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SOLAR DIFFERENTIAL ROTATION RATE AND
ACTIVITY CYCLE VARIATIONS
S.S.GUPTA 1, K.R.SIVARAMAN 2, R.F.HOWARD 3
1
Indian Institute of Astrophysics, Kodaikanal –624 103
Indian Institute of Astrophysics, Bangalore – 560 034
3
National Solar Observatory *, Tucson, AZ 85726, U.S.A
2
We have measured the umbral areas and positional coordinates of all
sunspots within 600 E – 600 W of longitudes from the Kodaikanal
photoheliograms for the period 1906 – 1987, using a digitizer pad that has a
resolution of 0.0025 cm (~ 0.2 arc sec on the sun). In this poster we describe
the optical telescope, white light observations, measurement method (Figure
1) and analysis of the data. From this we have calculated the differential
rotation rates for spots in three area categories: < 5, 5 – 15 and > 15 µ hemispheres (figure 4). The small spots are seen to rotate faster than the big
spots by about ~1.5 % (table 1). These measurements have an internal
accuracy of < 1 m sec -1 for the equatorial rotation rate.
The dependence of the rotation rates with the sizes of spots suggest
that these different rotation rates might correspond to the rotation rates of the
plasma layers at different depths in the solar interior at which the flux loops
of these spots are anchored.
The equatorial rotation rates of spots of the three area categories are
found to be higher during solar minimum (Figure 5) showing solar activity
dependence. If the surface rotation rates of spots of different sizes mimic the
rotation rates of the different plasma layers at their anchor depths, then, the
cycle dependence of the surface rotation rate implies a cycle dependence of
the internal dynamics of the Sun.
Many solar rotation studies from different data sets show
discrepancies in results (Howard, 1978) suggesting that systematic errors
may be the cause. Here, an attempt is made to determine the differential
rotation rate that takes into account of - i) the image distortion due to
telescope optics, ii) random selection of yearly data for digitization and iii)
single person digitization to avoid personal errors.
* Operated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with
National Science Foundation.
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1. The Telescope
The observational data consists of daily white-light photographs of the
full solar disk for the period 1906-87. The 20 cm diameter images of the sun
have been obtained on a photographic plate (till 1975)/film with a 15 cm
aperture refractor telescope that has a focal length of 2.44 m (Sivaraman,
Gupta and Howard, 1993; Gupta, 1994). The camera used to photograph the
solar image has a focal plane shutter which is a metal plate in the form of a
sector with a filter mount on it for mounting a broad-band filter. The shutter
is activated by releasing a metal spring, and the shutter then slides across the
aperture, providing an exposure time of about 0.001 s.
2. Data Digitization
A back illuminated digitizer of 91.5 cm X 61 cm size with a hand-held
cursor having a resolution of 0.0025 cm (~ 0.2 arc sec on the 20 cm solar
image) is used for the measurements. Each photoheliogram is oriented with
the axis of rotation along the Y-axis of the digitizing pad using the
ephemeris P-angle of the day. Positions are recorded in the two coordinates
of the pad with the hand-held cursor.
The measurement procedure we adopt is similar to that used for
Mount Wilson photoheliograms (Howard, Gilman and Gilman, 1984;
hereafter HGG). This is with a view to combine both the data sets so as to
follow more short lived spots than would be possible from a daily image
from one site and to obtain higher accuracies in the results.
The measurement on each day’s plate consists of digitization of 8
equally spaced apart limb points (for solar diameter), and measurements of
all spots present within 600 of the central meridian. Eight limb points are
necessary to determine the precise position of the solar disk in the
coordinates of the digitizing pad.
Two successive placements of the
cursor covering the spot umbra are made
(figure 1). The cursor’s cross hair is
positioned in such a way that the area
included in the quadrilateral formed by
the orthogonal cross-hair pattern equals
that of the umbra. The position of the
spot in the digitizing pad coordinate
system is the mean position of
successive vertices A and B in figure 1.
