Did you ever have the opportunity to observe the Sun through a filter with a telescope? If you did, you may have noticed little black marks on the Sun’s surface. Those are sunspots. And if you were to observe the Sun for a few consecutive days, you would notice that the sunspots “move” across the Sun’s surface. That is due to the Sun’s rotation. Even if you didn’t know that the Sun rotates around its axis until now, you can calculate its rotational period in a matter of minutes! Just take a look at this presentation, and you will be able, through a few calculations, to deepen your knowledge about the star that made life on Earth possible. #scichallenge2017

1. Open StemTopic
Author:Tanja Holc
Slovenia
#scichallenge2017

2.  In this project, the way of calculating the velocity of the apparent movement of sunspots is
presented, which is also an understandable way of determining the velocity of Sun's
rotation, and, consequently, determining the synodic and sidereal periods of Sun’s
rotation.
 A calculation of the velocity of a chosen sunspot with the aid of photographs, taken during
an observation of the Sun with a telescope from Murska Sobota, Slovenia, is presented as
well.
 Using the equations derived in this project, everyone can calculate the velocity of a
sunspot of their choice.This makes it interesting and usable for astronomy and mathematics
enthusiasts and people with little experience alike, and one‘s own photographs of the Sun
(with the aid of a telescope) can be used for the calculation.
 Key words: sunspots, rotation of the Sun, observation of the Sun, velocity of sunspots,
synodic and sidereal period.

3.  Sunspots areas on the Sun's
surface with lower temperatures
than the surrounding surface
 appear darker than their
surroundings
 appear to be moving across the
Sun's surface (as observed from
Earth) due to the rotation of
the Sun
Picture 1a,b:The
Sun with visible
sunspots and a
zoom-in on a
sunspot.

4.  For the calculation of the velocity of sunspots‘ apparent movement, two
photographs of the Sun are needed with a time interval of a few days. On the
photograph, we design a coordinate system, with heliographic longitudes on
the x axis and heliographic latitudes on the y axis.
Picture 2a,b:The
Sun on 24th and 27th
September 2016.

5.  With the aid of the pictures, we determine the following quantities:
 𝝋 heliographic latitude of the chosen sunspot
 𝒓 𝒚 Sun‘s radius on heliographic latitude 𝜑
 Δy distance between the sunspot and equator (x axis) in
the vertical direction
 Δx, Δx' distance between the sunspot y axis in the horizontal
direction (Δx – on the first day, Δx' - after a few days)
 the angles α, α, Δα (see Picture 7)
 Δs the distance the sunspot travelled between the
first and last day

6.  Here, the way of defining, deriving and calculating the aforementioned quantities
as well as the calculation of the velocity of a chosen sunspot (no. 2597) is presented.
 The photographs below were taken during an observation of the Sun with a
telescope from Murska Sobota, Slovenia, on 24th and 27th September 2016.
Picture 3a,b:The Sun on 24th and
27th September 2016, with the
Sunspot no. 2597 visible.

7.  Δy, Δx and Δx' are determined with the help of pixels (used was the graphic viewer
Irfanview):
Picture 4:The determination of Δy.
• Picture 4 shows the determination of Δy:
• Δy = 300 px
• Similarly, Δx and Δx' were determined:
• Δx = 110 px
• Δx' = 680 px
• I have also determined the Sun‘s radius in
pixels, which I will need for further calculation:
• R = 980 px

8.  Here, the equations for other needed quantities are derived with the aid of my
drawings:
 𝒓 𝒚 and 𝝋 : 𝑟 𝑦 = 𝑅 ∙ 𝑐𝑜𝑠 𝜑
(R is the Sun‘s radius)
𝑠𝑖𝑛 𝜑 =
𝛥𝑦
𝑅
𝜑 = 𝑠𝑖𝑛−1(
𝛥𝑦
𝑅
)
we derive the equation for the calculation of 𝑟 𝑦 :
𝑟 𝑦 = 𝑅 ∙ 𝑐𝑜𝑠( 𝑠𝑖𝑛−1
𝛥𝑦
𝑅
)
Picture 5

9.  Δx , Δx' , α and α'
sin 𝛼 =
Δ𝑥
𝑟(𝑦)
𝛼 = sin−1
(
Δ𝑥
𝑟(𝑦)
)
sin 𝛼′ =
Δ𝑥′
𝑟(𝑦)
𝛼′ = sin−1
(
Δ𝑥′
𝑟 𝑦
)
Picture 6: Δx and Δx'.
Picture 7: the relation between Δx , Δx' , α and α'.
*Picture 7 is drawn as a view of the Sun from below.The coordinates of the
sunspot no longer represent its heliographical longitude/latitude here.

