This is the second post of the series of four posts on this lab. In this post, we discuss how to find the rotational speed of the Sun using Doppler shifts due to the Sun's rotation. A Doppler shift is a change in the frequency of a wave due to motion of the emitting source relative to the observer. If the emitting source is moving towards the observer the frequency appears to be larger, and if the source is moving away from the observer, it appears to be smaller. This occurs because as the source moves with respect to the observer, the crests of each wave are emitted at a closer or further position from the observer than the previous one depending on the direction of the source's motion. This causes the second crest to reach the observer either before or after it would have had the source been stationary, thus giving the appearance of a different frequency than the one emitted. In astronomical terms with regards to light, an increase in frequency is called a blueshift and a decrease is called a redshift.
In the case of the Sun, since it is rotating, one side of the Sun is moving towards us and the other side is moving away from us, causing a redshift in the Sun's spectrum on one side, and a blueshift on the other, as shown in the image below.
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| Image Credit: http://a-levelphysicstutor.com/images/waves/dopp-redshift03.jpg |
If we find the Doppler shift in the spectrum of the Sun's light on either side of the Sun, we can find the relative speed of either side of Sun, which is the rotational speed of the Sun.
The Sun emits light at many wavelengths, however, so we must choose a specific region to observe. The NaD absorption lines in the Sun's spectrum are well known and are caused by sodium. They occur at 5889 and 5896 Angstroms. Using a sodium lamp, we can see where the sodium lines are without a Doppler shift, and use this knowledge to calibrate the spectrograph. Then using the Telluric absorption lines which won't be Doppler shifted since they are caused by the \(\text{H}_2\text{O}\) in the Earth's atmosphere, we can align the spectrum from the Sun to the non-Doppler shifted sodium spectrum and compare their positions to find the rotational speed of the Sun.
Equipment:
A spectrograph was used to measure the spectrum of the Doppler shifted sodium absorption lines and the Telluric lines. The spectrograph separates the light into a frequency spectrum which is then recorded by a CCD camera.
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| Spectrograph Image Credit: http://www.fas.harvard.edu/~astrolab/astrolab.html#heliostat |
We used the Heliostat to reflect the sunlight into laboratory and into the spectrograph. The mirrors had to be aligned correctly so that the reflected light was positioned into the spectrograph and then the Heliostat was set to follow the Sun's motion through the sky.
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| Heliostat Image Credit: http://www.fas.harvard.edu/~astrolab/astrolab.html#heliostat |
As already mentioned, a sodium lamp was used to calibrate the spectrograph and to have a comparison for the Doppler shifted spectrum of the Sun. Finally, the softwares, MaximDL and Excel were used to perform the analysis of the data.
Procedure:
First, for calibration, we placed the sodium lamp in from of the slit of the spectrograph and adjusted the slit width until we had a non-saturated, sharp image from the CCD camera. The camera itself was also moved so that the spectral lines were centered in the image. The quality of the image could be tested through short exposures on MaximDL until the desired image was found. It is important to note that no sunlight should get through the spectrograph.
After calibration, the sodium lamp was removed and the sunlight from the Heliostat was directed into the slit. A test image was taken to make sure everything was still aligned correctly and adjustments were made if necessary.
Since we don't know the direction of the Sun's rotation, we had to take measurements at four pairs of points: top, bottom; left, right; top left, bottom right; bottom left, top right. The pair of points with the largest difference in spectral lines would be the ones closest to the line of the equator, and therefore the direction of rotation.
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| Solar regions that were observed |
Next, the spectral images taken by the CCD camera were opened on MaximDL and a narrower box (about 10 pixels wide) was drawn across each of the eight images spanning the entire width of the image (from left to right). The data acquired from MaximDL through this process was then exported and saved for each image.
