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I learned as a child (and a common tale), that Aristarchus gave a method of measuring the sun-earth-moon angle to estimate the ratio of the earth-moon distance to the earth-sun distance. Indeed, during the first quarter moon phase, the sun-earth-moon reasonably form a right triangle at the moon.

I want to illustrate this to a math class that I am teaching, however I want some practical ways of measuring this sun-earth-moon angle. I reckon it will probably be a bad estimate anyway, as this angle might be nearly 90 degrees as the sun is quite far from the earth. But it might be a fun activity anyway. Are there any reasonable way to measure this angle with modest tools like a level and a plumb? (Or smart phones with a leveling tool.) Do I need to wait when both the sun and moon are visible in the sky during first quarter moon phase? Or, how did Aristarchus measured it, if he did at all?

Sorry I am not a practicing astronomer. Thanks for your insights.

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In summary, Aristarchus understood that at half moon (first or third quarter), given a triangle with vertices at Sun, Earth and Moon, angle Sun-Moon-Earth is nearly 90 degrees - a right angle triangle with the hypothenuse being the line between Earth and Sun.

The example below calculates the Sun-Earth-Moon angle and was calculated by getting respective Sun and Moon RA and Dec on Oct 28 (next 3rd Quarter). The angular separation function can be found here.

Note 1: I used Starry Night to get the values shown, which software can also give angular separation directly thus voiding the need to calculate the separation angle.

Note 2: Should you wish to do the actual measurment of the angles, a telescope will be necessary (WARNING: DO NOT look at the Sun directly through a telescope. If you are not familiar in telescope use for solar observation work with someone who does. Failing to do that WILL result in ocular damage up to and including blindness).

enter image description here

He also determined during a lunar eclipse the Moon is passing through the Earth’s shadow. A construct of the three bodies in line (as in the diagram below (from Wikipedia), allowed him to use similar triangles to estimate ratios for distances and body radii.

enter image description here

The Wikipedia article On the Sizes and Distances will possibly provide the information you need. Also refer to the Notes section at the bottom of the same article. The Turkish language YouTube video is quite handy (and I think it is the first time I really appreciated mathematics as being a universal language for I do not speak turkish).

Also look at these links

  1. this
  2. this
  3. this

I hope your students enjoy and appreciate this lesson.

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    $\begingroup$ Thanks! This is very useful, especially the video you recommended. It gives one way to establish this angle by looking at the ratio (first quarter moon phase to third quarter moon phase):(one period of the moon), as both first and third quarter moon phases give a sun-earth-moon triangle a right angle at the moon. Neat! $\endgroup$
    – bonsoon
    Oct 15 at 19:39
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    $\begingroup$ Wow, edit looks great. Thanks! $\endgroup$
    – uhoh
    Oct 16 at 7:53
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    $\begingroup$ Can you put the details of how you would go about measuring the various angles in your answer. Link-only answers may not be useful if the links change. $\endgroup$
    – ProfRob
    Oct 16 at 8:48

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