Rotational Characteristics of the Solar Transition Region Using SDO/AIA 304 Å Images

To date, the rotational characteristics of the solar transition region remain unclear. In this work, by applying the flux modulation method to the images derived from the Solar Dynamics Observatory/Atmospheric Imaging Assembly between 2011 and 2022 at 304 Å wavelength, we have studied the rotation of the solar transition region, and the results obtained are as follows. The solar transition region rotates differentially, while, from the perspective of the entire time interval, the rotation coefficients A and B are 14.39 (±0.08) and −1.61 (±0.15), respectively, and we find no prominent asymmetry in the average rotation rate of the northern and southern hemispheres. The solar transition region rotates fastest during the solar cycle maximum, and the average rotation rate follows the overall trend of solar activity. Both the equatorial rotation rate (represented by coefficient A) and the latitudinal gradient (represented by coefficient B) of the solar transition region are smaller than that of the solar chromosphere and the corona, indicating the solar transition region rotates more slowly and more rigidly than the other two layers, and we speculate that the solar chromosphere and corona seem to restrain the rotation of the solar transition region at the same time.


Introduction
Solar differential rotation is an essential basis for the theory of solar dynamo because it plays a key role in the transformation of the solar magnetic field (Babcock 1961;Gilman 1974;Ruždjak et al. 2017). It is universally acknowledged that spectroscopy and tracer methods are the two traditional tools to study the differential rotation of the Sun. The spectroscopy studies the solar rotation by calculating the Doppler shift of spectral lines (Howard & Harvey 1970;Snodgrass 1984;Snodgrass & Ulrich 1990), while the tracer method obtains the rotation rate by calculating the displacement of the tracer over a period of time. Recently, the flux modulation method has been widely used as the third method to study solar differential rotation. The autocorrelation analysis is used to calculate the flux time series data to obtain the rotation periods (Chandra et al. 2009(Chandra et al. , 2010Sharma et al. 2020aSharma et al. , 2020bSharma et al. , 2021. The solar rotation is very complex, with different layers of the solar atmosphere rotating at different rates. By comparing the equatorial rates obtained from observations of the C I 5380 line, Fe I 5233 line, and Hα line, Livingston (1969) found no appreciable difference through the photosphere, but the chromosphere is found to be moving 8% faster. Antonucci et al. (1979) proposed that the short-lived chromospheric features rotate at the same rate as the chromospheric plasma and faster than photospheric plasma. Li et al. (2020) found the high chromosphere rotates faster than the photosphere by investigating synoptic maps of He I intensity. Wan & Li (2022) reported the chromosphere filaments rotate faster than sunspots, the photosphere, and the medium-low chromosphere at middlelow latitudes. Chandra et al. (2009) reported the solar corona rotates less differentially than the photosphere and chromosphere by investigating solar full disk (SFD) images of Nobeyama Radioheliograph (NoRH) at 17 GHz. By tracing small bright coronal structures (SBCS) in images from the EUV imaging telescope (EIT) on board the Solar and Heliospheric Observatory (SOHO), Wöhl et al. (2010) obtained rotation coefficients A, B, and C of 14.499( ± 0.006), −2.54 ( ± 0.06), and −0.77( ± 0.09), respectively. They also reported a more differential rotation profile of the SBCS than the sunspots and sunspot groups. Li et al. (2019) revealed that the rotation rate in the corona is larger than that in the photosphere by analyzing solar spectral irradiances from the SORCE satellite at the spectral intervals 1-39 and 116-2416 nm.
To study coronal rotation more deeply, some scholars have attempted to reveal its relationship to latitude. For example, Hassler & Tomczyk (1996) analyzed the time series of whitelight polarized brightness data, and their results suggest more differential rotation and slightly faster mean rotation rates present at lower heights in the corona. Vats et al. (2001) investigated the solar flux at 11 radio frequencies in the range of 275-2800 MHz, and found the solar corona at a higher height (the outer corona) rotates faster than that at a lower height (the inner corona). Altrock (2003) analyzed synoptic photoelectric observations of the coronal Fe XIV and Fe X emission lines at 530.3 and 637.4 nm, and proposed that structures with lower temperatures rotate at a slower rate. Similar results were reported by Sharma et al. (2020a). They investigated the six different wavelengths of the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamics Observatory (SDO) space mission and presented that the sidereal rotation periods of different coronal layers decrease with increasing temperature (or height).
