How Far Away is the Moon – using a lunar eclipse

The distance to the moon is an easy thing to look up nowadays and with lasers and corner reflectors left on the moon from the Apollo missions the accepted values for the lunar orbit are very well defined. However, the distance to the moon has been known for thousands of years and with only a few assumptions you can still calculate this distance if you simply take a couple of measurements during a lunar eclipse.

Materials: This can vary somewhat depending on weather you have access to a telescope or graphics program or not but I’ll list the materials that could be used for either.

Way to record the appearance of the moon: Telescope, digital camera or ninja sketching skills
Way to measure the relative sizes of the moon, shadow of the earth and how far the moon moves: Compass and ruler or graphics program (Photoshop)

Experimental steps:

  1. Be prepared to watch a large portion of a lunar eclipse. There’s one coming up this sunday (September 27-28 2015) but there are usually two each year and are easy to predict. You can check your visibility here:
    http://www.timeanddate.com/eclipse/lunar/2015-september-28
  1. With Telescope: Point telescope toward the moon before the start of the eclipse. If you have a camera attachment simply latch onto the eyepiece and take pictures of the eclipse every 10 minutes or so. If you don’t have a camera attachment remove the eyepiece of the telescope and hold a piece of paper directly onto the eyepiece. The bright moon should be projected onto the paper clearly enough to trace. Make a new trace every 10 minutes noting the time for each picture. Be careful to keep the paper at the same angle relative to the telescope between each drawing or the distances may be affected.
    Just a camera: Take a relatively zoomed out picture of the moon every 10 minutes. It’s okay that the moon is small in the viewfinder as long as the shadow of the earth is visible. You may also try taping your camera onto one half of a pair of binoculars. Sight on the moon with one eyepiece and take a picture at the same time. Your camera should automatically record the time though you may need to look under the “properties” (get info on mac) of the .jpg when you import them into the computer to see this.
  1. Graphics program: Scan in your sketch or open up the photo’s you took in your preferred graphics program, it can be anything so long as there is a brush tool that can be scaled by the pixel. Use the brush tool to make a dot that matches the size of the moon and another for the shadow. Arrange these dots to show the start and finish positions of the moon and draw lines at each position. Adjust the size of the bruh tool again to fit between these lines and record the size. Also be sure to record the size of the earth dot.
    Compass and print out: First print out the starting image and the final image of the eclipse. Be sure to enlarge these somewhat to make measurement easier. Use a compass to draw out the size of the shadow of the earth based on the arc it cuts across the moon. Then use the compass to draw out the moon size. Finally align the shadows so that the distance the moon moved relative to the shadow can be measured (try stacking the papers). Take measurements with the ruler of the radius of the earth’s shadow and of the distance the moon traveled.
  1.     Convert the values of the measured earth radius (half the brush size if you’re using a graphics program) and the distance the moon moved, into kilometers.
  2.     Find the total time between your start and finish images and convert it into seconds since most speeds in earth orbit are measured in kilometers per second.
  3.     Take the distance traveled and divide it by number of seconds. You should get something near 1km per second. If you don’t check your data again.
  4.     Next multiply the speed of the moon by 27.3*24*60*60. This is the time is seconds that the moon takes to make one orbit of the earth. Just like when you multiply your speed driving a car by how long you traveled by this gives you the distance traveled.
  5.     Finally divide the total circumference of the lunar orbit you calculated by 2 pi and you have your distance.

Data:
lunareclipse-2015April
LunarEclipse

Math: Remember that the point of this is to be fun, as such the math in this one is pretty sloppy. For instance the value for the circumference of the earth is estimated and we are assuming a perfect circular orbit for the moon, which it doesn’t have. Regardless, it is possible to get a good approximation of the distance to the moon with this method. Start by recalling the relationship between the circumference of a circle and the radius of a circle:
c=2piR

Now rearrange it so that the radius is on one side
C/2pi=R

This is essentially the only equation we use. It may look different when other things get plugged in but it’s essentially the same thing over and over.

Still before we can get to plugging in the distances need to be converted. The best way to figure out a conversion factor between the values you measured and a kilometer in actual space is to set up this equation:
Earth-radius

The 40,000 is the rough size of the circumference (replacing the letter C in the earlier equation). THe 100 in my case is 100 pixels but it will be whatever value you measure for the radius of your earth shadow. That “x” is in there to act as the conversion factor, it’s the imagined ratio between the real size of the radius of the earth (6.371*10^3km) and the virtual size of the shadow you measured (either in your computer or on a printed piece of paper). You can either do the algebra yourself or simply set this problem up in wolfram alpha and have it solve for x for you. Here’s the conversion factor I got:
conversion-factor-x

Once you have a conversion factor, take the measured distance the moon traveled and devide it by the number you got for your conversion factor. In this case I got:
seconds

Now technically I should ignore the last 2 because of significant figures but for this project it’s okay to be a little lazy. With the distance traveled in hand all you have to do is divide by the time interval that the lunar eclipse took.
7002km

Then take the distance and divide it by the time to get a speed for the moon.

Speed-of-moon

Next you will need to turn back to the circumference formula but put in the circumference of the lunar orbit instead of the circumference of the earth. How do we know the circumference of the moon? Well we know the moon’s speed and we already know the moon’s period, that is how long the moon takes to go around the earth. This time is called the lunar cycle, you may have heard that it takes about 28 days and that’s good enough but if you want a bit more accuracy the actual time is more like 27.3 days. But an amount of time in days isn’t very useful since we have a speed in seconds, so you will need to convert this into seconds. First multiply the number of days by 24 hours in a day, then by sixty seconds in an hour and finally 60 seconds in a minute. Multiplying this by the speed you got fro the moon gives you the distance traveled by the moon in one orbit – in other words the circumference. Then you plug this into the circumference formula from earlier to get your final radius or DISTANCE TO THE MOON!

I did this is a single step as follows:
Final-eq-and-answer

You can check your value against the accepted distance here.

Error analysis: I was very lazy with my error monitoring with this one. Didn’t even bother with significant figures. If I did the values I would have gotten would have just rounded to 400,000km which is in the right ballpark still. However, there so many little errors and assumptions along the way (assuming circular orbit, assuming no falloff of earth’s shadow, rounding the circumference of the earth etc.) that worrying about these details would simply be frustrating and unless better recording methods could be applied isn’t really worth it. The whole point of this exercise is to get a broad sense of scale and the power of even loose mathematics.

Results and Interpretation:
So the final answer I got for the data I recorded in april 2015 was 377654km but if I were to round this to the correct number of significant figures (i.e. one) I would have gotten a distance of 400000km. The moon’s actual distance wobbles in and out throughout the year since the orbit of the moon isn’t perfectly circular. The distance ranges from about 360000km at closest approach and about 405000km at it’s farthest. If you get an answer in this range you’re doing great. There many more detailed ways to perform this calculation today but as a first step in understanding distances in the solar system it is a great start.

Linkography:

http://www.timeanddate.com/eclipse/lunar/2015-september-28 

http://www.wolframalpha.com/input/?i=distance+to+the+moon

http://www.wolframalpha.com/input/?i=40000%2F%282pi%29*x%3DR+solve+for+x

https://en.wikipedia.org/wiki/Lunar_distance_(astronomy)

https://www.youtube.com/watch?v=jYh0oe7Yozc

https://www.youtube.com/watch?v=uuqQ5GABCg8

https://en.wikipedia.org/wiki/Aristarchus_of_Samos

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