AU Lab: Rotational Velocity

Theoretical Background

Going from the doppler shifts at different spots on a rotating object to the objects rotational velocity is fairly easy.  You use this equation:

$\frac{\mathbf{source\: velocity}}{\mathbf{velocity\: of\: light}}=\frac{\mathbf{change\: in\: wavelength}}{\mathbf{rest\: wavelength}}$

Figuring out where to measure the doppler shifts is more difficult.  In order to use the doppler shift to accurately measure an object's rotational velocity, one has to measure the shifts at opposite points along the object's diameter.  To find measurements from the most accurate (farthest apart) locations, because we don't know where the equator of the sun is, we took doppler shift measurements at four different sets of opposite points.  Whichever set gives the biggest difference in shifts is the line closest to being a diameter of the sun.  

Supplies

• Heliostat - a device that uses mirrors to angle a reflection of the sun toward a target
• Spectrometer- a device that, in this context, is used to determine the wavelengths of light present in a light source

From the Wikipedia page on Spectrometers

• Sodium Lamp

Process

The first step is to calibrate the CCD to the sodium wavelength lines by shining the sodium lamp into the receiving slit of the spectrometer.  This is a very tedious process, involving meticulously adjusting the slit width, focal length of the camera, and exposure time to get the sharpest images of the sodium lines possible.  Once that is done, and you have an accurate measurement of the wavelengths of the sodium lines, remove the sodium lamp and get measurements using the sunlight.  

DO NOT move the CCD, because it is now perfectly aligned, and if you mess things up, your group will be very angry with you.  

Using the heliostat, angle the image of the sun into the spectrometer.  Use the motor to move the image reflected by the heliostat and take measurements of the sodium lines at eight different spots (shown below) on the sun's outline. 

Measure the shifts relative to the shifts of the water (telluric) lines, which shouldn't move because their measurement doesn't depend on the motion of the sun (the telluric shift should be 0).  

Analyze the shift data to see which set of points has the greatest difference in doppler shift, as this set of points will be closest to the sun's diameter.  

Analysis

We took two sets of measurements at each point so as to get more accurate results.  Measuring the doppler shifts at each of the points indicated in the figure above provided us with the following two graphs.  

There's little differnce between the two graphs, which is good, because that means there should be little difference between the two groups' results, which will be averaged at the end. 

The particular doppler shifts we're looking at are the shifts of the sodium lines, which are located at about 325 and 665 angstroms (along the x-axis).  When we zoomed into those two spots, we found that the lines representing the shifts at the top right and bottom left of the sun had the greatest difference.  To find the exact shift between those two locations at those wavelengths, we analyzed the following graphs.  

For lower-wavelength sodium line

For the higher-wavelength sodium line

I said before that we had to measure the shift of the sodium lines relative to the shift in the telluric lines, which should be 0.  Our group actually measured a small shift in the telluric lines,

which caused some issues.  The telluric lines shifted to the left, which decreased the relative shift of our sodium lines.  

Results

Our two different groups, after using the computer software to convert from the doppler shift between the two opposite spots to the rotational velocity using an equation similar to the one in the Theoretical Background section*, found the following velocities in km/s:

1.913

1.971

2.021

1.897

Averaging these four velocities gives a result of 1.951 km/s.  This is not the final rotational velocity of the sun, however, because the change in wavelength we observed by finding the difference in doppler shifts at the two opposite locations is actually twice the real difference.  One side of the sun is moving toward us at the same speed that the other side is moving away from us, so the relative speed between the two is twice the actual value.  In order to get the rotational velocity of the sun, one must divide the above value by 2, making the rotational velocity of the sun 0.975 km/s!

Discussion

The actual rotational velocity of the sun is about 2 km/s, so our team's result is less than half what it should be.  Why is our error so large? 

It's possible that our equipment--the camera we were using--was just really noisy.  Our results were also definitely affected by the fact that we measured a shift in the telluric lines.  Also, it's not guaranteed that we found the equator by measuring the shifts at the eight different spots on the sun.  Measuring the shift differences only guaranteed that we used the diameter line closest to the equator (compared to the others we tested).




*The process used by the computer program wasn't identical to the one described by the equation, because the program had to convert from pixels to wavelength by using the CCD's plate scale, which is the physical size subtended by one pixel.  The plate scale is a quantity measured in arcsec/pixel that can be found using this formula

$\Theta =\frac{P}{F}$

where P is the length of an indivual pixel and F is the focal length of the camera. This way, a shorter focal length gives you a wider field of view (bigger plate scale).  Theta has to be converted into arcseconds because this equation gives you radians.