Leap-seconds, the epsilon perspective

From: <Mark.Calabretta_at_ATNF.CSIRO.AU>
Date: Wed, 29 Jan 2003 10:31:30 +1100

On Mon 2003/01/27 18:21:02 -0000, Ed Davies wrote
in a message to: LEAPSECS_at_ROM.USNO.NAVY.MIL

>Everything would be a lot simpler and more reliable if all systems
>could work with a single simple "universal" time scale which:
>1. Has SI length seconds.
>2. Has minutes of 60 seconds, hours of 3'600 seconds and days of
> 86'400 SI seconds always, not just often enough to lull testers
> into a false sense of security.

Much of the problem boils down to the question of why we would want
to continue to pretend that a mean solar day has exactly 86400 SI
seconds when in fact, it has 86400+epsilon SI seconds.

Currently epsilon is small, approximately 2ms. Even so, after about
18 months the error in ignoring it accumulates to a full second - now
applied as a leap-second.

However, the issue addressed in the "GPS World" article of 1999/Nov
by McCarthy and Klepczynski which introduced this discussion (LEAPSECS
archive, http://rom.usno.navy.mil/cgi-bin/wa?A1=ind00&L=leapsecs&F=lf)
is that epsilon will continue to grow, although slowly, because of the
secular deceleration of the Earth's rotation rate caused by Earth-Moon
tidal interaction. Thus, in about 200 years time, epsilon will be
approximately 10ms and a leap-second would be required every three
months. Thereafter, matters will only get worse.

The thing to note here is that the pretence that a mean solar day has
exactly 86400 SI seconds leads sooner or later to a cumulative error
of the small quantity, epsilon to a point where it cannot be ignored
if UTC is to track UT1 to acceptable tolerance (currently 900ms).

A second point is that the physical cause of the secular deceleration
of the Earth's rotation rate is reasonably well understood and it
could, in principle, be predicted with reasonable accuracy well into
the future. Specifically, we know that it will have a *QUADRATIC*
effect on the difference between UT1 and TAI. This important point
seems to have been lost in the present discussion.

This suggests that, in order to eliminate leap-seconds, we should
quit pretending that a mean solar day has exactly 86400 SI seconds
and instead construct our clocks so that they measure its true length.

I note that this possibility was not canvassed in the GPS World
article, although the much more radical idea of changing the
definition of the SI second was (among others).

How would it be realized in practice?

   1) Any clock which keeps time to an accuracy of less than a few
      millisec per day (a few parts in 10^8) would not need changing.

      I suspect this would account for 99.9% of the world's clocks,
      including the clocks inside most computers, VCRs and microwave
      ovens; on your wrist; or next to your bed.

   2) All precision clocks (including, for example, NTP servers)
      would have to be given the capability of handling epsilon as a
      user-supplied input.

      The clock would count 86400+epsilon SI seconds before
      beginning a new day.

      The assumption here is that the basic "tick" of a precision
      clock is much less than a millisecond (I presume that this
      holds pretty well in practice). Supposing that all precision
      clocks tick at a microsecond rate (or faster), then the value
      of epsilon used in practice, call it Epsilon, would have to be
      an integral number of microseconds.

   3) Time-keeping software and algorithms, for example, that which
      transform between UTC and UT1, would have to be modified.

   4) The value of Epsilon in use would be held constant for as long
      as practical, but since the true value of epsilon changes with
      time, the time and frequency service providers would have to
      issue bulletins stating that the value of Epsilon will change
      to such-and-such a value at such-and-such a time. Clocks would
      be programmed with the new value of Epsilon to take effect at
      the appropriate time. Since epsilon changes very slowly, this
      would be a relatively uncommon event, certainly much less common
      than a leap-second insertion would become in the next few

      Although epsilon is reasonably predictable, its value could be
      adjusted slightly to correct for any non-secular variations
      which had accumulated over the period since the previous update.
      (Alternatively, non-secular variations might be ignored
      altogether, thus making Epsilon, and hence UTC, predictable -
      albeit with UTC diverging from UT1 by a few seconds over short

      It would be best if new values of Epsilon were issued on a
      fixed schedule.

   5) The current system where UTC has leap-seconds would be replaced
      at a pre-determined time by daily epsilon accounting, thus
      providing a smooth transition.

Why would this be a good idea?

   1) No leap-seconds - communication and navigation communities happy!

   2) UTC still tracks UT1 - astronomical community happy!

   3) Smooth, monotonic-increasing civil time scale - computer
      programmers happy!

   4) Cumulative effect of epsilon replaced by a small daily
      correction - grandchildren's grandchildren happy!

   5) No legislative changes required - politicians happy!

   6) No effect on religious or sporting calendars - religious and
      sporting communities happy!

   7) Conceptual simplicity - everyone happy!!

Why might this not be such a good idea?

   1) Leap-second updates replaced by Epsilon updates - user
      intervention still required (unless non-secular changes in
      epsilon are ignored).

      However, the error introduced by a missed, or delayed update
      would be much less, only a small fraction of a millisecond per
      day (cumulative).

   2) UTC still not predictable indefinitely into the future (unless
      non-secular changes in epsilon are ignored).

   3) The difference between TAI and UTC would no longer be an
      integral number of (leap) seconds.

      Historical time calculations would be a little more complicated
      (unless non-secular changes in epsilon are ignored) because the
      value of Epsilon and its date of introduction would have to be
      recorded, rather than just the date of introduction of a leap-

   4) While digital clock displays would handle this system easily,
      analogue clock faces (even if digitally driven) might have
      trouble once epsilon exceeds a few seconds - but that is a long
      way off.

   5) Burden falls on precision clockmakers, and timekeeping software.

Why wasn't this done from the start?

   I believe that the secular deceleration of the Earth's rotation
   rate was unknown, or poorly understood, in the early 1970s when
   UTC was introduced.

   Also at that time, clocks were not smart enough to handle this sort
   of correction, but now that they are driven by those new-fangled
   (in 1970s terms) micro-processors it should be relatively easy to

   Analogue clock faces would have been the norm in the early 1970s,
   adjusting them for leap seconds would have been simpler.

What would be the timescale for such a change?

   Many clocks (e.g. GPS Time and the clocks at most astronomical
   observatories) would maintain their best approximation of TAI
   and software would compute UTC from it using tabulated values of
   Epsilon (or computed values if non-secular changes in epsilon are
   ignored). Other clocks might maintain UTC directly using the
   current value of Epsilon. Either way, the clock and/or software
   would need changing.

   Costs would be minimized if the change was incorporated as part of
   the replacement cycle of precision clocks and timekeeping software.
   Thus, it depends largely on what the lifetimes of these are;
   ideally, all such should have been replaced by the time the change
   was introduced.

   Thus we may be looking at a multi-decade lead-in. However, there
   is no rush!

What would we tell the public?

   The Earth is slowing down (friction - sort of) and the days are
   getting very slightly longer, but you don't have to worry about
   it unless you need to know the time of day to within a few
   milliseconds. In particular, Ramadan, Diwali, Pesach, Easter,
   Hanamatsuri, the World Cup, etc. are not affected, nor are
   world timezones.

And the *really* long term?

   Millions of years hence, when the mean solar day is 25 SI hours
   long, epsilon would be one hour but UTC would still function
   properly. Notably, updates to Epsilon would be as infrequent
   as ever.

   However, the world's timezones, especially those furthest from
   Greenwich, would need a modest adjustment - half an hour at most.

Dr. Mark Calabretta
Australia Telescope National Facility
Received on Tue Jan 28 2003 - 15:42:17 PST

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