# User:Gamling/ShortMoonCycle

Andreas "Lalaith" Möhn used to have a website called *Lalaith's Middle-earth Science Pages*, which in October 2014 was removed and replaced by a webpage^{[1]} of purchasable books, including one called *The Moon in 'The Hobbit'*. Within the webpage (now within his book) was a section titled,"The Moon and Durin's Day". In this section Mr. Möhn claimed that the orbital period of the moon was about 20 minutes shorter at the end of the Third Age than it is today, which would bring the phases of the moon observed in *The Hobbit* into alignment with the lunar phases of *The Lord of the Rings*. Mr. Möhn justified this shortening by noting that the moon's orbital cycle has been lengthening over time and he left it to the reader to figure out how long ago the Third Age of Middle-earth was relative to our own current time.

Unfortunately, Mr. Möhn's hypothesis is flawed. When critical examination is applied to his claim, as explained below, it is found that his solution creates many more problems than it solves. The following "peer review" will examine Mr. Möhn's claim in detail and show why it should be rejected.

## Contents |

## [edit] Calculation of the Shorter Lunar Cycle

Mr. Möhn did not provide any calculations showing why shortening the moon's period by about 20 minutes would produce an alignment between J.R.R. Tolkien's two works. Here is the calculation in full:

When J.R.R. Tolkien wrote *The Lord of the Rings* he used the lunar phases of 1941 and 1942 for the years T.A. 3018 and 3019, with an offset of 6 days.^{[2]} This means that the full moon seen on 8 Afteryule 3019 corresponds to our 2 January 1942, which serves as a starting point. Knowing this information the phase of the moon can be calculated for any date in Middle-earth by finding the corresponding date in our current calendar and then looking up the moon's appearance for that date.^{[note 1]}

In *The Hobbit*, on midsummer's eve in T.A. 2941, Elrond read the moon-letters on Thrór's Map. At that time, as the text stated, "The moon was shining in a broad silver crescent."^{[3]} Midsummer's eve appears to correspond to Mid-year's Day, which is a specific date in the Shire Calendar, so the phase of the moon for that night can be determined. From 8 Afteryule 3019 back to Mid-year's Day 2941 is 28,315 days (this total takes into account leap days appearing in both calendars). This means that Mid-year’s Day 2941 corresponds to 24 June 1864 in our calendar. Unfortunately, it turns out that the moon was a waning gibbous moon on that date, not a broad silver crescent.

Mr. Möhn decided that the moon was 7 days past new on Mid-year's Day 2941, which would have been 14 days earlier than the actual phase of the moon. Since he says that the moon's orbital period was shorter, that means that there must be 28,315+14days or 28,329 days in the same number of lunar cycles between the two dates. Dividing 28,315 by 28,329 produces 0.99950580677. One synodic month is 29.530589 days so multiplying this by 0.99950580677 produces a shorter lunar month of 29.515995 days. The difference between these periods is 0.014594 days. Multiplying by 24 hours/day and again by 60 minutes/hour produces a lunar cycle that is 21.01547 minutes shorter than the current value, which is close to Mr. Möhn's "about 20 minutes".

## [edit] How Long Ago Was the Third Age?

Mr. Möhn mentions that a shorter lunar cycle would mean that the Third Age was sometime in the past, but again he provided no calculations and left determining when the Third Age was to the reader, which is not what one would expect from something called "science pages". Calculating when the moon's cycle was 21.0157 minutes shorter is difficult due to many factors, but can be determined with sufficient accuracy. The key value is that the moon is receding by 0.038247 meter/year (or 38.247 m/1,000 yrs):

The first step is to find the current radius of the moon's orbit using the equations *C = vt* and *r = C/2π* where

*C*is the circumference of the moon's orbit (the distance traveled in one full cycle)*v*is the velocity of the moon in its orbit (1.022Km/sec)*t*is the time it takes for a cycle to complete (27.321582 days, or 2,360,584.6848 sec)^{[note 2]}*r*is the radius of the moon's orbit^{[note 3]}

*C* = (1.022Km/sec)(2,360,584.6848 sec) = 2,412,517.548 Km
*r* =(2,412,517.548 Km) / (2)(3.14159) = 383,964.0930 Km or 383,964,093.0387 m

The second step is to find how much shorter the lunar cycle was 1,000 years ago. It is known that the distance to the moon, the radius of its orbit, is increasing by 38.247 m/1,000 years. A thousand years ago the radius of the orbit was 383,964,093.0387 m - 38.247 m = 383,964,054.7917 m or 383,964.0548 Km.

Using the equations in reverse, *C = 2πr* and *t = C/v*, the period of a lunar cycle 1,000 years ago was:
*C* = (2)(3.14159)(383,964.0548 Km) = 2,412,517.3076 Km
*t* = (2,412,517.3076 Km) / (1.022 Km/sec) = 2,360,584.4497 sec

Subtracting the older, shorter time from the current, longer time: 2,360,584.6848 sec - 2,360,584.4497sec = 0.2351 sec in 1,000 years

The third step is to find out what a 0.2351 second difference for 1,000 years becomes when the difference is increased to 21.01536 minutes:
(21.01536 min)(60 sec/1 min)(1,000 years/0.2351) = **5,363,342 years**.

