*Tips to the teacher:*

This part requires students to carefully follow steps of actions, so it is advisable to have a lab worksheet prepared to distribute to them.

If you have a beamer and an internet connection, we encourage the use of a google spreadsheet for students to list their results (see Discussion part). This enables them to work collaboratively and compare their results. It also allows you to easily average their results as well as to graph them. Otherwise, you can draw a large measurement table on the blackboard for students to write their measurements (and easily erase them if they find calculation errors).

*Observed student difficulties:*

In this activity, students want to compute the rotation period T by using the speed v of a sunspot and the length of its trajectory L: T = L/v. Some students are confused by the fact that neither L nor v is known (unlike in a conventional problem statement), and the teacher must insist that this is precisely why they should be measured.

This activity requires students to use twice the equation for average speed: To calculate the displacement speed of a sunspot, and to compute the rotation period. Some students confuse the two. To guide the students, the teacher can, for instance, write in big the equations on the blackboard with the link between them, as in Fig. 2 below. We suggest not to do so right at the beginning, for students to come up with those links themselves, but later in the activity, once they’ve figured it out.

*Fig. 2: Equations used in the activity and the link between them.*

**1. INTRODUCTION**

- Engage students in the historical context of this activity, by telling them how Galileo “surprisingly” discovered moving sunspots on the Sun.

- Tell them how, at the time of Galileo, the heavens were considered “still and immaculate”. But Galileo was curious and determined to test ideas by himself. When he pointed his telescope to the Sun, he discovered, to his great surprise, that the Sun had dark spots and that from one day to the next, these were moving across the solar surface.

- Distribute to the class copies of Galileo's drawings (file 'Galileo drawings.pdf'), also shown in Fig. 3

*Fig. 3: Galileo Galilei’s drawings from 24-27 June 1613. Note that the solar north pole is not oriented straight up, as it usually is in modern scientific images.*

- If you have a beamer, you can project the movie 'Galileo drawings'.

- Ask them what they think these sunspots could be, and tell them that Galileo himself was puzzled (we only know their “magnetic” nature since 1905, see Background).

- Ask them why they think sunspots appear to be moving.

- If you have a beamer, play a movie of the Sun rotating with the SDO images provided by the activity (in .jpg format, as these do not have latitude lines). This way the students can “see” and “feel” the solar rotation. You can use the free astronomy-for-classroom software SalsaJ (see “How to create an animation with several images” at http://eu-hou.net/index.php/salsaj-software-mainmenu-9/manual-salsaj-2)

- While they watch the movie, insist on the idea that everything is in motion in the Universe and everything spins!

- Make students come up with the idea of measuring the rotation period and how to do it.

Ask them: - Knowing that the Sun is rotating, what could we try to measure? - If you had these images (the ones shown in the movie) in your hands, how do you think you could determine the rotation period? *They might say that we could wait for a sunspot to make a complete turn and reappear at the same position. Unfortunately, most sunspots do not “live” that long.* - What if you could measure the average speed of a sunspot?

- Group students in pairs and give each pair an example of a printed dataset (at least two images) or show them how to find the images on the computer if doing the activity electronically (www.solarmonitor.org).

- Now that students are convinced that the Sun is a rotating celestial body, their task as scientists will be to calculate as precisely as possible the time it takes to complete a full rotation, that we call the rotation period T.

**2. MODELLING THE TRAJECTORY OF SUNSPOTS**

**Model** – We will assume that the Sun is perfectly spherical and that it rotates as a rigid body at a constant pace. In that case, the Sunspots on the solar surface are moving at a constant speed.

With this simple model, calculating the rotation period is equivalent to determining the time needed for a sunspot to complete a full turn around the Sun.

Students should first grasp how we can describe the motion of a sunspot on a spherical Sun and how the rotation period relates to the sunspot speed and trajectory length.