The cycle-related variations of the solar rotation have also aroused the interest of many authors. By tracing sunspots or sunspot groups, Hathaway & Wilson (1990) proposed that during the Maunder minimum, the Sun may have rotated slightly faster (by ∼0.5%) for the period 1921-1982. Gilman & Howard (1984) found a strong peak in rotation near the sunspot minimum and a somewhat weaker one near the sunspot maximum for the period 1921period -1982period . Brajša et al. (2006 found a higher-thanaverage rotation velocity in the minimum of activity in the years 1874-1981, applying the residual method. Ruždjak et al. (2017) reported the Sun rotates more differentially at the minimum than at the maximum of solar activity during the epoch 1977-2016, and there was a negative correlation between equatorial rotation and solar activity. Jurdana-Šepić et al. (2011) reported that during the maximum of the solar cycle 23 and just after it, the equatorial solar rotation velocity was lower than in other phases of the cycle in corona rotation, and their research was conducted by tracing SBCS in SOHO-EIT images.
However, some authors seem to have come to the opposite conclusion. Using the microwave low-brightness-temperature regions (LTRs) and Hα filaments as tracers, Brajša et al. (1997) presented that the Sun revealed a higher rotation rate on average during the maxima of the solar activity cycles 21 and 22, and the rotation coefficients A and B determined by tracing LTRs are 14.28( ± 0.06) and −1.82( ± 0.28), respectively. On the chromospheric rotation, Wan & Gao (2022) proposed the solar equator rotates slightly faster over all solar cycle maxima by analyzing the synoptic maps generated from the Ca II K line at the Mount Wilson Observatory . Sharma et al. ) 3.505 0.684 sin 2 , and found the solar filaments rotate at a higher rotation rate in the descending phase of the solar cycle than in the ascending phase by using the daily Ca II K3 observations. Therefore, the dependence of the solar rotation rate on the solar cycle phase is not clear and needs further study.
The solar photosphere, chromosphere, and corona have been paid more attention, but the rotation properties of the solar transition region (STR) are still unknown. By investigating the yearly time series of radio flux observed at the frequencies of 4995 and 8800 MHz during the years 1967-2010, Singh et al. (2021) found the STR has a significant 22 yr periodicity, and their investigation shows the rotation of STR is almost rigid. Nonetheless, there is a different result reported by Sharma et al. (2021), who found the STR rotates differentially and obtained the differential rotation profile. Similarly, by investigating the SFD images observed at wavelengths of 171 Å, Sharma et al. (2020a) proposed the upper STR rotates differentially as well and obtained the average sidereal rotation period is 27.03 days. At present, high-resolution images from satellite telescopes provide more possibilities for the study of the Sun, and they can generate images of different ion radiation and wavelengths, which conduce to researching the Sun at different depths in the visible spectrum or ultraviolet band. For example, coronal rotation can be studied by tracing X-ray bright points (XBPs) in observations from soft X-ray telescope (SXT) on board the Yohkoh, and by tracing coronal bright points (CBPs) in observations from satellites such as SDO, SOHO, Hinode, etc. (Brajša et al. 2002;Alipour & Safari 2015;Karachik et al. 2006;Kariyappa 2008). More importantly, on the basis of these images, the application of the flux modulation method can overcome some limitations of the tracer method, such as the inability to obtain high latitude rotation characteristics, which help to find the solar rotation patterns more efficiently and accurately.
The STR is a thin layer between the chromosphere and corona, which ranges from a temperature region of about 10 4 -10 6 K and has a strong connection with the coronal heating mechanisms (Doschek 2006;Feng & Gan 2006;Ptitsyna & Somov 2012;Schonfeld & Klimchuk 2020). Some studies on observational data or models of helium lines have suggested that the radiation of the He II (304 Å) line appears within the temperature range of the STR (Jordan 1975;Laming & Feldman 1992;). However, there is a Si XI line in this channel. Del Zanna et al.
(2015) measured the percentage contributions to the SOHO Solar EUV Monitor first-order band (SEM 1) observed count rates. Figure 6 in their paper shows the He II (304 Å) line contributes about 40%-70%; the Si XI 303 Å and Fe XV 284.1 Å lines alone contribute about 5%-20%. Moreover, Cushman & Rense (1978) measured the intensity ratios of He II and Si XI and even found it to be about 10:1 for a moderate level of solar activity and 30:1 for a quiet Sun. Hence, the Si XI line is weak and the He II (304 Å) line contributes a significant fraction. On the whole, it is appropriate to study STR with the He II (304 Å) line.
