When did the first annular eclipse happen?

A few hundreds of millions of years ago, the moon was closer to the Earth than it is today and hence was of a bigger apparent size. This made every solar eclipse either total or partial. Over the course of time, due to tidal forces between the moon and the Earth, the moon slowly drifted far from Earth decreasing its apparent size when viewed from Earth and hence causing annular eclipses to happen.

That would mean there must have been some day in history when Earth witnessed its first ever annular eclipse. Is there an estimate on when this happened?

• Please show your own efforts to find a solution. Jul 15 '19 at 23:16
• I initially thought that there would be a popular established answer to this question. But only after some research, I realised that the recession rate of the moon is influenced by so many factors and hence its historical variation is hard to estimate. This makes trying to get an accurate estimate for the first annular eclipse a difficult problem Jul 16 '19 at 3:58

A quick and dirty estimate is as follows.

An angular eclipse occurs first if the Moon is at apogee and Earth at perihelion. Assuming that these two events coincide (which happens every ~8.85 years owing to lunar apsidal precession), an angular eclipse requires that ($$a$$ denoting orbital semi-major axis and $$R$$ radius)

$$\frac{R_\supset}{(1+e_\supset)a_\supset} < \frac{R_\odot}{(1-e_\oplus)a_\oplus}$$

or

$$a_\supset > \frac{R_\supset}{R_\odot} \frac{1-e_\oplus}{1+e_\supset} a_\oplus\approx 348300\,\mathsf{km}$$

where I assumed that the orbital eccentricities, Lunar and Solar radii, and Earth's semi-major axis didn't change (over the period considered here) and are given by $$e_\supset=0.0549$$, $$e_\oplus=0.0167$$, $$R_\supset=1737.1\,$$km, $$R_\odot=695510\,$$km, and $$a_\oplus=149.6\times10^6\,$$km.

At the present we have $$a_\supset=384400\,$$km. If we assume an average annual change by 22 mm, we find a time of

$$1.6\times10^9\,\mathsf{years},$$

though it could have been more recent if the annual change of $$a_\supset$$ was larger some hundred million years ago (which may well have been the case).

In a similar way, one may also estimate when the last full Solar eclipse will occur. This time, the Moon is at perigee and Earth at apohelion, when

$$a_\supset < \frac{R_\supset}{R_\odot} \frac{1+e_\supset}{1-e_\oplus} a_\oplus\approx 400800\,\mathsf{km}$$

which last occurs in $$430\times10^6$$ or $$745\times10^6$$ years, depending on whether the current lunar recession rate of 38mm/yr or the long-term average of 22mm/yr is used.

• Wow, that's great. Thanks for answering. But you have used the number 22 mm/year which seems to be the average rate for the last 680 million years.But I just found out that the actual recession rate could vary over a large range with respect to that value and this would be the biggest contributor to the error in the final estimate. Would you agree to that? Can you suggest me a source or a method which I can work out to get a more accurate estimate? I am trying to get the best estimate I can. Thanks. Jul 16 '19 at 13:31
• @KevinSelvaPrasanna As you found out, it hinges mostly on the tidal recession rate. Estimating that accurately is near impossible. What do you need that for? Jul 16 '19 at 22:09