- Mean Length of Day (LOD)
- Linking the seen periodicity with possible geophysical events
- Mean Delta T
- The DeltaT formula
- Can eclipses provide a
clue around ndot
_{real}?

- New publication by Stephenson, et al.
- References

- Conclusion

- Observed mean changes [Morrison&Stephenson, 2004] in LOD (pink wavy line) (from 1650 CE is are quite accurate values)
- Tidal friction influence (straight light blue dotted line) of 2.3 [ms/cy] [Stephenson, 1997, page 513]
- Optimized mean changes in LOD (black crosses), this is as per
my optimized function (using Simplex method with one linear and
one periodic term).

The mean Solar Day function is also incorporated in the Excel XLA file for archaeoastronomy and geodesy functions.

- Average mean changes in LOD (yellow straight line), is around 1.72 [ms/cy], using my own Simplex optimized function. This is within the value range (1.70+/-0.05 [ms/cy]) as provided in Stephenson ([1997], page 514)

Interesting to see that there looks to be a periodicity of around 1440 and 1550 years (also mentioned by Stephenson ([1997], page 516), if people know such a cycle and have some proof due to an astronomical/geophysical process, let me know.

It could be this periodicity is caused by the splining methodology used (based on the ideas of Slutzky [1937]), but I doubt that.

Another reason for a spurious periodicity could be due to the sampling of only a limited amount of eclipses (Stephenson's pool of observations), but I think that that would not results in such a long term periodicity, because:

- I am thinking that the Nyquist-Shannon
theory
of
sampling and low pass (anti-aliasing) filtering could be
related to this idea. If this low pass filtering does not happen
at half the Nyquist sampling frequency one could get spurious
frequencies (f
_{sample}+/- f_{actual}) in the relating analysis.

- Assuming say an on average sampling of every 6 years an eclipse (pool of 400 observations over a period of 2300 years).
- to get a spurious periodicity of 1550 years in an analysis, an actual periodicity of some 5.98 or 6.02 years (1/1550 = 1/6 +/- 1/actual) must exist in the real LOD (which is not 'filtered out'). Such a natural(/actual) periodicity is not likely, IMHO.
- furthermore the splining in 5 knots would also not stimulate
such a long term periodicity over all knot intervals (the spurious periodicity
would even vary much more per knot interval; because the amount
of observations varies greatly per knot interval).

- In the above I used a uniform distribution of the observation, while in actual live it are random observations, but still the large variation of observation per knot interval would not give such a visible uniform long term periodicity over all knot intervals.
- A test, if this periodicity is spurious, is by deliberately changing the number of observations (larger spacing between observations) and see if the periodicity changes in duration. If it is spurious, the periodicity must change according to the (1/spurious = 1/sample +/- 1/actual) formula.

With
regard
to the quasi-periodic fluctuations on a timescale of some 1500
years, a possible mechanism would appear to be electromagnetic
coupling between the core and the mantle of the Earth. This is
the most likely cause of the decade fluctuations (Lambeck, 1980,
p. 247).

A possible other reason might be the Dansgaard-Oeschger (DO) warming events (with a possible solar origin), which are events spaced by 1473 years, see also Bond [1997] (which is close to my determined periodicity value of 1440 and 1550 years). These events can be seen in Greenland ice cores and relate to glacial periods. And I think that such climate events can impact the amount of ice and thus the momentum of the Earth; thus possibly its rotation. Some researcher also see the 1470 as a superimposition of the 210 and 87 year cycles that can be recognised in respectively the DeVries-Suess and Gleissberg solar cycles [Braun, et al, 2006]. Is it important to note that the DO events seem not to have an effect on the climate in the holocene (aka present past), but the 'clock' behind DO events still exists (Rahmstorf, Pers. comm. 2016).

The astronomical year of a DO warming event is on -9655 - 1473*event#

With: event# = an positive/negative integer

The periodicity of DeltaT seems to reach zero around 1820 CE, so that a DO warming event# would be between -7 or -8 (-7.8 to be specific).

