Eratosthenes

From the Earth to the Moon in 270 BC

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The brain is wider than the sky,

For, put them side by side,

The one the other will include

With ease, and you beside.

Emily Dickinson

How did human beings first work out the distance from the Earth to the Moon?

Aristrarchus of Samos (310 BC – 230 BC) figured out a way to do so in terms of the radius of the Earth in 270 BC. Combined with Eratosthenes’ measurement of the radius of the Earth (c. 240 BC) it enabled people to calculate the actual distance to the Moon. The ancient Greeks used a measurement of distance called stadia (singular: stadium) but we will present the measurements here in terms of kilometres.

Magic with a shadow, not with mirrors

Aristarchus used the fact that the Moon passes through the Earth’s shadow during a total lunar eclipse, which happen once every two to three years on average.

What does a total lunar eclipse look like? Watch this amazing 33 second time lapse video from astrophotographer Bartosz Wojczyński.

https://www.youtube.com/watch?v=LK_44AbfH2Q Note that Mr Wojczyński altered the exposure time of each shot to compensate for the reduced brightness of the Moon as it crossed into the shadow. For reference, the exposure time for the brightly lit Moon was 1/2500 second, and for the dim ‘Blood Moon’ (turned red by sunlight refracted by the Earth’s atmosphere) it was 6 seconds.

The video is sped up so that 1 second of video represents 8 minutes of real time. In the video, the Moon is in shadow for 24 seconds which equates to 8 x 24 = 192 minutes or 3 hours 12 minutes. We will use this later to model Aristarchus’ original calculation.

It’s always Aristarchus before the dawn…

Aristarchus began with the assumption that the Earth of radius r creates a cylinder of shadow that is 2r wide as shown in the diagram below.

The Moon orbits the Earth on a roughly circular path of radius R so it cover a total distance of 2πR. This means that its average speed over its whole journey is 2πR/T where T is the orbital period of the Moon, which is 27.3 days or 27.3 x 24 = 655.2 hours.

The average speed of the Moon as it passes through the Earth’s shadow is 2r / t where t is the time for a lunar eclipse (3 hours 12 minutes, in our example).

The average speed of the speed of the Moon is the same in both instances so we can write:

We can simplify by cancelling out the common factor of two:

Then we can rearrange to make R the subject:

Putting in values for t = 3 hours 12 minutes or 3.2 hours, T = 655.2 hours and Eratosthenes’ value for the radius of the Earth r = 6371 km (which was established a few years later):

So now they do it with mirrors…

Aristarchus’ value is just a shade over 7% too large compared with the modern value of the Earth-Moon distance of 384 400 km, but is impressive for a first approximation carried out in antiquity!

The modern value is measured in part by directing laser beams on to special reflectors left on the Moon’s surface by the Apollo astronauts and also the automated Lunokhod missions. Under ideal conditions, this method can measure the Earth-Moon distance to the nearest millimetre.

Quibbles, Caveats and Apologies

Aristarchus’ estimate was too large in part because of his assumption that Earth’s shadow was a cylinder with a uniform diameter. The Sun is an extended light source so Earth’s shadow forms a cone as shown below.

The value of t is smaller than it would if the shadow was 2r wide, leading to a too-large value of R using Aristarchus’ method.

Also, the plane of the Moon’s orbit is tilted with respect to the plane of the Earth’s orbit. This means that the path of the Moon during an eclipse might not pass through the ‘thickest’ part of the shadow. Aristarchus used the average time t calculated from a number of lunar eclipses.

When timing the lunar eclipse shown in Mr Wojczyński’s excellent video, I started the clock when the leading edge of the Moon entered the shadow, but I confess that I ‘cheated’ a little bit by not stopping the clock when the leading edge of the Moon left the shadow — the error is entirely mine and was deliberate in order to arrive at a reasonable value of R for pedagogic impact.

UPDATE: You could also watch this stunning visualisation of a lunar eclipse from Andrew McCarthy where the shadow of the Earth is tracked rather than the Moon.

This is part 2 of a series exploring how humans ‘measured the size of the sky’.

Part 1: How Eratosthenes measured the size of the Earth

Part 3: How Aristarchus measured the distance from the Earth to the Sun