# The Sun’s Mean Line-of-Sight Field

Neil R. Sheeley  Jr  Visiting Research Scientist
Lunar and Planetary Laboratory, University of Arizona
Tucson, AZ 85721, USA

###### Abstract

We regard the Sun-as-a-star magnetic field (i.e. the mean field) as a filter
for the spherical harmonic components of the photospheric field, and calculate
the transmission coefficients of this filter. The coefficients for each
harmonic, $Y_{l}^{m}$, are listed in three tables according to their
dependence on $B_{0}$, the observer’s latitude in the star’s polar coordinate
system. These coefficients are used to interpret the 46-yr sequence of daily
mean-field measurements at the Wilcox Solar Observatory. We find that the non-
axisymmetric part of the field originates in the $Y_{1}^{1}$, $Y_{2}^{2}$, and
a combination of the $Y_{3}^{3}$ and $Y_{3}^{1}$ harmonic components. The
axisymmetric part of the field originates in $Y_{2}^{0}$ plus a
$B_{0}$-dependent combination of the $Y_{1}^{0}$ and $Y_{3}^{0}$ components.
The power spectrum of the field has peaks at frequencies corresponding to the
${\sim}$27-day synodic equatorial rotation period and its second and third
harmonics. Each of these peaks has fine structure on its low-frequency side,
indicating magnetic patterns that rotate slowly under the influence of
differential rotation and meridional flow. The sidebands of the fundamental
mode resolve into peaks corresponding to periods of ${\sim}$28.5 and
${\sim}$30 days, which tend to occur at the start of sunspot maximum, whereas
the $\sim$27-day period tends to occur toward the end of sunspot maximum. We
expect similar rotational sidebands to occur in magnetic observations of other
Sun-like stars and to be a useful complement to asteroseismology studies of
convection and magnetic fields in those stars.

###### keywords:

Solar magnetic fields (1503)— Solar rotation (1524),—Solar cycle (1487)—
Stellar magnetic fields (1610)

##  1 Introduction

The Sun’s mean line-of-sight field is obtained by averaging the line-of-sight
component of the photospheric magnetic field over the (flat) solar disk. The
measurement is obtained from Earth, sometimes in integrated sunlight, and is
often called the ‘Sun-as-a-star’ magnetic field, as if the observation were
obtained from the even greater distance of another star.

In the early 1970s, John Wilcox proposed to build a new solar telescope in the
hills south of the Stanford University campus. The telescope would have
relatively coarse spatial resolution ${\sim}1$ arcmin and the capability of
measuring the Sun’s mean line-of-sight field. The idea was not to compete with
the telescopes at Mount Wilson and Kitt Peak, which were already obtaining
daily observations of the solar disk with much higher spatial resolution, but
instead to concentrate on factors like sensitivity and zero-point stability to
produce a long-term sequence of relatively precise, global observations that
could be used to study the Sun’s large-scale field.

Wilcox and Ness had discovered the interplanetary sector structure using
spacecraft data from the Interplanetary Monitoring Platform (IMP) (Wilcox &
Ness, 1965a, b; Ness & Wilcox, 1965). Also, by comparing those IMP spacecraft
data with photospheric magnetograms obtained at the Mount Wilson Observatory
(MWO), Wilcox and Robert Howard had shown that the sector structure originated
in long-lived, unipolar magnetic regions on the Sun (Wilcox & Howard, 1968).
Consequently, Wilcox thought that mean field observations would be important
for solar-terrestrial studies and, in particular, would help to improve the
${\sim}$4.5-day timing between the central meridian passage of a sector
boundary at the Sun and at the Earth. (See the discussion following Douglas
Jones’s talk at the Second Solar Wind Conference (Jones, 1972).) As pointed
out by Scherrer et al. (1977a), Kotov and Severny had already begun daily
observations of the mean field at the Crimean Observatory in 1968, and Robert
Howard began them at the Mount Wilson Observatory in 1970. So Wilcox’s
interest in what we now call ‘space weather’ was the motivation for building
the Stanford Solar Observatory111The observatory was renamed the Wilcox Solar
Observatory (WSO) in 1983 when John Wilcox died while swimming in Mexico.

Wilcox’s proposal was accepted and daily measurements of the Sun-as-a-star
field began on May 16, 1975. These measurements were obtained in ‘integrated
sunlight’ using a 2.7 m focal length objective lens that creates a 2.5 cm
solar image located 3.8 m above the entrance slit of the spectrograph
(Scherrer et al., 1977a). The new observations quickly confirmed that the
strength of the mean field is correlated with the central-meridian-passage
time of low-latitude coronal holes and with the associated pattern of
interplanetary sectors (Scherrer et al., 1976, 1977b; Sheeley et al., 1976).
In addition, 27-day Bartels displays of the WSO mean-field measurements
matched the corresponding displays of mean field calculated using the flux-
transport model. This provided one of the first verifications of the transport
model and showed that the mean field originated in flux that spread out from
its sources in active regions (Sheeley et al., 1985; Sheeley & DeVore, 1986a,
b). Many years later, we learned that the mean-field correlates with the
occurrence of coronal inflows seen with white-light coronagraphs on the Solar
and Heliospheric Observatory (SOHO) and Solar Terrestrial Relations
Observatory (STEREO) spacecraft (Sheeley & Wang, 2015).

The reason for these correlations is that the mean field is an approximate
measure of the Sun’s non-axisymmetric field, and, in particular, of its
horizontal dipole and quadrupole components, $Y_{1}^{1}$ and $Y_{2}^{2}$.
These non-axisymmetric fields are strengthened by the emergence of flux in
active regions, whose bright coronal extensions provide backgrounds for seeing
the much fainter inflows that rain downward from reconnection sites in the
outer corona (Sanchez-Diaz et al., 2017; Wang & Hess, 2018). The outward
components of these reconnections are sometimes observed as streamer blobs
moving out through the 30$R_{\odot}$ field of view like ‘leaves in the wind’
and gradually swept up by high speed streams to form regions of high density
(Sheeley et al., 2008a, b; Sheeley & Rouillard, 2010).

The purpose of this paper is to analyze an idealized Sun-as-a-star field into
its spherical harmonic components to determine the ones that ought to
contribute (both for the Sun and for a distant star), and then to decompose
(or demodulate) the observed WSO mean field to find out what those
contributions have been since the observations began in 1975\. Although
unknown to Wilcox in the early 1970s, the extension of these techniques to
other Sun-like stars may complement asteroseismology and exoplanet studies.

##  2 Theoretical Analysis of the Mean Field

####  2.0.1 Definition of the Field

Let’s begin with a definition for the mean line-of-sight field, $B_{m}$. In
general, it is just the average of the line-of-sight field over the flat solar
disk of radius, R:

$B_{m}~{}=~{}\int{B_{los}}dA_{disk}/{\pi}R^{2}.$ (1)

However, we would like to convert to an integral of the radial field, $B_{r}$,
over the surface area $A_{surf}$ of the Sun. In that case, we need two factors
of $\sin{\theta}\cos{\phi}$ – one factor to convert $B_{los}$ to $B_{r}$, and
the other factor to convert $dA_{los}$ to $dA_{surf}$. Also, we note that
${\theta}$ and ${\phi}$ are the usual polar and azimuthal angles in a
spherical coordinate system with the $x$-axis pointing toward Earth.
Therefore, Eq(1) becomes

$B_{m}~{}=~{}\int{B_{los}}dA_{disk}/{\pi}R^{2}~{}=~{}\int{B_{r}}(\sin{\theta}\cos{\phi})^{2}dA_{surf}/{\pi}R^{2}~{}=~{}(1/{\pi})\int_{-{\pi}/2}^{{\pi}/2}\int_{0}^{\pi}{B_{r}}(\sin{\theta}\cos{\phi})^{2}\sin{\theta}d{\theta}d{\phi},$
(2)

where ${\theta}$ runs from 0 to ${\pi}$, and ${\phi}$ runs from $-{\pi}/2$ to
$+{\pi}/2$. It is interesting to note that $\sin{\theta}\cos{\phi}$ is the
axisymmetric quantity that is usually called ${\mu}$ in theories of line
formation. So ${\mu}$ is 1 at disk center, 0 at the solar limb, and $B_{r}$ is
heavily weighted toward disk center as

$B_{m}~{}=~{}\int{B_{r}}{\mu}^{2}dA_{surf}/{\pi}R^{2}.$ (3)

As discussed by Scherrer et al. (1977a), the weighting toward disk center is
even greater for the WSO observations due to solar limb darkening and
diffraction from the entrance slit of the spectrograph. In the Appendix of
this paper, we consider the limb darkening in detail and find that the same
harmonic components contribute to the mean field for a so-called gray
atmosphere in the Eddington approximation as would occur in the absence of
limb darkening. However, the limb darkening reduces the strengths of the $l=1$
and $l=2$ components by 12%, and 3.75%, respectively, and increases the
strength of the much weaker $l=3$ components by 22%.

Up to this point, I have ignored the $7.25^{\circ}$ tilt of the Sun’s axis
away from the normal to the ecliptic plane. We can include this effect by
replacing $\sin{\theta}\cos{\phi}$ with
$\sin{\theta}\cos{\phi}\cos{B_{0}}+\cos{\theta}\sin{B_{0}}$, where $B_{0}$ is
Earth’s heliolatitude and varies annually from $-7.25^{\circ}$ in February-
March to $+7.25^{\circ}$ in August-September. In that case, $B_{m}$ becomes

$B_{m}~{}=~{}(1/{\pi})\int_{-{\pi}/2}^{{\pi}/2}\int_{0}^{\pi}{B_{r}}(\sin{\theta}\cos{\phi}\cos{B_{0}}+\cos{\theta}\sin{B_{0}})^{2}\sin{\theta}d{\theta}d{\phi},$
(4)

where $B_{r}$ depends on $({\theta},{\phi})$, but $B_{0}$ does not. Next, our
objective is to expand the binomial factor,
$(\sin{\theta}\cos{\phi}\cos{B_{0}}+\cos{\theta}\sin{B_{0}})^{2}$, and express
$B_{m}$ as the sum of three parts – one proportional to $\cos^{2}{B_{0}}$,
another proportional to $(\sin{2{B_{0}}})/2$, and the third proportional to
$\sin^{2}{B_{0}}$.

$B_{m}~{}=~{}(\cos^{2}{B_{0}})~{}\frac{1}{{\pi}}\int{B_{r}}(\sin{\theta}\cos{\phi})^{2}d{\Omega}~{}+~{}(\frac{\sin{2{B_{0}}}}{2})\frac{1}{{\pi}}\int{B_{r}}\sin{2{\theta}}\cos{\phi}~{}d{\Omega}~{}+~{}(\sin^{2}{B_{0}})~{}\frac{1}{{\pi}}\int{B_{r}}\cos^{2}{\theta}~{}d{\Omega},$
(5)

where $d{\Omega}=\sin{\theta}d{\theta}d{\phi}$ and the integral sign refers to
the double integral over ${\theta}$ and ${\phi}$, as indicated in Eq(4).

