# A long term multi-frequency study of solar rotation using solar radio flux
and its relationship with solar cycles

Vivek Kumar Singh,1 Satish Chandra,2 Sanish Thomas1 Som Kumar Sharma,3 and
Hari Om Vats4
1Department of Physics, Sam Higginbottom University of Agriculture Technology
and Sciences, Praygraj, India
2Department of Physics, Pt. Prithi Nath PG College, Kanpur, India
3Physical Research Laboratory, Ahmedabad, India
4Space Education and Research Foundation, Ahmedabad, India
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>(HOV)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

This paper examines long-term temporal and spatial fluctuations in the solar
rotation (more than four solar cycles) by investigating radio emission escapes
from various layers of the solar atmosphere during the years 1967-2010. The
flux modulation approach can also be used to investigate variations in solar
rotation, which is a contentious topic in solar physics. The current study
makes use of a time series of radio flux data at different frequencies
(245-15400 MHz) obtained at Sagamore Hill Solar Radio Observatory in
Massachusetts, USA, and other observatories from 1967 to 2010. The periodicity
present in the temporal variation of time series is estimated through Lomb
Scargle Periodogram (LSP). The rotation period estimated for five radio
emissions (606, 1415, & 2695 MHz; from corona, and 4995 & 8800 MHz; from
transition region) through statistical approach shows continuous temporal and
spatial variation throughout the years. The smoothed rotation period shows the
presence of $\sim$ 22-yrs periodic and $\sim$ 11-yrs components in it. The
22-year component could be linked to the reversal of the solar magnetic field
(Hale’s) cycle, while the 11-yrs component is most likely related to the
sunspot (Schwabe’s) cycle. Besides these two components, random components are
also prominently present in the analyzed data. The cross-correlation between
the sunspot number and the rotation period obtained shows a strong correlation
with 11-yrs Schwabe’s and 22-yr Hale cycle. The corona rotates faster or
slower than transition region in different epoch. The swap of faster rotation
speed between corona and transition region also follows the 22-yrs cycle.

###### keywords:

Sun: activity – Sun: corona – Sun: radio radiation – Sun: rotation

††pagerange: A long term multi-frequency study of solar rotation using solar
radio flux and its relationship with solar cycles–References††pubyear: 2020

##  1 Introduction

From long radio waves to short X-rays, the Sun releases radiation across a
wide variety of energy spectrum. It is common to record the solar emission as
a function of both frequency and time because the solar radio flux at
different frequencies has a higher degree of correlation with all other solar
phenomena. Thus, solar radio emissions are a very useful indicator of major
solar activity, such as sunspot cycle or magnetic reversal cycle (Chandra, &
Vats, 2011; Li et al., 2012; Xie, Shi & Zhang, 2017). The radio flux emissions
from Sun changes gradually in intensity. The emission of solar radio flux
probably originates from three different sources, namely, either from the
undisturbed solar surface or from initially active regions, and also from
short-lived enhancements above its daily level. The dependency of spectral
lines on various solar atmospheric regions has been studied mostly by
analyzing the time series of each individual spectral line, emanating from
various regions. Each region of the solar atmosphere could be a source of
radio flux at a definite frequency, and time-series of such emission could be
useful for spectral analysis (Kane, 2004, 2009).

The radio emission at various discrete frequencies originates from different
heights in the solar chromosphere, transition region and inner and outer
corona. Various models proposed on plasma density and radio emissions provide
approximate heights of its origin as a function of emissions frequency.
Aschwanden & Benz (1995) and Melendez et al. (1999) presented a generally
accepted model that predicted plasma density at various heights above the
solar surface. Using the model they proposed, the plasma frequency at various
heights can be calculated. We know that the radio emissions having frequencies
less than or equal to the plasma frequency will be absorbed, but radiation
with a second harmonic (at twice the plasma frequency) will propagate (Vats et
al., 2001). According to the plasma density models suggested, it is found that
usually higher frequencies can be supposed to be emanating from deeper
regions, i.e., from lower solar altitudes. Besides the plasma density model,
the temperature-height profiles given by Fontenla et al. (1999); Vats et al.
(1998, 2001) can also be used to determine the height from where the radio
flux at particular frequencies originates in the corona.

Solar radio fluxes emanated at different heights are also used to find out the
temporal and spatial change of coronal rotation (Vats et al., 1998, 2001;
Chandra et al., 2009; Chandra, & Vats, 2011; Li et al., 2012; Xie, Shi &
Zhang, 2017; Bhatt et al., 2017, 2018). The rotation of the solar corona can
also be used as an indicator of the differential rotation of subphotosperic
layers. Vats et al. (2001), uses disk-integrated simultaneous measurements of
solar radio flux for the frequency range of 275-2800 MHz and showed a strong
dependence of the coronal rotation period on the heights in the corona, i.e.,
the sidereal coronal rotation period increases with an increase in height in
the solar atmosphere. While Bhatt et al. (2017) contradict the result by using
the same data set and obtained that solar corona rotates slower at higher
altitudes. By analyzing the coronal rotation period as a function of latitude
between $\pm 60\circ$ for the period 1999-2001, Chandra et al. (2009) obtained
that as compared to the photosphere and chromosphere, solar corona rotates
less differentially. Similar results were confirmed by Chandra et al. (2010)
who evaluated the differential rotation of soft X-ray corona obtained from the
solar full-disk images of a soft X-ray telescope. Chandra, & Vats (2011) uses
2.8 GHz emitted solar radio flux data for the time span 1947-2009 and reported
that sidereal coronal rotation period varies from 19.0 days to 29.5 days with
an average of 24.3 days, and a 22-year component (which might be related to
the 22-year Hale magnetic cycle or magnetic field reversal) exists in the
long-term rotation. According to Xu, Gao & Shi (2020), variations in the
lengths of the solar atmospheric rotational period indicate several harmonics
of the solar cycle period, which is modulated by the solar activity cycle.

