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,
- 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 ),
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
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
- 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:
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,
A possible other reason might be the Dansgaard-Oeschger
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.
The astronomical year of a DO warming event is on -9658 -
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
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
- DeltaT due to tidal friction influence (straight light blue
dotted line) [Stephenson, 1997,
- 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]:
|Average LOD growth [msec/cy]
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.
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,
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 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
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