For the Sun, $|B_{0}|~{}{\leq}~{}7.25^{\circ}~{}{\approx}~{}0.126$ radians so
that the $\sin^{2}{B_{0}}$-factor is ${\leq}0.016$, and can be neglected.
Likewise, the $\cos^{2}{B_{0}}$-factor is ${\geq}0.984$ and can be replaced by
1. Finally, the $(\sin{2{B_{0}}})/2$-factor is approximately $B_{0}$, which
will vary between $-0.126$ and $+0.126$ during the year. Of course, for a star
whose tilt angle is large, we cannot make these approximations, and we may
need to keep all three terms.

####  2.0.2 Spherical Harmonic Components

Next, we consider the form of $B_{r}$. There are several ways that we could
represent this field. One way would be to express $B_{r}$ as a linear
combination of the ‘barber pole’ eigenfunctions of the flux-transport
equation, as DeVore (1987) did in his theoretical analysis of the Sun’s large-
scale field. This approach might help us to interpret the power spectrum of
the mean field in terms of the rigidly rotating patterns that are caused by
the latitudinal transport of flux (Sheeley et al., 1987; Wang & Sheeley,
1994a; Wang, 1998). Another way would be to represent the field in terms of
sectors of the form $B_{r}=f({\theta})\cos{m}\\{{\phi-{\omega}({\theta})t}\\}$
(where ${\omega}({\theta})$ is the angular rotation profile), as Sheeley &
DeVore (1986a) did in their analysis of the decay of the Sun’s mean field. In
this paper, I will try a third approach, representing $B_{r}$ as a linear
combination of spherical harmonic components, $Y^{m}_{l}({\theta},{\phi})$,
which are the familiar eigenfunctions of ${\nabla}^{2}B_{r}=0$ on the surface
of a sphere. (See Eqs (9) and (10) below.) With this understanding,

$B_{r}({\theta},{\phi},t)~{}=~{}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}{{\rho}_{lm}}(t)e^{i{\delta}_{lm}(t)}Y_{l}^{m}({\theta},{\phi}),$
(6)

where $i=\sqrt{-1}$. Also, ${\rho}_{lm}$ and ${\delta}_{lm}$ are the amplitude
and phase of each harmonic component, $Y_{l}^{m}$, and are defined so that
$B_{r}$ is real. Consequently,

$B_{m}~{}=~{}(\cos^{2}{B_{0}})\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}{\rho}_{lm}e^{i{\delta}_{lm}}I_{lm}~{}+~{}(\frac{\sin{2{B_{0}}}}{2})\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}{\rho}_{lm}e^{i{\delta}_{lm}}J_{lm}~{}+~{}(\sin^{2}{B_{0}})\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}{\rho}_{lm}e^{i{\delta}_{lm}}K_{lm},$
(7)

where the coefficients $I_{lm}$, $J_{lm}$, and $K_{lm}$ are given by

$\displaystyle I_{lm}~{}=~{}\frac{1}{{\pi}}\int
Y_{l}^{m}({\theta},{\phi})(\sin{\theta}\cos{\phi})^{2}d{\Omega},$ (8a)
$\displaystyle J_{lm}~{}=~{}\frac{1}{{\pi}}\int
Y_{l}^{m}({\theta},{\phi})\sin{2{\theta}}\cos{\phi}~{}d{\Omega},$ (8b)
$\displaystyle K_{lm}~{}=~{}\frac{1}{{\pi}}\int
Y_{l}^{m}({\theta},{\phi})\cos^{2}{\theta}~{}d{\Omega},$ (8c)

and the integral sign refers to the double integral in Eq(4). Thus, $I_{lm}$,
$J_{lm}$, and $K_{lm}$ (weighted by the respective $B_{0}$-dependent factor)
are real, and indicate the amounts by which the mean-field ‘filter’ reduces
the amplitude ${\rho}_{lm}$ of the field. The spherical harmonic functions,
$Y_{l}^{m}({\theta},{\phi})$, are defined in terms of the Associated Legendre
functions $P_{l}^{m}(\cos{\theta})$, given by Jahnke & Emde (1945), and the
normalization factor, $N_{lm}$, as follows

$Y_{l}^{m}({\theta},{\phi})~{}=~{}N_{lm}P_{l}^{m}(\cos{\theta})e^{im{\phi}}$
(9)

$N_{lm}~{}=~{}\sqrt{\frac{2l+1}{4{\pi}}\frac{(l-m)!}{(l+m)!}}.$ (10)

After some algebra, we can rewrite Eqs(8a)-(8c) as

$\displaystyle
I_{lm}~{}=~{}\frac{N_{lm}}{{\pi}}\int_{-1}^{1}P_{l}^{m}(x)(1-x^{2})dx\left[\frac{-4\sin{(m{\pi}/2)}}{m(m+2)(m-2)}\right],$
(11a) $\displaystyle
J_{lm}~{}=~{}2\frac{N_{lm}}{{\pi}}\int_{-1}^{1}P_{l}^{m}(x)x(1-x^{2})^{1/2}dx\left[\frac{-2\cos{(m{\pi}/2)}}{(m+1)(m-1)}\right],$
(11b) $\displaystyle
K_{lm}~{}=~{}\frac{N_{lm}}{{\pi}}\int_{-1}^{1}P_{l}^{m}(x)x^{2}dx\left[\frac{2\sin{(m{\pi}/2)}}{m}\right].$
(11c)

I used standard Mathematica software (Wolfram, 1999) to evaluate these
expressions and entered the results in Tables 1-3. However, Mathematica
defines the Associated Legendre functions using
$P_{l}^{m}(x)=(-1)^{m}(1-x^{2})^{m/2}d^{m}P_{l}(x)/dx^{m}$, which differ by a
factor of $(-1)^{m}$ from the Jahnke & Emde (1945) values. Therefore, I
changed the signs of the odd-m entries in Tables 1-3 to be consistent with the
Jahnke & Emde (1945) convention. In retrospect, I could have done this
automatically by including an extra factor of $(-1)^{m}$ in each integrand.

Tables 1 and 3 give non-zero contributions to the $Y_{0}^{0}$ magnetic-
monopole component. Ignoring this term, the main contributions to the
$\cos^{2}{B_{0}}$-part of the mean field come from the $Y_{1}^{1}$,
$Y_{2}^{2}$, and $Y_{2}^{0}$ components with smaller additional contributions
from $Y_{3}^{3}$ and $Y_{3}^{1}$. There are no contributions when $l=4$ and
the higher-order terms are less than 1%. Also, for the Sun,
$|B_{0}|~{}{\leq}~{}0.126$ and $\cos^{2}B_{0}~{}{\approx}~{}1$, so that the
contributions of these harmonics are not weakened appreciably by the
$B_{0}$-dependence.

However, in Table 2, the main contributions to the $(\sin{2B_{0}})/2$-part of
the mean field are from the $Y_{1}^{0}$ and $Y_{2}^{1}$ components, which are
antisymmetric across the equator. These contributions are +0.244 and +0.206,
respectively. For small $B_{0}$, $(\sin{2}{B_{0}})/2~{}{\approx}~{}B_{0}$,
which varies annually from -0.126 to +0.126. Consequently, the expected mean-
field contributions of the $Y_{1}^{0}$ and $Y_{2}^{1}$ components vary
annually and have peak amplitudes of $\pm$3.1% and $\pm$2.6%, respectively.
These contributions are comparable to the relatively small, but finite, 3.5%
and 2.7% contributions of the $Y_{3}^{3}$ and $Y_{3}^{1}$ components in Table
1. Thus, they ought to be noticeable, especially during sunspot cycles when
the polar fields are strong. The other contributions from Table 2 are
comparable to the contributions of the higher-order terms in Table 1, which we
have already chosen to neglect.

In Table 3, $K_{20}=+0.168$ and $K_{31}=+0.121$. However, these relatively
large values can be ignored because the $\sin^{2}{B_{0}}$ factor
($~{}{\sim}~{}0.016$) reduces their net mean-field contributions to less than
1%. Although these entries in Table 3 are unimportant for the Sun, they might
contribute appreciably to the mean-field of other stars whose rotation axes
may be directed closer to the line of sight. (Of course, for those distant
stars, the annual variations induced by Earth’s motion around the Sun would be
negligible.)

Table 1: Elements of $I_{lm}$ {for the $\cos^{2}B_{0}$-term} $l/m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
0 | +0.188 |  |  |  |  |  |  |
1 | 0 | +0.173 |  |  |  |  |  |
2 | -0.084 | 0 | +0.103 |  |  |  |  |
3 | 0 | -0.027 | 0 | +0.035 |  |  |  |
4 | 0 | 0 | 0 | 0 |  0 |  |  |
5 | 0 | -0.003 | 0 | +0.004 | 0 | -0.005 |  |
6 | 0 | 0 | 0 | 0 | 0 | 0 |  0 |
7 | 0 | -0.001 | 0 | +0.001 | 0 | -0.001 | 0 | +0.002
Table 2: Elements of $J_{lm}$ {for the $(\sin{2B_{0}})/2$-term} $l/m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
0 | 0 |  |  |  |  |  |  |
1 | +0.244 | 0 |  |  |  |  |  |
2 | 0 | +0.206 | 0 |  |  |  |  |
3 | -0.093 | 0 | +0.085 |  0 |  |  |  |
4 | 0 | 0 | 0 | 0 | 0 |  |  |
5 | -0.018 | 0 | +0.018 | 0 | -0.015 |  0 |  |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
7 | -0.007 | 0 | +0.007 | 0 | -0.007 | 0 | +0.006 |  0
Table 3: Elements of $K_{lm}$ {for the $\sin^{2}B_{0}$-term} $l/m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
0 | +0.188 |  |  |  |  |  |  |
1 | 0 | +0.086 |  |  |  |  |  |
2 | +0.168 | 0 |  0 |  |  |  |  |
3 | 0 | +0.121 | 0 | -0.017 |  |  |  |
4 | 0 | 0 | 0 | 0 |  0 |  |  |
5 | 0 | +0.045 | 0 | -0.034 | 0 | +0.007 |  |
6 | 0 | 0 | 0 | 0 | 0 | 0 |  0 |
7 | 0 | +0.026 | 0 | -0.023 | 0 | +0.017 | 0 | -0.004

In summary, the Sun-as-a-star field is dominated by the $Y_{1}^{1}$,
$Y_{2}^{2}$, and $Y_{2}^{0}$ components of the Sun’s field. Also, the mean
field has much smaller contributions from the $Y_{3}^{3}$ and $Y_{3}^{1}$
components, and from the $Y_{1}^{0}$ and $Y_{2}^{1}$ components whose
strengths are modulated by the annual variation of $B_{0}$ as Earth orbits the
Sun. Finally, it is important to realize that we have been describing the
‘transmission factors’ of the mean-field filter and that the real mean field
also depends on the amplitude, ${\rho}_{lm}$, and phase, ${\delta}_{lm}$, of
the radial field, $B_{r}$, that is being filtered. Next, we will use these
results to interpret mean-field observations obtained daily at the Wilcox
Solar Observatory during the 46-yr interval from May 16, 1975 to the present.