In this work, we used the SFD images at a wavelength of 304 Å observed from the SDO/AIA to investigate the rotation of the STR, and the flux modulation method was applied to our research. Section 2 contains the observational data, data processing, and research methodology. The method of extracting flux time series' from SFD images is described in Section 2.1, while we expound on the use of autocorrelation analysis to obtain sidereal rotation periods and the fit of rotation profile in Section 2.2. In Section 3, we discuss our results from three aspects, and finally, the conclusions are given in Section 4.

Data Processing
The SDO is designed to provide the data and scientific understanding necessary to predict solar activity, and its primary goal is to understand the physics of solar variations (Pesnell et al. 2011). The AIA telescope is one of three instruments on board the SDO, and it focuses on the evolution of the magnetic environment in the solar atmosphere and its interaction with embedded and surrounding plasma (Lemen et al. 2012). The data used in this work are narrowband imaging of EUV band passes centered on the He II (304 Å) line provided by AIA.
The data are publicly available on the website of SDO, and we downloaded the SFD images covering the time interval from 2011 to 2022. On each day, we selected one image at almost the same time. For the purpose of getting the information from the images, the complete solar disk region is segmented first, thus eliminating the interference of pixels outside the edge of the disk. From the equator to the northern or southern hemisphere, the rectangular bands are divided at 10°i ntervals from low latitudes to high latitudes. As shown in Figure 1, 16 rectangular bands were divided on the solar disk, covering the latitude range from 80°S to 80°N. Here, the EUV flux was extracted by calculating the average gray value of pixels in each rectangular band, and therefore we obtained the yearly average EUV flux time series for different latitudes.
Generally speaking, a time series consists of three parts, the seasonal component, the trend component, and the residual component. For the yearly average EUV flux time series, we noticed most of them show a significant upward or downward trend, which seems to imply the radiation intensity of the STR is changing over time. However, this kind of trend is not conducive to the periodicity analysis, hence a detrending processing is required. We used high-degree polynomial fitting to remove the trend of the EUV flux series, and the detrended series S was generated by the following formula: where i represents the day, and D i is the daily value of the EUV flux series, while f is the polynomial function fitted from the original series using the least squares method. We added the average value of D to restore the detrended series to the same level as before. An example of the average EUV flux time series at 60°N (2021) was plotted in Figure 2, where panel (a) shows the original series (black solid line) and its polynomial fitting curve (red dotted line), and panel (b) shows the detrended series that shows a steady variation, while panel (c) shows the results of the next processing step. Due to a variety of factors, the EUV flux time series contains many frequency components, but some of them are useless. As shown in the top panel of Figure 3, the horizontal lines in the spectrograms represent the frequencies, and the colors are the intensity (i.e., amplitude) corresponding to the frequencies. Thus, the large number of horizontal lines in the top panel suggests the presence of many different frequency components in the time series, and the higher frequency components  indicate the existence of noise. After our analysis, it was found that the frequencies with larger intensity are concentrated in the range of 0.02 day −1 -0.1 day −1 , which implies the main frequency components are in this range. Therefore, we attempt to remove the high-frequency components in the series through filtering, and a bandpass filter was designed with a range of 0.02 day −1 -0.1 day −1 , whose corresponding screening period is 10-50 days. From the bottom panel of Figure 3, we can clearly see the high-frequency components are removed, while the main frequency components are retained after bandpass filtering. The result of bandpass filtering is shown in panel (c) of Figure 2, where the smoother curve indicates that noise was removed from the data.