Figure 1: Temperature-frequency profile of solar atmosphere (Green colour -
Transition region and Yellow colour - Corona); emission frequencies and
temperatures are in deg. K and in MHz, respectively (after Kane (2009)).
Figure 2: Left Panels: The radio emission at 606, 1415, 2695, 4995 and 8800
MHz frequency is plotted against days for the year 1969. Right Panels: The LSP
of respective radio emission is plotted against period (FAP = 0.05).

In the present work flux modulation method is used to determine the rotation
period present in the time series of solar radio flux. The data sets used for
the study were available for a span of 44 years (1967-2010) at eight
frequencies 245, 410, 606, 1415, 2695, 4995, 8800 and 15,400 MHz, which
originating from different height from the solar atmosphere. The radio flux in
the 245-15400 MHz range is most likely escaping from the upper corona to the
upper chromosphere. (Kane, 2009). The height at which the radio flux at a
particular frequency emitted is estimated from the temperature-height profiles
(shown in Figure 1) and found that flux of 245 MHz emitted at the height of
$\sim$ 12,000 km, 410 MHz at $\sim$ 10,000 km, 606 MHz at $\sim$4,000 km, 1415
MHz at $\sim$2,500 km, 2695 MHz at $\sim$2,800 km approx), 4995 MHz from outer
transition region at $\sim$2,200 km, 8800 MHz from mid-transition region and
15400 MHz from inner transition region at $\sim$2,100 km above photosphere in
the solar atmosphere (Kane, 2004, 2005, 2009).

The fractal dimension are sometimes used as an index to quantify the time
variability of radio emission. For the years 1976–1990, Watari (1996) measured
the fractal dimensions of solar radio fluxes data at 245, 410, 610, 1415,
2695, 2800, 4995, 8800, and 15400 MHz. The sunspot number and radio emission
around 3 GHz have identical annual variations in fractal dimensions, implying
a close relationship between them.

According to the height-frequency profile of the plasma and gyromagnetic
frequencies given by Gary & Hurford (2004); Raulin & Pacini (2005), for free-
free opacity equal to unity ($f_{\tau}=1$), the highest frequency at which
plasma radio emission may have been detected in the solar corona is about 8000
MHz. Thus, emissions between frequencies 245-4995 MHz originates in the lower
corona (less than $0.2R_{\odot}$) whereas emissions at frequencies 8800 and
15400 MHz originate in the transition region. According to the Gary & Hurford
(2004) (based on Figure 4.1 given in the paper), we learn that the plasma and
bremsstr$\ddot{a}$hlung emission, both can coexist for the fluxes at 245, 410
and 606 MHz. Whereas, only bremsstr$\ddot{a}$hlung free-free emission will
prevail for the fluxes at 1415 and 2695 MHz. The gyro-emission mechanism will
dominate over free-free emission for the frequencies at 8800 and 15400 MHz.

The flux from active regions is normally dominated by gyro-resonance emission,
while plage regions displaced from sunspots emit thermal
bremsstr$\ddot{a}$hlung emission. As the number of sunspots changes during the
solar cycle, the contribution of the rotationally modulated portion should
change as well. Schmahl & Kundu (1994) discussed that the fraction of total
flux at intermediate frequencies (usually in the 1000-9400 MHz range) is much
greater than the fraction at lower and higher frequencies in the most of the
solar active years. As a result, the fraction spectrum peaks similarly to the
gyro-resonance spectrum, indicating that in active years, the free-free
contribution is a smaller fraction of the modulated portion. The lower
frequency fraction, on the other hand, increases to match the intermediate
frequency fraction at solar minimum. As a result, at solar minimum, the
rotationally-modulated component’s spectrum is flatter, suggesting a greater
contribution from plage-associated sources.

The slowly-varying components, or S-components, are those fraction of radio
emissions in the centimeter (15400, 8800, 4995, and 2695 MHz) and decimeter
(1415, 606, and 410 MHz) wavelength range whose variation shows the rotational
modulation of long-lived active regions. It’s either thermal
bremsstr$\ddot{a}$hlung from hot and dense coronal condensations in active
regions, which are correlated with plages, or gyroresonance emission whether
it’s originate above sunspots or intense magnetic fields. Around 1000-2000
MHz, plage-associated emission persists, while sunspot-associated emission
prevails above 5000 MHz. Between 2000 and 5000 MHz, plage-associated emission
and spot-associated emission, both may coexists (Schmahl & Kundu, 1994, 1995;
Watari, 1996).

High altitudes in the corona are the region from where the quiet-Sun emission
originates at meter (245 MHz) and decimeter (1415, 606, and 410 MHz)
wavelengths. Whereas, contributions from the chromosphere dominate the quiet-
Sun emission at centimeter (15400, 8800, 4995, and 2695 MHz) wavelengths. But
at decimeter wavelengths, both the chromosphere and the corona lead to the
quiet-Sun emission (Watari, 1996).

Estimated heights of origin of the flux at frequencies 245, 410, 606, 1415,
2695, and 4995 MHz through the model suggested by Gary & Hurford (2004) are
therefore, $\sim$1,90,000, $\sim$1,10,000, $\sim$53,000, $\sim$12,000,
$\sim$5,100 and $\sim$2,900 km, respectively (all in lower solar corona) and
$\sim$2,200 km (in upper transition region) for 8800 MHz and $\sim$2,100 km
(in lower transition region) for 15,400 MHz radio flux.

Different heights in the solar corona can emit radio emission at specific
frequencies. Radio emissions with frequencies less than or equal to the plasma
frequency are found to be absorbed, whereas second harmonic (at twice the
plasma frequency) radiation propagates. As a result, an electron density model
can be used to measure the location of the solar radio emission (Vats et al.,
2001). Zucca et al. (2014) has obtained the electron density as a function of
height, from $R_{\odot}$ to $2.5R_{\odot}$, for the active region, quiet Sun,
and coronal hole using radial profiles of electron density. The plasma
frequency ($\nu_{p}$) and the electron density ($n_{e}$) are related through
the following expression Gary & Hurford (2004),

$\nu_{p}=8.98\times 10^{3}\sqrt{n_{e}}$ (1)

whereas, the gyro-frequency ($\nu_{B}$), is expressed as

$\nu_{B}=2.6\times 10^{6}B$ (2)

where $B$ is the magnetic field strength in Gauss.

