Total solar eclipses will become impossible when the maximum apparent size the Moon can attain when seen by an observer on the Earth is less than the minimum size the Sun can attain. The apparent size of an object is determined by its size and its distance away from us: the latter can change not only because of changes in its mean orbital distance but also because of variations in the eccentricity of its orbit. The eccentricity of the Moon's orbit is normally quoted as varying from 0.026 to 0.0775 and that of the Earth from 0.0023 to 0.0577. However, while these variations are both caused by the same sort of effect - perturbation by other bodies - the timescales involved are radically different.
The eccentricity of the Moon varies in quite a simple way based on two, very short, periodicities - monthly and approximately six-monthly - both of which are caused by the influence of the Sun. The monthly period is due to its orbit round the Earth, which alters its distance from the Sun (and hence its influence) on the same timescale. The six-monthly period is due to successive alignments of the major axis of its orbit (the line joining the apogee and perigee points) with the Earth-Sun line: it is clearly easier for the Sun to increase the eccentricity if the long axis of the orbit is pointing towards it. The period is actually a little over six months (206 days, in fact) because the direction of the axis slowly rotates in space, once every 8.85 years. Importantly though, these short-term, periodic, fluctuations are superimposed upon a constant change in the mean value over time, similar to the change in mean orbital distance we have examined in the previous article: this "non-periodic" variation is called "secular" change.
The Earth's orbit is very much more stable than that of the Moon, as the influence of the Sun tends to balance out over time, leaving just the small perturbations caused by the closer (but not very close!) planets. The result is a very-long-period variation (of the order of 100,000yrs) whose amplitude varies rather chaotically between the extremes given above over timescales of millions of years. These changes may be calculated using a formula devised by Bretagnon [1], which was how I was able to quote these values. The formula is constructed using only periodic terms, so I must assume that any secular changes in the Earth's orbit are negligible.
Over the sorts of timescale involved in the extinction of total eclipses, therefore, the variation in the eccentricity of the Moon's orbit is characterised by the secular rather than the periodic changes while that of the Earth's orbit is the reverse. I thus feel it is reasonable to base my calculations on the extreme values for the Earth and on a "constant value plus secular change" formula for the Moon. This does beg the question of "which constant value?" though, not to mention "how big is the secular change?". For the moment, however, I shall simply use the current mean value of the lunar orbital eccentricity (0.0549) and ignore the secular change: this will at least allow direct comparison with the estimate of Meeus, who used the same approach. When these calculations have been discussed I shall consider the most reasonable constant value to use and factor in the effect of secular change.
The final variable to be allowed for is a change in the absolute size of the bodies concerned. This is something which is not usually considered, because of course the size of both the Earth and the Moon is not going to alter any time soon. However, unfortunately for the future of the Earth, this is not true in the case of the Sun. Although it is pretty stable at present it does, like all stars, have a natural life-cycle which will eventually affect both its composition and its size. The natural life-span for a star such as the Sun is about 10 billion years: it is thus currently literally "middle-aged". As it approaches the last third of its life it will begin to expand, becoming 10% greater than its present size in about another 1.1 billion years. This increase in size clearly has considerable consequences for the possibility of total eclipses.
So, there are four steps to go through before we can arrive at a conclusion: firstly, perform some basic calculations to establish the principles, then take into account the secular changes, next consider which is the most appropriate value of lunar orbital eccentricity to use, and finally factor in the increasing size of the Sun.
The Sun is at its smallest when the Earth is at aphelion (furthest from the Sun) and its orbital eccentricity is at its maximum value. Under these conditions the apparent diameter of the Sun is 30.24 arc-minutes (60th's of a degree). The Moon is the same apparent size when 394789km from the observer, or 401167km from the Earth's centre. The Moon is at its largest when at perigee (nearest to the Earth), when this distance equates to a semi-major axis of 424470km (at "standard" eccentricity). Referring to the model, this radius will be reached 3.22 x 109 years from now. This is the end-point for total eclipses, as after this time the apparent size of the Moon can never be greater than that of even the smallest-sized Sun. However, we must also consider the point where total eclipses start to become impossible - when the Earth is at aphelion but at minimum eccentricity.
When the Earth's orbit has its minimum eccentricity the apparent size of the Sun at aphelion is 31.92 arc-minutes. The Moon is this apparent size when 374110km from the observer, or 380488km from the Earth's centre. When at perigee, this equates to a semi-major axis of 402591km, which will be reached 1.21 x 109 years from now. Before this time the apparent size of the Moon can always be greater than that of the Sun and so some total eclipses are guaranteed. After that time its size cannot always exceed that of the Sun and so there will be periods when total eclipses are not possible, until the eccentricity of the Earth's orbit has decreased sufficiently to allow them to happen again. These periods will steadily become longer and longer until finally total eclipses are no longer possible under any circumstances.
