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The data of the mean magnetic field (MMF) of the Sun from the Wilcox Solar Observatory from 1975 to 2023 (49 years, number of daily measurements N=14516) were analyzed to identify the main periodicities over short time intervals. The wavelet-spectrum of these data shows that the MMF has significant rotational harmonics from the first to the third, although sometimes there are higher orders up to the sixth harmonic inclusively. Each of these harmonics corresponds to its own multiplet of the magnetic field (dipole, quadrupole and higher components). Rotation harmonics were identified in the original data series and an approximated time series was built from them. In this case, wavelet-spectrum data and refined parameters of sinusoids obtained using the least squares method were used. The constructed approximated data series contains more than 90 % of the power of the original series. The main components of the Sun's rotation change their amplitude, frequency and phase over time. Changes in the amplitude of oscillations occur depending on solar activity-they are usually maximal in the second half of the cycle. Rotational frequencies change over time, but do not show dependence on solar activity. The maximal amplitude of rotation component most often has a frequency of the main period of about $P\approx27$ day (55.4 % in time), as well as its half value $P/2\approx 13.5$ day (38.2 %); the share of periods $P/3\approx 9$ day is 6.1 %, the rest — no more than 0.3 %. This means that the Sun predominantly appears as a magnetic horizontal dipole ($\approx$55 % of the time), but the quadrupole component is also very significant and occupies 38 % of the time. In general, different multipoles are almost always present on the Sun, but the predominance of one or another component of the multipole does not reveal any patterns and is practically independent on solar activity.
Sun: magnetic fields, rotation
\documentclass{vak2024}
\usepackage{url}
\usepackage{hyperref}
\def\UrlFont{\rmfamily}
\begin{document}
\title{Magnetic multipoles of the Sun from observations of the mean magnetic field}
\titlerunning{Magnetic multipoles of the Sun} % abbreviated title (for running head)
\author{V.~I.~Haneychuk}
\authorrunning{Haneychuk V.I.} % abbreviated author list (for running head)
\institute{Crimean Astrophysical Observatory of the Russian Academy of Sciences, Nauchny, Crimea 298409, Russia}
\abstract{
The data of the mean magnetic field (MMF) of the Sun from the Wilcox Solar
Observatory from 1975 to 2023 (49 years, number of daily measurements $N=14516$)
were analyzed to identify the main periodicities over short time intervals.
The wavelet-spectrum of these data shows that the MMF has significant rotational
harmonics from the first to the third, although sometimes there are
higher orders up to the sixth harmonic inclusively. Each of these harmonics
corresponds to its own multiplet of the magnetic field (dipole, quadrupole and
higher components). Rotation harmonics were identified in the original data series
and an approximated time series was built from them. In this case,
wavelet-spectrum data and refined parameters of sinusoids obtained using
the least squares method were used. The constructed approximated data series
contains more than 90\% of the power of the original series.
The main components of the Sun's rotation change their amplitude, frequency and
phase over time. Changes in the amplitude of oscillations occur depending on
solar activity---they are usually maximal in the second half of the cycle.
Rotational frequencies change over time, but do not show dependence on solar
activity. The maximal amplitude of rotation component most often has a frequency
of the main period of about $P\approx27$~day (55.4\% in time), as well as its
half value $P/2\approx 13.5$~day (38.2\%); the share of periods
$P/3\approx 9$~day is 6.1\%, the rest---no more than 0.3\%. This means that the
Sun predominantly appears as a magnetic horizontal dipole ($\approx$55\% of the
time), but the quadrupole component is also very significant and occupies 38\%
of the time. In general, different multipoles are almost always present on the
Sun, but the predominance of one or another component of the multipole does not
reveal any patterns and is practically independent on solar activity.
\keywords{Sun: magnetic fields; Sun: rotation}
\doi{10.26119/VAK2024-ZZZZ}
}
\maketitle
\section{Introduction}
The mean magnetic field (MMF) of the Sun as a star is the average value of the
magnetic flux from the entire visible solar hemisphere. Its measurements began
in 1968 by Severny \citep{severny1969} at the Crimean Astrophysical Observatory,
and then continued in other observatories.
The periodicity of MMF has been studied by different authors, see, for example,
\cite{rivin1992}, \cite{haneychuk1999}, \cite{chaplin2003}, \cite{haneychuk2003},
\cite{kotov2020}, \cite{haneychuk2021}. As has been shown, the main changes
in the MMF are associated with the rotation of the Sun. This can be seen
directly in the measurement data themselves, and, of course, in the power
spectra or periodograms, where peaks in the region of 26--27 days dominate.