Using an Excel Template that was provided, the exported data for each of the eight images was transferred into their appropriate columns in the "Raw Data" sheet. On the "Normalized Plots" sheet, the data was plotted, and we could see the shifts between the pairs of points. The pair that had the largest separation was selected, and in this case, we found that the left and right points had the largest difference. The corresponding columns for the pair with the largest shift was then copied and pasted into the columns on the "Shift Data" sheet and larger plots of these two columns were made on the "Analysis" sheet. A second order polynomial was then fit to the plots and the range of the x-values was adjusted so that the polynomial and data points matched up as well as possible. The same thing was done for the plot of the Telluric lines and any shifts in the Telluric lines were accounted for in the shifts of the NaD lines. Additionally, a conversion factor for pixels to Angstroms was found using Excel. Finally, using the Doppler equation, we were able to solve for the difference in velocities on either side, and this difference was divided by two in order to account for the different frames of reference. And thus the solar rotational speed was found.
Results and Analysis:
The following graph is what was shown in the "Normalized Plots" sheet. It is the plot of the spectra of each image, where the x-axis is the pixel count, and the y-axis is intensity.
From this plot, we can see that the red lines have the largest difference, corresponding to the left and right images of the Sun. The two largest dips are spectral absorption lines associated with NaD. We chose to perform our analysis on the left-most absorption line. The following image is a close up of the left absorption line.
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| Left Absorption Line |
Here, we can see more clearly that the black lines, corresponding to the bottom and top readings, are the most separated and that the red lines corresponding to the left and right readings are the least separated. So we know that the Sun is rotating in the top-bottom direction.
Now we can do an closer analysis of both absorption lines in the NaD spectrum for the top and bottom images. The individual plots of the NaD absorption lines are shown below.
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| Left absorption line with corresponding 2nd order polynomials with a shift of 4.91 pixels |
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| Right absorption line with corresponding 2nd order polynomials with a shift of 4.78 pixels |
For calibration, we have to look at the shift in the Telluric lines, which come from the water in Earth's atmosphere, so that we can account for it in our calculation of the Doppler shifts. There should be no shift in the Telluric lines, however we do observe one. This is not caused by a Doppler shift since they Earth's atmosphere is not moving with respect to us. A possible explanation is that some of the light is being diffracted as it passes through the atmosphere, before reaching us.
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| Telluric line with corresponding 2nd order polynomials with a 1.99 pixel shift |
Finally, we need to find the conversion factor from pixels to Angstroms before we can continue our calculations. We know that the two sodium absorption lines should be separated by 5.97 Angstroms, and Excel shows that there are 332 pixels separating the absorption lines. So, on Excel, we calculated that there is a conversion factor of 0.018 Angstroms per pixel.
Now, to do the calculations, we first need to subtract the shift in the Telluric line from the shifts in the left and right NaD lines to account for the effects of the atmosphere. So we get shift of 2.91 on the left NaD line and 2.79 on the right NaD line. Then we convert these from pixels into angstroms to get shifts of 0.052 Angstroms on the left and 0.05 Angstroms on the right.
We are ready to apply the Doppler equation to solve for velocities:
\(\dfrac{\Delta v}{c}=\dfrac{\Delta \lambda}{\lambda}\)
Where c is the speed of light, \(\Delta v\) is the difference in velocity of the two sides of the Sun. \(\Delta\lambda\) is the difference in wavelengths between the two lines, and \(\lambda\) is the emitted wavelength. We want to find half of \(\Delta v\), which will give us the rotational speed:
\(v_\odot =\dfrac{\Delta v}{2}\)
So we get:
So we have an average solar rotational speed of 1.31 km/s.
Error Analysis and Discussion:
According to Wikipedia, the rotational velocity is 2 km/s. So our answer is off by about 0.7 km/s, which is a percent error of about 35%. There are several possible explanations for this large error. First of all, any bumping of the equipment or small mistakes in analysis could cause an error. Additionally, the sample size was small, and perhaps if we had done more trials, we would have had better results. We might have also been incorrect in our assumption that the top and bottom points of observation were on the equator. Also, the shifts in the Telluric lines, which were larger than one might expect, would have added more error to our data as well. Future experiments should be more careful in performing observations and should have more trials.
To calculate the standard error, we use the same method from our first post for this lab, which gives us a standard deviation of 0.057 km/s and a standard error of 0.04 km/s.
I would like to thank Charles Law and the TFs Allyson and Andrew for their help in completing this lab.
Citations:
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