[LEAPSECS] Leapin' on the Merry-go-round
Rob Seaman
seaman at noao.edu
Wed Jan 2 18:29:35 EST 2008
Clearly 2008 is going to be another chatty year :-)
John Cowan wrote:
> The world's slowest merry-go-round, certainly; the best guess seems
> to be
> that it will make a full turn in 28 ky, within the same order of
> magnitude
> as the precession of the equinoxes.
Not sure where you got that. Consider Steve Allen's Length of Day plot:
http://www.ucolick.org/~sla/leapsecs/ancient.pdf
The leap deficit accumulates as the area under the curve. If the
curve is approximated as linear (appropriate over multi-million year
periods of tidal slowing), then the area under the curve is
quadratic. Thus the "leap hour equivalents" accumulate more quickly
further from the origin (which falls in the early 19th century).
The equation to integrate is something like:
LOD = LOD(zero) + slope * centuries
But LOD(zero) can be identified as 86400 + delta(zero) since we
already have accumulated a leap debt since the 19th century. (The
circa 1970 zero point had an offset from the actual smoothed
LOD(1970).) So:
LOD = 86400 + delta(zero) + slope * centuries
But:
DUT1 = integral (LOD - 86400) from 0 to N days
Rearranging and integrating both sides:
DUT1 = integral (delta(zero) + slope * centuries) from 0 to N days
Delta(zero) is about 2 ms and slope is 1.7 ms/century (taking the
middle trend line - should be steeper and thus more rapid accumulation
over longer periods). Changing variables, etc:
DUT1 = 0.002 * N + 2.3e-8 * N^2
The accumulation of a leap deficit is thus roughly linear over the
next couple of centuries, but becomes dominated by the quadratic term
after that. Obviously our poor calibration will fail at some point,
but the *smoothed* effect has to be something like this unless the
tidal transfer of angular momentum is much more creative than
frictional braking. The very long term (gigaday) LOD information
seems consistent with something fairly linear in any event.
(The 600 year estimate for the first leap hour appears to have
resulted from setting DUT1 = 1800s and solving for N = 657 years.
Clearly if the ALHP were to be adopted, there would be a benefit in
seesawing a half hour behind and then a half hour ahead, rather than a
zero baseline sawtooth.)
This all adds up to a quadratically accelerating merry-go-round.
Neglecting the linear term, we have the first full turn of the
carousel completing in about 5,000 years.
As anybody with little kids knows, you only get a few turns of the
carousel before they kick you off and load the next batch of kids -
who would rather be on the Mad Hatter, anyway. The second turn of the
merry-go-round will take a bit over 2,000 years. The third turn,
significantly under 2,000 years with a leap hour equivalent every 60
years.
(The one thing I'm sure of is that my math will be corrected if I've
screwed it too badly :-)
Yes, the timelords can cheat mean solar time for a while, but the
burden (dozens of leap hour equivalents) loom more and more
frequently. This differs from any calibration errors in the Gregorian
calendar that maintain a decorous pace every few centuries or
millennia. (I can hear the suggestion now to fix it all with a
February 29.5 whenever noon becomes midnight :-)
Better to deal forthrightly with the natural requirement for tweaking
our clocks.
Rob
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