Magnetic field as a tracer for studying the differential rotation of the solar corona
aa r X i v : . [ a s t r o - ph . S R ] A ug Solar PhysicsDOI: 10.1007/ ••••• - ••• - ••• - •••• - • Magnetic field as a tracer for studying the differentialrotation of the solar corona
O. G. Badalyan · V. N. Obridko c (cid:13) Springer ••••
Abstract
The characteristics of differential rotation of the solar corona for theperiod 1976–2004 were studied as a function of the distance from the center ofthe Sun. For this study, we developed a method using the coronal magnetic fieldas a tracer. The field in a spherical layer from the base of the corona up to thesource surface was determined from photospheric measurements. Calculationswere performed for 14 heliocentric distances from the base of the corona up to2.45 R ⊙ solar radii (the vicinity of the source surface) and from the equatorto ± ◦ of latitude at 5 ◦ steps. For each day, we calculated three sphericalcomponents, which were then used to obtain the field strength. The coronalrotation periods were determined by the periodogram method. The rotationperiods were calculated for all distances and latitudes under consideration. Theresults of these calculations make it possible to study the distribution of therotation periods in the corona depending on distance, time, and phase of thecycle. The variations in the coronal differential rotation during the time interval1976–2004 were as follows: the gradient of differential rotation decreased withthe increase of heliocentric distance; the rotation remaining differential even inthe vicinity of the source surface. The largest rotation rates (shortest rotationperiods) were recorded at the cycle minimum at small heliospheric distances, i.e. small heights in the corona. The lowest rotation rate was observed at the middleof the ascending branch at large distances. At the minimum of the cycle, thedifferential rotation is most clearly pronounced, especially at small heliocentricdistances. As the distance increases, the differential gradient decreases in allphases. The results based on magnetic data and on the brightness of the coronalgreen line 530.3 nm Fe XIV used earlier show a satisfactory agreement. Since therotation of the magnetic field at the corresponding heights in the corona is probablydetermined by the conditions in the field generation region, an opportunity arises touse this method for diagnostics of differential rotation in the subphotospheric layers.
Keywords:
Magnetic fields, Corona; Rotation B V.N. Obridko [email protected]
O.G. Badalyan [email protected] Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio WavePropagation, RAS, 108840, Troitsk, Moscow, Russia
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1. Introduction
The present-day notion is that the rotation of the solar corona reflects therotation of its subphotospheric layers. So, the differential rotation of the coronacan provide us with additional information for the subsequent construction ofsolar dynamo models.However, finding the coronal rotation parameters is not an easy task. In thecorona, there are virtually no obvious tracers, such as, for example, sunspots orphotospheric faculae, which help us study the rotation of the photosphere bytracing their position on the disk and calculating their speed, i.e. , the synodicrotation rate of the Sun. The Doppler method cannot be used because the coronallines are too wide. Therefore, all existing studies of the differential rotation ofthe corona rely on the analysis of proxies, such as day-to-day changes in thebrightness of the coronal emission lines or the features related to the corona(bright and dark areas, coronal holes, etc. ).Most studies are based on out-of-eclipse observations of the brightness of thecoronal green line Fe xiv e.g. , Tlatov, 1997; Altrock, 2003) use also the red coronal lineFe x et al. , 2009; Jurdana-ˇSepi´c et al. , 2011), dark features and Hα filaments Brajˇsa et al. (1997), 10.7 cm radio emission Mouradian, Bocchia,and Botton (2002), and coronal holes (Insley, Moore, and Harrison, 1995; Nash,Sheeley, and Wang, 1988). Most of these studies lead us to a general conclusionthat the corona rotates differentially, the rotation parameters changing with thephase of the activity cycle.Unfortunately, the tracers mentioned above characterize the rotation of thecorona at relatively low heights above the limb. To study the rotation at sig-nificantly larger distances, we suggest using the fact that the structure of thecorona up to a few solar radii is fully determined by the magnetic field. So far asdirect magnetic measurements in the corona are impossible, the authors usuallyhave to extrapolate the field measured in the photosphere. There are variousmethods for extrapolating observed photospheric magnetic fields to the coronallayers that are based on quite simple and physically consistent assumptions.Comparison with eclipse observations of the corona carried out by many authorsshows a good agreement between the calculated and directly observed features.The agreement is also confirmed by theoretical comparison of the kinetic andmagnetic pressure in the corona.In this paper we use magnetic field calculations to study the rotation of thecorona over a large range of distances from the center of the Sun – from thebase of the corona to the source surface. In what follows, we mean by rotation ofthe corona (unless otherwise specified) the rotation of the calculated magneticfield. This method was proposed in Badalyan and Obridko (2015). It allows us SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 2 agnetic field and rotation of the corona to trace changes in the differential rotation of the corona with distance andwith the phase of the activity cycle and to compare them with the rotationparameters obtained from the study of other coronal tracers, e.g. , the brightnessof the coronal green line.This study covers the time interval from 24 June 1976 to 31 December 2004,i.e., Cycles 21, 22, and a part of Cycle 23. The magnetic field was calculatedat selected distances from the base of the corona to the source surface. Animportant distance is 1.1 R ⊙ , which is close to the distance the coronal green-line brightness data we were using are reduced to (see S´ykora, 1971b, Storiniand S´ykora, 1997, S´ykora and Ryb´ak, 2005). The database covers the intervalfrom 1939 to 2001, which allows us to compare the corona rotation parametersobtained from the green-line brightness and from the magnetic field at a givendistance.The main principles of our method for calculating the synodic rotation rateof the corona using the magnetic field as a tracer are described in Sections 2and 3. Further, in Sections 4 and 5, we consider the variation in the rotationcharacteristics depending on the phase of the activity cycle and heliocentricdistance. In Section 6, the results based on the magnetic data are compared withthose obtained earlier from observations of the coronal green-line brightness. TheConclusion dwells on the possibility of using the results obtained to study therotation characteristics of the subphotospheric layers of the Sun.
