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A SIMPLE INVESTIGATION OF THE SPATIAL
DISTRIBUTION OF ACTIVE LENGTHS IN THE SOLAR
PHOTOSPHERE
'
ARP '
AD CS'iK 1 , R '
OBERT ERD '
ELYI 2; \Lambda and MIKL '
OS MARIK 1
1 Department of Astronomy, E¨otv¨os L. University
Ludovika t'er 2, H­1083 Budapest, Hungary
and
2
Centre for Plasma Astrophysics, K.U.Leuven
Celestij\Lambdanenlaan 200B, B­3001 Heverlee, Belgi¨e
December 9, 1996
Manuscript ONLY. For submitting to Solar Physics
Abstract. In the present work a statistical investigation of the spatial distribution of
longitudinal coordinates of sunspot data has been carried out. For the analysis the maximal
area of umbra plus penumbra of sunspot groups has been taken into account.
Spatial distributions of longitudinal coordinates of sunspot data are obtained in different
coordinate systems (e.g. Carrington, equatorial and differential rotating).
Obtained spatial distributions of longitudinal coordinates of sunspots might reflect the
existance of large scaled active lengths on the surface of the Sun.
Key words: Sun -- Sunspots -- Active longitudes -- differential rotation
1. Introduction
Since 1958 it is a very well known observational fact that the latitude coordi­
nates of sunspots plotted with respect to time constract the so­called butter­
fly diagram (Fig 1). As the solar cycle progresses from maximum to minimum
the zone occupied by the sunspots moves steadily equatorwards over the 11
years. The first spots of a new cycle emerge around latitudes \Sigma25 \Gamma 30 ffi , while
the last sunspots at the end of the 11­year­cycle, by neglecting the varia­
tion of the magnetic polarities of the sunspots, appear around \Sigma5 \Gamma 10 ffi .
This spatial behaviour of sunspot data (e.g. area of umbra plus penumbra,
lifetime) has been investigated extensively statistically and theoretically as
well. The butterfly diagram can be considered as evidences of some distin­
guished active latitudes where dominant physical processes of the whole Sun
are presented.
However, one can have the following question: is there any reflection of active
\Lambda On leave from Dept. of Astropnomy, E¨otv¨os L. University, Ludovika t'er 2, H­1083
Budapest, Hungary

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A. CS ' IK ET AL.
longitudinal coordinates of sunspot data? Are there any distinguished lon­
gitudinal coordinates where some physical processes (if any) might have a
large scale influence on the whole behaviour of the Sun?
In the last few years there has not been too much interest in the search for
spatial distributions of the longitudinal coordinates of sunspots and to find
some (if any) active longitudes. Based on sunspot data most of the efforts
has been concentrated on the sophisticated description of the differential
rotation. An excellent review is given by e.g. Schr¨oter (1985).
In the recent work we analyse the sunspot data from the Greenwich
Photoheliographic Results observed in the period from 1967 to 1976. This
dataset is the first half period of solar cycle 21.
2. Method of investigation
In what follows, we briefly explain the method of our simple statistical study.
Let's suppose, that the dataset of sunspot groups have been observed at a
given, arbitrarily fixed interval of time. Let's coincide the whole time inter­
val for the studied dataset with the half­period of the total solar cycle 21.
The data were obtained, (by courtasy of P. McIntosh), from Greenwich Pho­
toheliographic Results observed for the period from 1967 to 1976. Let's cut
from this average half period an arbitrarily given time interval. An appro­
priate estimation for the length of this arbitrarily given time interval is
important. If it is too short, then there are not enough data for statistical
analysis. However, if the time interval is too long, then it can be much larger
than the characteristic lifetime of an active length (if any) and due to this
effect the characteristic spatial distribution may get indistinct.
To give the right value to this interval, we studied the shape of column
diagrams of the longitudinal coordinates to be discussed later. Finally, we
have choosen 13 total rotation periods of the solar equator. In this case, the
average number of sunspot groups which appear on the surface of the Sun,
are large enough (more than 500), wich is needed for a statistical approach.
In this arbitrarily given time interval, the statistical spatial distribution of
sunspot groups is studied.
Let's fix a spherical coordinate system at the center of the Sun. The sym­
metry axis of the this coordinate system coincides with the rotating axis of
the Sun. In this coordinate system positions of sunspot groups are given by
their longitudinal and azimuthal coordinates.
Let's devide the longitudinal axis of this coordinate system into 360 spa­
tially equal degrees. For each longitudinal degree, we calculate the maximal
umbral plus penumbral area of sunspot groups, that appear in the vicinity of
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ACTIVE LONGITUDES 3
n ffi around this band. In our study, we have taken n = 5 arbitrarily, because
the spatial expansion of a sunspot group can be as large as 10 ffi . This way, a
column diagram can constructed. The width of a band is 1 ffi and the height
is the sum of maximal umbral plus penumbral area of sunspot groups, which
appear in the above specified vicinity of the band. In this column diagram
there might be longitudinal bands with large and bands with small values of
summed area of sunspots. Large columns might reflect the position of active
lengths, small height of columns might reflect inactive lengths of the Sun.
