Estimating the radius of the Moon (and the Earth-Moon distance)

Graphic to follow!

Many years ago I timed a lunar eclipse and attempted to work out the diameter of the Moon compared with the Earth's from that information. Unfortunately, I didn't keep a record, but was pleased to note that the result was very roughly in the right general area.

With an estimate of the size and knowing that the Moon is about half a degree of arc in diameter, one can then estimate the distance from the Earth to the Moon. This is what the Ancient Greeks would have done.

One of my assumptions was that diameter of the umbra was equal to that of the Earth. Now, I knew this to be incorrect, but I didn't bother to find out by how much.

(I also assumed that the centre of the Moon went through the centre of the umbra, also wishful thinking, but I'll leave that for now).

Here is a very belated attempt to calculate the size of the Earth's umbra at the Moon:

• let's think of a solar eclipse that is momentarily total
• let the Moon be at distance d from the Sun
• let the radii of the Sun and Moon be rₛ and rₘ respectively
• let the length of the Moon's umbra be lₘ
• we can say that lₘ = d.rₘ/(rₛ - rₘ)
• now think of a lunar eclipse.
• this time the Earth, radius rₑ, is at distance d from the Sun
• let the length of the Earth's umbra be lₑ
• we can say that lₑ = d.rₑ/(rₛ - rₑ)
• hence the ratio of the length of the Earth's umbra to that of the Moon is 

     lₑ/lₘ = (rₑ/rₘ).(rₛ - rₘ)/(rₛ - rₑ) ~ rₑ/rₘ ~ 3.7                    (since rₛ >> rₑ, rₛ >> rₘ)

Now, at distance lₑ, the size of the Earth's umbra is 0 and at distance lₘ (the Earth-Moon distance) the diameter of the Earth's umbra is therefore about (3.7 - 1)/3.7 or 0.7 times the diameter of the Earth. The difference is not insignificant and if we ignore it, we will over-estimate the size of the Moon and also the Earth-Moon distance. While I didn't take this into account, I'm sure that the Ancient Greeks would have done so in order to estimate the Earth-Moon distance, as they did so quite accurately, at around 60 Earth radii IIRC.