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Parallax: distance

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.
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 78 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 something 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?

Want to join the conversation?

  • piceratops ultimate style avatar for user Elizabeth Rowe
    I see that there is error from using the length of the arc (aka surface distance) between the two points on earth rather than the chord length (as the mole digs), but the challenge question cites "sources" of error. What other sources can other students come up with?
    (6 votes)
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    • aqualine ultimate style avatar for user r.raina29
      Earth is curved. Distance measured over the surface, that is between the cities would be more than the actual distance between the two vantage points. if we take curvature into account and use the actual shorter underground distance the error would be less than 17 percent.

      Another point might be that the star in question is taken as a fixed point. Although negligible but it should undergo some parallax as well which might cause a slightly incorrect calculation.

      Also the calculations have been rounded off a lot to give a much higher value for the distance to the moon.
      (9 votes)
  • purple pi purple style avatar for user Αγνιβά Πολεμιστής Μαιτρα
    I am not getting how to measure the angle.
    (3 votes)
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    • blobby green style avatar for user riverstun
      Another questioner had the same basic question, so I will answer it: First, astronomers can measure it using the equivalent of a theodolite.. You can measure angles with instruments. However, there is a simpler way to think about it. Imagine a line running across the heavens like an equator that goes through the moon (its called the ecliptic, and defines which constellations form the zodiac... all planets and the moon move along this one track in the sky, more or less, because they are all in the same plane). Now, in your head imagine drawing loads of moon-sized circles along that line until you come back to where you started. Turns out you can fit 720 moons along such a line, which means that in each degree of sky, you can fit 2 moons; that is, each moon diameter fills 0.5 degrees in the sky. Now, look at the distance between the star images in the picture. How does that distance compare with the diameter of the moon? About 3/5, right? So the angle between the star images should be 3/5 of 0.5 degrees, or 0.3 degrees. So thats how you can measure it by eye. But really, its measured using instruments, like a sextant or something..., or by running the images through a computer. That is, its technical. The greeks probably measured it by comparing with the moon.
      (12 votes)
  • spunky sam green style avatar for user Vijay
    How did we calculate that tan(0.15)= 1/381.9?
    (5 votes)
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  • aqualine ultimate style avatar for user 2053647
    is there any other way to find the parallax angle? without using a reference behind the object?
    (3 votes)
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  • blobby green style avatar for user JOASH  SANTHOSH
    " We can simplify things by assuming the moon is exactly between the two points (isosceles triangle)." But wont the calculations all go wrong when we assume like that?
    (2 votes)
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  • blobby green style avatar for user aelakonda29
    can someone please explain me from where the equation base/2/tan(p/2) derived
    (2 votes)
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  • male robot hal style avatar for user Aryan Agrawal
    Isn't the reason the parallax occurs when we look into the distance is because we see from different angels. Your saying it's the same with the sun rays to the earth. That's why the moon shows up red.
    (2 votes)
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  • stelly blue style avatar for user aniketprasad123
    THIS IS THE STATEMENT WRITTEN IN ARTICLE

    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


    my question is how do we calculate the error percentage or what calculation is used so the answer is estimated by only 17% of actual distance ?
    (1 vote)
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  • purple pi purple style avatar for user shaunj536
    When measuring the distance between two vantage points on the Earth shouldn't the bottom line of the triangle be curved instead of a straight line since the Earth is round? Would a curved line change the math needed to correctly answer the question?
    (1 vote)
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  • hopper cool style avatar for user 🍕MBONKA🍕
    If they thought the world was flat, then does that mean that is why their measurement was off?
    (1 vote)
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