Let's suppose, that the height distribution of the column diagrams of
sunspot groups follow Gaussian distributions. In this case we can compute
the Gaussian probability, for a given height of a given column at a certain
longitudinal degree wich has a deviation of more than 2oe from the mean
value. As it is well­known, the probability of this 2oe deviation is very small,
approximately 4:55%. We might conclude that some physical effects could
cause such phenomena in the spatial distribution.
Let's determine the scattering of our sample of data. One has to keep
in mind that the height of columns laying close enough to each other are
not independent, because of the constracture of our diagram. Each sunspot
group is taken into account in 11 intervals laying next to each other (Fig.
2). The scattering of the spatial distribution of this given sample of data is,
as follows,
oe 2 =
P (x i \Gamma X) 2
N \Gamma d
(1)
where d = 2n + 1. This way we obtain a column diagram for the spa­
tial distribution of the area of sunspot groups for a given time interval. By
studing these column diagrams and their temporal evolution, we try to con­
clude the existence of some active lengths. A sample of this kind of diagram
is shown on Fig. 3.
3. The temporal evolution of the column diagrams
As we mentioned, we have arbitrarily fixed the time interval of the investigat­
ed data (13 total rotational periods of the solar equator), t, for constructing
the column diagrams. In what follows, we describe a procedure by wich we
might carry out a whole time­scan investigation for the full dataset, or with
other word, for the half solar cycle.
We devided the time interval of the whole dataset, T , into overlapping
time intervals with a length of t. The difference of the starting points between
intervals next to each other is a total rotation period for the equator of our
coordinate system. For later use, let's construct a new kind of diagram. On
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A. CS ' IK ET AL.
the horizontal axis, (x), we plot the longitudinal angle coordinates, while on
the vertical axis, y, we plot the time coordinate. The unit of the time axis
is the total rotation period of the equator of our coordinate system and the
origin is the beginning of the 10­year­long period.
We trace the temporal evolution of peaks of columns higher than X + 2oe
for every overlapping interval.
Let's plot a dot at any position (x,y), where peaks are higher than 2oe (Fig.
6.a). For convenience we compress this graph along the time axis, as shown
later (Fig. 6.b). In figures constructed this way, at a certain longitude the
height of a column is equal to the number of peaks above X + 2oe for the
whole 10­years­long period.
4. Coordinate systems on the Sun
The aim of our examination is to find active lengths on the Sun by using sim­
ple statistical methods. Unfortunatelly, we cannot know what is an appro­
priate coordinate system, where these active lengths are fixed in position.
We have carried out analysis in some simple cases. Let's call the coordinate
system, in wich the active lengths are fixed in time (or do not have torsion
with respect to time), ''natural'' coordinate systems. In what follows we are
lookong for this ''natural'' coordinate system.
We have completed our analysis in three different coordinate systems:
\Gamma Carrington coordinate system
\Gamma Heliographic equatorial coordinate system
\Gamma Differential rotating coordinate system
The first choice (Carrington coordinate system) is obvious, because this is
the original coordinate system for the observations. If one supposes that
the rotation profile inwards the center of the Sun is independent from the
heliographic latitude, then the active lengths rotate like a ''solid body'' with
constant angular velocity.
The second coordinate system (heliographic equatorial coordinate system)
is a solid body rotating with a constant angular velocity, where the angular
velocity of the system is equal to the rotational velocity of the equator of
the Sun.
?From observations it is well­known that the plasma on the solar surface
rotates in a differential mode. It might be interesting to examine the tempo­
ral evolution of the active lengths (if any), or with other words the temporal
behaviour of the spatial distribution of the longitudinal coordinates of sun­
pot data, in a coordinate system wich has this observational phenomenon.
The torsional function of the longitudinal circles of this coordinate system
has the same shape as the velocity profile of the sunspot groups on a certain
longitudinal circle. The mathematical description of such velocity profiles
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ACTIVE LONGITUDES 5
are given (by e.g. Ward 1966) as follows:
\Delta– = A+ Bsin 2 ' (2)
where
A = 13:53 ffi =day and B = 2:57 ffi =day.
5. Results
When we investigated the data in a differential rotating coordinate sys­
tem, we transformed the initial dataset due to the special rotation profile
determined by Equation (2). In this case, there are not any characteristic,
statistical structures (see Fig. 4a,b). Although sunspots have a differentially
rotating profile (e.g. Equation 2), the chaotic structure in Fig. 4a shows that
active lengths do not follow the profile of the differential rotation at all.
This means that the differential rotating coordinate system cannot be the
natural coordinate system of the large­scale longitudinal activities.
Second, we have corrected the initial dataset due to the angular velocity
of the Carrington coordinate system and the eqatorial coordinate system.