Even with numerous proposed height-frequency models, the exact height ranges
from which radio frequencies actually escape are still unknown, and all height
estimates are roughly correct. However, it is generally agreed in most
proposed models that emissions at higher frequencies will escape from lower
depths in a relative sense (lower solar altitudes). As a result, the
qualitative comparison should be appropriate, particularly when the data used
are averages over long periods of time.

##  2 Data Analysis and Methodology

The disk-integrated solar radio emissions observed at different frequencies
(245-15400 MHz) for each day are recorded from May 1966 to December 1987 at
Sagamore Solar Hill observatory, Massachusetts, USA and from Jan 1988 to
December 2010 at other observatories (Palehua, Hawaii; San Vito, Italy;
Learmonth, Australia; and also at Sagamore Hill, USA) (Kane, 2009). The noon
flux monitoring at 606, 1415, 2695, 4995, and 8800 MHz were first launched by
the Sagamore Hill Solar Radio Observatory in 1966. In 1967, the observational
frequency was augmented to 15400 MHz, then to 242 MHz in early 1969, and
finally to 410 MHz in early 1971. The noon flux disk-integrated data recorded
daily are available in the public domain through National Centers for
Environmental Information (NCEI-NOAA). To investigate the altitudinal
variation in solar rotation, these time series of disk-integrated radio flux
emanates at a particular frequency are taken from the year 1967-2010, each of
which belongs to a particular height in the solar transition region or lower
corona as shown in Figure 1. An annual time series are then generated for each
of the frequencies (245, 410, 606, 1415, 2695, 4995, 8800 and 15,400 MHz) for
the span of the year 1967 to 2010.

###  2.1 Data Analysis

After the initial review, it is found that the radio frequency data of 15,400
MHz originates from transition region have data gap more than permissible for
statistical analysis. Hence, we discarded the 15,400 MHz data and therefore,
we were not able to estimate any results from this region. Furthermore,
perhaps due to unsymmetrical structure in the higher corona region, from where
the radio frequency at 245 and 410 MHz originates, the variation in the
emission lacks any systematic rotational modulation which ultimately prevent
us to estimate the rotation period of this region. Bhatt et al. (2018) also
reported the similar observation for fluxes at 245 and 15,400 MHz. The
turbulence effect is inversely proportional to the increasing frequency (Vats
et al., 2001). It may be possible that due to the turbulence in the
intervening media from the Sun to the radio telescope the emissions at 245 and
410 MHz gets more randomized but other flux at higher frequencies propagates
unattenuated due to turbulence.

By determining the fractal dimension of a time series of radio flux, the
irregular aspects of solar radio flux variations can be quantified. The
specified time series has a fractal dimension of 1, indicating that it is
completely regular. As the time series’ variance becomes more irregular, the
fractal dimension increases. After calculating fractal dimensions for each
year and each frequency Watari (1996) found that, for frequency ranges less
than 1000 MHz and frequencies greater than 10,000 MHz, the fractal dimension
has a high value of 1.7. Around 3 GHz, the fractal dimension reaches a
minimum. The fractal dimension is greater than 1.5 at 245, 8800, and 15,400
MHz. However, for frequencies between 610 and 4995 MHz, the fractal dimension
is less than 1.3 and reaches a minimum around 3 GHz. This means that there is
a regular systematic rotational modulation in the solar atmosphere for the
time series of radio flux at frequencies between 610 and 4995 MHz (Watari,
1996).

Bhatt et al. (2018) also investigated fractal dimension of 410, 1415, 2695,
4995, and 8800 MHz solar radio flux measurements of two solar activity cycles
recorded on a daily basis and shown that with increasing frequency, fractal
dimension increases, implying that randomness increases towards the inner
corona. Their research also reveals that solar activity has a greater impact
on low frequencies, while solar activity has a lesser impact on higher
frequencies because they found that at lower frequency fractal dimension
difference between solar maximum and solar minimum is much more compared to
the difference at higher frequency.

Watari (1996) reported that in the distribution of radio-source lifetimes
simulated at 1.5 GHz, larger fractal dimensions were correlated with shorter
e-folding time constants. According to Gary & Hurford (2004), The cyclotron
frequency and its harmonics are represented by the gyro-emission. The gyro-
emission at the cyclotron frequency ($\nu=\nu_{B}$) is normally not relevant
for the solar corona within active regions, rather, the emission typically
occurs at the third harmonic, i.e., at ($\nu=3\nu_{B}$). The $3\nu_{B}$ line
stretches from 1–2 GHz just above $f_{\tau}=1$ level to 20 GHz below the
$f_{\tau}=1$ level. This is why, the gyro-emission dominates predominantly
over bremsstr$\ddot{a}$hlung free-free emission for radio emission $\sim$20
GHz, like 15.4 GHz emission.

According to the characteristic frequencies to height in the solar atmosphere
model, suggested by Gary & Hurford (2004), the plasma emission dominates at
frequencies ranging from 30 kHz to several hundred MHz. Plasma emission
dominates over thermal free-free emission in this frequency spectrum primarily
because the plasma level is above the $f_{\tau}=1$ level at these frequencies.
While plasma levels in active regions are usually below $f_{\tau}=1$ at
decimeter frequencies range (200 MHz–1 GHz), even though the plasma emission
may still be significant over free-free emission due to coronal inhomogeneity
in this height range. This may lead to the unsystematic variability found in
the data of radio flux at 245 and 410 MHz.

The rotational modulation will take place on every emission (irrespective of
the emission process) from the Sun. However, for estimating the rotation
period from these observations require the following conditions to be
fulfilled; (a) emission should originate from certain region/s of the Sun and
not uniformly from the entire Sun, and (b) emission should not be too much
modified by the irregular scattering in the intervening medium (from source to
receiver).

