Measuring the Distance from The Earth to The Sun

 

   Introduction
    To measure the distance between the earth and the sun (the AU), we used a heliostat to determine the angular size of the sun using the heliostat, measure the rotational speed of the sun using the spectrograph, and measure the rotational period of the sun using analysis of sunspots. The rotational speed ($v_{rot}$) and rotational period (P) will allow us to calculate the radius (R) of the sun using the equation $v_{rot}= \frac{2 \pi R}{P}$. Then, we can use triangle geometry to solve for the distance between the sun and the earth as shown in Figure 1.

Figure 1: Solving for distance between the earth and the sun given the radius of the sun and the angular diameter of the sun.



Methods

Angular Diameter:

A heliostat was used to measure how long it takes for the sun to move a length of its own diameter. A heliostat is a series of mirrors that reflects the light of the sun into the lab classroom and onto a table (Figure 2). Once lined up with the sun, tracking can be used so that the heliostat follows the sun, or it can be turned off so that it remains stationary. With the sun's light projected on the table we can measure how long it takes with the time recorded as shown in Figure 3.  To take these measurements, the outline of the sun's reflection is sketched onto paper so it is clear when the sun has travelled the length of its diameter, so we know when to start and stop the time (Figure 4). This experiment was repeated for a total of three trials and the average of the trials was calculated. These results are summarized in Table 1.

We know that the earth makes one rotation about every 24 hours so the sun appears to move 360 degrees or 2 $ \pi $every 24 hours. 

Therefore we can relate these proportions:

$\frac{24hrs \times 60 min/hr \times 60 sec/min}{2 \pi } = \frac{131.943 s}{\alpha } $
$ \alpha = \frac {263.9 \pi}{86,400} $
$ \alpha = 0.01 radians $

Figure 2: The heliostat uses mirrors to project the light from the sun inside the classroom [1].




Figure 3: The time is recorded as the reflection of the sun moves the length of its own diameter.

Figure 4: The edges of the reflection of the sun are sketched to measure angular diameter [1].




Table 1: The time it takes for the sun to travel the length of its diameter.





Rotational Speed:

    The doppler effect occurs because when a star is moving away from earth the light shifts towards a longer, redder wavelength and if the star is moving towards the earth, the light shifts towards a shorter, more violet wavelength [3]. This is useful because the sun is rotating so one side of the sun is moving towards earth and the other side is moving away from earth. 

    The only complication is that we do not know which axis the sun is rotating about. To clear this up, we can take measurements that account for four possible axes of rotation so four possibilities for which side is moving towards us and four possibilities of which side is moving away from us (Figure 5). We will analyze each of these data and determine which has the clearest observable doppler shift. To be sure what we are seeing is a doppler shift, we will also analyze telluric lines. Telluric lines are lines from the water in Earth's atmosphere so they should not have an observable doppler shift.

    To measure doppler shift, we can use a spectrograph and measure the sodium lines of the sun. We first measured the sodium lines from a sodium lamp to set up the spectrograph. We then removed the lamp and allowed the sunlight to shine through the slit of the spectrograph. We started by aligning the left side of the sun with the slit, then the right, top, bottom, bottom left, top right, top left, and bottom right sides of the sun to account for the possible axes of rotation. We then looked through the microscope lens and took pictures of each of the sodium lines (Figures 6 and 7). Python was used to plot the intensity of light across the pictures which allowed us to look at the doppler shift for each group of two pictures (left and right, top and bottom, top left and bottom right, and top right and bottom left) (Figure 8). We determined that top right and bottom left graphs showed the greatest doppler shift (Figure 9a). Each sodium line was zoomed in on to confirm the shift (Figure 9b and Figure 9c). A telluric line was found using [2]. This was plotted to confirm there was no doppler shift (Figure 9d).

     The curves were fit and minimum values were determined using python. We found $v_{rot}$ using the equation $v_{rot}= \frac {c \times \delta \lambda} {2 \lambda} $ 
$\lambda$ is a reference wavelength in this case the emission from a sodium lamp.
$ \delta \lambda$ is the difference in wavelength between the top right and bottom left data.


    We find that $v_{rot}$ on the top left is 1.978 km/s
    $v_{rot}$ on the bottom right is 2.010 km/s
    Average $v_{rot}$ = 1.994 km/s
    The shift on the telluric line is -0.042 pixels which is and should be close to 0
    Total offset on NaD left line is 4.291 pixels
    Total offset of NaD right line is 4.356 pixels

The full python colab can be found here: https://colab.research.google.com/drive/1upy9nNWSJJbXvzjCwgIgyPjx202aJxAG?usp=sharing
   

Figure 5: Position of sunlight into spectrograph.