##  3 Sun-as-a-Star Magnetic Field Measurements from WSO

###  3.1 27-day rotational modulation

Figure 1 shows daily measurements of the Sun’s mean field obtained at WSO
since 16 May 1975. Approximately

Figure 1: Daily measurements of the Sun’s mean line-of-sight field obtained at
WSO from 16 May 1975 to 16 November 2021. The blurred cloud of points shows
peaks of strength ${\gtrsim}$1G in each cycle, and valleys of strength
${\lesssim}$0.2G near sunspot minimum.

17,000 points over 46 years give a blurred distribution with peaks and valleys
around sunspot maximum and minimum in each of four sunspot cycles. This figure
is essentially the same as the one that is shown on the WSO web site
(http://wso.stanford.edu), and leaves us with the question of how to extract
information from these data. A big clue is contained in a plot of the data
obtained during the first year of observations, as shown in Figure 2. The
field

Figure 2: Mean-field measurements during the first year of observations at
WSO, showing the end of a 27-day recurrence pattern, that was associated with
the gradual demise of a long-lived coronal hole.

oscillates with a period of about 27 days (${\sim}$0.074 yr), before degrading
toward the end of the year, as the corresponding low-latitude coronal hole
gradually died (Sheeley et al., 1976). What we would like to know is how much
power is contained in this 27-day modulation and how that power varies with
time during the 46-yr interval. In effect, we are asking for the envelope of
this mean-field time series.

There are several approximately equivalent ways to produce this envelope. One
way is to divide the time base into 27-day segments and to compute the
maximum-minimum difference of the mean-field values on each segment. By
plotting the absolute values of these differences, we would obtain a display
of the mean field variation similar to the one that Sheeley & Wang (2015)
obtained in their paper describing the rejuvenation of the Sun’s large-scale
magnetic field. A similar result is obtained by computing the standard
deviation of the mean field on each 27-day segment (allowing for data gaps
when taking the averages), and then plotting that value as a function of time.
Another, nearly equivalent procedure is to set the mean-field data gaps equal
to zero before computing the standard deviations, and then to perform a 27-day
moving average of those standard deviations. This is the approach that I shall
use in the remainder of this paper.

Figure 3 shows this 27-day moving average, compared with the monthly averaged
sunspot number (divided by 2000) during cycles 21 - 24. As described
previously using the ‘max-min’ display (Sheeley & Wang, 2015), the mean field
originates in episodic bursts whose amplitudes often tend to be large as the
sunspot cycle enters its declining phase. Also, one can see the decrease of
the mean field strength during 1976 as the sunspot number reached its 11-year
minimum and the low-latitude coronal hole died (cf. Figure 2). Finally, note
that the amplitudes of the peaks in Figure 3 are about 1.4 times smaller than
those in Figure 1. This is close to the ${\sqrt{2}}$ that one might expect for
the difference between the standard deviation and the envelope of a curve.
(For example, if $f(t)=A(t)cos{\omega}t$, the envelope is ${\pm}A(t)$ and the
standard deviation is $A(t)/\sqrt{2}$.)

Figure 3: Root-mean-square deviation of the WSO mean field from its 27-day
average value (red), compared with the monthly averaged sunspot number from
the Royal Observatory of Belgium (SILSO) divided by 2000 (blue) during cycles
21 - 24. This comparison shows the tendency of the mean field to occur in
episodic bursts, often when the sunspot cycle begins its decline.

###  3.2 Using the Fourier transform approach

Next, we return to the WSO mean-field measurements that we displayed as a
function of time in Figure 1. After setting the missing field strengths equal
to 0, we take the discrete Fourier transform defined by

$f_{s}~{}=~{}\frac{1}{{\sqrt{N}}}\sum_{k=1}^{N}{B_{k}}e^{2{\pi}i(k-1)(s-1)/N}$
(12)

where $B_{k}$ refers to the individual mean-field measurements whose index,
$k$, runs from 1 to $N=16,987$, corresponding to the most recent measurement
on 16 November 2021. In this case, the frequency, ${\omega}$, in rad
$\textrm{day}^{-1}$ is given by

${\omega}=2{\pi}s/N.$ (13)

Because $f_{s}$ is a complex number, one typically plots the power,
$P({\omega})$, defined as the positive number, $f_{s}^{*}f_{s}$, versus
${\omega}$. However, to keep the units in Gauss, I plot the positive square
root of this power. Also, it is not necessary to include the full range
$(0,2{\pi})$ because the spectrum is symmetric about the point
${\omega}={\pi}$. Moreover, we do not even need to include all of the half
range $(0,{\pi})$ because the lines disappear after the $m=3$ peak around
${\omega}=0.7$ rad $\textrm{day}^{-1}$. This is consistent with our
expectations from Table 1, which gives 0 for $m=4$, and less than 1% for
$m=5$.

The power spectrum in Figure 4 shows three main peaks at frequencies of
approximately ${\omega}=0.231$, 0.460, and 0.702 $\textrm{rad~{}day}^{-1}$.
These frequencies are in the ratio of approximately 1:2:3, corresponding to a
fundamental rotation rate

Figure 4: Power spectrum of daily WSO mean-field measurements during 1975 -
2021, showing three main sectorial peaks at ${\omega}~{}{\approx}~{}0.231$,
0.460, and 0.702 rad $\textrm{day}^{-1}$ (corresponding to a rotation period
of 27.2 days and its second and third harmonics). The plot has been smoothed
in a moving average of 12 resolution elements, each of size
${\Delta}{\omega}=2{\pi}/N=3.70{\times}10^{-4}$ rad $\textrm{day}^{-1}$.

with $m=1$ and its first two harmonics with $m=2$ and $m=3$. The associated
periods are approximately 27.2, 13.6, and 9.0 days, respectively. Evidently,
we are seeing rigidly rotating recurrence patterns of two-sector, four-sector,
and six-sector fields (i.e. the dipole, quadrupole, and hexapole fields).

These regions are shown separately in the three panels of Figure 5. Each
spectrum is displayed with the same 12 unit

Figure 5: Segments of the power spectrum in Figure 4, showing the fine
structure of the $m=1$ (left), $m=2$ (middle), and $m=3$ (right) components
for the full time interval 1975 - 2021.

smoothing, which is 12 times the $2{\pi}/N$ resolution for $N{\sim}$ 16,987
points, and corresponds to 0.0044 $\textrm{rad~{}day}^{-1}$. On the low-
frequency side of the fundamental peak at
${\omega}=0.231~{}\textrm{rad~{}day}^{-1}$, there are peaks at 0.219 and 0.208
$\textrm{rad~{}day}^{-1}$, corresponding to periods of approximately 28.7 and
30.2 days, respectively. The 28.7-day period is comparable to the
${\sim}$28.5-day recurrence period of the slanted patterns seen around sunspot
maximum in Bartels displays of the interplanetary magnetic field observed by
in-ecliptic spacecraft and inferred from Earth-based magnetometers (Svalgaard
& Wilcox, 1975; Hoeksema, 1984; Sheeley et al., 1985; Sheeley & DeVore, 1986b;
Wang & Sheeley, 1994b). However, the 30.2-day peak has no in-ecliptic
counterpart, and therefore probably originates in rigidly rotating structures
at latitudes that are beyond the reach of the in-ecliptic measurements, as
discussed in Section 3.2.1 below.

Although the frequencies of the main peaks of the $m=1$, $m=2$, and $m=3$
distributions occur in the ratio of 1:2:3, the frequencies of the sidebands do
not occur in this ratio. In particular, the $m=2$ and $m=3$ sidebands are not
‘blurred out’ harmonics of the peaks at 0.208 and 0.219
$\textrm{rad~{}day}^{-1}$. Not only do the $m=2$ structures occur at different
frequencies than we would expect for second harmonics, but also these
structures are accompanied by additional features for which there is no
corresponding peak in the sidebands of $m=1$. For $m=3$, there are even more
fluctuations, crowding into a broad slope of nearly continuous intensity.

Finally, we note in Figure 4 that there is a noisy ‘ledge’ of strength
${\sim}$0.6 G at ${\omega}~{}{\sim}~{}0.017~{}\textrm{rad~{}day}^{-1}$. This
structure corresponds to a weak annual variation associated with the motion of
the Earth around the Sun. As discussed in Section 2, this annual variation is
introduced through $B_{0}$ \- Earth’s latitude in the Sun’s polar coordinate
system. At even lower frequencies, the spectrum rises steeply, and the
expected peak at ${\omega}~{}{\approx}~{}0.0016~{}\textrm{rad~{}day}^{-1}$
(corresponding to the 11-yr sunspot cycle) is not visible in this
$0.0044~{}\textrm{rad~{}day}^{-1}$ smoothed plot. Kotov (2019) has used mean-
field observations since 1968 to study this annular variation in greater
detail.

####  3.2.1 The temporal origin of the peaks in the power spectrum

The next problem is to find the temporal origin of these spectral peaks. We do
this by selecting the frequency range of interest and then taking the inverse
Fourier transform through that spectral window. We use the inverse transform

$B_{k}~{}=~{}\frac{1}{{\sqrt{N}}}\sum_{s=1}^{N}{f_{s}}e^{-2{\pi}i(k-1)(s-1)/N},$
(14)

but with $f_{s}$ multiplied by a function of $s$ (or equivalently ${\omega}$)
that is 1 on the interval of interest and 0 elsewhere. Note that $f_{s}$ is
the original complex Fourier transform given by Eq(12), and not the absolute
value that was used in Figure 4. In general, the inverse Fourier transform is
also a complex number, so we calculate the standard deviation, ${\sigma}$,
from the relation ${\sigma}^{2}=<|B_{k}|^{2}>-~{}|<B_{k}>|^{2}$, where in the
second term, we compute the 27-day average of $B_{k}$ before we take its
absolute value and square it. In this case, it is easy to show that
${\sigma}^{2}={\sigma}_{r}^{2}+{\sigma}_{i}^{2}$ (i.e. the standard deviation
of $B_{k}$ is the square root of the sum of the squares of the standard
deviations of its real and imaginary parts).

Figure 6 was created by selecting a disjoint interval consisting of the three
principal peaks of the power spectrum

Figure 6: Result of inverting the Fourier transform of the mean field back to
the temporal domain using all three frequency windows (0.20-0.25),
(0.42-0.50), and (0.65-0.73) $\textrm{rad~{}day}^{-1}$ (corresponding to the
$m=1$, $m=2$, and $m=3$ sectoral modes, respectively), and then displaying the
27-day moving standard deviation of this inverted transform.