2. Method for calculating the magnetic field in the corona
We calculated the coronal magnetic field in the potential approximation usingthe well-known method described in Hoeksema and Scherrer (1986); Hoeksema(1991) in its classical version without assuming the radial field in the photo-sphere. We used as the source data WSO (John Wilcox Solar Observatory)measurements of the longitudinal component of the photospheric magnetic field(http://wso.stanford.edu/synopticl.html) and on their basis built the synopticcharts for each Carrington rotation. The general method for extrapolating themagnetic field in the corona is to solve the boundary problem with the line-of-sight field component measured in the photosphere and strictly radial field atthe source surface. As a result, it becomes possible to calculate three magneticfield components in the spherical coordinates B r , B θ , B ϕ .The magnetic field components have the form: B r = X P mn (cos θ )( g nm cos mϕ + h nm sin mϕ ) × (cid:8) ( n + 1)( R ⊙ /R ) n +2 − n ( R/R s ) n − c n (cid:9) , (1) B θ = − X ∂P mn (cos θ ) ∂θ ( g nm cos mϕ + h nm sin mϕ ) × (cid:8) ( R ⊙ /R ) n +2 + ( R/R s ) n − c n (cid:9) , (2) SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 3 adalyan and Obridko B ϕ = − X m sin θ P mn (cos θ )( h nm cos mϕ − g nm sin mϕ ) × (cid:8) ( R ⊙ /R ) n +2 + ( R/R s ) n − c n (cid:9) . (3)In these equations, 0 ≤ m ≤ n ≤ N (in our case, N = 9), c n = − ( R ⊙ /R s ) n +2 , P mn are the Legendre polynomials, g nm and h nm are the coefficients of thespherical harmonics calculated in the course of the solution of the boundaryproblem, θ is the co-latitude (counted from the poles to the equator) ϕ is theCarrington longitude, and R s is the source surface radius. Usually it is assumedthat R s = 2 .
5. Hereinafter, the distances are measured in R ⊙ and are countedfrom the center of the Sun.In this work, we made use of a program that allowed us to calculate threecomponents of the magnetic field in a spherical layer from the photosphere to thesource surface Kharshiladze and Ivanov (1994). We performed summation over10 harmonics and introduced a polar correction to make allowance for insufficientreliability of magnetic measurements near the poles Obridko and Shelting (1999).The coefficients of the expansion into spherical harmonics were found by theleast square method without using the orthogonality of functions. The calculatedmagnetic field is limited to the latitudes ± ◦ .There are publications (see the discussion in Wang and Sheeley, 1992), whichpoint out the shortcomings of the classical method and propose the hypothesisof radial magnetic field in the photosphere. Our calculations Obridko, Shelting,and Kharshiladze (2006) have shown that when these two methods are used, thedifferences do exist and concern mainly the intensity of the magnetic field. At thesame time, the differences in the structure of the field lines are insignificant, es-pecially over large time intervals. Therefore, it can be assumed that the rotationcharacteristics we find do not strongly depend on the method applied. A slightdifference between the results obtained by these two methods are noticeable atlatitudes higher than 70 ◦ .
3. The rotation period as a function of distance from thecenter of the Sun
To study the variation in the differential rotation of the solar corona with dis-tance, the magnetic field was calculated at 14 selected distances from the baseof the corona to the source surface. These are the heliocentric distances from 1.0 R ⊙ to 2.2 R ⊙ with a step of 0.1 R ⊙ and a the distance of 2.45 R ⊙ . The distanceof 1.0 R ⊙ corresponds to the base of the corona and the distance of 2.45 R ⊙ ,to the coronal layers in the vicinity of the source surface. We did not use thefield on the source surface, where, in accordance with the boundary conditions,there only exists the radial field component. For each day in the period from 24June 1976 to 31 December 2004, three field components were computed for theheliolatitudes from − ◦ to +75 ◦ with a step of 5 ◦ . Then, the total magneticfield B was calculated as the square root of the sum of squares of the threecomponents. SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 4 agnetic field and rotation of the corona
After that, the method of periodogram analysis was applied. In this method,the correlation between the daily values of the calculated magnetic field andthe test harmonic function with a trial period T p is determined within the timewindow of a chosen length L . The correlation coefficient found shows the degreeof similarity between the function with the period T p and the distribution weare examining in this time window. After that, the window is shifted in timeby ∆ t and the whole procedure repeats. The periodogram method ensures quitea good resolution in period, which allows a detailed study of the time–latitudecharacteristics of the coronal rotation.In this work, periodograms were calculated with a window of 365 days (1year) and a step of 3 solar rotations (81 days) for each series of the magneticfield data obtained. The total number of steps (windows) within the time intervalmentioned above (24 June 1976 to 31 December 2004) was 125. The calculationswere performed at each selected distance for the latitudes from 0 ◦ to ± ◦ witha step of 5 ◦ . The periods of the trial harmonic functions T p varied from 22 to 36days with a step of 0.1 day, i.e. , the total of 140 values.The coronal rotation period T at a given distance and at a given specificlatitude was determined as follows. At each latitude, we obtained a sequenceof 125 time intervals (windows) for a given distance. Every such window con-tained 140 values of the periods of the trial harmonic functions with differentamplitudes, characterizing the degree of similarity of the trial function to theinitial distribution of the magnetic field intensity. At each step, we selected theoscillation period of the trial harmonic function with the maximum amplitudein the moving window. This means that the selected trial function had themaximum correlation (the greatest similarity) with the initial