From: msb@sq.sq.com (Mark Brader)
Subject: Re: Leap years (was Smoke cigarettes--leave early!)
Organization: SoftQuad Inc., Toronto, Canada
Date: Mon, 14 Dec 92 03:38:40 GMT

In an earlier article, someone wrote that the Gregorian calendar would
need adjustment in 1,650 years from now.  I responded:

>  Van Nostrand's Scientific Encyclopedia (6th ed., entry "Earth"), each
>  century the day is growing longer by about 1 millisecond, and the year
>  shorter by about 530 milliseconds.  Over a period like 1650 years,
>  these variations add up to something on the order of hours ...

And concluded that these hours would be enough to make a substantial
change to the number of years required between corrections.  Since then
there has been a series of 3 somewhat hotly worded articles as to whether
this is correct (as an approximation), or is (grossly) in error.  It would
be tedious to treat them point by point, but I will try to address here
the principal points of confusion.

What leap years are all about is making sure that January, say, is
always a summer month in the southern hemisphere and a winter month
in the northern, and not vice versa.  More precisely, the idea is to
make the vernal equinox always fall on or near the same calendar date.

Still more precisely, the idea is to make the *average* length of the
calendar year, in the long run, equal to the "tropical year", which
is the time from one vernal equinox to the next.

I will assume for purposes of calculation that the following figures
are currently *exact*: 1 day = 86,400 seconds, and 1 tropical year
= 365.242197 days = 31,556,925.821 seconds to the nearest millisecond.
(The length of the second nowadays is constant by definition, and no
longer related to the day.)

The Gregorian calendar provides a cycle where in each 400 years there
are 97 leap years.  Thus the "average length of the calendar year in the
long run" is 365+97/400 days, i.e. 365.2425 days or 31,556,952 seconds.
This number, of course, differs from the actual tropical year by 26.179
seconds, or 365.2425 - 365.242197 days.  This value is almost exactly
1/3300 of a day.  An error of 1/3300 day in one year amounts to 1 day
over 3300 years.

Thus, if the lengths of the day and the year were constant, then in
a period of just about 3300 years, the error in the calendar year
would accumulate to a single day, and a leap year would have to be
skipped.  The adoption of a system that was like the Gregorian calendar
except with one fewer leap year every 3300 years would bring the average
length of the calendar year down to 365+97/400-1/3300 days; to the
accuracy given, this agrees exactly with the figure 365.242197.

(Oh: the number 1650 mentioned above comes from the assumption that the
calendar was "correctly aligned" about 1650 years ago; therefore we are
now halfway to the needed adjustment.)

But all this is assuming constant day and year lengths, which is really
only an approximation.  In fact, in 1650 years...

|  16.5 centuries x 1 millisecond = 0.0165 seconds

... the day will be 86400.0165 seconds long.  This means incidentally
that if the length of the time units in everyday use remain the same,
then a leap second will be needed every 1/.0165 = 60.6+ days or so to
keep midnight where it belongs.

Also, in 1650 years from now...

|  16.5 centuries x 530 milliseconds = 8.745 seconds....

the tropical year will be down to only 31,556,917.076 seconds.  Thus
by that time there will be only 31,556,917.076/86400.0165 = 365.242026+
days in a tropical year.  The Gregorian calendar year will still be
365.2425 days, but they will be days of 86400.0165 seconds, so the
average calendar year will be 31,556,958.0265 seconds.  That is, it
will be too short by 40.9505 seconds or 365.242026+ - 365.2425 days.
This is just about 1/2110 of a day, so the interval between corrections,
which is 3300 years with today's figures, would be 2110 years by the
figures of 1650 years from now.

Is it now clear that a simple system of corrections every 3300 years
would be useless?


In my earlier posting I referred said that the variations "add up to
something on the order of hours", which seems to have caused some
misunderstanding.  I was referring to the following calculation:

!  16.5 cent* 1 millisec/day/cent *365 day/year    =  6.02 sec/year
!  16.5 cent* 530 millisec/year/cent               =  8.745 sec/year
!  difference per year = 6.02+8.745                = 14.765 sec/year
!  1650 years * 14.765 sec/year /2    =12181 sec   =  3.38 hours

I have shown above that the error in the Gregorian calendar amounts
to 1 day in a period of 3300 years, or 12 hours in the coming 1650 years,
assuming no variation in the day and year lengths.

Well, the last-quoted calculation shows the amount of *error* in the
other calculation, caused by the incorrect assumption.  The error over
1650 years is 3.38 hours, which is a substantial part of the 12 hours
that the naive calculation gave.  It does *not* mean that "1650 years
ago, the day was 20.32 hours long".  It means that if a calculation with
constant day and year lengths would put the vernal equinox of the year
3642 as coming at, say, 6:38 am UTC on March 21, and if leap seconds
are used over the intervening years, then the actual time of the vernal
equinox on that day would be 3:00 am UTC.  That was a rather complicated
specification, and I may have made some minor error in writing it, which
is why I tried to duck out of it by being vague in the original message.
But anyway, that's where the "on the order of hours" comes in.


!  Seriously, from memory the day was supposed to be about 16 hours long
!  about 2 billion years ago, according to tidal markings on fossilised
!  beaches.  Since the slowing is basically caused by tidal friction, my
!  understanding of the system is that as it gets slower the damping has
!  less effect....

Right in principle.  Asimov's New Guide to Science (1984) says the oldest
fossils are 570 million years old and that the days then "may have been
only a little over 20 hours long".  This implies a day-length change of
something over double the rate cited in Van Nostrand's, but then Asimov
doesn't give it as an exact and authoritative figure.

!  The lengthening of the DAY only affects leap SECONDS

The lengthening of the day will cause more and more leap seconds
to be required for the calendar day to continues to "track" the
physical day.  But since it will do so, there will be fewer physical
days in the tropical year, and that *does* affect the calendar.  

!  (and note that gravitational influences of planets, etc, swamps this factor).

Yes, but only in the short term, because those other effects do not
apply systematically in one direction.
-- 
Mark Brader			"All this government stuff, in other words,
SoftQuad Inc., Toronto		 is not reading matter, but prefabricated
utzoo!sq!msb, msb@sq.com	 parts of quarrels."	-- Rudolf Flesch

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