedure). Thus the
median absolute photospheric latitude of field lines connected
to equatorial coronal streamers does not correlate with the
equatorial coronal rotation rates—indeed, there is an antic-
orrelation. Figure 13(d)shows the fraction of equatorial
streamer field lines with footpoints within 5°of the equator
over the solar cycle. There are prominent peaks of this fraction
near 2009 January and during 2016/2017 that coincide with
the most obvious periods of fast coronal rotation. The most
obvious long period of slow rotation in 2012 to mid-2014
coincides with a period of generally low fraction of field line
equatorial footpoints. This largely qualitative result supports
the concept that the rotation rate of coronal streamers are
influenced by the latitudinal distribution of their magnetic
footpoints near the Sun. If a large fraction of footpoints are near
the equator, the streamer is more likely to rotate at the
Carrington rate or slightly higher. Note that this does not
explain the fast rotation rates of streamers at mid-to-high
latitudes. An important point here is that before fast subphoto-
spheric rotations can be used to interpret coronal rotation, we
must first understand how and where the corona is connected to
the photosphere.
One major uncertainty in this approach is the PFSS model. A
comparison of the tomography maps of Figure 12 panels (a)
and (b)with the PFSS convergence values of Figure 12 panels
(c)and (d)shows several similarities in the general distribution,
Figure 10. Histogram showing the percentage of time spent at a given rotation rate. The percentage is the fraction of time spanned by streamers measured at that
rotation rate relative to the total amount of time spanned by all measured streamers.
Figure 11. Composite images created using EIT/SOHO 193 Åchannel data (solar disk and lowest corona), MLSO MK4 coronameter daily average data (middle
corona)and LASCO/SOHO C2 data (outer corona)for dates (a)2009 January 17 and (b)2010 January 3. Regions above the disk have been processed using a point
filter and a normalizing radial graded filter (Morgan & Habbal 2005).
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but there are considerable disagreements in the actual
latitudinal position of structures. This has been shown in
previous studies (Morgan 2010,2011b). Despite the usefulness
of PFSS to analyze coronal structure, it is a model based on
several assumptions (including, most importantly, the height of
the source surface), and this leads to a poorly quantified
uncertainty in the actual position of structures (this is an
important consideration given the wide use of PFSS for
analysis, solar-wind modeling, and interpretation of in situ
measurements). Another uncertainty in our results is that we
have restricted the comparison to equatorial streamers only. At
times outside of solar minimum, the streamer sheets meander in
latitude and cross the equator. Our analysis takes only those
parts of the streamer sheets that are near the equator. Thus the
rotation of an extended streamer sheet may not be dominated
by the subset of field lines that are near the equatorial source
surface. There are obvious future improvements we can make
to this analysis through expanding the scope to streamers at a
broader range of latitudes.
4.2. Implications for Solar-wind Models
A key parameter for modeling the interplanetary solar wind
is the rotation rate of the lower boundary of models. This is a
fixed parameter based on the Carrington rotation period (25.38
days sidereal). Our study shows that this parameter should be
adjusted according to the estimated rotation rates of streamers
over yearly timescales. For a slow wind of 300 km s
−1
, the
Figure 12. (a)and (d): tomography density maps for a distance of 4 R
e
for mid-dates 2009 January 3 and 2010 January 3, respectively. (c)and (d): open field lines
from a PFSS model that are near the equator at the source surface distance of 2.5 R
e
are shown as gray or red for dates 2009 January 3 and 2010 January 3,
respectively. The green areas show regions of high convergence where high-density features are expected to reside. The blue line shows the polarity inversion sheet.
The red lines are field lines that are close to high-convergence regions at the source surface (that is, both close to the equator at the source surface and close to a high-
density streamer).(e)and (f): for the field lines colored red in (c)and (d)(i.e., those associated with high-density equatorial streamers), these histograms show the
latitudinal distribution of the photospheric footpoints for dates 2009 January 3 and 2010 January 3, respectively.
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travel time to Earth is approximately 5 days, and a coronal
rotation rate difference of −1°day
−1
compared to the
Carrington rate leads to a systematic error of 5°longitude in
the model’s output over this travel time period. In the context
of short timescales, this is only a minor correction that can be
considered negligible in the context of other, larger uncertain-
ties. However, for analyzing solar-wind models over the course
of longer periods (e.g., a whole rotation or longer), there is
obviously a large systematic error that can be corrected given
improved rotation rate estimates.
Therefore, for solar-wind modeling of long periods near the
solar equator, periods during 2007–2009 should remain at the
Carrington rotation rate or slightly higher. For periods from
2010 to 2016, a slower rate is recommended, at around
−1°day
−1
relative to the Carrington rate. From 2017 to 2020,
the Carrington rate is appropriate. For future solar-wind
forecasting, the most recent information on rotation rates
available from the tomography maps should be included. This
is a service we hope to provide to the community over the
coming years, with near-real-time updating of the tomography
density maps provided as part of the Space Weather Empirical
Ensemble Package (SWEEP)project detailed in the Acknowl-
edgments. We also wish to improve the cross-correlation
procedure detailed in the Appendix, which provides an
automated estimate of the coronal rotation rate.