distribution wewere examining in the given window. The period T p found in such a way canbe assumed closest to the “quasi-period” of the original observed distribution atthis step. This period was taken as the synodic period of the coronal rotation T at a given time at a given latitude. Thus, we obtained the time dependence ofthe coronal rotation period at a given latitude (for examples see Badalyan andS´ykora, 2005, Badalyan and S´ykora, 2006, Badalyan, Obridko, and S´ykora, 2006,Badalyan, 2010). We called this procedure the method of maximum amplitudes.Thus, we obtain a series of the coronal rotation periods at each latitude atdifferent distances as a function of time. Every such series allows us to findthe mean rotation period at a given latitude for the entire time interval underconsideration and to study its time variations. At each distance, we have 31such series. The totality of the series obtained for all heliocentric distances underconsideration demonstrates how the coronal rotation changes with distance fromthe center of the Sun.Figure 1 shows by way of example the maps (two-dimensional periodograms)for the distance 1.1 R ⊙ and the latitudes 10 ◦ and 55 ◦ North. The color on themaps characterizes the height of amplitude for a given rotation period. One cansee the selected periods form a kind of a broken band in the vicinity of a certainperiod characteristic of a given distance. We can observe the shift of this bandon the maps that represent the mean synodic period of the corona vs. latitude(the differential rotation). In Figure 1, this shift is noticeable when passing from
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Years
Years S y nod i c pe r i od , da ys Figure 1.
Two-dimensional periodograms for the latitudes of 10 ◦ ( left ) and 55 ◦ ( right ) Northand the distance of 1.1 R ⊙ . The lower panels show the periods with maximum amplitudes ina mowing time window. the latitude of 10 ◦ higher, to the latitude of 55 ◦ . It is seen that, on average, theband shifts to longer periods.The kinks of the band characterize the behavior of the rotation period withtime. This is illustrated on the lower panels, under the maps. The plots representthe periods with maximum amplitudes at each step in the moving time windowand illustrate the time variation in the coronal synodic period at the givenlatitude. The mean rotation period over the time interval under consideration is27.4 days at the latitude of 10 ◦ and 29.3 days at the latitude 55 ◦ . These valuesare shown on the plots with straight horizontal lines.Thus, the synodic rotation periods of the corona were determined at 14 dis-tances from the center of the Sun for all latitudes in the time interval underexamination. First, we found the mean period at every latitude from − ◦ to+75 ◦ (31 values), at each of the 14 distances. Figure 2 illustrates the meanlatitude dependence of the synodic period for a few distances. Note that thecurves in the figure are symmetric about the equator (the heliolatitude is zero).This means that the average periods for the northern and southern hemisphereswere calculated for each latitude, and then, the approximating polynomial wasdrawn over the points obtained. In fact, at low latitudes there is a noticeablenorth–south asymmetry of rotation, which can be seen in some figures below.Figure 2 reveals the following particularities of the relations obtained:1. The differential gradient of the coronal rotation decreases with the helio-spheric distance – as the distance increases, the curves become flatter.2. The synodic period at the equator ( ϕ = 0) increases gradually (the rotationrate decreases) with the increase of the distance.3. Even in the vicinity of the source surface (2.45 R ⊙ ), the rotation of the coronaremains differential.4. Some decrease of the rotation period is observed at high latitudes at shortheliocentric distances. This particularity was noted by Stenflo (1989). It can SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 6 agnetic field and rotation of the corona -80 -60 -40 -20 0 20 40 60 80 30.030.527.027.528.028.530.029.029.530.5 S y nod i c pe r i od Latitude -80 -60 -40 -20 0 20 40 60 8027.027.529.028.529.528.0
Latitude
Figure 2.
Synodic rotation period of the corona vs. latitude for some heliocentric distancesindicated on the panels. also be noticed when determining the rotation of the green-line corona (Letfusand S´ykora, 1982, Fig. 7).The series of latitudinal dependencies of the rotation periods at various dis-tances makes it possible to construct the general distribution of the coronarotation periods in the form of a map in the time–latitude coordinates. Figure 3represents such maps for the distances of 1.1 R ⊙ and 2.0 R ⊙ . The maps show thatthe rotation periods at both distances increase with latitude. For the convenienceof comparison, the same scale was used for both maps. The bottom panel showsthe monthly mean sunspot numbers in the new system (Version 2).As seen from Figure 3, the rotation period increases with distance (the ro-tation rate decreases) throughout the map. At low latitudes, the periods onboth maps do not exceed 28 days. However on the map for 2.0 R ⊙ , the totalarea of the regions where the rotation rate is maximum ( i.e. , the period is lessthan 27 days, red color) is smaller than on the map for 1.1 R ⊙ . It is interestingto note that at 2.0 R ⊙ , the rapidly rotating regions are usually observed afterthe cycle maximum, at the beginning of the descending branch. In Cycle 23,which was lower than the other two cycles, the periods less than 27 days arevirtually absent at 2.0 R ⊙ . At high latitudes, the maps do not display anyvisible periodicity in the appearance of slowly rotating regions (rotation periodsmore than 30 days, blue color), though there is a hint that they rather tend toform near the minimum of the cycle. Note also that on the map for 2.0 R ⊙ , suchregions are significantly fewer and the general range of the periods is smallerthan on the map for 1.1 R ⊙ . This agrees with Figure 2 (right panel), where thedependence on latitude for 2.0 R ⊙ is flatter than for 1.1 R ⊙ (left panel).
