Day Lab (Part 3): Rotational Period of the Sun

In the first two parts of the lab, we calculated the angular diameter of the Sun, $\theta_{\odot} = 0.5547°$ and the rotational velocity of the Sun, $v_{\odot} = 1.86 km/s$. Using these two values, we can calculate the time it takes for the Sun to make one full rotation, also known as it's period, $P_{\odot}$.

In order to calculate the full rotation of the Sun, it would make sense to use a marker on the Sun, and see how long it takes to see that marker again, which would indicate the period, $P_{\odot}$. Unfortunately, there are no permanent markers on the Sun, since the surface of the Sun is constantly changing. However, we do have temporary markers on the Sun, sunspots, which can be tracked as the Sun is rotating. Using sunspots, we can create a relationship between the the angular distance, $\theta$ a sunspot travels as it moves across the surface of the Sun over a given period of time, $t$, and the rotational period of the Sun, $P_{\odot}$ as it travels a full 360°.

This relationship can be modeled as follows:

\begin{align}
\frac{P_{\odot}}{360°} &= \frac{\Delta t}{\Delta \theta}\\
P_{\odot} &= \frac{\Delta t (360°)}{\Delta \theta}
\end{align}

Okay, so we have a model to calculate the period, but how do we measure the transit of the sunspots across the surface of the Sun?

Well, we could directly view sunspots by projecting an image of the Sun via mirrors on a piece of paper, and see how the sunspots that are visible over the course of a few days. However, when doing this lab, the day was rather cloudy, so we decided to use sunspot data from NASA in 2001.

Using this data, we can overlay a live recording of the surface of the Sun with graph of spherical coordinates, and measure the change in angular distance over a given period of time. In this lab, we recorded the transit of 3 sunspots, with the following data:

Sunspot 1: 

Trial 1: 03-04-2001 at 17:36 to 03-09-2001 at 11:12

$\Delta \theta = +70°$, $\Delta t = 408,960$ seconds 

Trial 2: 03-06-2001 at 00:00 to 03-08-2001 at 17:36

$\Delta \theta = +40°$, $\Delta t = 236,160$ seconds 

Trial 3: 03-05-2001 at 11:12 to 03-10-2001 at 00:00

$\Delta \theta = +67°$, $\Delta t = 291,680$ seconds 

Average: 

$\Delta \theta = 59°$, $\Delta t = 345,600$ seconds 

$P_{\odot} = \frac{\Delta t (360°)}{\Delta \theta} = \frac{345,600 (360°)}{59°} =$ 24.4 days

Sunspot 2: 

Trial 1: 03-22-2001 at 00:00 to 03-25-2001 at 11:12

$\Delta \theta = +50°$, $\Delta t = 299,520$ seconds 

Trial 2: 03-20-2001 at 17:36 to 03-27-2001 at 17:36

$\Delta \theta = +100°$, $\Delta t = 604,800$ seconds 

Trial 3: 03-22-2001 at 17:36 to 03-27-2001 at 00:14

$\Delta \theta = +60°$, $\Delta t = 369,480$ seconds 

Average: 

$\Delta \theta = 70°$, $\Delta t = 424,600$ seconds 

$P_{\odot} = \frac{\Delta t (360°)}{\Delta \theta} = \frac{424,600 (360°)}{70°} =$ 25.3 days

Sunspot 3: 

Trial 1: 04-06-2001 at 08:45 to 04-11-2001 at 00:00

$\Delta \theta = +60°$, $\Delta t = 400,500$ seconds 

Trial 2: 04-04-2001 at 19:12 to 04-10-2001 at 06:24

$\Delta \theta = +70°$, $\Delta t = 472,200$ seconds 

Trial 3: 04-07-2001 at 00:00 to 04-14-2001 at 11:12

$\Delta \theta = +90°$, $\Delta t = 645,120$ seconds 

Average: 

$\Delta \theta = 73.3°$, $\Delta t = 505,940$ seconds 

$P_{\odot} = \frac{\Delta t (360°)}{\Delta \theta} = \frac{505,940 (360°)}{73.3°} =$ 28.7 days

Having three measurements of rotational period, $P_{\odot}$, we can take the average of all three to get the rotational period of the Sun to be $P_{\odot} = 26.1$ days. The actual rotational period of the Sun at the equator is 24.7 days, but that is the sidereal rotation period. The method we used to get the period required following a fixed feature on the Sun, which is measured by synodic rotational period, which for the Sun is 26.24 days. The synodic rotational period of the Sun is very close to our measurements.