Day Lab (Part 2): Rotational Speed of the Sun

We said in the first part of the lab that in order to calculate the astronomical unit (AU), we will be using the Doppler shift technique. In order to use the Doppler shift technique, we need to know the angular diameter of the Sun, $\theta_{\odot}$, as well as its rotational speed. We got the angular diameter of the Sun in the previous part, which was $\theta_{\odot} &=  0.5547°$. Now, we need the rotational speed of the Sun.

Wait, what?!?! How can we tell how fast the Sun is rotating?

Well, we can use the spectral lines of the Sun, specifically the Sodium D line (NaD) spectral lines, to see if they are moving towards us or away from us using blueshift and redshift, respectively, of the light coming from the Sun as a result of the Doppler effect, and compare the movement of the NaD spectral light with the $H_2O$ Telluric line that is absorbed by the atmosphere, and therefore stationary.

In order to get the most accurate rotational speed of the Sun, we want to measure the movement of the spectral lines from the equator of the Sun. Unfortunately, we don't know what the orientation of the Sun is due to the tilt of the Earth, as well as the several mirrors that are used to project the solar disk into the spectrometer. Therefore, we will take 8 measurements of the NaD spectral lines, in the pairs that are opposite from one another:  (top, bottom), (left, right), (top-left, bottom-right), and (top-right, bottom-left). The orientation with the widest gap between the two pairs of NaD spectral lines would be at the solar equator.

After taking the picture of the NaD spectral lines from those 8 orientations, the resulting data showed the following:

Looking at the 8 measurements of the spectral lines, it is apparent that the greatest shift in the NaD spectra lines took place between the "top-right" and "bottom-left" orientation, indicated by the shift in the purple lines. Therefore, that is the orientation of the Sun that measures the spectral shift at the equator.

Knowing that the "top-right" and "bottom-left" orientation was the equatorial orientation, we can use the shift data to determine the exact redshift that occurs, and compare it to the Telluric spectral lines to calculate the rotational speed of the Sun.

Left sodium absorption line, with a pixel shift of 3.98 pixels.

Right sodium absorption line, with a pixel shift of 4.21 pixels.

In order to compare the pixel shift of absorption between the sodium absorption lines, we need to look at the Telluric spectral line of the Earth's atmosphere:

The Telluric Line has a pixel shift of 2.09 pixels. 

There should be no shift in the Telluric Line, because they are part of the Earth's atmosphere, and therefore are not redshifting. This is most likely due to the scattering of light in the Earth's atmosphere.

Based on the data from these graphs, the difference between the left and right sodium lines is 335.01 pixels separating the two, which is a separation of 5.97 angstroms, since the conversion factor is 0.017821 angstroms per pixel.

We can use the spectral separation of 5.97 angstrom to calculate the rotational velocity using the Doppler equation, which gives us an average rotational velocity between the two sodium spectral lines of the sun, $v_{\odot} = 1.86 km/s$.

Since the actual rotational speed of the Sun is 2 km/s, the calculated rotational velocity has a 7% error, which is rather good.