\- specifically, (0.20-0.25), (0.42-0.50), and (0.65-0.73)
$\textrm{rad~{}day}^{-1}$ \- and then by displaying the root-mean-square power
in the inverse transform. This plot is essentially the same as Figure 3
without the noise. Because we have not included power at the frequency of the
annual variation (0.0172 $\textrm{rad day}^{-1}$), we have removed potential
contributions from Table 2, so that the only contributions come from modes in
Table 1. This leaves only the $Y_{1}^{1}$ (and possibly the $Y_{3}^{1}$)
components as likely contributors from the $m=1$ sector, and only the
$Y_{2}^{2}$ and $Y_{3}^{3}$ components as contributors from the $m=2$ and
$m=3$ sectors, respectively. So, the red curve in Figure 6 indicates
contributions from the horizontal dipole, quadrupole, and hexapole components
($Y_{1}^{1}$, $Y_{2}^{2}$, and $Y_{3}^{3}$), and possibly a small contribution
from $Y_{3}^{1}$.

Next, we ask how this non-axisymmetric power is distributed among the $m=1$,
$m=2$, and $m=3$ sectorial modes. For this purpose, we use the individual
frequency ranges (0.20-0.25), (0.42-0.50), and (0.65-0.73)
$\textrm{rad~{}day}^{-1}$, which correspond to the fundamental, second, and
third harmonics shown in Figures 4 and 5. The 27-day running

Figure 7: Power in the two-sector (top), four-sector (middle), and six-sector
(bottom) patterns of the mean field. Referring to Table 1, we expect this
power to originate mainly in the $Y^{1}_{1}$, $Y^{2}_{2}$, and $Y^{3}_{3}$
components of the field.

averages are shown in Figure 7. In general, the $m=1$ component contributes
the most and the $m=3$ component contributes the least. With a few exceptions,
the mean-field is dominated by the $m=1$ and $m=2$ sectoral modes. The $m=1$
component has large peaks in 1982, 1991, 2003, and a small one in 2015 that we
recall from a nearly identical plot of the equatorial dipole, that was derived
from spatially resolved WSO observations (Sheeley & Wang, 2015). Also, the
$m=2$ component has moderately large, but narrow, peaks in 1981, 2000, and
2012. The $m=3$ component has a large, narrow peak in 1991 when the $m=1$-2
values are temporarily low. Also, in the relatively weak sunspot cycle 24, the
three components have nearly coincident peaks of approximately equal strength,
which combine to give a stronger peak of total mean-field power, as seen in
Figure 6.

Based on the factors in Tables 1 and 2, we expect that the $m=1$ power
originates primarily from the $Y_{1}^{1}$ horizontal dipole component of field
and secondarily from the $Y_{3}^{1}$ component. As mentioned above, by
filtering out annual variations, we have excluded contributions from the
$Y_{2}^{1}$ component, that would otherwise occur through the $B_{0}$ factor.
Likewise, the $Y_{3}^{2}$ component is excluded from the $m=2$ sector, which
indicates power in the $Y_{2}^{2}$ component alone.

Finally, we have the impression in Figure 7 that the temporal fluctuations
become systematically finer as $m$ increases from 1 to 3. This is consistent
with the increased coarseness of the power spectra in Figures 4 and 5 as $m$
increases: For $m=1$, the peaks were often resolvable; for $m=2$, they formed
coarser structures; and for $m=3$, they merged into a ‘bumpy’ continuum. In a
previous analytical study of the mean field, we noted that the field depended
on the product $mt$, not on $m$ and $t$ separately. This meant that the mean
field would decay as $1/m$ in the absence of meridional flow (Sheeley &
DeVore, 1986a). In particular, a 4-sector field would decay twice as fast as a
2-sector field, and a 6-sector field would decay three times as fast. When
flow was present, this analytical simplification did not occur. However, the
apparent trend in Figure 7 indicates that a monotonic relation may still be
present.

Next, we look for the origins of the three $m=1$ peaks at ${\omega}=0.231$,
0.219, and 0.208 $\textrm{rad day}^{-1}$, shown in the left panel of Figure 5.
To do this, we select relatively narrow spectral windows surrounding these
peaks, and then invert the Fourier transform and plot the running 27-day rms
averages. For the intervals, ${\omega}=0.226-0.237$, ${\omega}=0.215-0.225$

Figure 8:  Two-sector ($m=1$) power from 27-day (top), 28.5-day (middle), and
30-day (bottom) structures (blue) compared with their total $m=1$ power (red).
This figure shows that 27-day power originates in all four sunspot cycles, but
the 28.5-day and 30-day power come mainly from the stronger sunspot cycles 21
and 22\.

and ${\omega}=0.198-0.212~{}\textrm{rad day}^{-1}$, we obtain the blue curves
in Figure 8. Here, the red curve indicates the total $m=1$ power from the top
panel of Figure 7. So we are comparing the temporal origin of the individual
$m=1$ peaks with the temporal origin of the combined $m=1$ power.

Like Fourier transforms of continuous functions, these Fourier transforms of
discrete functions have peaks whose widths, ${\Delta}{\omega}$, are inversely
related to the lifetimes, ${\Delta}t$, of the corresponding temporal
structures. In fact, ${\Delta}{\omega}{\Delta}t~{}{\sim}~{}8$ for full widths
at $e^{-1}$ maximum and
${\Delta}{\omega}{\Delta}t~{}{\sim}~{}8~{}ln2~{}{\approx}~{}5.54$ for full
widths at half maximum. Consequently, the narrow $m=1$ peaks with
${\Delta}{\omega}~{}{\sim}~{}0.01~{}\textrm{rad day}^{-1}$ in the left panel
of Figure 5 correspond to long-lived features with
${\Delta}t~{}{\sim}~{}1-2~{}\textrm{yrs}$.

In the top panel of Figure 8 the blue curve refers to the power in the
spectral ‘line’ at ${\omega}=0.231~{}\textrm{rad day}^{-1}$ (corresponding to
a period of approximately 27.2 days). This blue curve tends to follow the more
rapidly fluctuating red curve, with appreciable contributions during each of
the four sunspot cycles. Thus, most of the two-sector power originates in
long-lived features that recur with a period of 27.2 days, and presumably
corresponds to quasi-vertical patterns in the 27.27-day Carrington stackplots
of mean-field observations.

In the middle panel, the blue curve refers to power in the spectral line at
${\omega}=0.219~{}\textrm{rad day}^{-1}$ (28.7days). Most of the 28.7-day
power originates in 1979-1985 and 1989-1993 coincident with large peaks of
27-day power. This overlapping trend did not continue into sunspot cycles 23
and 24 when the 28.7-day power was much smaller. This suggests that
${\sim}$28.5-day stackplot patterns may have been weaker or less frequent in
cycles 23 and 24 than in cycles 21 and 22.

In the bottom panel, the blue curve indicates power in the spectral line at
${\omega}=0.208~{}\textrm{rad day}^{-1}$ (30.2 days). Most of this
$\sim$30-day power occurred in 1989-1990, with lesser amounts during 1980 and
1982-1983, and only trace amounts in sunspot cycles 23 and 24.

To summarize the results of Figure 8, the power depends on the rotation period
with 27-day power coming from all four sunspot cycles (but with a relatively
small contribution from the weakest sunspot cycle 24). The 28.5-day power
originates mainly in sunspot cycles 21 and 22 with very small contributions
from cycles 23 and 24. The 30-day power comes mainly from the year 1989 in
cycle 22 and secondarily from small peaks in cycle 21.

The lack of substantial 30-day power after 1990 provides another way to
isolate the power during 1989 - 1990. We simply move the starting point of the
Fourier transform backwards in time through the year 1989 and watch the height
of the 30-day peak increase. Figure 9 shows a sample of the power spectra
obtained by moving the starting time backward from 02 February 1990 (CR1826)
in steps of 3 Carrington rotations (approximately 82 days) to 09 January 1989
(CR1811).

Figure 9: Power spectra of WSO mean-field measurements from time, t, to the
end of the data set on 16 November 2021, showing the emergence of the
${\sim}$30-day ($0.208~{}\textrm{rad day}^{-1}$) peak as t decreases from 22
February 1990 to 09 January 1989.

During this sequence, the 30-day ($0.208~{}\textrm{rad day}^{-1}$) peak
emerges from a continuum level at about 0.5 G to a maximum height of about 1.0
G. Although not shown here, a movie with 1-rotation time resolution indicated
that the 30-day peak emerged from the continuum at CR1824 (29 December 1989)
and strengthened until it reached about 1.0 G at CR1811 (09 January 1989),
corresponding to a lifetime of 13 rotations (1 yr and 11 days). Thus, the
30-day oscillation spanned the 1-year interval from 1989 to 1990.

We can continue this approach by moving the starting point of the Fourier
transform forward in time to successively exclude major contributions to the
30-day, 28.5-day, and 27-day periods. Referring to Figure 8, we select the
first starting time on 09 January 1989 when substantial power remained in all
three rotational periods. This is shown in the upper-left panel of Figure 10.
Then, we move farther in time to 01 June 1991, which is well after the peak of
30-day power, but still includes power at 28.5 days and 27-days, as shown in
the upper-right panel of Figure 10. Next, we choose 17 January 1996, which is
after the large peak of 28.5-day power. However, 17 January 1996 is still
before the occurrence of the large peak of 27-day power in 2003-2004, and this
contribution is shown in the lower-left panel of Figure 10. Finally, we select
02 January 2005 to remove this large peak of 27-day power. The remaining
27-day power comes from a small peak in 2015 -2016, as shown in the lower-
right panel of Figure 10. This contribution seemed very important when it
rejuvenated the large-scale field in sunspot cycle 24 (Sheeley & Wang, 2015).
In the next section, we shall compare these observations with spatially
resolved magnetograms and find that the peaks of 27-day power tend to occur
toward the end of sunspot maximum when the sunspot belts are closer together
and allow large unipolar magnetic regions to form at the equator.

Figure 10: Power spectra of the mean field, showing the $m=1$ fine structures
at 0.21, 0.22 and $0.23~{}\textrm{rad day}^{-1}$ as the starting time, t, of
the Fourier transform is increased. The 30-day and 28.5-day sidebands are
successively reduced as t shifts from 1989 to 1991 and then to 1996. The
27-day peak is reduced when t shifts to 2005, leaving a small contribution
from 2015 - 2016.

