Warning! Continuation of off-topic leap year posts!
Do not read if you are uninterested in the leap-year discussion!
1) The once per 10000 year leap-year exception referred to by Steinar IS
NOT part of the Gregorian calendrical system. As Ernst mentioned, it would
not even correctly fix the next order of error. Ernst suggested that 3
exceptions per 10000 years would come closer to "fixing" the next order of
error. But. . .
2) The main problem with trying to determine a permanent system of leap
year "fixes" is that the average number of days per tropical year IS
CHANGING with time. 3 leap-year exceptions per 10000 years would work out
to an average of 365.2422 days per year which is pretty close to the value
Ernst noted of 365.24219, but by the time 10000 years have elapsed this
value will be far from the actual case! I'm not sure what the current best
estimate of the rate of change is or even if it is a constant rate, but one
old reference puts it at minus 6.14 times 10^-8 days per year. This value
would add up to a cumulative need for 3 MORE leap-year exceptions over the
same 10000 year period. We could "fix" this problem by making ~6 exceptions
over 10000 years, or say one exception every 1600 years. But the
accumulated drift error would be greater than 1/2 a day at about the year
7000. To make a long-lasting rule that keeps the cumulative error below
1/2 a day for as long as possible, we should make an exception only every,
say 2000 years, rather than 1600 years. This rule would work until about
the year 10500 with less than 1/2 a day error over the whole period
(ASSUMING a constant value of rate of change of -6.14 x 10^-8 days per
year). BUT this works out on average to what we would get if the 400 year
exception rule were made to be a 500 year rule instead! IF ONLY THE
GREGORIAN CALENDAR would have made the 400 year rule to be a 500 year rule
instead it would have been SIMPLER AND MORE ACCURATE for a LONGER PERIOD of
time, until the year 10500!!
3) Global warming, which would expand the earth's atmosphere and change
earth's moment of inertia, and thus the length of a day, may make all of
the above irrelevant before the year 10500 anyway. . .
Cheers,
Todd Sauke
Ernst wrote:
>Steinar Gunderson <[EMAIL PROTECTED]> wrote:
>
>> OK, to complete the mess (I saw your message saying `ignore this',
>> but I want to throw in my own errors as well ;-) ):
>>
>> If year % 4 = 0 then leap year
>> If year % 100 = 0 then not leap year
>> If year % 400 = 0 then leap year
>> If year % 10000 = 0 then not leap year
>
>I'm not sure about your 10000-year correction, for reasons I'll
>give below (see the section about the anomalistic year below).
>
snip
>
>Of course all of these are simply corrections to account for
>the fact that the 'true' time for the Earth to complete one
>orbit relative to the 'fixed' stars, the so-called sidereal
>year, is slightly longer than 365 days, and is not (at least
>in general) a rational number.
>
>Whoops - what I really wanted was not the sidereal year (which
>astronomers prefer), but the tropical year, the length of time
>between successive vernal equinoxes (more interesting back
>during those predominantly agrarian economic times). Silly me!
>
>My Norton's Star Atlas (vintage 1950) lists the length of the
>tropical year as 365.24219... years (ellipses mine), so one
>can argue that Gregory could heve done much better by chucking
>out the old one-in-four rule and finding a parsimonious rational
>approximation to 0.24219..., rather than giving us the start
>of the recursive nightmare listed above, but from a practical
>point of view, it is desirable to be able to add corrections
>to the pre-existing ones, i.e. one wants each higher modulus
>to be divisible by all smaller ones. From a human-nature point
>of view, one wants things to be in terms of numbers people (or
>at least 16th-century popes :) are comfortable with, hence the
>bias toward base-10. So you calendrically oriented recreational
>math enthusiasts can work out the next few corrections, starting
>with:
>
>- the civil year, of necessity a whole number of days, 365;
>- adding one leap day every 4th year gives year = 365.25 days;
>- subtracting one leap day every 100 years gives year = 365.24 days.
>- adding one leap day every 400 years gives year = 365.2425 days.
>
>Now, if we subtract one leap day every 10000 years we get
>year = 365.2424 days, but if we instead subtract three leap
>days every 10000 years, we get year = 365.2422 days, which is
>much closer to 365.24219 days. So either my Norton's has it wrong
>or we need to ... yes, you guessed it ... modify the calendar yet
>again!
>
>What does any of this have to do with Mersenne primes? Let's
>see if I can whip up a quick ex post facto connection:
>
>- Pope Gregory was a Catholic; so was (arguments about which
>sect and such aside); so was friar Mersenne;
>
>- Gregory devised his modifications to the Julian calendar
>in 1582; Mersenne was born in 1588, which is just six (the
>smallest perfect number!) years (civil years, that is) later.
>
>Gosh, what an amazing set of coincidences.
>
>Gosh, look how much time I just wasted composing this.
>
>-Ernst
>
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