Figure 1. Schematic representation of the method of measuring the spot positions and areas.
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For the smallest spots the width of the cross hair was comparable to
the diameter of the spot, and such spots (spot area < 0.1 mm2) were
arbitrarily assigned an area of 0.05 millionths of a hemisphere.
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Measurements were made for the whole year from Jan. to Dec., in
sequence. The years were chosen randomly, so that any gradual changes in
the measuring instrumentation or in the expertise of the person who
measured would not result as a secular change in some solar characteristic.
3. Data Reduction
The measured ‘raw’data consists of individual spot umbral position
and area for each year. From the ‘raw’data set, each photoheliogram’s eight
limb positions, individual spot umbral areas and position measurements
were corrected for atmospheric refraction (HGG). The corrected limb
positions are used in a least-squares solution for a circle, and then sunspot
latitudes, CMDs, and areas are calculated from the sunspot measurements,
using the geometrical method (Smart, 1977). In addition, corrections for
optical aberrations were made to the position measurements (Howard, Gupta
and Sivaraman, 1999). All the areas are converted to disk center areas.
4. Data Analysis
Spot groups are defined by the proximity of individual spots (HGG),
i.e., spots that are within 50 in heliocentric longitude and 30 in latitude would
be included in the same spot group. This way there is little ambiguity in
identifying the group in a subsequent observation. The ‘second’ data set of
files, one for each year, consists of heliocentric positions and umbral areas
of individual sunspots and, separately, positions and total umbral areas of
sunspot groups. Group positions are area-weighted, using the areas of the
individual sunspots. Observation dates and times are given as fractional day
numbers. The ‘second’ data set is used to determine the motions of
individual sunspots and sunspot groups.
The method of identification of individual sunspots described in HGG
has been modified (Howard, Gupta and Sivaraman, 1999) for the
Kodaikanal, Mount Wilson sites and merged data sets to include more
criteria to determine an identification of a group return than just the number
of returned sunspots in the group. The criteria now are the average
individual sunspot position difference, the average individual sunspot area
difference, the number of returned sunspots, and the group position
difference (from that expected from the average differential rotation rate for
groups at that latitude). These criteria are weighted, with the most weight
assigned to the number of spots returned, and only a few percent of the total
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weight given to the spot and group position distances and less than 1% given
to the area differences. This means that in practice the difference from the
earlier method is small.
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Sunspot group motions in longitude (rotation) and latitude (meridional
motion) are determined by calculating the longitude or latitude difference
between adjacent observations and divided by the time difference between
the two observations. This is done for the individual sunspots as well. The
‘sunspot’ data set that is made up from this contains the position and area of
each returned sunspot for each of the two consecutive observations, as well
as the sidereal rotation rate, meridional drift, and other information.
The (separate) data set of each ‘returned’ sunspot groups contains
rotation rate, meridional motion of each group, ‘tilt angle’ of the group and
other information.
5. Results
5.1. Sunspot Rotation Rate
Figure 2 shows the differential rotation for individual sunspots
(averaged over 50 intervals in heliocentric latitude) for the 82 year (1906-87)
Kodaikanal data. The least-squares solution for the latitude dependence in
the expression ω = A + B sin2 φ, where φ is the latitude, is shown as the
smooth line in figure 2. In this and in all least-squares calculations in this
paper the solution is done for the individual sunspots or groups, and not for
the averaged values shown in the plots.
Figure 1. Rotation rate in deg. /day for all spots for the period 1906-87. The smooth line shows
the least-square solution.
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The derived values of the coefficients, A and B in deg day-1 sidereal
and the number, N of individual sunspots are:
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Site
A
B
Kodaikanal
Mt.Wilson
14.456 ± 0.002
14.456 ± 0.002
-2.89 ± 0.02
-2.88 ± 0.02
N
Period
121361
113112
1906-87
1917-85
5.2 Sunspot Group Rotation Rate
Figure 3 shows the differential rotation for the sunspot groups. The
coefficients and the number of groups are:
Kodaikanal
Mt.Wilson
14.471 ± 0.005
14.470 ± 0.005
-2.99 ± 0.05
-2.97 ± 0.05
44050
41224
1906-87
1917-85
The agreement between the two sites is very good for both spots and
spot groups though the time interval differs.