5. Conclusions
Tomographical maps of the coronal density structure give
detailed information on the distribution of high-density
streamers. In this study, maps from 2007 to 2020 show
longitudinal drifts of streamers over time, giving an accurate
measurement of coronal rotation rates over a whole solar cycle.
The bulk of rotation rates are between −1 and 0°. 5 day
−1
relative to the Carrington rate, with values measured from
−2°.2 to 1°.6 day
−1
. Rotation rates can change abruptly at all
latitudes, showing that the concept of a rigidly rotating corona
(compared to the photosphere)is oversimplistic. The rigid
rotation is only found when rotation rates are averaged over
long periods, e.g., a whole solar cycle.
We find a strong north–south asymmetry in rotation rates,
with the southern corona rotating more rigidly than the north.
Between latitudes of 50°and 65°north we find the consistently
slowest rotation rates over the solar cycle. Our results in the
north agree well with rotation rates of coronal holes made by
Bagashvili et al. (2017), but disagree at many latitudes in the
south. This can be explained by measurements made at
different periods during the solar cycle, and with the large
difference in height of the different measurements.
Using the tomography maps and a PFSS magnetic model, we
interpret the rotation rates at the equator in terms of the
magnetic connection between the coronal streamer belt and the
lowest corona. Periods of the cycle with fast equatorial rotation
are periods when there are a larger fraction of field lines
connected to the equatorial photosphere, and long periods of
slow rotation coincide with a period where there is a smaller
fraction of equatorial connection. This interpretation explains
abrupt changes in the rotation rate in terms of abrupt changes in
the coronal structure. For example, as the corona changes from
a solar minimum dipole-dominated configuration to a more
quadrapolar-dominated configuration in the ascending phase to
solar maximum, starting in year 2009, we see an abrupt change
from faster to slower rotation at the equator.
The corona is more likely to be rotating faster than the
underlying photosphere at the same latitude. This supports the
concept of a subphotospheric influence on large-scale coronal
structure, but, as highlighted by our analysis of Section 4.1, the
magnetic connection between the corona and the photosphere
needs to be better understood. The mechanisms (e.g.,
interchange reconnection or more rapid reconfigurations)that
allow the corona to rotate faster than the photosphere also need
improved understanding, and the use of global simulations such
as magnetofrictional models may be useful in this context.
Routine measurements of the coronal magnetic field will also
be crucial to gain a full understanding.
This study provides improved estimates of the coronal
rotation rates that can be adopted by historical solar-wind
models of the past cycle in order to correct for systematic errors
on the order of 5°in longitude or larger. For equatorial regions,
periods during 2007–2009 should be slightly faster than the
Carrington rotation rate. For periods from 2010 to 2016, a
slower rate is recommended, at around −1°day
−1
relative to
the Carrington rate. From 2017 to 2020, the Carrington rate is
appropriate. We plan to provide improved and updated
estimates of equatorial rotation rates in the future, as part of
the SWEEP project (detailed in the Acknowledgments).
We acknowledge STFC grants ST/S000518/1 and ST/
V00235X/1, Leverhulme grant RPG-2019-361, and the
excellent facilities and support of SuperComputing Wales.
STFC project ST/V00235X/1 is the Space Weather Empirical
Ensemble Package (SWEEP)project, funded to provide an
operational space weather forecasting package for the UK
Meterological Office: a collaboration between Aberystwyth
University, University of Reading, Durham University, and
Northumbria University. The STEREO/SECCHI project is an
international consortium of the Naval Research Laboratory
(USA), Lockheed Martin Solar and Astrophysics Lab (USA),
NASA Goddard Space Flight Center (USA), Rutherford
Appleton Laboratory (UK), University of Birmingham (UK),
Max-Planck-Institut für Sonnen-systemforschung (Germany),
Centre Spatial de Liege (Belgium), Institut Optique Théorique
et Appliqúee (France), and Institut d’Astrophysique Spatiale
(France). MKIV coronameter data (DOI:10.5065/D66972C9)
is courtesy of the Mauna Loa Solar Observatory, operated by
the High Altitude Observatory, as part of the National Center
for Atmospheric Research (NCAR). NCAR is supported by the
National Science Foundation. The SOHO/LASCO data used
here are produced by a consortium of the Naval Research
Laboratory (USA), Max-Planck-Institut für Aeronomie (Ger-
many), Laboratoire d’Astronomie (France), and the University
of Birmingham (UK). SOHO is a project of international
cooperation between the ESA and NASA.
