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How far away is the Moon?

In the second century BC Hipparchus derived a very good estimate of the distance to the moon using lunar parallax. It is based on how much the moon shifts relative to the background stars when we observe it from different vantage points on earth.

Image Credit: Ernie Wright

To develop our measurement we first need to setup a triangle. Think of the moon as a point in space with two straight lines connecting it to points on Earth:

Image Credit: Ernie Wright

In this example the two vantage points are Selsey, UK and Athens, Greece which are separated by 2360 km.

Image Credit: Ernie Wright

This gives us a triangle. We can simplify things by assuming the moon is exactly between the two points (isosceles triangle).

Not to scale

Now we need to determine the angle p using the parallax effect.

Finding parallax angle

Here are two photos taken at the same time from Athens and Selsey. We can assume the star (Regulus) near the moon is fixed since it’s 79 million light years away.

The Moon has appeared to shift position. Our goal is to figure out the angular distance of this shift. To do so we combine these images as a stereo pair. To see this, simply cross your eyes (using above images) until the Moons overlap.

If you do this correctly you will see some like this:

The angular distance between the stars turns out to be approximately 1100 arcseconds, or 0.30 degrees. This looks about right since we know the Moon has an angular diameter of 0.5 degrees. We now have the angle needed. The Moon appears to shift 0.3 degrees when we observe it from two vantage points 2360 km apart.

Not to scale

Finally, we can split our triangle in half to create a right triangle. This allows us to apply our trigonometric functions to find the distance d directly.

Not to scale

tan(angle) = opposite/adjacent

tan(0.15) = 1180/distance

1/381.9 = 1180/distance

distance = 1180*381.9

 

this gives us our estimated distance to the Moon:

Estimated lunar distance = 450 642 km

 

This estimate is off by only 17% of the actual distance, which is pretty good for a rough estimate! Compare this to the actual average value: 384,000 km

Challenge question: What were the sources of error in our method above? How could we improve?