####  3.2.2 Corresponding solar images

Next, we look for these magnetic patterns in spatially resolved solar
observations. We begin with Figure 11, which shows Carrington maps of the
photospheric magnetic field obtained at the National Solar Observatory
(NSO)222https://nispdata.nso.edu/ftp/kpvt/synoptic/mag. At NSO, each of these
maps was divided by ${\mu}=\sin{\theta}\cos{\phi}$ to convert the observed
line-of-sight component to a radial component, assuming that the fields are
radial at the photosphere where they are measured. Thus, we regard these maps
as displays of the radial component of photospheric field.

Figure 11: NSO Carrington maps of the radial magnetic field during sunspot
cycles 21 - 23 (top, middle, and bottom panels, respectively), showing flux
streaming poleward and eastward from the sunspot belts during the initial
phase of sunspot maximum (left panels) and flux accumulating in the equatorial
region between the sunspot belts at the end phase of sunspot maximum (right
panels). Longitude runs left to right from 0∘ to 360∘, and sine latitude
ranges from -1 to +1 from bottom to top of each map. Positive polarity is
white and negative is black. Dates refer to the starting times at the right
edge of each map.

This figure contains images from the start of sunspot maximum (left column)
and the end of sunspot maximum (right column) in sunspot cycles 21 (top row),
22 (middle row), and 23 (bottom row). In the left panels, each sunspot cycle
has progressed far enough that several large active regions have emerged to
form activity belts with flux streaming poleward and eastward (leftward) from
those belts. However, the sunspot cycles have not progressed far enough for
flux to spread equatorward to fill in the wide gaps between the two belts, as
has happened at the times of the images in the right panels. In fact, by the
end of the sunspot-maximum era, the two activity belts have reached
sufficiently low latitudes and narrow separations that the equatorward
diffusion of flux dominates the poleward convection by meridional flow,
causing flux to accumulate around the equator.

We have encountered these phenomena before. First, in numerical simulations of
the mean field, Sheeley & DeVore (1986b) found that the 28-29-day recurrent
patterns originated in flux that was migrating poleward from its sources in
the sunspot belts. The dramatic eastward drift of these patterns is well known
to many of us from viewing time-lapse movies of Carrington maps like the ones
in Figure 11. Second, Sheeley & Wang (2015) found that a juxtaposition of
northern-hemisphere and southern-hemisphere active regions during the second
half of 2014 created a large region of positive-polarity flux at the equator,
which produced a major rejuvenation of the Sun’s large scale field. As shown
in the top panel of Figure 8, the 27-day power in the two-sector component of
the mean field reached a peak at this time. Also, Sheeley & Wang (2015) noted
that this rejuvenation of the large-scale field was not an isolated
characteristic of sunspot cycle 24, and that similar enhancements of the
equatorial dipole field in 1982, 1991, and 2003 have marked the end of the
sunspot maximum era (or the start of the declining phase) of cycles 21, 22,
and 23.

Another reason for selecting the images in Figure 11 was to relate these flux
distributions to the profiles of spectral power, especially for the two-sector
($m=1$) plots in Figure 8. Thus, the maps in the top panel of Figure 11 occur
in 1979 and 1982 when there were peaks in the spectra for 27 days, 28.5 days
and, to a lesser extent, 30 days. The maps in the middle panels occurred in
1989 when the 30-day power reached its maximum, and in 1991 when the 27-day
and 28.5-day power reached their maximum values. The map in the bottom-right
panel was chosen because it occurred in 2003 when the 27-day power dominated
the two-sector spectrum. As one can see, it shows a large two-sector pattern
of equatorial flux with positive-polarity flux left of center and negative-
polarity flux right of center.

In choosing these images, I looked for strong poleward streams in each sunspot
cycle. However, I did not always find them. Whereas the maps in the upper-left
and middle-left panels show major streams, the map in the lower-left panel
(CR1955, 11 October 1999) shows relatively weak streams, despite the fact that
it was at the same phase of the sunspot cycle. Those 1999 streams are only
slightly more impressive than the weak streams in 1982, 1991, and 2003 in the
right column. This favoring of sunspot cycles 21 and 22 over cycle 23 is
consistent with the power spectra in Figure 8, which show more 28.5-day and
30-day power during cycles 21 and 22 than during cycle 23.

If the 28.5-day power originates in poleward migrating flux from large active
regions, as previously reported (Sheeley & DeVore, 1986b), then it seems
plausible that the much rarer 30-day power in 1989 is a statistical
fluctuation caused by the emergence of an especially large, high-latitude
active region at that time. The strong, northern-hemisphere stream in the
middle-left panel of Figure 11 originated in such an active region.

The right panel of Figure 12 shows the evolution of this region during
CR1811-1818 (09 January 1989 - 19 July 1989) when sunspot cycle 22 reached the
start of its 3-4 years of high sunspot activity333J.W. Harvey recently
reminded me that during CR1813, this region (5395) was the source of many
X-ray flares and coronal mass ejections (CMEs), two of which were responsible
for the blackout of the Hydro-Québec power system on 13 March 1989 (Boteler,
2019).. Each image is the northern-hemisphere part of a Carrington map that
has been cropped at the equator. Thus, longitude runs left to right from 0∘ to
360∘ and sine latitude runs bottom to top from 0 to +1. The faint, yellow
lines provide a reference drift for a 30.2-day rotation, corresponding to the
frequency ${\omega}$ = 0.208 $\textrm{rad~{}day}^{-1}$ that we obtained from
the power spectra in Figures 4 and 5.

For comparison, the left panel of Figure 12 shows the evolution of a much
smaller region during CR1788-1795 (22 April - 30 October 1987) at the

Figure 12: Carrington maps, cropped to show northern-hemisphere magnetic
fields as a function of longitude (0∘ to 360∘ on the horizontal axis) and sine
latitude (0 to +1 on the vertical axis). (left): CR1788-1795 (22 April - 30
October 1987), showing the northward and eastward drift of flux from a new-
cycle active region. (right): CR1811-1818 (09 January - 19 July 1989), showing
a stronger stream from a bigger active region near sunspot maximum. Faint
yellow lines indicate a 30.2-day rotation.

start of cycle 22. Both regions emerged at latitudes ${\sim}$35∘, and their
fluxes evolved into similar patterns at comparable speeds. (An interesting,
and probably coincidental, similarity is that both streams were replenished by
flux from a second active region that emerged 3-4 rotations later toward the
left side of each panel.) The major difference is that the 1989 source was
larger and stronger than the 1987 source. Presumably, this difference in
strength was responsible for the difference in spectral power shown in Figure
8. The 1989 stream coincided with a large peak of 30-day spectral power,
whereas the 1987 stream did not.

At the ${\sim}$35∘ latitude of these two active regions, the rotation period
of small-scale magnetic tracers is about 29 days (Snodgrass, 1983). However,
within a few Carrington rotations, these streams of trailing, positive-
polarity flux had drifted poleward into the 45∘ latitude range where the
rotation period is 30 days. Although one can confirm this by measuring the
vertical locations of the streams in these images, one can also refer to the
faint yellow lines, which provide a 30.2-day reference drift. These lines
track the trailing streams of positive-polarity flux fairly well until the
tails of the streams reach latitudes above 45∘ and also begin to merge with
flux from other active regions.

Multi-latitude stackplots of the WSO-derived source-surface field support the
identification of the 1989 pattern as the source of the 30-day power (Wang &
Sheeley, 1994b). Calculated for 1977 - 1993, these stackplots show a rare,
30-day, two-sector recurrence pattern over a wide range of latitudes from 20∘N
to 80∘N during 1989. It is the most prominent 30-day pattern during that
16-year interval. In addition, its positive-polarity sector moves from right
to left across the Carrington frame during 1989, so that the phase of this
two-sector pattern also agrees with that of the field in Figure 12.

The ‘1989 pattern’ continued its poleward and eastward migration well after
the end of 1989 when the 30-day power ended, according to Figure 9. By CR1830
(11 June 1990), the trailing end of the stream extended to about 68∘ latitude
where the Snodgrass (1983) rotation period is 35.3 days. However, the rotation
period of the mean field did not increase beyond 30 days. Thus, for latitudes
above 45∘, even this relatively strong field was too weak to overcome the
${\mu}^{2}$-dependence of the mean-field integral in Eq(3), and (as we shall
see in the Appendix), an extra factor of $\mu$ due to limb darkening. In the
next section, we shall see that this result provides a clue for understanding
some puzzling associations between the axisymmetric component of the mean
field and the Sun’s polar magnetic fields.

####  3.2.3 Power in the Axisymmetric Component of the Sun’s Mean Field

To display the non-axisymmetric part of the mean field, $B_{m}$, we used the
27-day running mean of the standard deviation, defined by
$B_{m}^{rms}=(<B_{m}^{2}>-<B_{m}>^{2})^{1/2}$, where the brackets refer to
averages over a 27-day moving window. In this subsection, we are interested in
$<B_{m}>$, the 27-day moving average of the mean field (i.e. the axisymmetric
component that we squared and subtracted from $<B_{m}^{2}>$ to get that non-
axisymmetric component).

Another way to compare the axisymmetric and non-axisymmetric components of
$B_{m}$ is to express Eq(7) in terms of real variables and separate the $m=0$
and $m{\neq}0$ terms. To first order in $B_{0}$, we obtain

$B_{m}~{}{\approx}~{}\left\\{\sum_{l=1}^{\infty}{\rho}_{l0}I_{l0}~{}+~{}2\sum_{l=1}^{\infty}\sum_{m=1}^{l}{\rho}_{lm}I_{lm}\cos{\delta_{lm}}\right\\}~{}+~{}B_{0}\left\\{\sum_{l=1}^{\infty}{\rho}_{l0}J_{l0}~{}+~{}2\sum_{l=1}^{\infty}\sum_{m=1}^{l}{\rho}_{lm}J_{lm}\cos{\delta_{lm}}\right\\}.$
(15)

Averaging over ${\delta}_{lm}$ (which we assume varies linearly with time,
$t$, or more precisely with $mt$), we get

$B_{m}^{rms}~{}{\approx}~{}\sqrt{2}~{}\sqrt{{\rho^{2}_{11}}I_{11}^{2}~{}+~{}{\rho^{2}_{22}}I_{22}^{2}~{}+~{}{\rho^{2}_{33}}I_{33}^{2}~{}~{}+{\rho^{2}_{31}}I_{31}^{2}}$
(16)

for the root-mean-square deviation of $B_{m}$ from its average value, and

$<B_{m}>~{}{\approx}~{}{\rho}_{20}I_{20}~{}+~{}B_{0}\left\\{{\rho}_{10}J_{10}~{}+~{}{\rho}_{30}J_{30}\right\\}$
(17)

for the 27-day moving average of $B_{m}$. In Eq(16), we have omitted
$J_{lm}$-dependent terms because they occur with the second-order factor,
$B_{0}^{2}$, and we have omitted terms with $I_{21}$ and $I_{32}$, which
vanish, as indicated in Table 1. Likewise, in Eq(17), we have omitted terms
with $I_{10}$, $I_{30}$, and $J_{20}$, which also vanish as indicated in
Tables 1 and 2. Thus, $B_{m}^{rms}$ has non-axisymmetric ($m~{}{\neq}~{}0$)
contributions from $Y_{1}^{1}$, $Y_{2}^{2}$, $Y_{3}^{3}$, and $Y_{3}^{1}$ for
which $I_{11}=+0.173$, $I_{22}=+0.103$, $I_{33}=+0.035$, and $I_{31}=-0.027$,
as obtained from Table 1. In that case, we expect the non-axisymmetric
component of $B_{m}$ to be dominated by contributions from $Y_{1}^{1}$ and
$Y_{2}^{2}$ with much smaller contributions from $Y_{3}^{3}$ and $Y_{3}^{1}$.