Appendix
Automated Correlation Analysis for Rotation Rates
For a given latitude, the tomography data map has
dimensions time and Carrington longitude (see, for example,
Figure 4for the equator). The concept for the correlation
analysis is to calculate the cross-correlation between a long-
itudinal slice of the density map at a given time and a slice at a
later time. The maximum peak in the cross-correlation profile
gives the longitudinal shift and an estimate of rotation rate.
This can be applied to multiple time steps throughout the map
to give a time series of estimated rotation rates. In practice,
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noise and errors in the tomography map, plus actual structural
changes in the corona, lead to high variations and incoherence
in the resulting rotation rate time series. To gain meaningful
results the following steps are made:
1. Prior to cross-correlation, the time–longitude map is
smoothed along the time direction over 11 time bins, or
approximately 16 days.
2. The cross-correlation is made for five selected time
intervals: 30, 55, 82, 108, and 135 days. Note that results
for the final 135 days of the time series are discarded,
corresponding to the longest time interval for cross-
correlation.
3. For each of the five time intervals, the peak cross-
correlation is recorded at each time step, and the
longitudinal lag of the peak is converted into a rotation
rate based on the longitudinal lag and time interval.
4. For each of the five time intervals, the time series of lags
is smoothed with a median sliding window of width
125 days.
5. The final rotation rate value is calculated as an average of
the five values given by the five time intervals.
6. For further analysis, a sliding-window smoothing is
applied to the rotation rate time series. For the example
shown in Figure 13(c), we median-smooth initially with a
sliding-window width of 118 days, then average-smooth
over a width of 44 days. The choice of these widths
provides a reasonably coherent and smooth time series of
rotation rates.
Figure 14(a)shows an example set of cross-correlation
curves, as a function of longitudinal lag, for date 2009 April 7.
This date corresponds to the base date, with cross-correlations
calculated between the base date and the five subsequent time
intervals. This example shows the difficulty of automatically
estimating rotation rates. The peaks are broad, which reflects
high uncertainty in choosing the lag at the point of maximum
Figure 13. (a)The median absolute photospheric latitude of field lines
associated with equatorial streamers from 2007 to 2020. The error bars show
the standard deviation of latitudes from the median for each date. (b)The
photospheric rotation rate at the field line footpoints, assuming the sunspot-
based rotation rates of Howard (1984). The error bars relate to the variance of
latitudes found for each date. The values are relative to the Carrington rate,
with zero equal to the Carrington rate. (c)Rotation rates calculated from a
correlation procedure (see the Appendix)applied to the equatorial time–
longitude tomographical density distribution. The light gray crosses show the
unsmoothed rotation rates, and the black line shows the smoothed rates (see the
Appendix). The red (blue)horizontal lines show positive (negative)rotation
rates (compared to Carrington)gained from the manual measurement of
structures in the tomographical maps. (d)The fraction of field-line footpoins,
relative to the total number of footpoints for that date, that are situated near the
equator (within 5°latitude)over time.
Figure 14. (a)Example of cross-correlation curves between the equatorial
densities for base date 2009 April 7. The different colors show the cross-
correlation with densities measured at subsequent dates 30 (purple),55
(orange),82(blue), 108 (green), and 135 (red)days after this date, as shown in
the legend to part (b). The point of maximum cross-correlation is indicated by
the triangles, with the vertical lines showing the corresponding longitudinal lag
in degrees. (b)The estimated rotation rates, in degrees per day relative to the
Carrington rate (Carrington rate is zero), as a function of time as calculated for
each choice of time interval. The black points and line show the mean over all
five time intervals.
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cross-correlation. Often two or more peaks are seen, which
shows that different density features may be moving at
different rates, or that there are changes in density not
associated with rotation. For this example, the 30 day interval
is at a negative lag, while the other four intervals are tightly
grouped at around 50°lag. For the optimal detection of
rotation, the lags should all be the same sign, and should also
increase in magnitude linearly with increasing time interval—
this is clearly not the case for this example, and is a further
reflection of the difficulties involved in this kind of automated
analysis.
Figure 14(b)shows the estimated rotation rates as a function
of time for each choice of time interval. There are times when
the five time intervals agree well (for example, years
2008–2009), and this is probably due to the low activity of
the Sun during solar minimum. At this time, changes in the
configuration of the coronal density structure are slow, and the
method can more reliably detect the dominant rotation rates. At
other times, values can vary widely, particularly during solar
maximum in years 2012 to 2014. Despite this variation, all five
time intervals show a reduced rotation rate. The 30 day interval
shows only small deviations from zero throughout the whole
period.
ORCID iDs
Liam Edwards https://orcid.org/0000-0002-9222-8648
David Kuridze https://orcid.org/0000-0003-2760-2311
Thomas Williams https://orcid.org/0000-0002-2006-6096
Huw Morgan https://orcid.org/0000-0002-6547-5838
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