The outcome of Watari (1996); Vats et al. (2001); Gary & Hurford (2004); Bhatt
et al. (2018) justifies the use of annual time series of radio frequency 606,
1415, 2695, 4995 and 8800 MHz. For the reasons stated therein and above, the
metric emission at 245 and 410 MHz and centimeter emission at 15,400 MHz radio
data found to have more noise and hence excluded for further study. The
rotation period of inner corona and transition region, from where the radio
frequency 606, 1415, 2695, 4995 and 8800 MHz originates, could have been
estimated successfully with few data gaps years. A typical examples of such
annual time series of disk-integrated flux of year 1967 are plotted against
the days of year in Figure 2 (left panel) for each of the frequencies under
investigation. The intensity variation shows the dominating periodically
varying component throughout the period of study (annual data set in our
case). But, it is also found that in some cases the varying component is quite
weak and random during some part under study. This may be because of the
presence of short-lived solar features like radio noise, radio burst, etc..
Such time regions in the time series have to be discarded from the time series
under study.

Figure 3: Scattered plot shows sidereal rotation period (SRP) estimated
between year 1967-2010 at five radio frequencies and smooth curves are three
years smoothed SRP. The vertical dotted (pink) and dashed (green) lines
represents the positions of the years of solar minima and maxima,
respectively. Figure 4: Space-time diagram of sidereal rotation period (3-yrs
smoothed) obtained from solar emissions at five radio frequency 606-8800 MHz
observed during 1967-2010. The colour codes (shown on top) represent the
sidereal rotation period, which ranges from 24.0 to 26.5 days. The horizontal
dotted and dashed lines represents the positions of solar minima and maxima
years, respectively.

###  2.2 Methodology

Vats et al. (1998) found that the time series of radio frequency at 2.8 GHz
shows particularly significant modulation owing to coronal rotation,
indicating that radio emissions are less impacted by irregularities and
unsystematic features inside the solar corona. Hence, time series of such
radio flux/radiation can be used to estimate rotation period of solar corona.
In the present work, flux modulation is used to estimate the rotation period
by radiation/flux emitted from the Sun at different frequencies. This method
has been employed with the disk-integrated simultaneous daily measurements of
solar flux at different frequencies (such as radio flux 2.8 GHz) to measure
differential rotation as a function of altitude in the solar atmosphere (Vats
et al., 1998, 2001; Chandra, & Vats, 2011; Li et al., 2012; Xie, Shi & Zhang,
2017; Bhatt et al., 2017).

The radio flux can itself illustrate a rotational modulation (as shown in left
panels of Figure 2) in its variation. But, determining rotation period
directly from modulated variation in radio flux will not be possible directly
and hence some time series statistical analysis technique is required to find
statistically plausible period of rotation. The rotation periods were
determined mostly using auto-correlation analysis or continuous wavelet
transformation analysis in previous works (Vats et al., 2001; Chandra, & Vats,
2011; Li et al., 2012; Xie, Shi & Zhang, 2017). But in the present work we
employed the Lomb-Scargle Periodogram (LSP) to get probable rotational
periodicity present in the multi-frequency time series of radio emissions.

This type of periodogram is used for a frequency/period analysis of such data
which are not collected at a regular time interval or have any data missing in
time-series (Lomb, 1976; Scargle, 1982; Horne & Baliunas, 1986; VanderPlas,
2018). The Lomb-Scargle periodogram is widely used in the field of
astrophysics to lessen aliasing from unequally spaced data. Unlike Fast
Fourier Transformation (FFT), LSP also gives accurate calculation of a
spectrum estimator from unevenly sampled time-series and is analogous to
least-square fitting of sinusoidal curves, i.e., it searches the sinusoidal
periodicity and find out the best fit of a sinusoidal function, as compared to
the time-series, for a selected period or frequency. The location of dominant
peaks of the exponential probability distribution function of power P(T) on
the period of oscillation, T gives rotation periods of all periodic components
present in that time series. After observing power spectrum of periodogram,
FAP (false alarm probability) measurement is essential to verify the
probability that the dominating peak is due to any true periodic signal and is
not the outcome of any noise present randomly in the time series. The FAP
Assessment of a FAP level of power numerically requires at least the
computation of 10/power (Chowdhury et al., 2013; Delisle, Hara & Segransan,
2020; Mancuso et al., 2020).

##  3 Results and Discussion

Our investigation depends on picking out any statistically significant
periodic signals present in the multi-frequency time series of radio flux
observed during year 1968-2009 at Sagamore Solar Hill observatory,
Massachusetts, USA. The Lomb-Scargle periodogram (LSP), along with the auto-
correlation function has widely been used for detecting periodicity in radio
flux time series (Vats et al., 1998, 2001; Chandra, & Vats, 2011; Li et al.,
2012; Xie, Shi & Zhang, 2017; Bhatt et al., 2017).

###  3.1 Spatial variation of solar rotation with height

The right panels of Figure 2 shows the periodogram power plotted as a function
of the period of rotation for five observed radio frequencies at 606, 1415,
2695, 4995 and 8800 MHz for the year 1969. The plots clearly shows the
statistically significant prominent peaks power are well above the $95\%$
confidence level (FAP) and which shows that the peaks are due to presence of
true periodic signal and are not because of any inherent noise in the radio
data. The synodic rotation period can be determined by single out the position
of statistically significant prominent peak on the axis of period of rotation.
The initial investigation shows that the locus at which the peaks are situated
in the plots of Figure 2 (left panel) are almost same for all the frequencies
under study, which means that the period of rotation are invariant with
respect to the frequency. Such invariance have been found in most of the years
during 1967-2010, with few exceptions. The emissions at these frequencies
emanate from different heights in inner solar corona and transition region.
So, our investigation shows that the rotation of inner solar corona, as well
as transition region, is almost rigid with respect to altitude above solar
surface (Chandra et al., 2009; Morgan, 2011).