4. Rotation period vs. the phase of the activity cycle
To study the variations in the coronal rotation period during an activity cycle,we use the notion of the phase of the cycle. According to Mitchell (1929), thephase is determined as Φ = ( τ − m ) / ( | M − m | ) . (4) SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 7 adalyan and Obridko -60-40-200204060 1980 1985 1990 1995 2000 2005-60-40-200204060 La t i t ude -60-40-200204060 -60-40-200204060 La t i t ude Years W o l f nu m be r s
21 22
Figure 3.
Time–latitude maps of the rotation periods for the distances 1.1 R ⊙ ( top ) and 2.0 R ⊙ ( middle ). The lower panel gives sunspot numbers in the new system (Version 2). Here, τ is the current time, M and m are the times of the nearest maximum andminimum of the 11-year cycle, respectively. As follows from this definition, thephase is 0 at the minimum of each cycle and ± − SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 8 agnetic field and rotation of the corona -60-3003060 0.50.0-0.5 -1.0-1.0 -60-3003060-60-3003060-60-3003060 1.01.01.00.0-0.5 -1.0-1.0
PhasePhase -60-3003060 0.50.0-0.5-60-3003060-60-3003060 0.50.5 1.00.0-0.5 L a t i t u d e -60-3003060 Figure 4.
Distribution of the rotation periods of the solar corona on the phase–latitude mapscalculated for various heliocentric distances in the range of 1.0 R ⊙ – 2.45 R ⊙ . The distancesin solar radii are indicated on the right of each map. The scale of the synodic rotation periodsof the corona are given at the bottom. The first thing that catches the eye is that the maps are similar up to thedistance of 1.3 R ⊙ – 1.4 R ⊙ . This particularly concerns the regions of relativelyfast rotation observed in the descending branch until the minimum (periods of26-27 days). As the distance increases, these regions disappear abruptly. Thefast-rotating region splinters and afrer that its characteristics dimensions donot exceed 0.1 in phase. This may mean that such a particularity of the coronalrotation is most likely associated with large activity complexes that often appearin the descending branch. The characteristic size of the complexes is 30 ◦ –40 ◦ , i.e. , about 0.1 radius of the Sun. The contribution of the magnetic field of suchobjects to the general magnetic structure at 1.4 R ⊙ decreases by at least anorder of magnitude.The regions of medium rotation rates (periods of ≈
28 days), on the contrary,retain their structure in the equatorial zone up to ≈ . R ⊙ independent ofthe phase of the cycle, and only at larger distances on the ascending branch ofactivity they are replaced by even lower rotation rates (longer periods).And finally, the slowest rotation (colored blue on the maps) is observed inthe polar zones at the minimum of activity and the beginning of the ascendingbranch. Note that the lifetime of these large periods somewhat differs in the twohemispheres, being much longer in the southern hemisphere. SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 9 adalyan and Obridko - 0.4 -60 -30 0 30 602627282930313233
Latitude -60 -30 0 30 60
Latitude -60 -30 0 30 60 2627282930313233 S y nod i c pe r i od , da ys Latitude
Figure 5.
Synodic rotation periods of the corona as a function of latitude for different dis-tances and phase intervals of the cycle. The mean phase values are indicated on the panels;the scale of distances in solar radii is given on the right top panel.
Now, consider the dependence of the coronal rotation on the phase of theactivity cycle in more detail. For this purpose, we divided the phase interval from − − . .
2. Then, the rotation periodwas plotted as a function of latitude for each of the five “phases” at all 14distances under consideration.Figure 5 illustrates the mean distributions of the rotation periods vs. latitudeat five selected distances. The mean phase values are given on the panels. Thescale on the right shows the color of the curve for a given distance. According tothe definition Equation 4, phase Φ = − . − . . . − . . i.e. , the differential gradientis small. The steepest (lowest) curves on the lower left-hand panel for distancesnot exceeding 1.5 R ⊙ refer to the minimum of the cycle; in other words, at theminimum, the coronal rotation period changes most abruptly (the differentialgradient is the largest) when passing from the equator to higher latitudes. Thehighest curve (largest rotation periods; i.e. , smallest rotation rates) refers tophase +0 .
4, which is close to the middle of the ascending branch. This agrees
SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 10 agnetic field and rotation of the corona -60 -30 0 30 60 2627282930313233 -60 -30 0 30 60 2627282930313233 S y nod i c pe r i od , da ys Latitude - 0.8 - 0.4 0.0 0.4 0.8 -60 -30 0 30 60
Latitude -80 -60 -40 -20 0 20 40 60 802627282930313233
Latitude
Figure 6.