Methodology
The autocorrelation function (ACF) describes the degree of correlation between a random signal at different times, and the autocorrelation coefficients ρ(k) were calculated by the ACF (Chandra & Vats 2011). Assuming that there is a time series {X t , t = 0, 1, 2, L ,n − 1}, and it is autocorrelated up to a lag of k, then the ACF can be expressed as In this study, the autocorrelation analysis is utilized to investigate the average EUV flux time series after data processing. We set the lag of ACF to 150 days and subsequently calculated the autocorrelation coefficient. Figure 4 shows the autocorrelogram of the series at latitude 60°N (top) and 60°S (bottom) in 2021, and the black dotted lines show that the series at 60°N and 60°S stand for two situations with more noise and less noise. The comparison between the black dotted line and the blue solid line after bandpass filtering shows that the bandpass filtering has good adaptability and can preserve the main frequency components of the series. It can be seen that an obvious periodic peak structure appears in the curve of the blue solid line, and each peak contains different periodic information. The first secondary peak shows a clearer and higher structure, which reveals this peak contains the main periodic information in the series. As a result, Gaussian fitting of the first secondary peak can obtain a more accurate synodic rotation period (Sharma et al. 2020a(Sharma et al. , 2020b. The formula of Gaussian fitting is as follows: where x 0 is the center coordinate of the Gaussian peak, which represents the synodic rotation period, a is the height of the peak, and c is the standard variance. The coefficient of determination (represented by R 2 ) was imported to evaluate the degree of fitting, and R 2 0.95 is needed. Figure 5 shows the Gaussian fitting at 60°N and 60°S in 2021, and the corresponding solar synodic rotation periods are 30.06 and 30.18 days, respectively. The orbital period of SDO is 365.26 days, thus converting the synodic rotation period into the sidereal rotation period can be calculated by the following  and subsequently, the solar rotation rate can be obtained from the following formula: where ω is the angular velocity (deg/day) of the solar rotation. It is widely confirmed that the Sun exhibits differential rotation, meaning the Sun rotates fastest at the equator, and its rotation rate decreases with increasing latitudes. The solar rotation rate as a function of latitude can be described by the following traditional formula (Newton & Nunn 1951;Pulkkinen & Tuominen 1998a, 1998bWan & Li 2022): where f is latitude, and A represents the equatorial rotation rate (deg/day), while B represents the latitudinal gradient of rotation. We used the least square method to fit Equation (6) and finally obtained the solar rotation profile for each year. Moreover, we used the average rotation rate of all years at different latitudes to fit the profile of the entire period (2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020)(2021)(2022). The coefficient A, coefficient B, standard errors (SE), and rms errors are listed in Table 1.

Differential Rotation of the STR
We observed that the angular velocity of the STR varies with the latitude. As shown in Figure 6, the black scatter points represent the average rotation rates in the entire time interval (2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020)(2021)(2022), and their corresponding error bars are also shown. The average rotation rates decrease in the northern hemisphere from low to high latitudes in the range 14.54 deg day −1 -12.59 deg day −1 . For the southern hemisphere, they also decrease from low to high latitudes in the range 14.67 deg day −1 -12.99 deg day −1 . We got a high precision fitting (rms error = 0.18), which proves the STR rotates differentially. The blue solid line shown in Figure 6 is the differential rotation profile. The coefficients of A and B are 14.39 ( ± 0.08) and −1.61 ( ± 0.15), respectively. Sharma et al. (2021) obtained the differential rotation profile of STR, and their coefficient A is 14.7, which is larger than our result; their coefficient B is −1.26, whose absolute value is smaller than ours. The discrepancies may be due to different data. Sharma et al. (2020a) studied the rotation of the upper STR and obtained that the average sidereal rotation period was 27.03 days during the period from 2012 to 2018. In our work, the average sidereal rotation period from 2011 to 2022 was 26.37 days.  The north-south asymmetry has been well-studied using different indicators (Carbonell et al. 1993;Zhang et al. 2015;Sharma et al. 2020b). As mentioned above, we defined a parameter AS to measure the north-south asymmetry in rotation rates, which is given as where ω N and ω S represent the rotation rate (deg/day) of the northern and southern hemispheres, respectively. We calculated the coefficient AS from 2011 to 2022 and listed them in Table 2. We found the absolute value of AS is small, indicating that the north-south asymmetry of rotation rates is not significant. In the entire period from 2011 to 2022, AS = − 0.002, which manifests that the southern hemisphere rotates slightly faster than the northern hemisphere (by ∼0.44%). Some authors have reported similar results. Hathaway & Wilson (1990) proposed the southern hemisphere rotates significantly faster than the northern one for the period 1921-1982. Pulkkinen & Tuominen (1998a) found the southern hemisphere is at most cycles more rapid than the northern one in cycles 10-22. Wöhl et al. (2010) presented that the southern solar hemisphere rotates more rapidly for the period 1998-2006. Wan & Gao (2022) observed the southern hemisphere rotates faster at most cycles during the years 1915-1985 in the study of chromospheric rotation. However, our result did not find any specific variations between the north-south asymmetry of rotation rate and solar activity.

Cycle-related Variations of the Rotation Rate and Solar Activity
It is a common practice to use sunspot numbers to represent solar activity (Jurdana-Šepić et al. 2011;Sharma et al. 2021;Wan & Gao 2022). In order to explore the dependence between the solar rotation rate and solar activity, we compared the average rotation rate of the STR to the annual sunspot numbers from 2011 to 2022. The time period can be divided into the majority of solar cycle (SC) 24 from 2011 to 2018 and the beginning of SC 25 from 2019 to 2022. The average rotation rate is calculated based on the sidereal rotation rate at different latitudes, and the yearly mean total sunspot numbers came from the World Data Center SILSO, Royal Observatory of Belgium, Brussels 5 (Clette et al. 2014).