The result differs completely with the study reported by Vats et al. (2001)
and Bhatt et al. (2017), who used approximately two years radio flux data
observed at Cracow Astronomical Observatory, Poland. According to Vats et al.
(2001) the coronal rotation period increases with the frequency of emissions
(275-1755 MHz) emanate from the solar corona, whereas Bhatt et al. (2017)
reported with the same data that period of rotation decreases with increase in
frequency of the coronal radio flux. The differential rotation of solar inner
corona with respect to the height in solar atmosphere reported by Vats et al.
(2001) and Bhatt et al. (2017) is completely ruled out in our study.

The observed almost rigid rotation with respect to altitude in transition
region and inner corona may be understand considering the similar quasi-rigid
rotation behaviour reported with low latitude coronal holes. Coronal holes are
cool, low-density regions on the Sun that can be found at low latitudes as
well as near the polar caps (Raju et al., 2005). Many coronal holes have a
surprisingly rigid rotation, indicating that field-line reconnection happens
regularly in the corona. Magnetic interactions between active regions and
coronal holes have been investigated as a potential explanation for coronal
holes’ rigid rotation. Magnetic reconnection between open and closed magnetic
fields (known as interchange reconnection) must occur at the boundary of
coronal holes, contributing to the rigid rotation of coronal holes. Magnetic
reconnection at the coronal holes boundaries (CHBs) appears to be the cause of
rigid rotation, as reported in many observations (Kahler & Hudson, 2002;
Madjarska et al., 2004; Raju et al., 2005; Hiremath & Hegde, 2013).

So we can conclude that, our and others observations suggest that the large-
scale features of corona rotates more rigidly than the underlying photosphere
in general, and that the rotation becomes more rigid as the height is
increased up to about $2.5R_{\odot}$, where the plasma is dominated by the
magnetic field. The rapid decrease of higher order multipoles of the magnetic
field above the photosphere can be attributed to this nature (Antonucci et
al., 2020).

Figure 5: Cross-correlograms are plotted between two time-series, one is
sunspot numbers (SSN) and other is 3-yr smoothed sidereal rotation period
(SRP), at radio frequency 606, 1415, & 2695 MHz (emission from inner corona)
and 4995 & 8800 MHz (emission from transition region).

###  3.2 Temporal variation of solar rotation with years

The synodic rotation period, $T$ obtain through LSP analysis (as shown in left
panels of Figure 2) is actually apparant rotation period. The Sun rotates
little extra due to the simultaneous rotation of Earth along with Sun during
each of its rotation. So, we have to find the actual period known as sidereal
rotation period, $T_{o}$, which can simply be calculated using the well known
formula (Chandra et al., 2009; Deng et al., 2020),

$T_{o}=\frac{\left(T\times 365.26\right)}{\left(T+365.26\right)}$ (3)

where, $T$ is the synodic rotation period.

The sidereal rotation period calculated, by the formula given above, for the
radio emission at five discreet frequencies obtain from each annual time
series between the years 1967-2010 are plotted in five panels and shown in
Figure 3 as scatter plots along with their statistical errors. The years of
sunspot cycle maxima (by dashed vertical lines) and years of sunspot cycle
minima (by dotted vertical lines) are also shown in each panels of Figure 3
(Chandra, & Vats, 2011; Xie, Shi & Zhang, 2017).

The scatter plots in five panels of Figure 3, shows the temporal variation in
sidereal rotation period of five particular layers of solar lower corona and
transition region. The variation in rotation period seems quite random in
nature, at first sight, unless they are smoothed by taking running mean of 3
years. The 3-yrs. smoothed rotation period data for each frequencies are also
plotted along with the scatter plots of rotation in Figure 3. These smoothed
curves of rotation period show the presence of longer duration periodic
components into it, which seems to wax and wane along with the sunspot cycle.
The curves shows maxima around two alternate solar maximum years, i.e., around
year 1969 and 1990. The findings are quite similar to the study investigated
with 2800 MHz radio data by Chandra, & Vats (2011); Xie, Shi & Zhang (2017).
It is therefore pertinent to analyze further the interrelationship between the
solar activity cycles and rotation period of inner corona and transition
region.

###  3.3 Space-Time variation in sidereal rotation period

The smoothed sidereal rotation period (by taking 3-yrs running mean) shown as
smooth curves in plots of Figure 3 are again plotted in Figure 4 as space-time
diagram for the multi-frequencies (606-8800 MHz) flux emitted during the years
1968-2009. The smoothed period varies between 24.0 - 26.5 days, which have
been depicted through the six different colour codes, from violet to red, and
separated by 0.5 days from each other. The contour plot is overlaid by
horizontal lines, dotted and dashed, which represents the years of solar
activity minima and maxima falls during the years of observation, i.e.,
1968-2009.

The diagram along with dotted lines of solar minima clearly show that the
rotation period changes its wax and wane at the dawn of any solar cycle. For
example, in 1976 and 1997 (beginning of cycle-21 and -23, respectively) the
rotation periods touches its least value across the frequencies, which
originates from different layers of solar atmosphere, whereas in 1986, when
cycle-22 kicks off, the rotation periods of lower corona and transition region
are almost at its peak. It is evident from the Figure 4, that rotation periods
rises from one solar minima to other consecutive minima, and then falls till
next solar minima. This systematic rise and fall in sidereal rotation periods
have been followed by all the five frequencies, although the contrast in
highest and lowest value of period is more for those frequencies, which
emerges from transition region, in comparison to that originates from lower
corona. The wax and wane of sidereal rotation period prominently shown in
Figure 4 suggests the linkage of rotation period with the phases of 11-yrs and
22-yrs solar cycles.

Figure 6: Space-time diagram of sidereal rotation period (11-yrs smoothed)
obtained from solar emissions at five radio frequency 606-8800 MHz observed
during 1972-2001. The colour codes (shown on top) represent the sidereal
rotation period, which ranges from 24.4 to 25.8 days. The horizontal dotted
and dashed lines represents the positions of solar minima and maxima years,
respectively.