The profile of the curves in relation to distance in different phase intervals. Thedistance in solar radii is given on the panels; the phase scale is provided on the right-hand toppanel. with the distribution of the periods in Figure 4 for large distances, where a broadvertical band of large periods is seen in the middle of the rise phase.Figura 6 compares the profile of the curves for the same phase intervals atdifferent distances. At small distances (lower panels) the steepest curves refer tothe minimum of the cycle. The closer we are to the maximum both along theascending and along the descending branch, the flatter become the curves. Atlarge distances (upper panels), all curves are rather flat. Here, the highest curves(largest periods) at both distances refer to phase Φ = +0 . i.e. , to approximatelythe middle of the ascending branch of the activity cycle. At this time, the rotationof the corona at large distances is the slowest, and the differential gradient (thechange of the rotation period with heliolatitude) is small.The general distribution of the corona rotation periods with the distance forfive phase intervals during a cycle is represented in more detail in Figure 7. Eachcycle is divided into five intervals so that the duration is each interval is abouttwo years. Figure 7 shows that the field distributions differ in different phases,the maps changing asymmetrically with respect to the activity minimum. Forexample, the map for Φ = − . . − . . ◦ North,while on the latter, such a band is absent, but there are two bands of fast rotationmore or less symmetric about the equator that do not go beyond 2 R ⊙ . At theminimum, the N–S asymmetry is not observed. The band of fast rotation islocated at equatorial latitudes and goes from the base of the corona up to thesource surface. While moving to higher latitudes at the minimum of the cycle,the periods increase significantly; the differential gradient on the map is thelargest. SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 11 adalyan and Obridko -60-40-200204060 1.6 1.41.4 1.21.2 -60-40-200204060 L a t i t u d e -60-40-200204060 1.81.81.81.6 1.41.41.2 1.21.01.01.01.0 Distance -60-40-2002040602.4 2.42.42.42.2 2.22.22.22.0 2.02.02.01.8 1.6 + 0.4 -60-40-200204060 1.6 + 0.8
Figure 7.
Rotation periods in the distance–latitude reference frame. The cycle phases aregiven on the right of the maps.
Immediately after the maximum at the beginning of the descending branch(left highest panel), the zone of small rotation periods at short distances ( < . R ⊙ ) is located near the equator. At the same time, the zone of moderaterotation rates (periods of 26.9–27.4 days), which exists throughout the descend-ing branch including the minimum (left lowest panel), extends to large heights.During the entire descending branch, a zone of small rotation rates (periodsof 29.6-30.0 days), forms gradually at high latitudes. Immediately after themaximum, this zone is only seen in the north hemisphere at relatively smallheights (left highest panel), but at the end of the descending branch (left lowestpanel) and particularly, at the beginning of the ascending branch (right highestpanel), it extends to all heights in both hemispheres. By this moment, the zonesof moderate rotation rates at large heights disappear, being only observed upto the height of 1.6 R ⊙ . In the equatorial region, the smallest periods are notobserved at all. As the maximum of the cycle approaches, the velocity field atmid latitudes is restored and the zones of large periods at the poles decrease.
5. Characteristics of differential rotation of the corona atdifferent distances
To obtain the parameters of differential rotation and consider its cycle variation,we used the traditional Faye formula:
SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 12 agnetic field and rotation of the corona H e li o c en t r i c d i s t an c e Phase 2.42.22.01.81.61.41.21.0 0.0-0.4 -0.4 -0.25 -- 0.00 -1.25 -- -1.00 -0.50 -- -0.25 -1.50 -- -1.25 -0.75 -- -0.50 -1.75 -- -1.50 -1.00 -- -0.75 -2.00 -- -1.75
Phase min maxmax min maxmax
Figure 8.
The phase–latitude distribution of a ( left ) and b ( right ). The scale at the bottom provides the coefficients in degrees per day. ω = a + b sin ϕ. (5)Here ω is the angular synodic rotation rate in degrees per day; a is the coeffi-cient that characterizes with some approximation the angular rotation rate of theSun near the equator; and b is the change in the rotation rate with latitude. Forthe Sun, the latter coefficient is negative; i.e. , the rotation rate decreases (periodincreases) with latitude. Coefficient b is often called the differential gradient. Inthe general case, a and b depend on the height of the object under examination( e.g. , coronal magnetic field) in the solar atmosphere and on its variation withthe phase of the cycle.To compute a and b , the data for all latitudes in each of the five phase intervalswere combined in a single series and the periods were converted to synodicangular velocity. Each series contains 31 points at a given distance. They wereused to determine a and b from the slope of the ω = f (sin ϕ ) line. Computationswere carried out up to latitudes 60 ◦ inclusive. At higher latitudes, the Faye lawrequires that the fourth degree of the sine of latitude be included. The diagramsof the phase-latitude distribution of a and b are represented in Figure 8.Figure 8 shows that coefficient a (left panel) in the time interval under con-sideration is the largest near the cycle minimum, where its value reaches 13 . ◦ per day ( i.e. , the synodic period is 26.4 days).Note that analyzing daily Doppler measurements for the period 1967–1976,R. Howard arrived at the conclusion that a reached its maximum exactly at thecycle minimum in 1976 Howard (1976). The relation obtained by Howard is closeto a similar relation for sunspots. In a later work Howard, Gilman, and Gilman(1984), the authors analyzed dependencies for the rotation characteristics of SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 13 adalyan and Obridko individual sunspots inside a group. It turned out that the equatorial rotationrate decreases with the increase of the spot area. The equatorial rotation rateincreases noticeably in the vicinity of the cycle minimum, while the differentialgradient decreases abruptly 1-2 years before the minimum. In Belvedere et al. (1977), the authors make even a more general conclusion that at all levels inthe solar atmosphere, the objects with smaller sizes and shorter lifetimes rotatefaster than large-scale and long-lived objects.Such a high equatorial rotation rate is only observed at the minimum ofthe cycle at distances no more than 1.4 R ⊙ . The smallest coefficient a on themap ( i.e. , the lowest rotation rate) is seen at large distances at the middle ofthe ascending branch. One can also see a horizontal band (zone) at a distance ∼ . − . R ⊙ , in which a does not virtually change and is approximately 13 . ◦ per day in all phases of the cycle. It is interesting to note that this velocitycorresponds to the Carrington period of 27.2753 days the Carrington referenceframe 360 / . . b has thelargest absolute value at small distances near the cycle minimum approximatelywhere the largest values of a are observed. At larger distances, b is small. Thedifferential gradient here is small, the rotation rate almost does not change withlatitude. On the same panel, we can also see a band of relatively increasedvalues of coefficient b (slightly increased differential gradient) in the middle of theascending branch of activity. This band exists at all distances and the differentialgradient somewhat decreases with distance.
