The Geometry of a Solar Eclipse

Updated 23 May 26

Geometry of a Solar Eclipse

The diagram shows a simplified representation of a particular type of solar eclipse in which the Sun is obscured for an instant only. If the Moon were closer, the eclipse would be total for a period of time and if it were further away, the eclipse would be annular.

At right is a red vertical bar representing the Sun, radius r₃; in the middle is a green vertical bar representing the Moon, radius r₂; at left is a black vertical bar representing the extent of the observed  partial eclipse (penumbra), radius r₁. 

Point 1 represents the location of an observer (on the Earth) of this momentary total eclipse. 

The distance between point 1 and (the centre of) the Moon is d₁ (known); the distance between (the centre of) the Moon and point 2 is d₂ (unknown) and the distance between point 2 and (the centre of) the Sun is d₃ (also unknown). The total distance between the point 1 and (the centre of) the Sun is therefore d₁ + d₂ + d₃.

Marked, but in the end not used, are angles θ (half the angle subtended by the Moon at point 1), and φ (half the angle subtended by the Sun at point 2).

From similar triangles:    

1.    (d₁ + d₂)/r₁ = d₂/r₂ = d₃/r₃

and:    

2.    d₁/r₂ = (d₁ + d₂ + d₃)/r₃.

From 1:

3.    d₂ = r₂.d₁/(r₁ - r₂).

From 2 & 3:

4.    d₃/(d₁ + d₂ + d₃) = d₂/d₁ =  r₂/(r₁ - r₂).

So: 

5.    (d₁ + d₂)/ (d₁ + d₂ + d₃) = 1 - d₃/(d₁ + d₂ + d₃) 

=  (r₁ - 2r₂)/(r₁ - r₂). 

(Note:  r₁ > 2r₂)

Hence, the ratio of the Earth-Moon distance to the Astronomical Unit is (minus the radius of the Earth):

6.    d₁/ (d₁ + d₂ + d₃) = ((r₁ - r₂)/r₁).(d₁ + d₂)/ (d₁ + d₂ + d₃)

= (r₁ - 2r₂)/r₁.

So, all we need to do is measure r₁, plug it into the above equation along with the known value for r₂ and we are done! However, there is a problem. Let's say the ratio in the above expression is 1/n, then:

7.    r₁ = 2r₂.(n - 1)/n. 

This means we must measure r₁ with an accuracy (much) greater than 1/n - an estimate will not do.

I had hoped that this could form the basis of a simple method to estimate the AU, but that does not seem to be the case. And I've not even considered yet just how to estimate r₁ from the timings of a solar eclipse on the spinning Earth. Not simple then and I can only think that this was never a practical method either for finding the size of the AU. However, for high n, r₁ to all intents and purpose equals 2r₂, so this is instead one way that the diameter of the Moon could be measured.