Let's drive the discussion forward into a region that we haven't
touched here before:

What happens with a leap-second and leap-hour free time scale (as the
TI proposed at Torino) in the long run (>> 1 kiloyear into the future)
and what interactions with the length of the year emerge there?

Let's say, we get used to the idea of dropping all leap seconds from
UTC from today and we rename the resulting new uniform atomic time
scale into International Time (TI). This would put the point (or
meridian) on Earth where International Time coincides with local time
(currently for UTC this point wanders around somewhere near the
Greenwich meridian) onto a slowly accelerating course eastwards,
detaching International Time from London and making it truely
international.

Fine so far (apart from short-term concerns with legacy software over
the next 10-20 years).

How will local times have to be adjusted?

Stephenson and Morrison give

  delta_T(T) = TAI - UT1 = (31 s/hy^2) T^2 - 52 s

where T = (t - 1820-01-01T00:00) / (100 * 365.25 days) is the number
of centuries (hectoyears, hy) since 1820. For Temps International
TI = TAI - 32 s (if introduced today), we get equivalently

  delta_T(T) = TI - UT1 = (31 s/hy^2) T^2 - 84 s

The offset between local civilian times would have to be adjusted by
one hour when delta_T(T) = 0.5 h, 1.5 h, 2.5 h, ..., i.e. at about the
years

  sqrt((1 * 1800 s + 84 s) / (31 s/hy^2)) * 100 + 2000 = 2780
  sqrt((3 * 1800 s + 84 s) / (31 s/hy^2)) * 100 + 2000 = 3330
  sqrt((5 * 1800 s + 84 s) / (31 s/hy^2)) * 100 + 2000 = 3712
  ...

   2780, 3330, 3712, 4023, 4292, 4533, 4752, 4956, 5146,
   5326, 5496, 5658, 5814, 5963, 6107, 6246, 6380, 6511,
   6638, 6762, 6882, 6999, 7114, 7227, 7337, 7444, 7550,
   7654, 7755, 7855, 7954, 8050, 8146, 8239, 8332, 8423,
   8513, 8601, 8689, 8775, 8860, 8944, 9027, 9109, 9191,
   9271, 9350, 9429, 9507, 9584, 9660, 9735, 9810, 9884,
   9957, 10030, 10102, 10173

The adjustment of a civilian time by one hour can easily be
accomplished without particular disruption as part of the summer time
arrangements (assuming that this 1% electricity saving measureis still
of concern in the far future). This would have to be done for the
first time near the year 2780, and then every few hundred years, and
from the year 7700 on even several times per century.

What I am still struggling with is the long-term perspective. At
present, the maximum difference between any civilian local time and
the international reference time (currently: UTC) is limited to +/- 13
h. That limit would be dropped if we replaced UTC with TI, and at
about the year

  sqrt((12 h + 84 s) / (31 s/hy^2)) * 100 + 2000 = 5736

the point where International Time corresponds to local time will
cross the International Date Line. What do we do then? Having
International Time and local civilian time several days apart sounds
rather unpractical for doing mental arithmetic and could lead to
confusion far more severe than anything leap seconds might ever cause.

I had briefly hoped that we can play around with 29 February and
remove from the civilian time zones a 29 February (compared to what
pope Gregory dictates) near the year 5700, in order to keep the
maximum offset between any civilian time zone and International time
at least limited to +/- 25 hours. International Time would continue to
strictly follow the Gregorian rules, as it must be uniform and
long-term predictable. A Gregorian "leap year" in TI that is ignored
in local civilian times would bring civilian times back into sync with
TI without much disruption.

This would at first glance of course mess up the date of the spring
equinox (the reason for the Gregorian calendar reform), and who knows
whether people still worry about when Easter Sunday is by then. We
need leap years, in order to compensate for the fact that the
rotational frequency of the earth around the sun and around its own
polar axis have a non-integer relationship. However, if the length of
the tropical year were highly constant, the need to keep local
civilian times and TI from diverging by more than a day and the need
to keep the spring equinox on the same date every year would lead to
compatible requirements for scheduling leap years in civilian time
zones beyond the period when the Gregorian rule works.

Unfortunately, the rotation of the Earth around the Sun accellerates
far too fast for this idea to work:

Newcomb's formula for the geometric mean longitude of the Sun is

  L = 279 41' 48".04 + 129602768".13 C + 1".089 C^2

where C = (t - 1900-01-01T12:00Z) / (100 * 365.25 days) is the number
of (Julian) centuries since 1900. Newcomb's third term represents the
acceleration of the Earth's mean angular velocity around the sun. This
term will grow to 0.5 days or equivalently a mean-sun longitude offset
of 360/(2 * 365.25) at

  C = sqrt(360/(2 * 365.25) * 60*60" / 1".089) = 40.36 centuries = 4036 years

In other words, the longterm frequency stability of the annual
oscillation of the earth is not significantly better than that of the
daily oscillation (even worse: one speeds up and one slows down),
therefore leap-days (29 February) cannot solve both problems at the
same time.

What else can we do? Shall we give up the long-term stability of the
date of the spring equinox? Shall we hand over the authority of
deciding, whether a year that is a multiple of 400 shall be a leap
year in civilian time zones to someone like the IERS, while the
insertation of leap days into TI will strictly follow the Gregorian
rule in the interest of long-term uniformity?

In other words, can we live with the spring equinox (and therfore
religious dates) moving over the next few thousand years? Is the date
of the spring equinox more important to people than having an upper
limit for the offset between a uniform atomic International Time and
local civilian time?

(Perhaps it is time to get the theologians back into this discussion
after so many centuries ... ;-)


Reference:

  - Nelson, McCarthy, et al.: The leap second: its history and
    possible future. Metrologia, Vol. 38, pp. 509-529, 2001.
    http://www.cl.cam.ac.uk/~mgk25/time/c/metrologia-leapsecond.pdf

--
Markus Kuhn, Computer Lab, Univ of Cambridge, GB
http://www.cl.cam.ac.uk/~mgk25/ | __oo_O..O_oo__

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