This main harmonic of the rotation of the MMF is associated with the magnetic
horizontal dipole of the Sun. The second harmonic of rotation has a period of
about 13.5~days and is associated with the quadrupole component of the magnetic
field. Higher harmonics reflect the presence of more high multipoles.
This article analyzes changes in the MMF over 49 years of observations in order
to identify the main multipoles {\bf or their horizontal components} present
on the surface of the Sun.
\section{Observation}
To analyze changes in the MMF, an observational data of the Wilcox Solar
Observatory \citep{scherrer1977} was used, because it is homogeneous,
the longest and most numerous. Observations have been carried out on the same
instrument since 1975 and continues to this time. One measurement per day
is made, taking into account the zero position of the magnetograph.
The data is published on the Internet on the observatory
website\footnote{\url{http://wso.stanford.edu/}}.
The observation period from 1975 to 2023 was taken for the study, the number of
measurements was $N$ = 14516, the total duration of the series $T$ was about 49
years. The measurements cover more than four cycles of solar activity.
The original values of the MMF usually do not exceed 1~Gauss in modulus,
and at a minimum they are no more than 0.5~Gauss. Positive values correspond to
the northern polarity of the magnetic field, negative ones---to the southern
polarity.
\section{Spectra of observational data}
%Fig1
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig1a.png}
\includegraphics[width=0.5\textwidth]{fig1b.png}
\caption{a) Power spectrum of the MMF, along the $x$ axis---frequency in $\mu$Hz,
along the $y$ axis---power in Gs$^2$. b) Amplitude spectrum of the MMF, along the
$x$ axis---frequency, along the $y$ axis---amplitude in Gs. c) Wavelet spectrum
of the MMF, along the $x$ axis---time in years, along the $y$ axis---frequency
in $\mu$Hz, the horizontal bar at the top shows the oscillation amplitude in Gs.}
\label{FigSp}
\end{figure}
Fig.~\ref{FigSp}a shows the power spectrum of the studied series of the MMF.
Three groups of peaks associated with the rotation of the Sun are clearly
visible on it. The highest peak corresponds to the main rotation harmonic with
a period $P$ of about 27~days. The second most powerful group of peaks
corresponds to the second harmonic of rotation with periods in the region
of $P/2 \approx 13.5$~day. The third, smallest in power, group of peaks in the
region of $P/3 \approx 9$~day is also noticeable. No higher rotation harmonics
are observed. For a more detailed search for high harmonics, the amplitude
spectrum of this series of MMFs was constructed (Fig.~\ref{FigSp}b).
The first three fundamental harmonics of rotation are marked on the graph.
Higher harmonics do not show themselves here either.
Fig.~\ref{FigSp}c shows the wavelet spectrum of the studied series, which
shows the change in the amplitude of MMF oscillations with time.
The window width was taken to be 60~days. The graph clearly shows that
the maximum amplitudes of oscillations are observed during the maximum
of solar activity, which can be seen also directly from observational data.
It can also be seen that there are three horizontal bands of increased power,
which correspond to the first three harmonics of rotation. Here, however,
it is noticeable that there is a certain oscillation power in the region
of higher frequencies, which corresponds to harmonics from the fourth to
the sixth. They are not preserved in the spectra of the entire data series
(Fig.~\ref{FigSp}ab), since they have a random phase and small amplitudes.
From Fig.~\ref{FigSp} it follows that the studied MMF series contains mainly
the first three harmonics of rotation. This means that we can try to
approximate the original data with sinusoids of these harmonics.
In this way they can be separated and then examined in detail.
\section{Approximation of the original data series}
%Fig2
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig2a.png}
\includegraphics[width=0.5\textwidth]{fig2b.png}
\caption{a) An example of approximation of the original data series for May 8,
1979: cyan color---original data, red color---approximation by sum of three
sinusoids from the power spectrum, blue---sum of three sinusoids calculated by
the least squares method; the vertical line marks the point for which the
approximation was carried out.
b) Comparison of the original data series (cyan color) and the
artificial series constructed from the main harmonics of rotation (red color).}
\label{FigApp}
\end{figure}
For each point of the original time series, a region around it was selected in
the interval $\pm$30~days (two rotation periods), in which the data were
approximated by sinusoids $y(t)=A_0+A\cos(\omega t-\varphi)$. The initial
values of the sinusoid parameters $A_0,A,\omega,\varphi$ were determined
from the spectrum of this section for the maximum oscillation power.