## [edit] Over 5 Million Years Ago?

If you are a Darwinist and believe that Middle-earth is our world and subject to evolution through natural selection then this result is nonsense. Over 5 million years ago there were no modern human beings, just early hairy hominids whose idea of a "war" would have been two packs throwing rocks and bones at each other. If you are a Creationist then the result is still nonsense because Creationists believe in a young earth – calculations among Creationists vary but all of them would reject an earth that is more than several thousand years old. From an "anthropological" point of view, if humans had iron, swords, armor, ships, castles, and other Medieval technology over 5 million years ago, why, 5 million year later, did humans only have stone age technology?

In addition to violating the readers' sense of when the Third Age of Middle-earth was, a time over 5 million years ago disagrees with J.R.R. Tolkien's historical placement of the age. In one letter (to Miss Beare, dated 14 October 1958) Tolkien said, "I imagine the gap to be about 6,000 years"^{[4]} between then and now. In *The Lord of the Rings* he also said, "The year no doubt was of the same length now, for long ago as those times are now reckoned in years and lives of men, they were not very remote according to the memory of the earth."^{[5]}

## [edit] Another Major Consequence

In addition to omitting the calculation of his "about 20 minutes" and hinting at but not determining how long ago the moon's orbital period was short enough to align Tolkien's two stories, Mr. Möhn made no mention as to *why* the moon's orbital period was longer in the past and what effect that may have on his hypothesis. The reason the moon's orbital period is getting longer (and thus was shorter in the past) is due to tidal acceleration, which both increases the radius of the orbit and causes the rotation of the earth to slow.^{[note 4]}

Currently the earth's rotation is slowing by 2.3 milliseconds per century. Multiplying by the number of years ago when the Third Age existed determines how much shorter the day was at that time:

- (5,363,342 years)(2.3 msec/100 years) = 123.357 seconds, or 2 minutes 3.357 seconds

This does not seem like much but it adds up over a year. The length of the year is not changing; the Tidal Acceleration between the earth and the sun is so small it can be ignored. However, over 5 million years ago there were more short days within each year. This can be determined by this equation:

- Current day = 24 hours = 86,400 seconds
- Shorter day back then = 24 hours – 2 minutes, 3.357 seconds =86,400 – 123.357 = 86,276.643 seconds
- Current length of the year = 365.242199 days = 365 days, 5 hours, 48 minutes, 46 seconds
- Length of the year over 5 million years ago = (365.242199 days)(86,400 sec/86,276.643 sec) = 365.76442 seconds =
**365 days, 18 hours, 20 minutes, 46 seconds**

What this means it that you cannot wait 4 years to add a leap day; a calendar for over 5 million years in the past would have been off by more than two days every three years. What this further means is that such a year does not match the Shire Calendar, the Reckoning of Rivendell, the King's Reckoning, the Revised Calendar, or most of the text in Appendix D of *The Lord of the Rings*.

## [edit] Conclusion

Andreas Möhn's solution for the discrepancy between the descriptions of the moon's phases in *The Hobbit* and *The Lord of the Rings* does not hold up. Introducing a 20-minute shorter lunar cycle places the time of the late Third Age too long ago to agree with what readers would expect and is directly contradicted by Tolkien's own statements placing the Third Age thousands but not millions of years ago. The shortening of the length of the day, unconsidered by Mr. Möhn, means that his solution violates the majority of Tolkien's Appendix D of *The Lord of the Rings*. Therefore Mr. Möhn's solution for reconciling the moon's phases between the two stories must be rejected.

## Notes

- ↑ For looking up the moon's appearance on any date, see Sun and Moon Data for One Day. For the location data, use Longitude: east, 19 degrees, 1 minute; Latitude: north, 54 degrees, 0 minutes; and Time Zone: 1.27 hours east of Greenwich - This corresponds to the location of Rivendell.
- ↑ For this calculation the moon's orbital period rather than the synodic month is used.
- ↑ The moon's orbit is not circular but its eccentricity is low enough for these calculations.
- ↑ For an explanation of tidal acceleration see Wikipedia|Tidal acceleration.

## References

- ↑ http://www.alice-dsl.net/lalaith/M-earth.html
- ↑
J.R.R. Tolkien, Christopher Tolkien (ed.),
*The Treason of Isengard*, "XVII. The Great River", Phases of the Moon - ↑ J.R.R. Tolkien,
*The Hobbit*, "A Short Rest" - ↑ J.R.R. Tolkien; Humphrey Carpenter, Christopher Tolkien (eds.),
*The Letters of J.R.R. Tolkien*, Letter 211, (dated 14 October 1958) - ↑ J.R.R. Tolkien,
*The Lord of the Rings*, Appendix D, "The Shire Calendar"