Figure 2. Rotation rate in deg./day for all spot groups, for the period 1906-87.
5.3. Sunspot Sizes
The spots have been categorized into 3-area (in µ hemispheres) sizes
of <5, 5<A<15 and >15 and they show difference in rotation rates (figure 4
and table 1).
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Kodaikanal data (1906-87).
Mount Wilson data (1917-85).
Figure 3. Rotation rate in deg./day for all spots divided into 3-area categories.
Solid lines represent sunspots of areas (in µ hemispheres) < 5,
short dashed lines for spots of areas 5<A<15 and
long dashed lines for spots of areas >15.
Site
Kodaikanal
Area <5
5 < Area <15
Area >15
A
B
N
14.491 ± 0.003
14.380 ± 0.004
14.279 ± 0.005
-2.85 ± 0.03
-2.84 ± 0.04
-2.83 ± 0.04
87050
25522
8789
Mount Wilson
Area <5
5 < Area <15
Area >15
14.477 ± 0.003
14.363 ± 0.006
14.248 ± 0.009
-2.80 ± 0.03
-2.65 ± 0.05
-2.61 ± 0.09
84460
20490
5799
Table 1. Rotation rates for spots of 3-area categories. A and B are the coefficients in the equation,
ω = A + B sin2 φ, and N is the number of spots.
From table 1, we notice that the differential rotation rates derived
from Kodaikanal data (1906-87) and Mount Wilson data (1915-85) agree
well within the error bars, indicating that small area spots rotate faster than
large spots by ~1.5%. This result confirms the previous results (HGG;
Howard, 1984; Sivaraman, Gupta and Howard, 1993; Balthasar, Vazquez
and Wohl, 1986; Howard, 1996).
5.4. Solar Cycle Dependence of Rotation
Figure 5 is a superposed-epoch plot of the residual rotation rate in
deg./day of all the spots on the disk over a cycle. The residual rotation is the
rotation rate of each spot minus the average rate for all spots over the full
time interval at that latitude. This eliminates the large effect that would
otherwise be present because of differential rotation.
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Figure 5. Residual sidereal rotation rate in deg./day for all spots from Kodaikanal data (solid lines)
for the period 1906-87 and for Mount Wilson data for the period 1917-85.
From figure 5, it is seen that the results from both Kodaikanal (solid
lines) and Mount Wilson (dashed lines) show a tendency of rotation at solar
minimum to be faster than at solar max, confirming the earlier results from
different data sets (Balthasar and Wohl, 1980; Gilman and Howard, 1984;
and many others). Similar plots for all the three area categories shows a
marginal cycle dependence of rotation but is found to be more pronounced
for the small spots (Gupta, Sivaraman and Howard, 1999).
References
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Gilman, P.A. and Howard, R.F.:1984, Astrophys. J. 283, 385
Gupta, S.S.: 1994, Ph.D. Thesis, Pandit Ravishankar Shukla University
Gupta, S.S., Sivaraman, K.R., and Howard, R.F.: 1999, Solar Phys. 188, 225
Howard, R.: 1978, Rev. Geophys. Space Phys. 16, 721
Howard, R.F.:1984, Ann. Rev. Astron. Astrophys. 22, 131
Howard, R.F.:1996, Ann. Rev. Astron. Astrophys. 34, 75
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Howard, R.F., Gupta, S.S. and Sivaraman, K.R.: 1999, Solar Phys. 186, 25
Sivaraman, K.R., Gupta, S.S., and Howard, R.F.: 1993, Solar Phys. 146, 27
Smart, W.M.: 1977, Textbook on Spherical Astronomy, Cambridge Univ. Press, London