6. Comparison with the corona rotation as inferred fromgreen-line observations
The differential rotation of the solar corona inferred from the coronal green-linebrightness was studied by many authors (for some references, see Introduction).We have examined this issue in detail in Badalyan and S´ykora (2005, 2006);Badalyan, Obridko, and S´ykora (2006); Badalyan (2010). The studies were car-ried out using the database by J. S´ykora (Slovak Republic). J. S´ykora was thefirst to take over a difficult task of reducing observational data from differentcoronal stations to a single photometric system. The discussion of the arisingproblems and the first results were published in 1971 S´ykora (1971b). As theamount of data was increasing, the work was continued (S´ykora, 1980, 1992,1994, Storini and S´ykora, 1997, S´ykora and Ryb´ak, 2005).The database contains the results of measurements of the brightness of thecoronal green line reduced to a single photometric scale with a step of 5 ◦ inlatitude and ≈ ◦ in longitude (1 day). It covers the interval 1939-2001. Duringthe first few years, the observations were irregular, therefore the data are mainlyused since 1943. Original daily measurements taken separately on the easternand western limb were used to derive brightness on the central meridian on eachparticular day. It was obtained as the mean of the values measured on the easternlimb 7 days before a given date and on the western limb 7 days after it ( i.e. ,approximately at the moments when a meridian corresponding to the central SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 14 agnetic field and rotation of the corona -60 -30 0 30 6027.027.528.028.529.029.530.0 S y nod i c pe r i od , da ys Latitude
Figure 9.
Synodic periods of the differential rotation of the solar corona as determined fromthe green-line brightness ( blue curve ) and magnetic-field data at the distance 1.1 R ⊙ ( redcurve ). meridian on the given day was passing through the eastern and the westernlimb, respectively). The green–line brightness is adjusted to the height of 60”above the limb (the height of Pic-du-Midi measurements Trellis (1957). This isclose to the distance of 1.1 R ⊙ from the center of the disk.Latitudinal variations in the synodic periods of the coronal differential ro-tation determined from the data on the green-line brightness (blue curve) andmagnetic field at 1.1 R ⊙ (red curve) are compared in Figure 9. As in Figure 2,the curves are symmetric with respect to line X = 0 (with respect to theequator); i.e. , the data for the northern and southern hemispheres are averaged.The green-line curve covers the period 1943-2001; the magnetic-field curve, theperiod 1976-2004.As follows from Figure 9, both the green-line brightness and the magnetic fielddata confidently reveal the differential rotation of the corona. At the same time,there are noticeable differences between the two curves. The green-line curveis mainly located inside the curve of the magnetic field and displays smallergradient to higher latitudes. The magnetic-field curve shows that short periodspersist without significant change up to the latitudes of 30 ◦ − ◦ . As the latitudeincreases further, a sharp increase in the rotation period occurs, and at thehighest latitudes, the magnetic field shows a lower rotation rate than the green-line brightness.Figure 10 represents the maps of distribution of the corona rotation periodsderived from the green-line (top) and magnetic-field (bottom) data in the phase-latitude coordinates. The figure reveals the similar and different features on thetwo maps. Thus, the low-latitude band of high rotation rates (periods less than28 days) occupies approximately the same latitude range. There is also somesimilarity in the distribution of the fastest rotation rates (periods less than 27days). At higher latitudes, one can see the zones of slower rotation, but their SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 15 adalyan and Obridko -60-3003060 -60-30030600.0-0.5 0.5-1.0 max minmax La t i t ude -60-3003060 -60-30030600.0-0.5 0.5 1.0 1.0-1.0 La t i t ude Phase
Figure 10.
Cycle distribution of the rotation periods based on the green-line ( top ) andmagnetic-field ( bottom ) data. location on the maps is different. The map based on the green-line brightnessshows that, in the middle of the descending branch of activity, the rotation ratesat high latitudes are close to those at low latitudes (the band of small differentialgradients at Φ ≈ − . − . SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 16 agnetic field and rotation of the corona -60 -30 0 30 602627282930313233 - 0.8 - 0.4 0.0 0.4 0.8 S y nod i c pe r i od , da ys Latitude -60 -30 0 30 60 2627282930313233 - 0.8 - 0.4 0.0 0.4 0.8
Latitude
Figure 11.