As shown in Figure 7, the average rotation rate is represented by the black scatter points on the blue solid line, and its associated error bars are also shown. It can be clearly seen that the average rotation rate of the STR shows an upward trend in the increasing phase of the 24th SC (2011)(2012)(2013)(2014), and reaches the maximum in 2014 (14.18 deg day −1 ), then shows a downward trend in the decreasing phase of SC 24 (2014-2020), reaching the minimum in 2020 (13.22 deg day −1 ). Meanwhile, at the end of SC 24 and the beginning of SC 25 (2020-2022), the rotation rate of the STR shows a continuous upward trend, which corresponds to the increasing phase of annual sunspot numbers. These behaviors suggest that the rotation rate of the STR follows the same trend as solar activity and is modulated by the solar cycle as well. Some authors have reported the Sun rotates faster during the solar cycle maxima by analyzing different data sets (Japaridze & Chargeishvili 2016;Sharma et al. 2021;Wan & Gao 2022;   Wan & Li 2022), and our finding is in agreement with their results. Especially for the result of Sharma et al. (2021), who studied the rotation of the STR as well. They reported the average rotation rate of the STR follows the solar activity cycle from 2008 to 2018. Our results also confirm that there is a dependence between the rotation rate of the STR and solar activity, and we speculate that this relationship is positively correlated.

Comparison of Rotation Properties in Different Solar Layers
The analysis of the rotation in different solar layers is helpful in understanding the transition process of the solar magnetic field. In consequence, we compared the rotational coefficients of this work with the results reported by other scholars to analyze the relationship between the solar chromosphere, the STR, and the corona. For decades, many scholars have obtained the rotation profiles of the solar chromosphere and corona using different data and methods, some of which are listed in Table 3. From the comparison in the table, we found that the coefficient A of the STR obtained by our study is smaller than that of the solar chromosphere and the corona, except for Hara (2009) and Bertello et al. (2020). This result reveals the equatorial rotation rate of the STR is smaller than that of the solar chromosphere and the corona. Meanwhile, the absolute value of the coefficient B of the STR was found to be smaller than that of the solar chromosphere and the corona, except for Chandra et al. (2010), which manifests that the STR rotates less differentially than the solar chromosphere and the corona.
Our result is consistent with that reported by Sharma et al. (2021), who found the STR rotates less differentially than the corona as well. The comparative results show that the solar transition region rotates more slowly and more rigidly than the solar chromosphere and the corona. However, the reasons for this phenomenon are manifold. The rotation rate of the Sun varies with the time, that is, it is different in different solar cycles and in different phases of a given solar cycle. Meanwhile, the solar rotation rate seems to vary with altitude or temperature, and it has been demonstrated in the rotation of the STR and the corona (Vats et al. 2001;Altrock 2003;Sharma et al. 2020a). Another plausible explanation is the influence of the magnetic field structure. Smallscale magnetic structures of the coronal and large-scale magnetic structures, such as the plages of the chromosphere, seem to restrain the rotation of the STR at the same time.

Conclusions
In this work, we studied the rotation of the STR by analyzing the SFD images at a wavelength of 304 Å observed from SDO/ AIA from 2011 to 2022. The images contain information about the long-lived structure of the STR, and we obtained the yearly EUV flux time series for different latitudes by dividing the  latitudinal rectangular bands. However, the flux series needs to be detrended and high-frequency noise removed to generate obvious periodic information through autocorrelation analysis. By applying flux modulation, we obtained the synodic rotation periods at different latitudes between 75°S and 75°N, with an interval of 10°. We observed that the rotation rate of the STR decreases with increasing latitude, and this confirms that the STR rotates differentially. Moreover, the rotation profile of the entire period of 2011-2022 was fitted with a high precision fitting (rms error = 0.18), and the rotational coefficients A and B are 14.39 ( ± 0.08) and −1.61 (±0.15), respectively. We also noted that there is a slight north-south asymmetry by comparing the average rotation rates of the northern hemisphere (13.63 deg day −1 ) and southern hemisphere (13.69 deg day −1 ), and the southern hemisphere rotates slightly faster than the northern hemisphere (by ∼0.44%).
In addition, we discussed the correlation between the rotation rate and the solar activity represented by the yearly mean total sunspot numbers, and found the rotation rate of the STR follows the overall trend as solar activity, which supports the results reported by Japaridze & Chargeishvili (2016), Sharma et al. (2021), Wan & Gao (2022), and Wan & Li (2022). In the comparison of the rotational coefficients, we found the STR rotates more slowly and more rigidly than the solar chromosphere and the corona.