###  3.4 Correlation with solar activity cycles

To investigate it further we performed cross-correlation (CC) analysis between
sidereal rotation period and annual sunspot numbers after taking the 3-years
running mean of both the time series. Although, cross-correlation are used to
determine the degree of fit between two time series of same data size. But,
the CC analysis can also be used to see the mutual correlation between solar
rotation and other observed solar indices (Chandra, & Vats, 2011; Xu, Gao &
Shi, 2020).

The cross-correlograms between smoothed sidereal rotation period and sunspot
numbers for five different frequencies 606, 1415, 2695, 4995 and 8800 MHz. are
plotted in Figure 5. The curves plotted are smooth, cyclic and prominent in
most of the cases. They rises and falls periodically with almost same time
period for the emissions originate from inner corona, but shows ambiguity in
some parts in the curves belongs to transition region. Even-though, the
consecutive maximas or minimas of cross-correlation coefficient are almost 11
- 13 years apart from each other for all the five frequencies. The smoothness
of curve, in some cases, is affected by the presence of gap of data of some
year’s. So we can conclude, that when shorter duration components (less than 3
year) are suppressed than the periodic components of $\sim$11 yrs, which are
also embedded in the analyzed rotation period data and their presence is
perhaps due to the 11 yrs sunspot activity cycle, which can be seen in the
temporal variation of rotation period of any solar layer (Chandra, & Vats,
2011; Li et al., 2012; Mei, Deng, & Wang, 2018; Xu, Gao & Shi, 2020).

Taking the lead from Figure 3 and to find the presence of periodic components
of longer duration, the evaluated rotation period profiles for all five radio
emissions are once again smoothed by taking 11 years running mean. Such
smoothing has eliminated all those periodic components which have periodicity
shorter than $\sim$11 yrs. After smoothing, the rotation periods are plotted
again in space-time diagram (as shown in Figure 6) for the observation years
from 1972 to 2001. Due to smoothing up to 11-yrs, the rotation period varies
less in comparison to raw data or 3-yrs smoothed data (as shown in Figure 3
and 4). It ranges between 24.4 - 25.8 days, which are colour coded in Figure 6
as violet to red colour. The space-time diagram in Figure 6 confirms the
presence of 22-yrs cycle in sidereal rotation period of lower corona and
transition region. The horizontal dotted lines drawn over the contour diagram
reveals that any solar cycle begins either when the rotation periods of solar
atmospheric layer are at its minimum (as evident in solar cycle 21 & 23) or at
maximum (as in solar cycle 22) of its value. The colour distribution for solar
cycle 21 and 22 appears like a mirror image of each other. This variation is
quite similar to the behaviour of solar magnetic field reversal phenomenon
which happens after every $\sim$22 years. In the same way, our investigation
establishes that the rotation period of the layers under study repeats its
successive minimum or maximum period after every $\sim$22 years (Chandra, &
Vats, 2011; Xie, Shi & Zhang, 2017; Xu, Gao & Shi, 2020).

Figure 7: Cross-correlograms are plotted between two time-series, one is
sunspot numbers (SSN) and other is 11-yr smoothed sidereal rotation period
(SRP), at radio frequency 606, 1415, & 2695 MHz (emission from inner corona)
and 4995 & 8800 MHz (emission from transition region).

The 11-yrs smoothed rotation period time series are cross-correlated with the
similarly smoothed annual sunspot numbers (SSN). The cross-correlograms, so
obtained, are plotted for each frequency in Figure 7. The CC coefficients are
prominent at peaks and valley and curves shows almost cyclic behaviour in its
variation. Although the curves in some cases are not so smooth (like in case
of 1415 MHz emission), even though the difference between two consecutive
peaks and valley of cross-correlation coefficient have years difference of
almost 22-26 yrs. This peak to peak or valley to valley difference may be
because of the presence of longer duration periodic components in rotation
period which may have the some influence on 22 years magnetic field reversal
cycle (Chandra, & Vats, 2011; Deng et al., 2020; Xu, Gao & Shi, 2020).

As we have seen that the variations of the rotation period of the layers of
inner corona and transition region obtain through the different radio
wavelengths are in tandem with the solar cycle. There are similar views about
the rotation profiles of coronal holes. Navarro-Peralta & Sanchez-Ibarra
(1994) and references therein, reported that the rotation profile of coronal
holes is also dependent on the phase of the solar cycle. As we know that the
Sun is characterized by coronal holes near two polar caps with opposite
magnetic polarities at the minimum of the solar activity cycle. There are two
types of coronal holes found in the mid-latitudes. Coronal holes that are
disconnected from polar coronal holes are often found near active regions and
have an incidence rate that closely corresponds to the solar activity cycle.
The equatorial extension of a polar coronal holes is the second type of
coronal holes. During times other than solar maximum, however, their rate of
incidence is reasonably consistent. These rotate nearly rigidly, as contrasted
to disconnected coronal holes, which rotate differentially with latitude,
although slightly more rigidly than the photospheric differential rotation
rate (Navarro-Peralta & Sanchez-Ibarra, 1994; Insley, Moore & Harrison, 1995;
Wang, Hawley & Sheeley, 1996; Kahler & Hudson, 2002; Prabhu et al., 2018).

###  3.5 Interchangeability of rotation speed of transition region and corona
with solar cycles

The contours of 3 yrs and 11 yrs running mean (Figure 4 and 6, respectively)
show that there is a significant 22 year periodicity in the temporal variation
of sidereal rotation periods. In order to see this feature even more clearly,
we plot the average 11 yrs running mean of (a) all the five frequencies; (b)
two frequencies of transition region (4995 and 8800 MHz), and (c) three
coronal emission frequencies (606, 1415 and 2695 MHz). These three averages
are shown in Figure 8. All three curves show a clear 22 yrs periodicity.
Additionally, this variation also show an interesting outcome as follows:

(1) During 1972 (or even before) to 1977 corona rotates faster than transition
region.

