Measuring the Rotation Rate of the Sun Using Sunspots

Galileo was the first to observe the Sun through a telescope in 1609 and identify dark regions
on the solar surface, which are referred to as sunspots. It is now confirmed that sunspots
have a strong magnetic field of order of 3000 G, which tends to suppress convection, thus
reducing the energy flux being transmitted in that region. That is why they look dark as
compared to the surrounding photosphere. Although, there were some earlier reports of
sunspots, it was believed that these were some planets which come between the Earth and
the Sun. Galileo also found that the sunspots change the position on the solar disk from day
to day (Fig. 1). By carefully, recording their position every day he showed that these are
spots on the surface of the Sun. This was based on the fact that if the spots are on solar
surface the transit speed would depend on the distance from the central meridian because
of projection effects. The aim of this experiment is to repeat this measurements, using the
daily images of the Sun, obtain by Global Oscillation Network Group (GONG) station at
the Cerro Tollolo Interamerican Observatory, Chile. The data can be downloaded from the
web-site
https://gong.nso.edu
The images for the month of January 2002 are supplied as Flexible Image Transport System
(FITS) files with the date and time of observation determining the file name. The fits file
have data in binary format, along with a header, which occupies the first few records of the
binary file. The header contains some useful information about the data. These images can
be viewed through ds9. Before studying the image look at the header, apart from general
information like the number and type of entries it also contains some keywords specifying the
centre and radius of image all in terms of the pixels in the image
• Centre: (FNDLMBXC, FNDLMBYC) (xc , yc )
• Radius: (FNDLMBMA+FNDLMBMI)/2 R
Please note that the images are rotated through 90◦ and the North pole of the Sun is along
the positive x-axis. Thus the Sun is rotating about the x-axis and the sunspots migrate
upwards in the image. The image is a projection of the Sun and if θ is the latitude and φ
the longitude measured from the central meridian in the image (x-axis), then if (x, y) are
coordinates of the sunspot in the image:

x − xc = R sin θ, y − yc = R cos θ sin φ.

Using this the latitude and longitude of the sunspot on a given day can be calculated. If
the same sunspot is followed for a few successive days then it can be seen that the latitude
does not change significantly, while longitudes drifts with time. The sunspots are not rigidly
attached to the Sun and they also move with time. Thus we need to take average over several
sunspots to get a reliable estimate of solar rotation. Once φ(t) is measured for a period of
5–7 days then the slope of straight line fit would give the dφ/dt = Ω = 2π/P , where P is the
rotation period, which is about 25–27 days. If the spots were due to transiting planets, then
y − yc would be a linear function of time, instead of φ. Check for this also. For the case of
planets transiting the Sun, their velocities would also be different.
To find the period perform a least squares fit of the form φ = c + m(t − hti) using the points
(ti , φi ) obtained from images. Use
X X s1
sj = tji , tj = φi tji , hti = ;,
i i
s0

1
Figure 1 : A copy of the original drawings of Galileo showing positions of sunspots on different
days.

to get
s0 t1 − s1 t0 t0
m= , c= .
s0 s2 − s21 s0

Repeat the exercise for a few sunspots and plot P vs θ. It is well known that the Sun has
differential rotation with equator rotating faster than the poles, but with limited precision
it may not be possible to estimate the differential rotation. This gives the Synodic rotation
rate, i.e., that observed from the Earth which is orbiting the Sun. To get the sidereal rotation
rate, i.e., with respected to distant stars, add the annual frequency, (360/365.25)◦ d−1 , or
2π/365.25 d−1 to it. The sidereal differential rotation can be approximated by:

ω = A + B sin2 θ + C sin4 θ

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where

A = (14.713 ± 0.049)◦ d−1 , B = (−2.396 ± 0.188)◦ d−1 , C = (−1.787 ± 0.253)◦ d−1 .

To get a reliable measure this exercise is to be repeated for thousands of sunspots. This can
be done using the data for all sunspots. The file ARs 1996-2022.txt gives the daily positions
of all sunspot groups during 1996–2022. This directly gives the longitude and latitude of the
sunspot group every day, one per line. Each group is identified by a unique number assigned
to it, which starts from 7933 which is the NOAA serial number assigned to every sunspot
group. The date of observation is given in the first 3 columns, yy, mm, dd. The 4th column is
the NOAA number, the 5th is the number of spots in the group. The 6th and 7th column are
the heliographic latitude and longitude. These are the coordinates in the rotating Sun. The
last entry in each row is the CM longitude, that is the longitude as measured with respect
to the central meridian as viewed from the Earth. This can be used to calculate the rotation
rate. The calculated rotation rate can be plotted against the latitude to check for differential
rotation.
Another exercise is to plot the latitude against time to get the butterfly diagram. The time
can be approximated in year by
Year + Month/12+date/365.25.
This shows the equatorward migration of sunspot positions with time.