*Fig. 4: Sketch of a sunspot’s trajectory as seen from Earth (front view), and the same trajectory if we could see the Sun in top view, evidencing a circular trajectory (in green).*

- Draw schematics of the Sun along with a sunspot and its trajectory as viewed from the front (i.e. as we would see it from Earth). See Fig. 4.
- Let students complete the top view (i.e. if we were looking at the Sun from above its North pole) and add the trajectory of the Sunspot represented in the front view (in green in the image below). We will note the length of this trajectory as L. What is the shape of this trajectory? Figure 4 shows the correct circular trajectory as a dashed-green line.
- Ask students: If they knew the speed of a sunspot v, and the length of the trajectory L, how could they compute the rotation period of the Sun? We will note this period as T.
- Let students write their equation for T: they should find
**T = L/v.** - Write their equation in big letters on the board.
- Explain to students that in this problem, L and v are unknowns. How to find them? They’ll need to measure them from the images.
- Ask the students to check the units in the equation: To compute the rotation period with their equation, do they need to know the scale of the images (i.e. to have distances in km), or can they simply use a ruler and measure distances in cm and speeds in cm/h? Students should find that the distance units cancel out in the equation for T. Although images are smaller spatial representations of reality, the time dimension is unchanged.

**3. MEASURING THE SPEED OF A SUNSPOT**

To benefit the final discussion, it is best that each student group works with a different sunspot, in order to compare the measurements performed on different spots.

- Depending on the number of students and the activity timing, you can either let them choose a sunspot number (indicated on the images) within the complete dataset or assign a sunspot number to each student pair from the start. In the latter case, it is sufficient to give them two consecutive images featuring that spot near the disk centre.
- If you did not assign a specific sunspot to each group, tell students to look at the dataset, choose a sunspot and find its 5-digit number. Make sure that different student pairs chose different sunspots (write them on the blackboard or on a google spreadsheet if you can project it);
- Tell them to look at consecutive images of that spot and think about how they could measure the average speed of that spot in [cm/h].
- Assuming that sunspots move at a constant speed on the surface of the Sun, their instantaneous speed at any instant in time is equal to their average speed. We can then simply apply the equation of the average speed between two images :

**v = d/tel =** (distance travelled between the two images) / (elapsed time between the images) [cm/h]

- By using a flexible ruler on a spherical ball (simulating the Sun) as in Fig. 5, ask students where is it easier to read the ruler to estimate distances: near the centre or near the edge of the ball? This should guide the students to the idea of taking two consecutive images of a sunspot near the disk centre. Otherwise, the distances will appear foreshortened. For the same reason, it is best that the images are close in time (in the case of this dataset, the minimum separation is one day).

*Fig. 5: A flexible ruler on a ball illustrates that in an image of a spherical object, only the distances measured near the disk centre are accurate because the distances nearer the limb are foreshortened.*

From here, student teams can follow these steps:

1) Pick two consecutive images of their numbered sunspot near disk centre.

2) Think about a method to measure as precisely as possible the distance travelled d (cm) by the spot between the two consecutive images, with the help of a ruler. As an example, Fig. 6 shows the displacement of sunspot 12218 between the 29th and 30th November. (With the paper version of the activity, students can use the left edge of the images as a reference to place the ruler, or the white longitude lines on the images as help to place their rulers.)

3) Find the elapsed time t (h) between the two consecutive images (the time at which an image was taken is indicated at the top of each image).

4) Compute the speed v.

5) Write the measurements on the blackboard or in a google spreadsheet.

*Fig. 6: Measuring displacement. Letter “d” represents the distance travelled by a spot between two consecutive images (see text for more information).*

**4. MEASURING THE LENGTH OF THE TRAJECTORY**

Students should :

- Draw the diameter D of the trajectory on the front and top views of the sketches of they did at the beginning of the activity (see Fig. 7).

*Fig. 7: Sketch of the full diameter travelled by a sunspot.*

- Measure the diameter D of the sunspot’s trajectory on the images as shown in Fig. 8. How can this measurement be used to compute the length of the trajectory L? Students are expected to use the perimeter-diameter relationship for a circle: L = πD

*Fig. 8: Measuring the diameter D of the spot trajectory (in this case sunspot 12236)*

**5. COMPUTING THE ROTATION PERIOD**

- Students can use their previous measurements to compute the rotation period T in hours and in terrestrial days (1 day = 24 h) using the formula: T = L/v.
- When they finish, students should write all their values on the blackboard table or the google spreadsheet.
- Using the blackboard or a google spreadsheet will make all their results visible to the class and facilitate the discussion (see Table 1 for an example).