10.  Δα and Δs
 Δs – the distance the sunspot travelled between the first and last day
 the Sun is a sphere, therefore I used the following equation for the calculation of Δs:
𝑙 =
𝜋𝑟𝛼
180°
𝛥𝑠 =
𝜋∙𝑟(𝑦)∙Δ𝛼
180°
𝛥𝛼 = 𝛼 − 𝛼′ = sin−1(
Δ𝑥
𝑟(𝑦)
) − sin−1(
Δ𝑥′
𝑟 𝑦
)
𝛥𝑠 =
𝜋∙𝑟 𝑦 ∙(sin−1(
Δ𝑥
𝑟 𝑦
) −sin−1(
Δ𝑥′
𝑟 𝑦
))
180°
 velocity of the sunspot: 𝑣 𝑠𝑢𝑛𝑠𝑝𝑜𝑡 =
Δ𝑠
𝑡
 t – time interval between photographs 1 and 2

11.  To calculate the velocity of our chosen sunspot, we first have to
determine Δy, Δx and Δx‘ in kilometres
 I have previously determined Sun‘s radius in pixels: R = 980 px
 Sun‘s radius in kilometres: R = 695.700 km
 980 px ………. 695.700 km
 1 px …………... 709,9 km
Date Sunspot y x x' 2R R
24. in 27. 9. 2016 2597 300 px 110 px 680 px 1960 px 980 px
- - 212.970 km 78.089 km 482.732 km 1.391.400 km 695.700 km
Table 1:The
quantities in
pixels and in
kilometres.

12.  Secondly, we calculate , r(y), α, α', α and s, using the previously derived equations
and Δy, Δx and Δx‘ in kilometres (the example of calculating s is shown):
𝜑 = 𝑠𝑖𝑛−1
(
𝛥𝑦
𝑅
)
𝑟 𝑦 = 𝑅 ∙ 𝑐𝑜𝑠 𝜑
𝛼 = sin−1(
Δ𝑥
𝑟(𝑦)
)
𝛼′ = sin−1
(
Δ𝑥′
𝑟 𝑦
)
𝛥𝑠 =
𝜋 ∙ 𝑟 𝑦 ∙ 𝛼′
− 𝛼
180°
=
𝜋 ∙ 𝑟 𝑦 ∙ (sin−1
(
Δ𝑥′
𝑟 𝑦
) − sin−1
(
Δ𝑥
𝑟 𝑦
))
180°
=
𝜋∙662.285 𝑘𝑚∙(sin−1(
482.732 𝑘𝑚
662.285 𝑘𝑚
) −sin−1(
78.089 𝑘𝑚
662.285 𝑘𝑚
))
180°
= 462.615 𝑘𝑚
 r(y) α α' α s
17,83° 662.285 km 6,8 ° 46,8° 40° 462.615 km
Table 2:The remaining
quantities calculated.

13.  To calculate the sunspot‘s velocity, we use the equation 𝑣 =
𝑠
𝑡
,
where
 𝑠 = ∆𝑠
 𝑡 = 3 𝑑𝑎𝑦𝑠 = 72 ℎ (the time difference between 24th and 27th September,
when the photographs were taken)
 This way, the velocity equals:
𝑣 =
𝑠
𝑡
=
462.615 𝑘𝑚
72 ℎ
= 6425
𝑘𝑚
ℎ
= 𝟏, 𝟕𝟖𝟓
𝒌𝒎
𝒔

14.  With a sunspot‘s velocity calculated, we can determine the Sun‘s synodic period = the
time in which the Sun (or in our case, the parts of the Sun with heliographic latitude 𝜑 = 17,83°, which is
the latitude of our sunspot) rotates once around its axis
 In this time period, the distance the sunspot travels equals:
𝑠 = 2𝜋𝑟 = 2𝜋 ∙ 𝑟 𝑦 = 2𝜋 ∙ 662.285 𝑘𝑚 = 4.161.259 𝑘𝑚
(this is circumference of the Sun on 𝜑 = 17,83°)
 𝑣 = 6425
𝑘𝑚
ℎ
= 1,785
𝑘𝑚
𝑠
therefore, for 1 rotation, the Sun needs:
 𝑡1 =
𝑠
𝑣
=
4.161.259 𝑘𝑚
6425
𝑘𝑚
ℎ
= 647,7 ℎ ≈ 𝟐𝟕 𝒅𝒂𝒚𝒔

15. In the calculation on the previous slide, we didn‘t take the Earth‘s movement around the Sun
into account.This is the synodic period, as calculated from the Earth, which differs from the
sidereal period, as measured relative to distant stars.
Picture 8:The difference
between the synodic and
sidereal period.
direction of distant stars