Another possible reason given by Lamb is the change of sea-level variations (Lamb, 1982); "as significant long-term alterations in climate have been detected in the last few milennia". In earlier publications Lamb (1972, pp 220) is more specific: quoting Stacey (and Pettersson) about maximum tide-generating forces happening with an approximate period of 1670 years due to a perihelion-node-apside cycle [Pettersson, 1914, pp2].

This can be checked using the FDM method, described here, looking at Anomalistic Year (related to Earth's perihelion), Draconic Month (related to Lunar nodal cycle) and Anomalistic Month (related to Lunar apse cycle). This gives a beat period of around 1250 Years (for epochs: around 1000 to 2000). So this is a different periodicity than mentioned by Pettersson.

Is this perihelion-node-apside cycle another reason for the possible periodicy in DeltaT?

So there are some different periods mentioned: DO events; 1473Years or perihelion-node-apside cycle; 1670Years [Pettersson/Stacey] and 1250Years [Reijs].

- Observed mean DeltaT [Morrison&Stephenson, 2004] (pink line)
- DeltaT due to tidal friction influence (straight light blue dotted line) [Stephenson, 1997, page 513]
- Optimized mean DeltaT (black crosses), this is as per the
integral of my change in LOD function

This mean DeltaT function is also incorporated in the Excel XLA file for archaeoastronomy and geodesy functions.

- The average mean DeltaT (yellow line) based on LOD change of around 1.7 [ms/cy] is also depicted

StartYear = 1820 [year]

Average = 1.80 [msec/cy]

Periodicity = 1443 [year]

Amplitude = 3.76 [msec]

Y2D = 365.25

OffSetYear = (JDutfromDate(StartYear, 0) - JDNDays) / 365.25

DeltaT

Solar/lunar eclipses can not provide the real-life value of the
lunar accleration (ndot_{real}). The argumentation is as
follows:

- An ephemeris is made using a certain physical model and that
physical model will result in the (likely) locations of
celestial bodies.

- One has to start with a reasoanble accurate physical mode. Furthermore ephemerii will evolve over time (an iterative process).
- From that physical model one can determine if a celestial
object has an accelaration over a long period. In that way one
can determine e.g. the ndot
_{eph}(secular accelaration) of the Moon for that particular physical model/ephemeris. - So an ephemeris has such secular acceleration parameter (most of the time implicitly) included.
- Using any ephemeris with a certain ndot
_{eph}, one can calculate when the eclipse happens by varying the DeltaT (only valid for that for that ephemeris). This will give a time interval when it can be witnessed (this has been redone by the author for some 80% of Stephensons' documented eclipses (Stephenson, 1997)). - The DeltaT value/formula always needs to be accompanied by an
ndot
_{ref}value. Without such an ndot_{ref}value a DeltaT formula is worthless.

- Remember the ndot
_{ref}belonging to DeltaT formula is just a reference parameter, nothing more. One can always convert a DeltaT formula to another ndot_{convert}(Morrison&Stephenson, 2012; updated formula from McCarthy&Babcock, 1986):

`ΔT`_{convert}`= ΔT`_{ref}`- 0.910747 * (ndot`_{convert}`- ndot`_{ref}`) * t`^{2}

`where: t = (decimalisedyear-1955.5)/100`

- If the ndot in real life or in an ephemeris is different than
ndot
_{ref}, one would still have an eclipse when one converts the DeltaT values to this different (possible real-life) ndot. - DeltatT formula evolves over time (due to finding more eclipse events e.g. (Stephenson, 1997) and (Huber, 2004); and/or finding a more accurate physical model behind DeltaT).
- For eclipse determination: it is not important to know the
ndot
_{real}, as long as one converts the DeltaT value to the ephemeris' ndot. - It is essential to realise that the above methodology does not
provide the ndot
_{real}, the above methodology only provides an ndot based on a physical model and a DeltaT based on a specific ndot.

- The ndot
_{real}is only reductable from a theory (like within the underlaying ephemeris model) and/or matching on recorded observations.

- I doubt if eclipse events can be the tool to determine the
ndot
_{real}(as one can match an eclipse with many combinations of physicalmodel/ephemeris/ndot and DeltaT formula).