(2) During 1978 to 1988 transition region rotates faster than corona.

(3) During 1989 to 2000 corona rotates faster than transition region.

(4) Both rotate at the same speed in the years 1979, 1989 & 2000 (almost at
solar maximas).

The curves show a tendency of similar change over before 1972 and also after
2000. This is a new characteristic of rotational profile in the solar
atmosphere; that corona rotates faster or slower than transition region in
different epoch. This also has $\sim$22 years periodicity.

##  4 Conclusion

We investigated multi-frequency yearly time series of radio flux observed at
five frequency 606, 1415, 2695, 4995 and 8800 MHz during year 1967-2010 at
Sagamore Solar Hill Observatory, USA (Figure 1). The continuous temporal
variation in synodic rotation period obtains through Lomb-Scargle Periodogram
(LSP) analysis for each frequency during study period confirms the presence of
prominent random component in the solar rotation (Figure 2). When scattered
sidereal rotation period are smoothed to suppress the random component by
taking 3-yr running mean, it hints the presence of longer duration periodic
cycle, perhaps $\sim$11-yr Schwabe cycle (Mei, Deng, & Wang, 2018) and
$\sim$22 yrs Hale cycle (Xu, Gao & Shi, 2020) (Figure 3).

The space-time diagram in Figure 4 and 6, are the smoothed sidereal rotation
period (by taking 3-yrs and 11-yrs running mean) plotted against radio
emissions at five different frequencies for the period under study. The
diagram reveals that any solar cycle kicks off at the inflexion point of
rotation period of solar layer. In these diagrams, we can see that the solar
cycle 21 & 23 starts when a particular solar layer rotates fastest and the
cycle 22 begins when it rotates at its slowest pace. Such behaviour seems to
be in phase with the 22-year magnetic field reversal cycle (Chandra, & Vats,
2011; Li et al., 2012; Xie, Shi & Zhang, 2017).

To see any relation with the phases of 11 yrs solar cycle, the cross-
correlation (CC) analysis between 3-yr running mean of sidereal rotation
period and sunspot number (SSN) has been performed. The CC analysis confirms
the presence of $\sim$11yr periodic component in the obtained rotation period.
The position of peaks in cross-correlograms (Figure 5) show that the rotation
period of inner corona and sunspot number are in-phase but in transition
region, it is leading slightly ($\sim$1 year).

The presence of all the component having periodicity less than 11 yrs has been
eliminated by taking 11-yrs running mean. When time series of 11-yrs running
mean of sidereal rotation period are cross-correlated with time series of
annual sunspot numbers, the cross-correlograms show peaks at interval of
almost 22 years. The peaks for rotation of inner corona is lagging for $\sim$3
yrs, whereas for upper transition layer, it is almost in phase with the
sunspot cycle (Chandra, & Vats, 2011; Deng et al., 2020).

When, average of 11 yrs running mean rotation period, which emanates from
corona is compared with those emanates from transition region (as shown in
Figure 8), the variation in rotation period of corona and transition region
show the prominent presence of 22-yrs Hale’s cycle. Besides that, the curves
show that the rotation period of corona is more than transition region near
the dawn of solar cycle 21 & 23, whereas it is less almost at the start of
cycle 22. It means corona rotates faster in comparison to transition region in
alternate solar cycle, and $\it{vice-versa}$. It is evident from Figure 4,
that their speeds interchange its track at the time of solar maxima. Thus, we
can also conclude that swapping of rotation speed of corona and transition
region essentially follows the 22-yrs periodicity.

Thus, the period of rotation for five radio frequencies at 606, 1415, 2695,
4995 and 8800 MHz originates from lower corona and upper transition layer
during the year 1967-2010 shows the presence of 11-yrs and 22-yrs component
besides the dominating random components.

Figure 8: 11 yrs running mean of average sidereal rotation period (SRP)
estimated for all the five radio emissions (as black line) plotted against
years is compared with average SRP of three emissions originated from corona
(red line) and two emisions from transition region (blue line). The vertical
dotted and dashed lines represents the positions of solar minima and maxima
years, respectively.

## Acknowledgments

The authors wish to acknowledge the numerous observers and data archivers of
the radio flux and sunspot data used in this present work. The sunspot number
data are credited to the source; WDC-SILSO data, Royal Observatory of Belgium,
Brussels and the radio data to the source; National Centers for Environmental
Information (NCEI-NOAA), USA. This research was supported by the CSJM
University, Kanpur, India under its financial research grant scheme. We
appreciate the anonymous reviewers’ thorough reading of our manuscript as well
as their numerous insightful comments and suggestions.

## Data Availability

The data-sets used in this present work were derived from the sources freely
available as public data. The data underlying this article are available at
National Centers for Environmental Information (NCEI-NOAA) at
https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-
radio/noontime-flux/sagamore-hill/ and World Data Center - SILSO, Royal
Observatory of Belgium, Brussels at http://www.sidc.be/silso/datafiles.