One can also consider the opposite question - when did all eclipses stop being total? This happened when the minimum apparent size the Moon could attain became less than the absolute maximum size the Sun could attain. We must therefore consider the case of the Moon at apogee and the Earth at perihelion with maximum orbital eccentricity, when the apparent diameter of the Sun is 33.95 arc-minutes. The Moon is the same apparent size when 351715km from the observer, or 358093km from the Earth's centre. When at apogee this distance equates to a semi-major axis of 339457km. Referring to the model, this radius was reached 1.70 x 109 years ago. As before, the limiting case is when the Earth's orbital eccentricity is minimum at perihelion: the Sun's apparent size is then 32.06 arc-minutes. The apogee Moon has the same apparent size when 372393km from the observer, or 378771km from the Earth's centre. This distance equates to a semi-major axis of 359059km, which was reached 1.14 x 109 years ago.
The basic analysis thus tells us that up to 1.70 billion years ago all solar eclipses were total. From then on there were some periods when annular eclipses were possible but annulars were not certain until 1.14 billion years ago. We are now in an epoch when both types are possible at all times, but the percentage of totals is decreasing and the percentage of annulars is increasing. Looking into the future, the calculations show that eventually, in 1.21 billion years time, there will be periods when totals are not possible until, eventually, they will disappear entirely in 3.22 billion years from now.
Note the highly non-linear timescale of this progression - just 560 million years to go from all total to a mixture but 2.01 billion years to go from a mixture to all annular. This is a reflection of the shape of the distance/time graph, which is the main reason I believe Jean Meeus' estimate is wide of the mark - he assumed that the recession rate of the Moon will remain constant at 3.8cm/yr, which we now know is most unlikely to be true. In the period from now into the future a typical rate would be 1.3cm/yr, which immediately pushes out his "no totals" estimate to 3537 million years, quite close to mine.
I said above that the long-period variation in the eccentricity of the Moon's orbit was characterised by its secular change. The best number I have been able to find for the current value of this change is an increase of 0.0143 per billion years. It can be seen that, over the timescales indicated by the simplified calculations, this would cause a significant increase over the current mean value of 0.0549. But is it typical of the Moon's history? Answer - yes and no! As with the variation in its orbital distance, the variation in eccentricity since the formation of the Moon is highly unlikely to have been linear, given that is it caused by the same sorts of processes - transfer of angular momentum due to tidal interactions. However, over the time-period given by the simplified calculations (1.70 billion yrs before the present to 3.22 billion yrs after), the distance-time graph is "reasonably flat". I thus feel that it is justifiable to perform some extended calculations where the constant rate of secular increase quoted above is added to the current eccentricity: this will at least give us a plausible indication of how things might turn out, even if not a definite answer.
Re-calculating using a secular increase results in a possibly surprising conclusion: in the future, the decrease in perigee distance due to the increase in orbital eccentricity in fact more than compensates for the increase in mean orbital distance. In other words, although the Moon gets steadily further away on average, its minimum distance actually decreases (and so it looks larger than it would have done with no secular change). This means that under this scenario total eclipses will always be possible! (though only under favourable circumstances). The point at which they cease to be possible with the Earth at more than minimum eccentricity also moves out considerably, to 2.83 billion yrs (from 1.21 billion). Going back in time, the eccentricity of course will steadily decrease, so the apogee distance also decreases and again the Moon looks larger than it would have done with no secular change. This moves the point at which annular eclipses become possible to 1.57 billion yrs ago (rather than 1.70 billion) and certain to 0.86 billion yrs (rather than 1.14 billion).
I said above that the eccentricity of the Moon's orbit varies in quite a regular way, with periodicities of a lunar month and about six calendar months, between the extremes of 0.026 and 0.0775. Thus far, I have simply used the mean value (0.0549) in my calculations, but we must now consider whether this is justifiable.
Two facts are relevant here. Firstly, in his book 'Mathematical Astronomy Morsels', Jean Meeus states (on page 11) that the greatest values of the eccentricity occur when the major axis of the Moon's orbit (the line joining the apogee and perigee points) is directed towards the Sun. Secondly, we saw above that the limiting cases for eclipses being possible or not happen when the Moon is at either apogee or perigee. For there to be an eclipse at all the Moon must be sitting on the Earth-Sun line and so for the limiting cases it is both at apogee/perigee and also on the Earth-Sun line. It therefore follows from Meeus' statement that at the limiting cases the Moon's orbit must have the maximum value of eccentricity.
