# Determining mean Length of Day and DeltaT formula

Using observed data for 500 BCE to 1600 CE from [Morrison&Stephenson, 2004] and for 1650 CE to 1990 CE from IERS, in the below section the change in LOD and Delta T are described.

## Mean Length of Day (LOD)

The following mean changes in Length of Day (LOD) are depicted in the below figure:
• 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 (, page 514) Interesting to see that there looks to be a periodicity of around 1440 and 1550 years (also mentioned by Stephenson (, 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 spliing methodology  used (based on the ideas of Slutzky ), 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 (fsample +/- factual) 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.

## Linking the seen periodicity with possible geophysical or celestial events

Stephenson [Stephenson, 1997, page 513] gives an possible explanation what causes the periodicity:
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 1470 years, see also Bond   (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 -9658 - 1473*event#
With: event# = an positive/negative integer
The periodicity of DeltaT seems to reach zero around 1820 CE, so that would be a DO warming event# around -7 or -8 (-7.8 to be specific).

Another possible reason he gives 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). Is this perihelion-node-apside cycle another reason for the possible periodicy in DeltaT.
So there are some different periods mentioned (DO events: 1470Years or perihelion-node-apside cycle 1800/1670Years [Pettersson/Stacey] and 1250Years [Reijs]).

## Mean Delta T

The below mean DeltaT graph is determined using the my above change in LOD formula. The following graphs are depicted:
• 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 ## The DeltaT formula

Using Stephenson&Morrison  as the basis (n.dot = -26"/cy^2):
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
DeltaTVR = (OffSetYear ^ 2 / 100 / 2 * Average + Periodicity / 2 / Pi * Amplitude * (Cos((2 * Pi * OffSetYear / Periodicity)) - 1)) * Y2D [msec]

## New publication by Stephenson, et al.

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

 Parameter DeltaTVR [Reijs, 2006] Stephenson  StartYear [CE] 1820 1825 Average LOD growth [msec/cy] 1.80 1.78 Periodicity [Year] 1443 1500 Amplitude [msec] 3.76 4.0

## References

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

## Conclusion

The parabolic formula provided by Stephenson to calculate the mean DeltaT is perhaps lacking enough elements to predict the DeltatT (or LOD) accurate enough over the whole time period of 500 BCE to say 1300 CE (differences smaller than 10%). This additional periodic term in the formula gives a better mapping to the table of Morrison&Stephenson than only a parabolic formula.
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