The axisymmetric quantity, $<B_{m}>$, has contributions from $Y_{2}^{0}$ with
$I_{20}=-0.084$, plus $B_{0}$-dependent contributions from $Y_{1}^{0}$ and
$Y_{3}^{0}$, with the somewhat larger values of $J_{10}=+0.244$ and
$J_{30}=-0.093$. However, the factor of $B_{0}$ reduces these contributions
and modulates them with a 1-year period. When $B_{0}$ has its maximum value of
$7.25^{\circ}$ (0.1265 rad), $B_{0}J_{10}$ is only 0.031 (37% of $I_{20}$) and
$B_{0}J_{30}$ has the even smaller value of -0.0118 (38% of $B_{0}J_{10}$).
Thus, in rough terms, we can regard $Y_{2}^{0}$, $Y_{1}^{0}$, and $Y_{3}^{0}$
as making three monotonically decreasing contributions to the axisymmetric
part of $B_{m}$, in which each contribution is about 40% of the previous one.
Now, let us see how well these terms fit the WSO mean-field measurements.

The top panel of Figure 13 contains a plot of $<B_{m}>$ (red) and the monthly
averaged sunspot number from the Royal Observatory of Belgium (SILSO) (blue)
during sunspot cycles 21 - 24. The sunspot number has been divided

Figure 13: (top) 27-day running average of the WSO mean field (red), compared
with the monthly averaged sunspot number from the Royal Observatory of Belgium
(SILSO) (blue). (bottom) 365-day moving average of the red curve, showing the
residual $Y_{2}^{0}$ component of the axisymmetric field (purple) after the
annually varying $Y_{1}^{0}$ and $Y_{3}^{0}$ contributions have been removed.

by 2000 and shifted downward by 0.55 units on the vertical scale. At first
glance, the red curve seems to be a meandering collection of relatively
uninteresting noisy wiggles. However, closer inspection reveals a rough trend
for the signal to be negative during the early years of sunspot maximum in
sunspot cycles 22 and 24 and positive during the declining phase of those
cycles. The polarities are reversed during the odd-numbered cycles 21 and 23.
In addition, there are well defined annual variations during the 1976, 1986,
and 2019 sunspot minima, which are presumably the most visible indications of
the $B_{0}$-dependence of the axisymmetric component of the mean field as
Earth orbits the Sun during the year.

These $B_{0}$-induced variations are removed in the bottom panel of Figure 13,
whose purple curve is the 365-day running mean of the $<B_{m}>$ values
associated with the red curve in the upper panel. We suppose that this purple
curve indicates the $Y_{2}^{0}$ contribution given by
${\rho}_{20}I_{20}=-0.084{\rho}_{20}$ in Eq(17)444Note that $Y^{0}_{2}$ has
the sign of its polar region, which is negative when its equatorial region is
positive, according to the definition
$P_{2}(\cos{\theta})=(1/2)(3\cos^{2}{\theta}-1)$. This accounts for the
negative sign of $I_{20}$ in Table 1.. Thus, the mean field reduces the
contribution of the $Y_{2}^{0}$ component by a factor of about 12 and has a
negative value corresponding to the sign of the equatorial part of a
positively directed quadrupole.

The temporal profile of the purple curve in Figure 13 is similar to the
profile of the $Y_{2}^{0}$ component calculated from spatially resolved
observations at both WSO and MWO (but not shown here). The main difference
occurred during 1992-1996 when the spatially resolved measurements give a much
larger positive field than the mean field in Figure 13. This stronger field
would have strengthened the rough, alternating-polarity rule noted from the
red curve in the top panel of Figure 13.

As pointed out previously by Wang & Robbrecht (2011) and Robbrecht & Wang
(2012), this polarity rule reflects the tendency of equatorial flux to
originate from the leading parts of active regions together with the greater
activity in the southern hemisphere than in the northern hemisphere during the
declining phases of sunspot cycles 21-24. Likewise, the northern hemispheres
were more active during the rising phases of those cycles. However, Wang &
Robbrecht (2011) also found that this alternating-polarity rule broke down
during the very high activity of sunspot cycle 19 when the northern hemisphere
tended to be more active than the southern hemisphere throughout the cycle.
(See also Figure 1 of White & Trotter (1977) who found no systematic
variations during 1874-1971555The reader may have to go to the hardcopy
edition of this paper because the figure was a large foldout that was not
scanned into the online edition..)

The annually varying part of the axisymmetric field is shown by the purple
curve in Figure 14.

Figure 14: The axisymmetric field, $<B_{m}>$, with its $Y_{2}^{0}$ component
removed and a 75-day smoothing applied to the remainder. The resulting purple
curve shows annular modulation, especially during the 1976, 1986, and 2019
sunspot minima. The black curve shows the monthly averaged sunspot number from
the Royal Observatory of Belgium (SILSO).

We obtained this curve by subtracting the 1-year averaged field (purple curve
in the bottom panel of Figure 13) from the total axisymmetric field (red curve
in the top panel of Figure 13), and then taking a 75-day running average to
remove the noise. Referring to Eq(17), we expect this $B_{0}$-induced field to
be $B_{0}({\rho}_{10}J_{10}+{\rho}_{30}J_{30})$, where $B_{0}$ is given by

$B_{0}(t)~{}{\approx}~{}0.126\sin\left\\{2{\pi}\left(\frac{t-157}{365}\right)\right\\}$
(18)

whose amplitude vanishes on day-of-year 157 (June 6) and reaches 0.126 rad
(7.25∘) on day-of-year 249 (September 6), and where $J_{10}=+0.244$ and
$J_{30}=-0.093$, as given in Table 2. The vertical meandering of the
$Y_{2}^{0}$ component, that was present in the top panel of Figure 13, is
clearly gone. The annual variation is strongly visible around the 1976, 1986,
and 2019 sunspot minima, but only weakly visible around the 1997 and 2009
minima. Stronger non-periodic bursts occur during the intervening sunspot-
maximum intervals.

Faint dotted lines have been drawn at the even-numbered years and the panels
have been enlarged to help show the phase of the annual variation. For
example, during 1976 and 1977, the positive peaks occurred in the fall and the
negative peaks occurred in the spring, as expected for a positive axisymmetric
field. In contrast, during 1985-1987, the sharp negative peaks occurred in the
fall and the blunted positive peaks occurred in the spring, consistent with a
negative axisymmetric field. This means that the $B_{0}$-induced axisymmetric
component of the mean field changed its sign from plus to minus in going from
sunspot cycle 21 to 22, in agreement with the signs of the axisymmetric dipole
and the Sun’s polar magnetic field. The amplitude of the $B_{0}$-induced
component was much weaker during the 1997 and 2009 solar minima, but the
alternation of signs was still detectable. Then in 2019, the field was strong
again, and its sign was positive as expected for the continued 11-yr
alternation of polarity. Thus, during the past 5 sunspot minima from 1976 to
2019, the $B_{0}$-induced component of the mean field reversed its polarity in
phase with the polarity of the Sun’s polar magnetic field and the Sun’s
axisymmetric dipole component.

Figures 15 and 16 provide more graphic displays of this phase alignment. In
Figure 15, plots of the WSO polar

Figure 15: WSO measurements of the Sun’s north (black) and south (blue) polar
fields, superimposed on the smoothed plot of the $B_{0}$-induced part of the
axisymmetric field (purple), and compared with the monthly-averaged sunspot
number (black). Annual variations of the purple curve are in phase with the
annual variations of the polar fields, but the polar fields had to be divided
by separate factors of 20 and 30 to improve the agreement between the long-
term envelopes of these curves.

field strengths are superimposed on the $B_{0}$-induced part of the
axisymmetric mean field (shown in purple again). To obtain the best overall
agreement with the envelope of the mean field, I reduced the north and south
polar field strengths by factors of 20 and 30, respectively, before plotting
them. Now, the envelopes agree fairly well in the years around sunspot
minimum, but not around sunspot maximum when the polar fields were reversing
and the mean field had several large bursts. A detailed inspection of these
overlapping curves shows that the annual variations of the polar fields are in
sync with the annual variations of the $B_{0}$-induced field. The north polar
field and its mean-field ripple reached their greatest absolute magnitudes in
the fall of each year, and the south polar field and its corresponding mean-
field ripple reached their greatest magnitudes in the spring.

Figure 16 provides another comparison between the $B_{0}$-induced mean field
and the polar fields. In this case,

Figure 16: Weighted sum of WSO north and south polar fields (blue),
superimposed on the $B_{0}$-induced part of the axisymmetric field (purple),
and compared with the monthly-averaged sunspot number (black) during cycles 21
- 24. Annual variations of this sum approximately match those of the purple
curve, except in the years around sunspot maximum when the polar fields
reverse and the purple curve has large spikes.

the weighted values of the north and south polar fields are added, to form a
single oscillating blue curve similar to the overlapping purple plot of the
mean field variations. Also, the weighting is changed slightly so that the
contribution of the north pole is increased by a factor of
$18/13~{}{\approx}~{}1.4$, instead of the factor of 30/20 = 1.50 that was used
in Figure 15. The overall trend is essentially the same as we found in Figure
15 with consistent agreement in phase and fair agreement in magnitude, except
in the years around sunspot maximum when the polar fields changed sign and the
mean field had several large purple spikes.

These graphical comparisons between the $B_{0}$-induced component of the mean
field and the amplitude of the polar field clearly indicate that these two
fields are in phase and that they reverse their polarities together from one
sunspot cycle to the next. We might expect this phase synchronization because
both of these axisymmetric fields depend strongly on $Y_{1}^{0}$, which
reverses the polarity of its contribution from one sunspot cycle to the next.