16. Picture 9: Position of the Earth
and the sunspot on the Sun, on
the first day and after 27 days.
 The sidereal period of Sun‘s rotation can be determined
with the aid of Picture 9 and the following quantities:
• t1 = 27 days – synodic period of Sun‘s rotation
• t0 = 365,25 days – orbital period of Earth‘s movement around the Sun
• δ = 360° = 2π radians – the angle for which the Earth moves around the
Sun in 1 year
 We determine the Earth‘s angular velocity:
ω 𝐸 =
2𝜋
365,25 𝑑𝑎𝑦𝑠
= 0,0172 𝑑𝑎𝑦−1
 From Picture 9, the following can be derived:
𝜔 𝐸 =
𝛼
27 𝑑𝑎𝑦𝑠
𝛼 = 𝜔 𝑧 ∙ 27 𝑑𝑎𝑦𝑠 = 0,4644
π radians … 180°
0,4644 radian … α = 26,6°

17. Picture 9: Position of the Earth
and the sunspot on the Sun, on
the first day and after 27 days.
It can be seen from Picture 9 that the sunspot was on the same line with the
Earth on the first day and after 27 days. In this time, the Earth moved around
the Sun for the angle α, whereas the Sun made 1 rotation plus rotated for α.
The Sun rotated for:
β = 360°+ α = 386,6°
In t1 = 27 days, the Sun rotated for 386,6°.We are aiming to calculate t2, the
time in which the Sun rotated for 360°, which is the sidereal period of Sun‘s
rotation (or the Sun‘s rotational period as „observed“ from distant stars).
386,6°… 27 days
360°… t2 = 𝟐𝟓, 𝟏 𝒅𝒂𝒚𝒔
The sidereal period of Sun‘s rotation (of the parts on the Sun with heliographic
latitude 𝜑 = 17,83°) is, as follows, 25,1 days and is shorter than the synodic
period.

18.  In this project, I have presented a simple and understandable way of calculating the
velocity of a sunspot‘s apparent movement across the Sun‘s surface.The sunspot I chose
for the calculation was Sunspot no. 2597. My calculations:
 𝑣 𝑠𝑢𝑛𝑠𝑝𝑜𝑡 =
Δ𝑠
𝑡
= 6425
𝑘𝑚
ℎ
= 1,785
𝑘𝑚
𝑠
 synodic period of the Sun‘s rotation: 𝑡1 = 647,7 ℎ ≈ 27 𝑑𝑎𝑦𝑠
 sidereal period of the Sun‘s rotation: 𝑡2 = 25,1 𝑑𝑎𝑦𝑠
 The correctness of my calculations can be confirmed with data from this source:
 „The Sun has an equatorial rotation speed of ~2 km/s; its differential rotation implies that the angular
velocity decreases with increased latitude.The poles make one rotation every 34.3 days and the
equator every 25.05 days, as measured relative to distant stars (sidereal rotation).“
▪ (Differential rotation of the Sun. https://en.wikipedia.org/wiki/Differential_rotation#Differential_rotation_of_the_Sun (2017-04-22).
of parts of the Sun with
heliographic latitude
𝜑 = 17,83°

19.  Cover picture: https://www.nasa.gov/content/goddard/one-giant-sunspot-6-substantial-flares/ (2017-04-22)
 Differential rotation of the Sun. https://en.wikipedia.org/wiki/Differential_rotation#Differential_rotation_of_the_Sun
(2017-04-22).
 Author of pictures 3a,b: Bojan Jandrašič (2016).
 AGUILAR, David A. Planeti, zvezde in galaksije : ilustrirana enciklopedija našega vesolja. Ljubljana: Rokus Klett, 2008 (str.
34).
 Sončeve pege. http://fizika.dssl.si/Galileo/indexe648.html?option=com_content&view=article&id=52&Itemid=68 (2016-
11-04).
 EMMERICH, M., Melchert, S. Astronomija : Čudovito vesolje, opazovanje planetov, zvezd in galaksij. Kranj : Narava, 2006
(str. 30, 31).
 Spaceweather Archive. http://spaceweather.com/archive.php?view=1&day=08&month=10&year=2016, (2016-11-05).
 Sunspots. http://galileo.rice.edu/sci/observations/sunspots.html (2016-11-14).
 Source of pictures 1a,b; 2a,b: http://spaceweather.com/archive.php (2016-11-05).
 Source of picture 8: http://astro.unl.edu/naap/motion3/sidereal_synodic.html (2017-04-22).
 Author of pictures 5, 6, 7, 9:Tanja Holc (2016, 2017).
With thanks to dr. Simon Ülen and Bojan Jandrašič.