Comparing the above derived parameters with the published values
by Stephenson [2016, formula 5.1]:

Parameter |
DeltaT_{VR} [Reijs,
2006] |
Stephenson [2016] |

StartYear [CE] |
1820 |
1825 |

Average LOD growth [msec/cy] |
1.80 |
1.78 |

Periodicity [Year] |
1443 |
1500 |

Amplitude [msec] |
3.76 |
4.0 |

Bond, Gerard, Showers William, Chesby Maziet, Lotti Rusty, Almasi
Peter, deMenocal Peter, Priore Paul, Cullen Heidi, Hajdas Irka,
and Bonani Georges, 'A pervasive millennial-scale cycle in North
Atlantic holocene and glacial climates', Science 278, no. 5341
(1997): 1257. http://ruby.fgcu.edu/courses/twimberley/EnviroPhilo/BondPap.pdf

Braun, Holger, Marcus Christl, and Stefan Rahmstorf. 2005.
'Possible solar origin of the 1,470-glacial climate cycle
demonstrated in a coupled model', Nature, Vol 438: pp. 208-11.

Chapront, J. et al.: A new determination of lunar orbital
parameters, precession constant and tidal acceleration from LLR
measurements. In: Astron. Astrophys 387 (2002), pp. 700-709.

Henriksson, Göran The acceleration of the Moon and the universe:
The mass of the graviton. In: Advances in Astrophysics 2 (2017),
issue 3, pp. 184-196. http://www.isaacpub.org/images/PaperPDF/AdAp_100064_2017041816162617572.pdf

Huber, Peter J. and Salvo de Meis: Babylonian eclipse observations
from 750 BC to 1 BC. Trans. by, Mimesis 2004.

Lamb, H.H. 1972. Climate: Present, past and future (London:
Methuen & Co).

McCarthy, Dennis D. and Alice K. Babcock: The length of day since
1656. In: Physics of the Earth and Planetary Interiors, 44 (1986),
pp. 281-292.

Morrison, L.V. and R.W. Stephens: Observations of secular and
decade changes in the Earth's rotation. In. Anny Cazenave (ed):
Earth rotation: Solved and unsolved problems. Springer
Netherlands 2012. pp. 69-78.

Morrison, L.V., and F. Richard Stephenson. 2004. 'Historical
values of the Earth's clock error Delta T and the calculation of
eclipses', Journal for the History of Astronomy, Vol 35: pp.
327-36.

Nautical Almanac Office: The Astronomical Almanac for the year
2014: And Its companion the Astronomical Almanac Online. Trans.
by, U.S. Government Printing Office 2013.

Pettersson, O. 1914. "Climatic variations in historic and
prehistoric time." in Svenska Hydrografik-Biologiska Kommissions
Skrfiter, Vol 5. Göteborg.

Rahmstorf, Stefan. 2003. 'Timing of abrupt climate change: A
precise clock', Geophysical research letters, Vol 30: pp. 17-1,
17-4.

Reijs, Victor M. M. 2006. 'Determining mean Length of Day and
DeltaT formula', in http://www.archaeocosmology.org/eng/DeltaTeval.htm
[accessed 30 May 2006].

Slutzky, Eugen. 1937. 'The summation of random causes as the
source of cyclic processes ', Econometrica, Vol 5: pp. 105-46.

Stephenson, F. Richard. 1997. Historical eclipses and Earth's
rotation (Cambridge University Press).

Stephenson, F. Richard., Leslie V. Morrison, and Catherine Y.
Hohenkerk. 2016. "Measurement of the
Earth's rotation: 720 BC to AD 2015." in Proceedings of the
Royal society, Vol 472. in http://rspa.royalsocietypublishing.org/content/472/2196/20160404.

The periodic formula published [Stephenson, 2016] compares very well with the earlier derived formula [Reijs, 2006]: there is not that much difference in these parameter values. Reijs gives a possible reason for this periodicity: Dansgaard-Oeschger (DO) warming events.

The periodic formula has also been implemented in Stellarium since 2013. If you want to test the formula (one can use Excel XLA file), let me know.

Major content related changes: May 3, 2006