Table 1: Measurements taken for all sunspots that were near the disk centre for two consecutive days. The mean value of T is 26.8 terrestrial days for this dataset.

**6. COMPARING AND DISCUSSING THE MEASUREMENTS**

Now the student teams will compare their results between themselves and with the official value. They should obtain rotation period values close to 26 or 27 days, otherwise, check for calculation errors.

Also, pay attention to whether students have chosen images where their spot is near the disk centre, otherwise they will underestimate the distance travelled by the spots between the two images, and therefore underestimate the velocity and overestimate the rotation period (see below).

- Show them the official value from Wikipedia (https://en.wikipedia.org/wiki/Solar_rotation) or NASA (https://www.nasa.gov/mission_pages/sunearth/science/solar-rotation.html), or tell them to search for it if they are on a computer. You have to search for the “synodic” rotation period (about 27 days).

*How close are they from the official value?*

*You can make them compute a relative error.*

- Ask them to examine all results of T:
*Did all student pairs agree on the same value? Let’s find out why!* - Each team shall explain to the rest of the class how they performed their measurement (especially of the distance travelled d).

*Can you find some measurements errors or calculation errors?*

*As scientists, it is crucial to discuss why we did not obtain a single “true” answer to our question. The experimental nature of science means that we never get a unique answer: Measurement results fluctuate either because the phenomenon itself fluctuates or because WE fluctuate in our way of measuring things! So let us consider explanations for the discrepancies.*

- If you recorded the measurements of students in a spreadsheet, you can plot their values of T against sunspot number to make them see the dispersion in their results (see Fig. 9).

*Fig. 9: Rotation period vs. Sunspot number for all the spots measured in Table 1. The horizontal line is the mean value of the dataset.*

- Make students think about the phenomenon of solar rotation and the model they used to describe it:

*Think about the model we considered to start with… can we really model the Sun as a rigid body? Why or why not?*

*How about the sunspots, do they look exactly the same on two consecutive days? Why or why not?*

*Scientists have shown that the rotation period is longer near the poles (sidereal rotation period of up to 38 days) than at the equator (about 25 days), a phenomenon called differential rotation or non-rigid rotation. Yet students won't be able to clearly see this on your dataset because most spots are close to the equator.*

- Ask students: How can we combine all measurements to compare them with the official value?

*We can compute an average value and its uncertainty. At school level, the uncertainty can simply be taken as a “dispersion” or “rank”, taking care of removing outlier measurements: (max value - min value)/2. Here we find a dispersion of ±2 days (a more proper standard deviation yields 1.4 days).*

- Average the measurements of T obtained by all teams to get a class average (see Table 1 and Fig. 9). Compare again with the official value.

*How close is it to the official value?*

*Do they agree within the uncertainty of our measurement? That’s great teamwork!*

- Make students realize that they succeeded in measuring the rotation period of a star using only one Kinematics equation (average speed)!
- Make students reflect about the meaning of their average value of T:

*Compare the rotation period of the Sun you calculated with the rotation period of the Earth: is it shorter? longer? Does it make sense to you (think about the relative sizes)?*

*Suppose you are a solar astronomer and find a new sunspot appearing near the left border of the solar disk. How long do you have to observe this spot before it disappears on the right side?*

- Finally, let us question the observer’s perspective on the average rotation period. Does it really represent the time that the Sun actually takes to rotate on itself, or is it also a matter of the observer's perspective? Think about the fact that the images were taken from Earth, which revolves around the Sun.

Since we have not taken into account the motion of the Earth around the Sun in our treatment, all we can say is that as viewed from the Earth (in this case, from the SDO satellite that orbits the Earth), a feature on the Sun's surface completes one rotation on around the Sun in about 27 Earth days (which is called “synodic” rotation period). Yet because the Earth orbits the Sun in the same direction as the Sun rotates (see Fig. 10), this rotation period as seen from the Earth is longer than the rotation period of the Sun seen from a static observer, which is about 25 days (called the “sidereal” rotation period).

*Fig. 10: Sketch showing how the Earth revolves around the Sun in the same direction as the Sun’s rotation.*