Dependence on the phase of the cycle for the differential curves based on green-linedata ( left ) The same for the curves based on magnetic data at the distance of 1.1 R ⊙ ( right ).One can see that the curves themselves as well as their cycle variations are different. in Figure 11 is the parabola for Φ = 0 on the right panel (magnetic data). Thesame effect is seen on the lower map in Figure 10 at the minimum of the cycle.Thus, as expected, there isn’t and cannot be complete correspondence betweenthe rotation characteristics determined by the green line and by the magneticfield. It is well known that the green-line emission depends on the temperatureand density of plasma. The problem of the corona heating has been consideredby many authors (see references in Mandrini, D´emoulin, and Klimchuk, 2000,Badalyan and Obridko, 2007). All heating mechanisms existing nowadays dependin different ways on the strength and spatial dimensions of the magnetic field.The relative contribution of these mechanisms varies depending on the latitudeand the phase of the solar cycle. Nevertheless, there is a certain similarity be-tween the rotation characteristics, which may be an additional argument in favorof the applicability of the proposed method.
7. Conclusion
In this paper, we have considered the possibility of using magnetic data to studythe rotation of the solar corona. Magnetic field controls the structure of coronalobjects. Therefore when studying the differential rotation of the magnetic field,we actually use the latter as tracers to analyze variations in the rotation parame-ters with distance and with the phase of the activity cycle. The coronal magneticfield is calculated by extrapolating the fields observed in the photosphere. Thus,the method proposed in our work makes it possible to study the characteristicsof differential rotation of the corona in a wide range of heights and in differentphases of the cycles.Our study covers the time interval from 24 June 1976 to 31 December 2004; i.e. , Cycles 21, 22, and part of Cycle 23. It is shown that the differential gradient(the range of rotation rate variations with latitude) decreases with distance fromthe center of the Sun. The rotation rate at the equator decreases gradually ( i.e. , SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 17 adalyan and Obridko the rotation period increases) with the increase of the distance. However therotation remains differential even at the source surface, that is at about 2.5 R ⊙ from the center. The latter is very important for calculating the spatial structureof the solar wind and possible changes in the sector structure when moving awayfrom the ecliptic plane.Variations in the differential rotation of the corona with the phase of activitycycle have been considered in detail. In the time interval under consideration,the largest differential gradient is detected at short distances (no more than 1.4 R ⊙ ) from the center in the equatorial zone. Both at the ascending and at thedescending branch of activity, the rotation of the corona is the less differentialthe closer we are to the maximum of activity. At the same time, the rotationparameters in the phases symmetric with respect to the cycle minimum aredifferent. Besides that, the north–south asymmetry of rotation noticeable inother phases of the cycle is not observed at the minimum.The time interval selected for examination allowed us to compare the rotationparameters obtained by our method based on magnetic data with the parametersobtained earlier in the photosphere from observations of various tracers andin the corona from observations of the green-line brightness at low altitudes.The results show a satisfactory agreement, though one can also notice somedifferences.Our results show that when going to higher coronal levels, the essentially dif-ferential rotation is becoming increasingly rigid. As follows from the calculationprocedure itself, this is accompanied by disappearance of high-order harmonics.The energy contribution of different harmonics depends on distance as a power-law function with exponent β = − n + 2), see Equations 1– 3. Thus, as theheliocentric distance increases, we encounter objects of increasingly large scales.Besides, non-radial components disappear more rapidly, and at 2.5 R ⊙ the fieldbecomes strictly radial.This reminds us of the change in the rotation characteristics with depth inthe subphotospheric layers. There is certainly a great difference between thebasic processes in the corona and in the subphotospheric layers. The magneticfield is not generated in a stationary solar corona, which allows us to use thepotential approximation in our calculations. The calculated magnetic field in thecorona is completely determined by the conditions in the photosphere and sub-photospheric layers. In up-to-date models, the characteristic spatial scales of themagnetic fields depend on their generation region. The fields of higher scales aregenerated deeper under the photosphere. So, the rotation characteristics of thecorona can reflect variations in the plasma rotation rate under the photosphere.Thus, according to modern concepts, the coronal rotation reflects the rotationof the subphotospheric layers ( e.g. , see Kitchatinov, 2013). The higher layers ofthe corona reflect the rotation of the deeper layers of the Sun. The proposedmethod allows us to expect that the study of the corona rotation at distancesfrom its base up to the source surface will make it possible to “look” into thesubphotospheric layers and calculate the rotation parameters therein. The resultsobtained in this work suggest that either the generation depth of magnetic fieldsof different scales or the generation process itself and its amplitude change duringan activity cycle.In future, we are going to apply the results obtained to the studyof rotation of deep sub-photospheric layers of the Sun. SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 18 agnetic field and rotation of the corona
8. Acknowledgements
The work was supported by the Russian Foundation for Basic Research, Project17-02-00300. We are grateful to the WSO team for the data available on theInternet site
WSO.stanford.edu/forms/prsyn.html . Disclosure of Potential Conflicts of Interest
The authors declare that theyhave no conflicts of interest.
