<p>A quick and dirty estimate is as follows.</p>

<p>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 (<span class="math-container">$a$</span> denoting orbital semi-major axis and <span class="math-container">$R$</span> radius)</p>

<p><span class="math-container">$$
      \frac{R_\supset}{(1+e_\supset)a_\supset} &lt; \frac{R_\odot}{(1-e_\oplus)a_\oplus}
$$</span></p>

<p>or</p>

<p><span class="math-container">$$
      a_\supset &gt; \frac{R_\supset}{R_\odot} \frac{1-e_\oplus}{1+e_\supset} a_\oplus\approx 348300\,\mathsf{km}
$$</span></p>

<p>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
<a href="https://en.wikipedia.org/wiki/Orbit_of_the_Moon" rel="noreferrer"><span class="math-container">$e_\supset=0.0549$</span></a>, <a href="https://en.wikipedia.org/wiki/Earth%27s_orbit" rel="noreferrer"><span class="math-container">$e_\oplus=0.0167$</span></a>, <span class="math-container">$R_\supset=1737.1\,$</span>km, <span class="math-container">$R_\odot=695510\,$</span>km, and <span class="math-container">$a_\oplus=149.6\times10^6\,$</span>km.</p>

<p>At the present we have <span class="math-container">$a_\supset=384400\,$</span>km. If we assume an <a href="https://en.wikipedia.org/wiki/Orbit_of_the_Moon#Tidal_evolution" rel="noreferrer">average annual change by 22 mm</a>, we find a time of </p>

<p><span class="math-container">$$1.6\times10^9\,\mathsf{years},$$</span></p>

<p>though it could have been more recent if the annual change of <span class="math-container">$a_\supset$</span> was larger some hundred million years ago (which may well have been the case).</p>

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

<p><span class="math-container">$$
      a_\supset &lt; \frac{R_\supset}{R_\odot} \frac{1+e_\supset}{1-e_\oplus} a_\oplus\approx 400800\,\mathsf{km}
$$</span></p>

<p>which last occurs in <span class="math-container">$430\times10^6$</span> or <span class="math-container">$745\times10^6$</span> years, depending on whether the current lunar recession rate of 38mm/yr or the long-term average of 22mm/yr is used.</p>

CC BY-SA 4.0
2019-07-16T13:31:11.340Z
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.
Kevin Selva Prasanna
CC BY-SA 4.0
2019-07-16T22:09:27.753Z
@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?
Walter
CC BY-SA 4.0
2019-07-16T00:12:51.190Z
2019-07-16T00:25:05.817Z
2019-07-16T00:25:05.817Z
Walter
<p>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.</p>

<p>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? </p>

CC BY-SA 4.0
2019-07-15T23:16:59.070Z
Please show your own efforts to find a solution.
Walter
CC BY-SA 4.0
2019-07-16T03:58:15.623Z
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
Kevin Selva Prasanna
<earth><solar-eclipse><history>
When did the first annular eclipse happen?
Kevin Selva Prasanna
# When did the first annular eclipse happen?

(Asked by: username_0 on 2019-07-15)

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?

**Tags:** earth, solar-eclipse, history
> **Comments:**
> **username_1**: Please show your own efforts to find a solution.  
> **username_0**: 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  

---

## 1 answers

### Answer by username_1 on 2019-07-16. Score: 5 | Accepted answer

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
[$e\_\supset=0.0549$](https://en.wikipedia.org/wiki/Orbit_of_the_Moon), [$e\_\oplus=0.0167$](https://en.wikipedia.org/wiki/Earth%27s_orbit), $R\_\supset=1737.1\,$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](https://en.wikipedia.org/wiki/Orbit_of_the_Moon#Tidal_evolution), 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.

> **Comments:**
> **username_0**: 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.  
> **username_1**: @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?  

---