However, the $B_{0}$-induced field and the polar field have differences that
might cause their amplitudes to differ. The $Y_{3}^{0}$ component contributes
to both fields, but with opposite signs. We have seen in Eq(17), that
$Y_{3}^{0}$ makes a negative contribution to the $B_{0}$-induced mean field
because the coefficient, $J_{30}$, is negative. On the other hand, the topknot
character of the polar field requires a weighted sum of the $Y_{1}^{0}$ and
$Y_{3}^{0}$ contributions (and smaller contributions from $Y_{5}^{0}$ and
$Y_{7}^{0}$) in order to strengthen the field at the poles and weaken the
field around the equator (Svalgaard et al., 1978; DeVore et al., 1984; Sheeley
et al., 1989a, b). Also, the north and south polar fields often differ in
strength due to symmetry-breaking contributions from harmonic components of
even $l$, especially the $Y_{2}^{0}$ component. As shown in the second column
of Table 2, these even-order components do not contribute to the
$B_{0}$-induced axisymmetric component of the mean field. This explains why we
had to equalize the strengths of the polar fields in order to bring their
plots into better agreement with the envelope of the $B_{0}$-induced
axisymmetric field in Figures 15 and 16.

##  4 Summary and Discussion

In this paper, I have regarded the Sun as an unresolved source of light like
that from a distant star, and calculated the transmission that this ‘mean-
field filter’ would have for the spherical harmonic components, $Y_{l}^{m}$,
of the field. This transmission depends on the mode numbers, $l$ and $m$, and
$B_{0}$, the observer’s latitude in the star’s polar coordinate system. The
transmissions fell into three separate classes, proportional to
$\cos^{2}B_{0}$, $(\sin 2B_{0})/2$, and $\sin^{2}B_{0}$, as given in Tables
1-3, respectively, and described in Eqs(5) and (7). For the Sun, $B_{0}$
varies between $-7.^{\circ}25$ and $+7.^{\circ}25$ during the year, so that
$\sin^{2}B_{0}$ is small and Table 3 can be neglected. In that case, we expect
the axisymmetric part of the mean field to originate from the $Y_{2}^{0}$
component and the first few harmonic components of odd $l$ (mainly $Y_{1}^{0}$
and $Y_{3}^{0}$), which are related to the polar fields and are coupled to the
mean field via $B_{0}$. On the other hand, we expect the non-axisymmetric part
of the mean field to originate from the $Y_{1}^{1}$ and $Y_{2}^{2}$ components
with occasional small contributions from the $Y_{3}^{3}$ and $Y_{3}^{1}$
components.

For our observations of the Sun, $B_{0}$ is small and the contribution of the
$Y_{2}^{0}$ component comes from its equatorial band, which is negative for a
positively defined $Y_{2}^{0}$ component. This is why the sign of $I_{20}$ is
negative in Table 1. On the other hand, for a distant star observed nearly
head-on, $B_{0}$ would be ${\sim}90^{\circ}$, and the $Y_{2}^{0}$ contribution
would come from one of its two polar regions of positive polarity. This is why
the sign of $K_{20}$ is positive in Table 3.

I applied these ideas to the 46-year series of WSO daily measurements of the
Sun’s line-of-sight mean field, first plotting the 27-day running average of
the rms deviation of the mean field from its average value. This rms variation
provided a measure of the power in the non-axisymmetric components of the
field, and showed peaks already familiar to us from plots of the occurrence of
coronal inflows, of the Sun’s open flux and interplanetary magnetic field, of
the Sun’s equatorial dipole and quadrupole field, of the non-axisymmetric
component of the Sun’s source-surface magnetic field, and of the mean field
represented by a running 27-day average of its maximum-minus-minimum values
(Sheeley & Wang, 2014, 2015). In retrospect, this similarity of plots should
be no surprise because they all show peaks corresponding to the envelope of
the non-axisymmetric component of the Sun’s large-scale field as it is
modulated by solar rotation, analogous to the audio component of a high-
frequency radio wave.

The second step was to take the Fourier transform of the mean-field
measurements and display power as a function of frequency. The result was a
series of peaks at frequencies corresponding to the 27-day synodic solar
rotation period and its first two harmonics. In addition, each peak showed
fine structure at slightly longer periods, corresponding to the Sun’s rotation
at higher latitudes. By inverting this Fourier transform using broad windows
around these three peaks, I obtained a relatively noise-free version of the
non-axisymmetric mean field. In addition, temporal plots of the $m=1$, $m=2$,
and $m=3$ components of the non-axisymmetric field were obtained by selecting
each of the three windows separately. By comparing these plots with the
temporal plot of their sum, we learned that nearly all of the large peaks were
provided by the combination of $m=1$ and $m=2$ plus a few residual
contributions of $m=3$. In other words, the mean field is dominated by the
$Y_{1}^{1}$ and $Y_{2}^{2}$ components of the field, consistent with our
analysis of Eq(16).

The third step was to find the source of the fine structure in the power
spectrum, and in particular of the peak whose frequency corresponded to a
period of ${\sim}$30 days. This narrow peak was particularly interesting
because it had no counterpart in the interplanetary sector structure inferred
from Earth-based magnetometer measurements. Those in-ecliptic measurements
often show 28.5-day recurrence patterns, but never 30-day patterns.

By inverting the Fourier transform of the mean field through a narrow window
around the 30-day peak, I found that most of the power originated in 1989-1990
when photospheric magnetograms showed elongated patterns of magnetic fields
migrating to high latitudes in the northern hemisphere. This reminds us that
the mean-field is sampling large-scale magnetic patterns, which gradually
rotate rigidly as supergranular diffusion and meridional flow carry their flux
across latitudes (Sheeley et al., 1987; DeVore, 1987; Wang, 1998). Thus, mean-
field measurements give the pattern rate, which depends on the meridional
transport parameters as well as the rate of differential rotation, and not
just the rotation rate itself. We are still left with the interesting
quantitative question of why the 1989-1990 fields gave a rotation period of
${\sim}$30 days, rather than a longer period associated with the high-latitude
tail of the migrating stream or the shorter, ${\sim}$28.5-day period found for
so many other migrating streams.

If the answer lies in the strength of the active region that emerged at 35∘N
latitude in 1989, then we might ask if similar (or even stronger) active
regions may have emerged at high latitudes in previous sunspot cycles, like
cycles 18 and 19, that were more active than cycles 21 - 24. A large northern-
hemisphere pattern was visible in Ca II K-line maps and Fe I 5250 Å
magnetograms obtained at the Mount Wilson Observatory during Carrington
rotations 1417 (8 August - 4 September, 1959) and 1419 (2-29 October 1959)
(Sheeley et al., 2011). Another occurred in the southern hemisphere during
rotations 1259 (21 October - 17 November 1947) and 1261 (14 December 1947 - 11
January 1948). Perhaps those migrating fields would have produced $m=1$
sidebands with rotation periods of at least 30 days in the power spectrum of
the mean field.

The fourth step was to look at the axisymmetric component of the mean field.
This component consisted of the $Y_{2}^{0}$ component plus the strongest
$B_{0}$-induced components of odd $l$, mainly $Y_{1}^{0}$ and $Y_{3}^{0}$.
Because $B_{0}$ varies annually due to Earth’s orbital motion around the Sun,
it was possible to remove the $B_{0}$-induced term, and display the annually
averaged $Y_{2}^{0}$ component separately. Although not shown here, its
temporal profile was similar to those obtained from spatially resolved
observations at both WSO and MWO, except for the interval 1992-1996 when the
spatially resolved measurements gave a much larger positive field. Resolving
this discrepancy will be a challenge for the future.

Once the $Y_{2}^{0}$ component was found, it was then possible to extract the
$B_{0}$-induced part of the axisymmetric field. Its annual variation was in
phase with the annual variation of the polar magnetic fields. This means that
the $B_{0}$-induced field was oriented in the same direction as the polar
field, reversing its direction from one sunspot cycle to the next. A remaining
puzzle is why the annual variations were strong around the 2019 sunspot
minimum when the polar fields were weak, and why the annual variations were
weak around the 1997 minimum when the polar fields were stronger.

Finally, we note that this approach can be used to find spherical harmonic
components of the fields in other stars. For example, the $B_{0}$-dependence
of the strengths of these harmonic components may complement asteroseismology
determinations of the orientations of the rotational axes of these stars
(Gizon & Solanki, 2003). Also, the appearance of low-frequency sidebands in
the WSO spectra suggests that similar sidebands may occur in observations of
other stars, providing information about differential rotation and meridional
flow in those stars. This information may help to remove ambiguities in the
inferences of large scale convection and magnetic cycles in asteroseismology
studies of Sun-like stars.

###### Acknowledgements.

I am grateful to Phil Scherrer (WSO/Stanford) and Todd Hoeksema (WSO/Stanford)
for helpful comments about the WSO telescope and its data, and Yi-Ming Wang
(NRL) for numerous discussions of the Sun’s large-scale magnetic field. I am
grateful to Jack Harvey (NSO/LPL/UA) for helping me find archived Carrington
maps of observations obtained with the Kitt Peak Vacuum Telescope. This work
evolved from a talk that I gave at an LPL/UA Heliophysics Group Zoom session
where Joe Giacalone, Jack Harvey, and John Leibacher provided useful comments
and ideas. Wilcox Solar Observatory data used in this study were obtained via
the web site http://wso.stanford.edu courtesy of J.T. Hoeksema. NSO data were
acquired by SOLIS instruments operated by NISP/NSO/AURA/NSF. Sunspot numbers
were obtained from WDC-SILSO, Royal Observatory of Belgium, Brussels. I am
grateful to the referee for several helpful comments, including one about limb
darkening which motivated the calculations in the Appendix.

##  Appendix A Limb Darkening

In Section 2, we converted the integral of the line-of-sight field over the
(flat) solar disk to a surface integral of the radial field over the visible
hemisphere. As shown in Eq(3), this conversion introduced two factors of
${\mu}$ into the integrand, causing the surface integral of $B_{r}$ to be
weighted toward the disk center by a factor of ${\mu}^{2}$. However, we did
not include the natural weighting that is produced by the limb darkening of
the Sun and stars in the visual region of the spectrum. This limb darkening
occurs because light from disk center originates in deeper, hotter, and
brighter layers than light from positions toward the limb. As described by
Scherrer et al. (1977a), the central weighting of the WSO measurements is
mainly due to this natural limb darkening plus a contribution of diffraction
from the entrance slit of the spectrograph.

It is relatively easy to include limb darkening in our equations for the mean
line-of-sight magnetic field. We simply insert the limb darkening intensity
profile, $I({\mu})/I(0)$, next to the ${\mu}^{2}$ in the integrand of Eq (3).
With the definition $F({\mu})=I({\mu})/I(0)$, Eq(3) becomes

$B_{m}~{}=~{}\int{B_{r}}{\mu}^{2}F({\mu})dA_{surf}/{\pi}R^{2}.$ (19)

Then, we replace ${\mu}$ by
$\sin{\theta}\cos{\phi}\cos{B_{0}}+\cos{\theta}\sin{B_{0}}$, and perform the
integration over the variables ${\theta}$ and ${\phi}$ as indicated in Eqs(4)
and (5). If $F({\mu})$ is a simple first- or second-order polynomial in
${\mu}$, one can do the integration analytically, but if the profile is more
complicated, then it might be more convenient to integrate numerically.