References
Altrock R.C.: 2003,
Solar Phys. , , 23. DOI. ADS.Antonucci, E., Svalgaard, L.: 1974, Solar Phys. , , 3. DOI. ADS.Badalyan, O.G.: 2009, Astron. Zh. , , 295 (English translation 2009, Astron. Reports , ,262). DOI. ADS.Badalyan, O.G.: 2010, New Astron. , , 135. DOI. ADS.Badalyan, O.G., Obridko, V.N.: 2007, Pis’ma Astron. Zh. , , 210 (English translation 2007, Astron. Lett. , , 158). DOI. ADS.Badalyan, O.G., Obridko, V.N.: in Stepanov, A.V. and Nagovitsyn, Yu.A. (eds.), Solarand solar-terrestrial physics 2015 , St.-Petersburg, Astron. Obs. RAS at Pulkovo, 13 (inRussian), .Badalyan, O.G., S´ykora, J.: 2005,
Contrib. Astron. Obs. Skalnat´e Pleso , , 180. ADS.Badalyan, O.G., S´ykora, J.: 2006, Adv. Space Res. , , 906. DOI. ADS.Badalyan, O.G., Obridko, V.N., S´ykora, J.: 2006, Astron. Zh. , , 352, (English translation2006, Astron. Reports , , 312). DOI. ADS.Belvedere, G., Godoli, G., Motta, S., Paterno, L., Zappala, R.A.: 1977, Astrophys. J. Lett. , , L91. DOI. ADS.Brajˇsa, R., Ruˇzdjak, D., Vrˇsnak, B., Pohjolainen, S., Urpo, S., Scholl, A., W¨ohl, H.: 1997, Solar Phys. , , 1. DOI. ADS.Hoeksema, J.T.: 1991, Solar magnetic fields – 1985 through 1990 , Report CSSA-ASTRO-91-01.Hoeksema, J.T., Scherrer, P.H.: 1986,
The Solar Magnetic Field – 1976 through 1985 , WDCAReport UAG-94, NGDC, Boulder. ADS.Howard, R.: 1976,
Astrophys. J. Lett. , , L159. DOI. ADS.Howard R., Gilman P.I., Gilman P.A.: 1984, Astrophys. J. , , 373. DOI. ADS.Insley, J.E., Moore, V.I., Harrison, R.A.: 1995, Solar Phys. , , 1. DOI. ADS.Jurdana-ˇSepi´c, R., Brajˇsa, R., W¨ohl, H., Hanslmeier, A., Poljanˇci´c, L., Svalgaard, L.,Gissot, S.F.: 2011, Astron. Astrophys. , , A17. DOI. ADS.Kharshiladze, A.P., Ivanov, K.G.: 1994, Geomagnetizm i Aeronomia , , 22 (in Russian).ADS.Kitchatinov, L.L.: 2013, Solar and Astrophysical Dynamos and Magnetic Activity , Proc. IAUSymposium, , 399. DOI. ADS.Letfus, V., S´ykora, J.: 1982,
Atlas of the Green Corona Synoptic Charts for the Period 1947-1976 , Veda Publ. House, Bratislava. ADS.Makarov, V.I., Tlatov, A.G.: 1997,
Astron. Zh. , , 615. (English translation 1997, AstronRep. , , 543). ADS.Mandrini, C.H, D´emoulin, J., Klimchuk, A.: 2000, Astrophys. J. , , 999. DOI. ADS.Mitchell S.A.: 1929, Handb. Astrophys., , 231.Mouradian, Z., Bocchia, R., Botton, C.: 2002, Astron. Astrophys. , , 1103. DOI. ADS.Nash, A.G., Sheeley, N.R. Jr., Wang, Y.-M.: 1988, Solar Phys. , , 359. DOI. ADS.Obridko V.N., Shelting B.D.: Solar Phys. ., , 187. DOI. ADS.Obridko, V.N.; Shelting,, B.D.; Kharshiladze, A.F.: 2006, Geomagnetizm i Aeronomia , , 294(English translation 2006, Geomagnetism and Aeronomy , , 294). DOI. ADS.Ryb´ak, J.: 1994, Solar Phys. , , 161. DOI. ADS.Ryb´ak, J.: 2000, Hvar Obs. Bull. , , 135. ADS.Sime D.G., Fisher R.R., Altrock R.C.: 1989, Astrophys. J. , , 454. DOI. ADS.Stenflo, J.: 1989, Astron. Astrophys. , , 403. ADS.Storini, M., S´ykora, J.: 1997, Nuovo Cimento , , 923. ADS.S´ykora, J.: 1971a, Solar Phys. , , 72. DOI. ADS. SOLA: PAPER_1.tex; 17 August 2018; 0:23; p. 19 adalyan and ObridkoS´ykora, J.: 1971b,
Bull. Astron. Inst. Czechosl. , , 12. ADS.S´ykora, J.: 1980, in Dryer, M. and Tandberg–Hanssen, E. (eds.), Solar and InterplanetaryDynamics , Reidel, Dordrecht, 87. ADS.S´ykora, J.: 1992,
Contrib. Astron. Obs. Skalnat´e Pleso , , 55. ADS.S´ykora, J.: 1994, Adv. Space Res. , , 73. DOI. ADS.S´ykora, J., Ryb´ak, J.: 2005, Adv. Space Res. , , 393. DOI. ADS.Tlatov, A.G.: 1997, Astron. Zh. , , 621. (English translation 1997, Astron. Reports , , 548).ADS.Trellis, M.: 1957. Ann. d’Astrophys. , Suppl. No. 5. ADS.Wang, Y.M., Sheeley, N.R.: 1992,
Astrophys. J. , , 310. DOI. ADS.Zaatri, A., W¨ohl, H., Roth, M., Corbard, T., Brajˇsa, R.: 2009, Astron. Astrophys. , , 589.DOI. ADS., 589.DOI. ADS.