# The Year 2100 without Leap Seconds

New -- Computers can handle leap seconds correctly. There is one working and deployed scheme which can be distributed using a protocol in RFC 7808.

## If leap seconds are abandoned in the year 2020, how far might the clock and calendar drift away from the sun?

Discussions on the LEAPSECS mail list during 2014 indicated that the year 2100 was of concern to some. In particular, the concern was about the magnitude of the difference that might be expected between purely atomic time and mean solar time (Universal Time) as of the year 2100. This web page examines various estimates for how far the boundary of a calendar day measured using cesium-133 hyperfine transitions could differ from the boundary of a calendar day determined by measuring the rotation of the earth. As seen in the sibling web page Extrapolations of the difference ( TI - UT1 ) the difference being discussed will be called "DUTC".

This web page is basically a rework of "The Leap Minute" paper presented by J. Seago at the Future of UTC colloquium in Charlottesville Virginia during 2013 May. The curves plotted in the graphs here are based on notable papers by authors who have studied the rotation of the earth over history. Estimating the magnitude of Delta T was not the goal of most of these papers, so the results here may not represent the opinions of those authors. In addition to the estimates in these papers, these plots also include the three parabolic extrapolations found in the web pages sibling to this one.

The first plot presents the estimates for Delta T found in the various papers. The values for Delta T at year 2100 are not representative of the value of DUTC because some of these papers were written without the expectation that leap seconds might be abandoned. Other papers supposed that leap seconds would be abandoned, but they chose different dates for when that might happen.

In the above plot most of the curves are a parabola; that is the expected result for a linear change over time in Length of Day (LOD). Some of the curves are straight lines; that is the expected result for a value of LOD that is constant with time.

The Espenak+Meeus curve differs from the others because they chose to represent Delta T with piecewise polynomials. It appears that Espenak+Meeus decided to guess that the recent lack of change in LOD might end in year 2050, and that the earth rotation would then decelerate more.

The simplest way to adjust the various estimates of Delta T so that they agree on a date for the supposed final leap second is to choose that year and then offset their values such that all of them pass through zero in that year. The ITU-R WRC-15 will occur during 2015 November. Previous drafts proposing changes to the radio broadcast time signals have indicated that the change would go into effect five years after the ITU-R adopts the change. Therefore it is convenient to choose the year 2020 as the supposed date of the final leap second. The following plot shows the result when all of the estimates of Delta T are adjusted to pass through zero in the year 2020.

The McCarthy (2012) curve is based on the deceleration for earth rotation found using the models of mantle and ocean presented in Mathews+Lambert (2009).

When looking at the above plot (and the one below) it is relevant to keep in mind the historical values that have been measured for LOD. The values of LOD can be seen in the plots on the sibling web page on DUTC. A quick glance at those LOD plots shows that LOD has never been well represented by a line of any slope, therefore Delta T (the integrated values of LOD) in the plots here is not well represented by any parabola. A longer glance at those LOD plots shows the decadal fluctuations during the past two centuries have involved huge and unpredictable torques which accelerated and decelerated earth rotation enough to change LOD by more than 4 ms.

The paper by Huber (2006) includes a statistical analysis of the fluctuations of LOD over time. Huber determined estimates of the sizes of various components of the LOD variations and produced an equation for how large of an error should be expected in any prediction of Delta T. Over an extrapolated interval of 80 years the equation by Huber predicts that one standard error in a prediction for Delta T would be 37 seconds.

The plot above shows a one sigma error bar of size +/- 37 seconds which could be applied to the end point of any of the predictions. It should be expected that any prediction of Delta T for year 2100, even one which uses exactly the right model for the long-term changes in LOD, will be uncertain by 37 seconds because of the effectively stochastic short-term variations in LOD. Deviations of two sigma, three sigma, or more from any prediction are possible.

Several of the Delta T curves in the above plots were specifically tailored to match the observations immediately surrounding the 20th century. The plots below show all of the curves along with the historic values of Delta T for the past 1500 years. It is clear that the linear predictions of Delta T are not a good match for the historic measurements.

## References

• D.D. McCarthy & A.K. Babcock, 1986, Physics of the Earth and Planetary Interiors, 44, 281-292
• D.D. McCarthy & W.J. Klepczynski, 1999, GPS World, 10(11), 50-57
• R.A. Nelson, et al., 2001, Metrologia, 38, 509-529
• L.V Morrison & F.R. Stephenson, 2004, Journal for the History of Astronomy, 35, 327-336
• L.V Morrison & F.R. Stephenson, 2005, Journal for the History of Astronomy, 36, 339
• P.J. Huber, 2006, Journal of Geodesy, 80, 283-303
• F. Espenak & J. Meeus, 2006, Five Millennium Canon of Solar Eclipses: -1999 to +3000, NASA Technical Publication TP-2006-214141, 14-18, http://eclipse.gsfc.nasa.gov/SEcat5/deltatpoly.html
• P.M. Mathews & S.B. Lambert, 2009, Astronomy & Astrophysics, 493, 325-330
• D.D. McCarthy, 2012, "A Convention for Coordinated Universal Time", IAU Joint Discussion 7, Presentation 2, http://referencesystems.info/uploads/3/0/3/0/3030024/2-07.pdf
• J.H. Seago, 2013, "The Leap Minute", Paper AAS 13-510 from Requirements for UTC and Civil Timekeeping on Earth, American Astronautical Society Science and Technology Series, Vol. 115, Univelt Inc., San Diego
• D. Matsakis, 2014, "Comments on the Debate over the Proposal to Redefine UTC", URSI General Assembly, presentation A01.2, http://tycho.usno.navy.mil/papers/ts-2014/Matsakis-LeapSecondComments.URSI-2014.pdf
• C. Bizouard & D. Gambis, 2014, IERS EOP C04 Combined Series, http://hpiers.obspm.fr/eop-pc/products/combined/C04.html

Steve Allen <sla@ucolick.org>
\$Id: year2100.html,v 1.20 2017/11/24 18:58:48 sla Exp \$