## References

  * Antonucci et al. (2020) Antonucci, E., Harra, L., Susino, R., & Telloni, D., 2020, Space Sci. Rev., 216, 117
  * Aschwanden & Benz (1995) Aschwanden, M. J., & Benz, A. O., 1995, ApJ, 438, 997
  * Benz (1993) Benz, A. O., Plasma Astrophysics: Kinetic Proceses in Solar and Stellar Corona, Kluwer Academic Publishers, Dordrecht, 1993, Vol. 184, 127
  * Bhatt et al. (2017) Bhatt, H., Trivedi, R., Sharma, S. K., & Vats, H. O., 2017, Sol. Phys., 292(4), 55
  * Bhatt et al. (2018) Bhatt, H., Sharma, S. K., Trivedi, R., & Vats, H. O., 2018, MNRAS, 475 (3), 3117
  * Chandra et al. (2009) Chandra, S., Vats, H. O., & Iyer, K. N., 2009, MNRAS, 400(1), L34
  * Chandra et al. (2010) Chandra, S., Vats, H. O., & Iyer, K. N., 2010, MNRAS, 407, 1108
  * Chandra, & Vats (2011) Chandra, S., & Vats, H. O., 2011, MNRAS, 414(4), 3158
  * Chowdhury et al. (2013) Chowdhury, P., Jain, R. & Awasthi, A. K., 2013, ApJ, 778, 28
  * Delisle, Hara & Segransan (2020) Delisle, J.-B., Hara, N., & Segransan, D., 2020, A&A, 635, A83
  * Deng et al. (2020) Deng, L. H., Zhang, J. X., Deng, H., Mei, Y., & Wang, F., 2020, MNRAS, 491(1), 848
  * Donnelly (1987) Donnelly, R. F., 1987, Sol. Phys., 109, 37
  * Donnelly & Puga (1990) Donnelly, R. F., & Puga, L. C., 1990, Sol. Phys., 130, 369
  * D’Silva & Howard (1994) D’Silva, S., & Howard, R. F., 1994, Sol. Phys., 151, 213
  * Fontenla et al. (1999) Fontenla, J., White, O. R., Fox, P. A., Avrett, E. H., & Kurucz, R. L., 1999, ApJ, 518, 480
  * Gary & Hurford (2004) Gary, D. E. & Hurford, G. J., Radio Spectral Diagnostics, Solar & Space Weather Radiophysics, Kluwer ASSL, Dordrecht, 2004, Vol. 303, 71
  * Hiremath & Hegde (2013) Hiremath, K. M., & Hegde, M., 2013, ApJ, 763(2), 137
  * Horne & Baliunas (1986) Horne, J. H., & Baliunas, S. L., 1986, ApJS, 302, 757
  * Howard (1984) Howard, R., 1984, ARA&A, 22, 131
  * Howard & Harvey (1970) Howard, R. F., & Harvey, J., 1970, Sol. Phys., 12, 23
  * Insley, Moore & Harrison (1995) Insley, J. E., Moore, V., & Harrison, R. A., 1995, Sol. Phys., 160, 1
  * Kahler & Hudson (2002) Kahler, S. W., & Hudson, H. S., 2002, ApJ, 574, 467
  * Kane & Trivedi (1985) Kane, R. P., & Trivedi, N. R., 1985, J. Geomagn. Geoelect., 1071, 37
  * Kane (1999) Kane, R. P., 1999, Sol. Phys., 189, 217
  * Kane (2004) Kane, R. P., 2004, Sol. Phys., 219, 357
  * Kane (2005) Kane, R. P., 2005, Sol. Phys., 277, 155
  * Kane (2009) Kane, R. P., 2009, Annales Geophyicae, 27(4), 1469
  * Li et al. (2012) Li, K. J., Shi, X. J., Feng, W. et al., 2012, MNRAS, 423, 3584
  * Li et al. (2019) Li, K. J., Xu, J. C., Yin, Z. Q., & Feng, W., 2019, ApJ, 875(2), 90
  * Lomb (1976) Lomb, N. R., 1976, Ap&SS, 39, 447
  * Madjarska et al. (2004) Madjarska, M. S., Doyle, J. G., & van Driel-Gesztelyi, L., 2004, ApJ, 603, L57
  * Mancuso et al. (2020) Mancuso, S., Giordano, S., Barghini, D., & Telloni, D., 2020, A&A, 644, A18
  * Mei, Deng, & Wang (2018) Mei, Y., Deng, H. & Wang, F., 2018, Ap&SS, 363, 84
  * Melendez et al. (1999) Melendez, J. L. M., Sawant, H. S., Fernandes, F. C. R., & Benz, A. O., 1999, Sol. Phys., 187, 77
  * Morgan (2011) Morgan, H., 2011, ApJ, 738, 189
  * Navarro-Peralta & Sanchez-Ibarra (1994) Navarro-Peralta, P., Sanchez-Ibarra, A., 1994, Sol. Phys., 153, 169
  * Pap, Tobiska & Bouwer (1990) Pap, J., Tobiska, W. K., & Bouwer, S. D., 1990, Sol. Phys., 129, 165
  * Prabhu et al. (2018) Prabhu, K., Ravindra, B., Hegde, M., & Doddamani, V. H., 2018, Ap&SS, 363, 108
  * Raju et al. (2005) Raju, K. P., Bromage, B. J. I., Chapman, S. A., & Del Zanna, G. 2005, A&A, 432, 341
  * Raulin & Pacini (2005) Raulin, J. P., & Pacini, A. A., 2005, Advances in Space Research, 35, 739
  * Scargle (1982) Scargle, J. D., 1982, ApJ, 263, 835
  * Schmahl & Kundu (1994) Schmahl, E. J., & Kundu, M. R., 1994, Sol. Phys., 152(1), 167
  * Schmahl & Kundu (1995) Schmahl, E. J., & Kundu, M. R., 1995, J. Geophys. Res., 100(A10), 19851
  * VanderPlas (2018) VanderPlas, J. T., 2018, ApJS, 236(1),28
  * Vats et al. (1998) Vats, H. O., Deshpande, M. R., Shah, C. R., & Mehta, M., 1998, Sol. Phys., 181, 351
  * Vats et al. (2001) Vats, H. O., Cecatto, J. R., Mehta, M., Sawant, H. S., & Neri, J. A. C. F., 2001, ApJ, 548, L87
  * Xie, Shi & Zhang (2017) Xie, J. L., Shi, X. J., & Zhang, J., 2017, ApJ, 841, 42
  * Wang, Hawley & Sheeley (1996) Wang, Y.-M., Hawley, S. H., & Sheeley, N. R., Jr., 1996, Science, 271, 464
  * Watari (1996) Watari, S., 1996, Sol. Phys., 163(2), 371
  * Xu, Gao & Shi (2020) Xu, J. C., Gao, P. X., & Shi, X. J., 2020, ApJ, 902, 64
  * Zucca et al. (2014) Zucca, P., Carley, E. P., Bloomfield, D. S., & Gallagher, P. T., 2014, A&A, 564, A47