Figure 6: Looking into the spectrograph through the microscope lens to look at NaD lines[1]





Figure 7: Photographs taken from the spectrograph of sodium lines from different sides of the sun.


Figure 8: Plot of normalized intensity v pixels for each of the photographs taken on the spectrograph.



Figure 9a: The greatest doppler shift was observed between the top right and bottom left sides of the sun.




            Figure 9b: Clear shift in the left Na D.                    Figure 9c: Clear shift in the right Na D line. 


Figure 9d: No significant observable shift in the telluric line.



Rotational Period:


The sunspots can be measured by reflecting the light from the sun into the classroom using the heliostat. The sunspots are visible from the reflection so they can be marked on a sheet of paper and observed over the course of a few hours/days. This was not possible for our lab group because there were too many clouds so the heliostat could not be used consistently enough to measure the path of the sunspots. Therefore, we used archived data (Figure 10) [5]. We measured the time it took for a certain sunspot to move across the sun by overlaying a grid that helped us measure the movement (Figure 11). We repeated this experiment for a total of three trials. Knowing how far the sunspots move over a certain amount of time, we can determine how long it would take for the sunspots to move 360 degrees which is the period of rotation (Table 2a and Table 2b).


Figure 10: Sunspots move across the sun overtime (Archived data [5]).




Figure 11: Grid to overlap on the sunspots to measure movement over time.




Table 2a: Change in position of sunspots over time.



 Table 2b: Rotational period of the sun based on movement of sunspots



Average Period = 2140320 seconds
Average Period = 24.77 days

Analysis

The useful values that we have found so far are:

$ \alpha = 0.01 radians $
Average $v_{rot}$ = 1.994 km/s
Average P = 2140320 s

Now this equation can be used to find R $v_{rot}= \frac{2 \pi R}{P}$
$R = \frac {v_{rot} \times P}{2 \pi}$
$R = \frac {1.994 km/s \times 2140320 s}{2 \pi}$
$R= 679,241km $

Finally, we can return to Figure 1 to find the distance between the sun and the earth.



Using the small angle approximation, we can approximate that $tan(\alpha) = \alpha$

$\frac{\alpha}{2})= \frac{R}{d}$
$d= \frac{R}{\frac{\alpha}{2}}$
$ d=135,848,200 km= 1.36 \times 10^{13}$

The actual value is AU=1.49597870 x 1013 cm 
So the actual value of about $ 1.5 \times 10^{13}$ cm is very close to the calculated value of about $1.4 \times 10^{13}$

Uncertainty
Uncertainty of timing device: +/- 0.01s
Angular Diameter: $ \alpha = 0.01 radians +/- 0.01$

Uncertainty of velocity: +/- 0.001 km/s
Rotational Velocity: Average $v_{rot}$ = 1.994 km/s  +/- 0.001 

Uncertainty of movement of sunspots: +/- 60 s/1 degree
Rotational Period: Average P = 2140320 s +/- 60 

Distance between sun and earth $1.36 \times 10^{13}$ +/- 12,000

Possible error:
When calculating angular diameter, one student was operating the heliostat, one student was timing the movement of the sun, and one student was sketching the outline of the sun. The student operating the heliostat was outside while the other students were inside so there was some delay in turning off the tracking of the heliostat, and starting the timer which could lead to recorder times being greater than the actual time.

When calculating the rotational velocity, we took into account the complication that we did not know which axis the sun was rotating about. To correct this, we took eight measurements and used the two that produced the greatest doppler shift. This likely decreased error, but it was still only an estimation because a greater doppler shift could have been occurring on sides of the sun that we did not measure.

When calculating the rotational period of the sun, we ignored the fact that the sun has differential rotation.

Acknowledgements

I would like to thank Allyson Bieryla for instructing my groups first two lab classes when we measured angular diameter and used the heliostat and spectrograph. I would like to thank Jea Adams for instructing our last lab class where we completed the rotational velocity python calculations. I would like to thank my fellow lab members including Callie Garcia, Larom Segev, Max Christopher, Mekhi Moore, Nyle Garg, Owen Bae, and Tovi Sonnenberg for participating in lab class with me.

Sources

[1] Gupta, S.: Sky is the limit, https://news.harvard.edu/gazette/story/2016/07/sky-is-the-limit/.

[2] LESIA, P.O.-: Solar spectrum, http://bass2000.obspm.fr/solar_spect.php.

[3] Doppler effect, https://www.britannica.com/science/Doppler-effect.

[4] Harvard University, https://astrolab.fas.harvard.edu/AU_Ay16.html.

[5] Solar and heliospheric observatory homepage, https://soho.nascom.nasa.gov/home.html.

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