Results are in Fig. 5. and Fig. 5b. In Fig. 5a there are parallel oblique lines.
If the selected coordinate system has a constant angular velocity, which is
different from the angular velocity of the Carrington system, there are no
large peaks on Fig. 5b, i.e. there are no evidences of active lengths.
Finally the results by completing analysis in a Carrington coordinate system
are plotted in Fig. 6a and Fig. 6b. In Fig. 6a one can observe closed 'for­
mations' orientated parallel to the time axis and there are no considerable
disturbing noises observable. The compressed diagram (in Fig. 6b.) shows
very large peaks. These peaks can be identified to the active lengths in the
Sun. It is worth to note that there are some longitude intervals between the
peaks, wich handly contain any active points at all.
6. Conclusions
The aim of the present short paper was to examine the existence (if any at
all) of active lengths in large spatial scales on the Sun. The recent study
shows that the active lengths are existing not only on small scales, but on
the large scales as well.
First, although sunspots have a differentially rotating profile, Fig. 4a shows
that active lengths do not follow the differential rotation at all. This means
that differentially rotating coordinate systems are not natural coordinate
systems for large scale longitudinal activities.
Second, results in Fig. 5. and Fig. 5b. show that the equatorial coordinate
system moves with a constant angular velocity with respect to the natural
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A. CS ' IK ET AL.
coordinate system of active lengths.
By analyzing Fig. 6a,b we suggests as a natural coordinate system for
active lengths the Carrington coordinate system. Spatial formations parallel
to the time axis in Fig. 6a let us to estimate the extension of active lengths in
time and space. Characteristic lifetime of active lengths are approximately
between 10 and 40 Carrington rotation periods. The characteristic width of
these lengths are around 10 degrees.
If one supposes that the place of the appearance of sunspot groups reflect
the origin of flux tubes at lower levels of the convective zone (flux tubes are
bounded to the core of the Sun), then one might conclude that the angular
rotational velocity of the solar core equals to the average Carrington angular
velocity. On the other hand, this speculation also means that there has to
be an acceleration mechanism which speeds up the surface close to the solar
equator, and a slow down mechanism which strongly breaks down equatorial
motions at the solar surface close to the poles.
7. Acknowledgements
'
A. CS thanks to prof. M. Goossens the warm hospitality at K.U.Leuven,
and to prof. Muzslay for financial supporting. The authors thank to Dr. K.
Petrovay and Dr. A. Ludm'any for their comments and advises meanwhile
preparing the manuscript.
References
Abarbanell, C. and W¨ohl, H.: 1981, Solar Phys. 70, 197
Abuzeid, A. and Petrovay, K.: 1990, Publ. Debrecen Heliogr. Obs. 7, 98
Becker, U.: 1955, Z. Astrophys. 37, 47
Castenmiller, M.J.M., Zwaan, C. and Van Der Zlam, E.B.J.: 1986, Solar Phys 105, 237
Petrovay, K. and Abuzeid, A.: 1991, Solar Phys 131, 239
Schr¨oter, E.H.: 1985, Solar Phys. 100, 141
Ward, F.: 1966, Astrophys. J. 145, 416
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ACTIVE LONGITUDES 7
Figure caption
Fig.1. The distribution of sunspot groups on a) ' \Gamma t, b) – \Gamma t diagram for
period 1967 to 1976. There is the well known butterfly diagram on Fig 1a,
while Fig 1b does not show any recognizable pattern.
Fig.2. The horizontal axe of this diagram is identified with the longitudinal
axe of the coordinate system on the Sun. For each 11 degrees longitudinal
interval, we calculated the maximal umbral plus penumbral area of sunspot
groups, that appeared in a given period. The height of the central column
of the specified longitudinal interval is equil to this sum. The picture shows
us, that there can be sunspot groups wich belong only to column signed A,
signed B, or both. The height of columns laying close enough each other are
not independent.
Fig.3. This column diagram shows the longitudinal distribution of sunspot
groups in the Carrington coordinate system, for a 13 Carrington rotation
period. The vertical axe of the diagram represents the sum maximal umbral
plus penumbral area of sunspot groups.
Fig.4. The distribution of the active lengths in differential rotating coordi­
nate system can be seen on Fig.4.a. The position of activities in time and in
longitude is shown on the vertical and the horizontal axe respectively. The
unit of the vertical axe is one total rotation period of the equator of the
coordinate system we used.
Diagram 4.b. helps to identify the active lengths, that appeared in the period
1967 to 1970, in case of differential rotating coordinate system. The graph
is the vertical compression of the diagram on Fig.4.a.
Fig.5. The distribution of the active lengths in equatorial coordinate sys­
tem. The meaning of the axes are the same like on Fig.4.a. and Fig.4b.
Fig.6. The distribution of the active lengths in the Carrington coordinate
system. The meaning of the axes are the same like on Fig.4.a.
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