Then the first component was subtracted from the original series and
the parameters of the second component were determined from the spectrum
of residuals, etc. In this way, this region of the time series was approximated.
Fig.~\ref{FigApp}a shows an example of approximation for data on May 8, 1979.
The sum of three sinusoids determined from the power spectrum is shown in red.
To refine the parameters of the sinusoids, the least squares method (LSM)
was used. The blue curve shows the LSM sinusoids approximation.
For most points in the series, three sinusoids quite satisfactorily describe
changes in the MMF. However, for some points this was not enough, and then
higher components were used---up to the sixth inclusive. The final criterion
for selecting the approximation curve was the minimum deviations of the
constructed curve from the original point.
Fig.~\ref{FigApp}b shows a comparison of the approximated MMF series (red)
with the original (cyan). The standard deviation of the two series is almost
the same: $\Delta_o=0.361$~Gs for the original series and
$\Delta_a=0.360$~Gs---for the approximated one. The original series in peak
field values is larger than the approximated one. However, there are few
such points. More than 90\% of the power of the original data is contained
in the approximated series.
\section{Basic multipoles of the Sun}
%Fig3
\begin{figure}
\includegraphics[width=0.5\textwidth]{fig3a.png}
\includegraphics[width=0.5\textwidth]{fig3b.png}
\caption{a) Frequency of the first rotation component (black, left scale)
as a function of time; red color---Wolf numbers curve (right scale).
b) Amplitude of the first rotation component (black, left scale);
red color---Wolf numbers curve (right scale).
c) Frequency distribution histogram for the first rotation component:
$N$---number of occurances, left scale; percentage---right scale.}
\label{FigCmp}
\end{figure}
Fig.~\ref{FigCmp}b shows the time course of the amplitude of the main (first)
component of the rotation of the MMF (black color) in comparison with changes
in the Wolf numbers $W$ (red color). It can be seen that the amplitude follows
the course of solar activity. It can be noted, however, that it lags somewhat
behind $W$, especially in odd cycles 21 and 23, and has its maximum closer to
the beginning of the decline phase of solar activity.
Fig.~\ref{FigCmp}a shows the change over time in the frequency of the first
component of rotation of the MMF (black color) and the Wolf number $W$
(red color). Three horizontal stripes are noticeable here, which correspond to
the main harmonics of rotation or multipoles of the MMF.
The lowest band is the basic harmonic (dipole), the second is the quadrupole,
the third, weakest, corresponds to the period $P/3$.
The change in the frequency of the main oscillation (or change in the multipole)
occurs in an indefinite manner and does not depend on solar activity.
In the 21st cycle of activity on the Sun, both dipole and quadrupole components
were present approximately equally in time, but at its end---in 1984---the
longest interval of the dipole field began
%, which occupied almost the entire
%22nd cycle and lasted 11 years---
until the end of 1995.
After that, at a minimum of 1996--1998 a predominance of high multipole
components was observed, up to the sixth harmonic inclusive, which is very
unusual for the Sun. In subsequent cycles 23--25, the Sun appeared alternately
as a dipole and quadrupole with some more noticeable proportion of the octupole
component. In general, different multipoles are almost always present on the Sun,
but the predominance of one or another component of the multipole does not reveal
any patterns and is independent on solar activity.
Fig.~\ref{FigCmp}c shows a histogram of the frequency distribution for the first
rotation component. Each vertical bar shows the number $N$ of occurrences of
the main rotation components or multipoles (left red scale in percentage).
The frequency of the basic rotation period $P\approx27$~day
occurs in 55.4\% of cases, the second harmonic---in 38.2\% of cases;
the share of periods with $P/3\approx9$~days is 6.1\%, the rest---no more
than 0.3\%. This means that the Sun appears predominantly as a magnetic
horizontal dipole ($\approx55$\% of the time), but the quadrupole component
is also very significant and occupies 38\% of the time.
\section*{Funding}
This work was supported by the Ministry of Higher Education and Science of
the Russian Federation within the framework of the state assignment on
the topic ``General and local characteristics of the Sun'',
state registration number 122022400224-7.
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