I thus re-did the basic calculations using an eccentricity value of 0.0775 rather than 0.0549. The much larger eccentricity (and therefore a much reduced perigee distance) results in total eclipses becoming always possible (i.e. irrespective of the eccentricity of the Earth's orbit) in the future. Going back in time the "annulars possible/certain" dates revert to almost where they were under the basic calculation assumptions: this is because the larger starting value of the eccentricity makes up for the decrease due to secular change. The actual numbers are:- possible 1.69 billion yrs ago, certain at 1.20 billion yrs.
That is not quite the end of the story though, as the inclination of the Moon's orbit also undergoes secular change: it decreases by about 0.25deg per billion yrs. It is the fact that the orbit is inclined to the plane of the Earth-Sun line that prevents there being a solar eclipse every new Moon and a lunar eclipse every full Moon, and so a decrease in inclination will lead to there being more eclipses of each type. As described in my article on Eclipse Limits, because of the finite size of the Earth and Moon the line-up does not have to be perfect for an eclipse to be possible: this is why we get frequent eclipses now, despite the inclination. In fact, under favourable circumstances a total solar eclipse can still be possible despite the Moon being misaligned by over 11deg in longitude. However, this limit depends on both the perigee distance and the inclination and so will vary as each of these undergoes secular change and, in the case of the perigee, change due to the increase in the mean Earth-Moon distance with time.
To give an idea of the magnitude of the change in the ecliptic limit for total solar eclipses due to the secular change in orbital inclination I calculated the limit as at 1 billion yrs in the future (i.e. about the time when total eclipses would disappear under the "mean value of eccentricity" scenario), firstly for the case of no secular change and then with the change of 0.25deg per billion years quoted above. The first value was 10deg 45min, the second 11deg 37min: these should be compared with the current value of 11deg 19min. One can see that although a decrease in eclipse numbers of 5% would result from the change in perigee distance alone, the secular change in the inclination restores this to an increase of 2.7%. So, while total eclipses will constitute a smaller percentage of all eclipses in the distant future, the overall numbers of eclipses will actually slightly increase - every cloud has a siver lining!
The major problem with all the above calculations is that they assume that the Sun itself will remain unchanged, particularly in size. As mentioned in the introduction, this is not actually true: it will become 10% greater than its present size in about another 1.1 billion years. If this size increase is linear from the present epoch, and taking into account secular changes in the eccentricity (presumed to take its maximum value), total eclipses could start to become endangered in a mere 0.65 billion years rather than being always possible. Whatever assumption is made about the rate of increase, they will definitely become impossible after just 1.17 billion years. The drastic reduction in timescales for an apparently slight increase in size is a reflection of the quite delicate situation in force at present, where the apparent sizes of Sun and Moon are only a few percent different from each other. In this context, a size increase of 10% is actually very considerable and ends up dominating the calculations (something which has been overlooked in similar calculations performed by other authors).
Bizarrely, therefore, despite the fact that he used the wrong value for the lunar recession rate and the wrong value for the lunar orbital eccentricity at the limiting cases, ignored secular change and did not consider the effect of an expanding Sun, Jean Meeus' estimate of "no totals after 1210 million years" turns out to have been remarkably accurate! A classic case of the right answer having been achieved for totally the wrong reasons.
The Sun will continue to expand after total eclipses become impossible, and so after a while the remaining annular eclipses might perhaps more correctly be called "transits" rather than "eclipses", because of the much greater size of the Sun. However, this may be something of a moot point as the extra heat generated by the larger Sun will begin to dry up the Earth, killing off many species of life - probably including us! By the time we get to 3.5 billion years the Sun will be 40% larger (and much hotter), which will probably mean that all life on Earth will cease to exist. Finally, at about the 5 billion year mark, it will expand very rapidly and turn into a Red Giant star - a lot cooler but very much larger: possibly large enough to engulf the planet Mercury, in fact. This is the effective end of its life, and that of the Solar System will follow soon afterwards as firstly, after an initial contraction, another expansion will increase its size by 100 times (swallowing up Venus and maybe the Earth/Moon system as well) and then successive shells of material will be expelled into space leaving only a small but dense core called a White Dwarf: that really is the end.
| References (Return to text) | |
| [1] | Eccentricity of the orbit of the Earth from 1,000,000 BC to 1,000,000 AD - Giesen, J. |