Let’s begin by considering some of the limb darkening relations that are used
to describe the Sun and stars. Classically, the natural limb darkening has
been described by a linear function of ${\mu}$ of the form

$\frac{I({\mu})}{I(0)}~{}=~{}1-{\gamma}(1-{\mu})=(1-{\gamma})+{\gamma}{\mu}.$
(20)

Here, ${\gamma}$ is a wavelength-dependent limb-darkening coefficient that
indicates how dark the limb is relative to the disk center at that wavelength.
For the so-called gray atmosphere in the Eddington approximation (Foukal,
1990; Schwarzschild, 1906), ${\gamma}=0.6$ and $I({\mu})/I(0)=0.4+0.6{\mu}$.
According to Maxted (2018), Kopal’s (1950) quadratic limb darkening relation
of the form

$\frac{I({\mu})}{I(0)}~{}=~{}1-c_{1}(1-{\mu})-c_{2}(1-{\mu})^{2}$ (21)

is the most commonly used profile in modern exoplanet studies. Stellar limb
darkening is important in these studies because it affects the light curve
produced by the transiting exoplanet and therefore the accuracy with which the
exoplanetary radius can be determined. Consequently, other profiles are
sometimes used to match the high-precision light curves of transiting
exoplanet systems (Knutson et al., 2007; Espinoza & Jordán, 2015). In
particular, Maxted (2018) considered another two-parameter limb-darkening
relation of the form

$\frac{I({\mu})}{I(0)}~{}=~{}1-c(1-{\mu}^{\alpha}).$ (22)

Although any of these limb darkening relations can be substituted into Eq(A1)
above, I will illustrate the procedure using the linear relation given by
Eq(A2). In this case, $B_{m}$ can be obtained from proportionate parts of
Eq(4) and of the modified version of Eq(4) when an extra factor of ${\mu}$ is
included in its integrand. This latter quantity, which I call $B_{\mu}$ to
distinguish it from $B_{m}$, is given by

$B_{\mu}~{}=~{}(1/{\pi})\int_{-{\pi}/2}^{{\pi}/2}\int_{0}^{\pi}{B_{r}}(\sin{\theta}\cos{\phi}\cos{B_{0}}+\cos{\theta}\sin{B_{0}})^{3}\sin{\theta}d{\theta}d{\phi}.$
(23)

The third-power expansion of this binomial expression for ${\mu}$ gives four
terms with factors of $\cos^{3}{B_{0}}$, $3\cos^{2}{B_{0}}\sin{B_{0}}$,
$3\cos{B_{0}}\sin^{2}{B_{0}}$, and $\sin^{3}{B_{0}}$, respectively, instead of
the three terms that we obtained from the second-power expansion of that
binomial in Eq(4).

It would be easy to evaluate all four integrals and put the results in tables
as we did in the main text. However, to estimate the effect of limb darkening
on the mean field of the Sun, it is sufficient to expand the $B_{0}$-dependent
factors in powers of $B_{0}$ and retain only the zeroth-order and first-order
terms. This means that we need to keep only the $\cos^{3}{B_{0}}$ and
$3\cos^{2}{B_{0}}\sin{B_{0}}$ factors, which reduce to 1 and $3B_{0}$,
respectively. Then, the limb darkened mean field becomes

$B_{m}~{}=~{}(1-{\gamma})(I_{lm}+B_{0}J_{lm})~{}+~{}{\gamma}(I_{lm}^{\mu}+B_{0}J_{lm}^{\mu}),$
(24)

where the terms that originate from Eq(A5) are given by

$\displaystyle
I_{lm}^{\mu}~{}=~{}\frac{N_{lm}}{{\pi}}\int_{-1}^{1}P_{l}^{m}(x)(1-x^{2})^{3/2}dx\left[\frac{12\cos{(m{\pi}/2)}}{(m+1)(m-1)(m+3)(m-3)}\right],$
(25a) $\displaystyle
J_{lm}^{\mu}~{}=~{}3\frac{N_{lm}}{{\pi}}\int_{-1}^{1}P_{l}^{m}(x)(1-x^{2})xdx\left[\frac{-4\sin{(m{\pi}/2)}}{m(m+2)(m-2)}\right].$
(25b)

Next, we rearrange Eq(A6) as a power series in $B_{0}$ to obtain

$B_{m}~{}=~{}\\{(1-{\gamma})I_{lm}+{\gamma}I_{lm}^{\mu}\\}~{}+~{}B_{0}\\{(1-{\gamma})J_{lm}+{\gamma}J_{lm}^{\mu}\\}.$
(26)

I used the conventional Mathematica software to evaluate $I_{lm}^{\mu}$ and
$J_{lm}^{\mu}$ for $l$ and $m$ in the range $0-7$, again reversing the signs
of the odd-m entries to be consistent with the Jahnke & Emde
(1945)-convention. Then I set ${\gamma}=0.6$, and combined these results with
the values of $I_{lm}$ and $J_{lm}$ given in Tables 1 and 2 to obtain the limb
darkened intensities for the gray atmosphere in the Eddington approximation.
The results are given in Tables 4 and 5.

Table 4: Limb-Darkened Elements of $B_{lm}$ {zeroth-order in $B_{0}$} $l/m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
0 | +0.160 |  |  |  |  |  |  |
1 | 0 | +0.152 |  |  |  |  |  |
2 | -0.081 | 0 | +0.099 |  |  |  |  |
3 | 0 | -0.033 | 0 | +0.043 |  |  |  |
4 | +0.006 | 0 | -0.006 | 0 |  +0.008 |  |  |
5 | 0 | -0.001 | 0 | +0.002 | 0 | -0.002 |  |
6 | +0.001 | 0 | -0.001 | 0 | +0.001 | 0 |  -0.001 |
7 | 0 | -0.000 | 0 | +0.000 | 0 | -0.000 | 0 | +0.001
Table 5: Limb-Darkened Elements of $B_{lm}$ {first-order in $B_{0}$} $l/m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
0 | 0 |  |  |  |  |  |  |
1 | +0.215 | 0 |  |  |  |  |  |
2 | 0 | +0.198 | 0 |  |  |  |  |
3 | -0.114 | 0 | +0.104 |  0 |  |  |  |
4 | 0 | -0.026 | 0 | +0.023 | 0 |  |  |
5 | -0.007 | 0 | +0.007 | 0 | -0.006 |  0 |  |
6 | 0 | -0.004 | 0 | +0.004 | 0 | -0.003 | 0 |
7 | -0.003 | 0 | +0.003 | 0 | -0.003 | 0 | +0.002 |  0

Comparing Table 1 and Table 4, we see that the elements have the same sign and
nearly the same magnitude for $l<4$. For larger values of $l$, there are some
sign differences, especially for $l=4$ and $l=6$, but the magnitudes of these
higher-order harmonic components are less than 0.01 and can be neglected. A
comparison between Tables 2 and 5 gives a similar result. This means that the
same harmonic components contribute to the mean field with or without limb
darkening, but that the strengths of these contributions differ slightly,
depending on the values of ${\gamma}$ and $l$ (as we shall see next).

We can gain further insight by computing the relative differences between
these limb-darkened and non-limb-darkened intensities. Doing this separately
for the terms that are zero-order and first-order in $B_{0}$, we obtain

$(\frac{{\Delta}B}{B})_{0}~{}=~{}\frac{\\{(1-{\gamma})I_{lm}+{\gamma}I_{lm}^{\mu}\\}-I_{lm}}{I_{lm}}~{}=~{}-{\gamma}(1-\frac{I_{lm}^{\mu}}{I_{lm}})$
(27)

and

$(\frac{{\Delta}B}{B})_{1}~{}=~{}\frac{\\{(1-{\gamma})J_{lm}+{\gamma}J_{lm}^{\mu}\\}-J_{lm}}{J_{lm}}~{}=~{}-{\gamma}(1-\frac{J_{lm}^{\mu}}{J_{lm}}).$
(28)

Thus, the zeroth-order change, $({\Delta}B/B)_{0}$, depends on the ratio
$I_{lm}^{\mu}/I_{lm}$, and the first-order change, $({\Delta}B/B)_{1}$,
depends on the ratio $J_{lm}^{\mu}/J_{lm}$. And both changes are proportional
to the limb darkening coefficient, ${\gamma}$. Moreover, if we use Eqs(A7ab)
and Eqs(11ab) to evaluate the ratios, $I_{lm}^{\mu}/I_{lm}$ and
$J_{lm}^{\mu}/J_{lm}$, we obtain the remarkable result that the values of
these ratios are the rational numbers 4/5, 15/16, and 48/35, for $l=1$, 2, and
3, respectively, independent of the values of $m$. (Of course, this applies
only for the non-zero values of $I_{lm}$ and $J_{lm}$ in Tables 1 and 2,
respectively.)

In other words, for the $Y_{1}^{0}$ and $Y_{1}^{1}$ harmonics, the ratios,
$J_{10}^{\mu}/J_{10}$ and $I_{11}^{\mu}/I_{11}$, are both equal to 4/5.
Likewise, for the $Y_{2}^{0}$, $Y_{2}^{1}$, and $Y_{2}^{2}$ harmonics, the
ratios are all equal to 15/16. And for the four harmonics with $l=3$, the
ratios are 48/35. Subtracting these ratios from 1, we obtain 1/5, 1/16, and
-13/35, as the fractional differences in Eqs(A9) and (A10) for $l=1$, 2, and
3, respectively, before the limb darkening factor, ${\gamma}$, is applied.

If we let ${\gamma}=0.6$, we obtain changes of -12%, -3.75%, and +22.2%, for
$l=1$, 2, and 3, respectively, independent of the value of $m$. Thus, for an
ideal gray atmosphere in the Eddington approximation, the amplitudes of the
$Y_{1}^{0}$ and $Y_{1}^{1}$ components decrease by 12%. The three components
with $l=2$ decrease by only 3.75% and the four relatively weak components with
$l=3$ go in the opposite direction, increasing by 22.2%.

The nice aspect of Eqs(A9) and (A10) is that we can increase ${\gamma}$ to
obtain the fractional changes for a greater amount of limb darkening in the
violet part of the spectrum, or decrease ${\gamma}$ to obtain the smaller
changes expected in the infrared. We simply return to the fractions 1/5, 1/16,
and -13/35 and multiply them by $-{\gamma}$.

I did this calculation to learn how solar limb darkening in the simple
Eddington approximation might affect the harmonic components of the mean
field. However, for more complex limb darkening profiles and the more inclined
rotational axes, that might occur in exoplanet or asteroseismic studies, one
could relax the small-$B_{0}$ approximation used for the Sun, and do the
integration numerically for specific values of $B_{0}$. We might expect limb
darkening to have a more complicated effect for a star whose rotational axis
makes an oblique angle to the line of sight.

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