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Off-Topic: Tropical Year Length (in calendric context)



In the course of a thread on another list, I browsed the following URL,
and queried the author for amplification, which he provided (below)
Although somewhat off topic, I felt it might be of interest to physical
astronomers here.

Brian W

Author:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Simon Cassidy <scassidy@EARTHLINK.NET>
<http://home.earthlink.net/~scassidy>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Date: Mon, 26 Nov 2001 08:59:37 -0800

Sender: History of Astronomy Discussion Group <HASTRO-L@WVNVM.WVNET.EDU>
From: Simon Cassidy <scassidy@EARTHLINK.NET>
Subject: Re: Tropical Year Length (was Rolleston's "Mazzaroth"...)
To: HASTRO-L@WVNVM.WVNET.EDU

Brian Whatcott wrote:
> I thought it would be interesting to verify Simon's URL
> My attention was drawn to his discussion of the tropical year length.
>
> I am probably missing his point which includes strong sentiments like
these:
> "...COMPLETELY ERRONEOUS ANALYSES of calendar accuracy
> IN GENERAL REFERENCE WORKS have resulted from this continuing scandal..."
>
> The cause of Simon's righteous dismay seems to be a few parts per million
> discrepancy in the quoted tropical year on account of the averaging method
> generally used.

Simon responds:
Those few parts per million, may sound negligible but have been the basis for
very many errors in encyclopaedias, web-sites and calendrical texts. Astronomy
texts (such as the Calendars chapter of the Supplement to the Ephemeris) which
mistake the mean interval between vernal equinoxes for the astronomers' mean
tropical year, do a great disservice to calendar scholars and historians (who
tend to have great faith in the expertise of astronomers).

The "averaging method generally used" to achieve the astronomer's mean
tropical
year is not the correct method to use to achieve the mean interval between
vernal equinoxes and it is simply mistaken to state that the astronomers
mean "tropical year is defined as the mean interval between vernal equinoxes"
when it is actually defined as the tropical period of the mean longitude of
the sun.

The difference in question is currently about seven seconds per year and it
is growing (since the astronomers mean tropical year will continue to drop
for the next few millenia while the mean interval between vernal equinoxes
should remain relatively constsnt when measured in calendar days).

Those few seconds make the difference between the truth and falsehood of
such statements as:

The Gregorian calendar-year is a day too long in 3300 years. (less than
half a day)
The Revised Julian calendar is more accurate than the Gregorian
calendar.(not so)
The Iranian calendar's (observationally based) leap-days are predicted better
with an 128-year pattern than with a 33-year pattern. (not so)

Yours sincerely
Simon Cassidy.


Date: Mon, 26 Nov 2001 11:29:11 -0600

From: Brian Whatcott <inet@INTELLISYS.NET>
Subject: Re: Tropical Year Length (was Rolleston's "Mazzaroth"...)
To: HASTRO-L@WVNVM.WVNET.EDU


Thank you for this temperate amplification of your web offering.
I acknowledge that a mistake is still a mistake if the error amounts only
to a few parts per million, if it illustrates the use of a mistaken concept.
And so I am happy to be made aware of it.

Sincerely

Brian Whatcott



Date: Mon, 26 Nov 2001 09:28:52 -0800

Sender: History of Astronomy Discussion Group <HASTRO-L@WVNVM.WVNET.EDU>
From: Simon Cassidy <scassidy@EARTHLINK.NET>
Subject: Tropical Year Length (in calendric context)
Comments: cc: "P. Kenneth Seidelmann" <pks@spica.usno.navy.mil>
To: HASTRO-L@WVNVM.WVNET.EDU

I have presented the following proposal to P. Kenneth Seidelmann
(editor of the Explanatory Supplement) with the approval of
Jean Meeus and Duncan Steel. I would appreciate anyone's thoughts
on this matter. If you wish to let Seidelmann know your thoughts
also then you could cc to pks@spica.usno.navy.mil.

thanks,
Simon Cassidy.

There would be no impact on any of the USNO's clients, except scholars of
calendrics and the only impact on the business of the USNO would be to have
the next issue of the "Explanatory Supplement" correct the mistake in the
"Calendars" chapter in some such fashion as follows:

Current text:
"The tropical year is defined as the mean interval between vernal equinoxes;
it corresponds to the cycle of the seasons. The following expression, based
on the orbital elements of Laskar (1986), is used for calculating the length
of the tropical year:

365.2421896698 - 0.00000615359 T - 7.29E-10 T^2 + 2.64E-10 T^3 [days]

where T = (JD - 2451545.0)/36525 and JD is the Julian day number. However,
the interval from a particular vernal equinox to the next may vary from
this mean by several minutes.
...
In the preceding formulas, T is measured in Julian centuries of Terrestrial
Dynamical Time (TDT), which is independent of the variable rotation of the
Earth. Thus, the lengths of the tropical year and synodic month are here
defined in days of 86400 seconds of International Atomic Time (TAI)."

Suggested replacement text:
"The tropical year is defined as the time taken for the mean solar
longitude to progress 360 degrees with respect to the dynamical equinox;
it corresponds to the cycle of noon-altitudes of the sun. The following
expression, based on the orbital elements of Laskar (1986), is used for
calculating the length of this mean tropical year:

365.2421896698 - 0.00000615359 T - 7.29E-10 T^2 + 2.64E-10 T^3 [days]
where T = (JD - 2451545.0)/36525 and JD is the Julian day number.

However the mean interval between instants of vernal (northwards}
equinox, which is the target year-length of the Gregorian calendar and
other widely used solar calendars, varies cyclically from the above
mean tropical year by about a minute with a period near 20 millenia.
The following expression, based on the work of Jean Meeus (1991-2000)
can be used to calculate the mean interval between vernal (northwards)
equinoxes for the period from about 500 B.C. to A.D. 4500:

365.2423748 + 0.00001034 T - 1.243E-7 T^2 - 2.263E-9 T^3 + 1.31E-11 T^4
[days]

where T = (JD - 2451545.0)/36525 and JD is the Julian day number. However,
the interval from a particular vernal equinox to the next may vary from
this mean vernal-equinox year by several minutes.
...
In the preceding formulas, T is measured in Julian centuries of Terrestrial
Dynamical Time (TDT), which is independent of the variable rotation of the
Earth. Thus, the lengths of the mean tropical year, mean vernal-equinox
year and synodic month are here defined in days of 86400 seconds of
International Atomic Time (TAI).

However the basic unit of calendars is the calendar day which is best
approximated as the mean solar day of Universal time. Thus a correction for
the estimated change in the rotation of the Earth should be used when
applying the above formulas to a calendar's performance in the past or
future..." [Simon's note: an expression for such an estimated correction
should probably be supplied here in the text].

This last extra paragraph (expanding on the definition of "day") is an
additional suggestion for the edification of scholars of calendrics and has
minor importance compared to the need to correct the mistake in the definition
of the tropical year.


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~




Brian Whatcott
Altus OK Eureka!