Talk:Tropical year/Archive 4

Meeus's calculation of equinox and solstice tropical years
There were discussions in Talk:Tropical year/Archive 3 about the difference between the modern definition of a tropical year, based on the mean mean ecliptic longitude of the Sun, versus the length of a tropical year based on two successive equinoxes or solstices. Meeus and Savoie, as discussed in the article, published mean values of these equinox-to-equinox or solstice-to-solstice tropical years. In the section Meeus's calculation of equinox and solstice tropical years there was a discussion of how Meeus and Savoie might have calculated these mean values.

I have obtained a book published in 2002 by Meeus (see the Further reading section of the article for bibliographic details). After reading this, I have been able to reproduce Meeus's results to 6 decimal places (about 0.1 second of time) which is the precision Meeus presented in his 2002 book. (I checked the March & September equinoxes and June & December solstices for the years 0 and 2000.)

Chapter 63 Meeus's 2002 book, "The Gregorian calendar and the tropical year" specifically addresses the concept that the Gregorian calendar was designed to keep the vernal equinox on March 21. He concludes the Gregorian calendar does a better job of keeping the equinox on March 21 than generally acknowledged, and estimates it will not be off by more than a day until AD 7200.

This chapter also explains Meeus's methodology. He recapitulates the expressions published by Simon et al. in 1992 to find the Keplerian mean orbital elements of the Earth's orbit from the year -4000 to the year 8000. Using well-known procedures (including solving Kepler's equation), described in chapter 30 of Meeus's 1998 book, from these elements one can find a mean longitude that does change its rate of change during the course of the year, faster at perihelion and slower at aphelion. But periodic variations due to factors such as the gravitational force of the orbiting Moon and gravitational forces from other planets are smoothed out.

For a chosen year, one can iterate the date (terrestrial time) to find the dates and time when the longitude of the Earth is 0°, 90°, 180°, and 270°. I did this for the years -1, 1, 1999, and 2001. I then found the time between corresponding events in -1 & 1, and also 1999 & 2001. I divided these intervals by 2 and compared them to the values Meeus published in 2002 (p. 362, table 63.A). The results agreed to 6 decimal places, which is the precision of this table. The differences from the values published in 1992 were negligible.

Also discussed in Talk:Tropical year/Archive 3 was how the "mean" equinox date expressions in table 27.A of Meeus's 1998 book were found. For 1999 to 2001, the equinoxes and solstices given by these expressions are about 8 minutes later than the dates I described above, based on mean orbital elements. I still don't know the basis of the table 27.A expressions. Jc3s5h (talk) 15:16, 7 February 2019 (UTC)

"Complex equation"
Yesterday I added the following after the cubic equation for the length of the tropical year over the period 8000 BC to AD 12000: The above is in the Taylor series form traditional for orbital mechanics. A sine series having the same first, second, and third derivatives is


 * $$365.242 189 669 8-6.153 59\times 10^{-6}(\sin \omega T)/\omega-4\times 7.29\times 10^{-10}(\sin \omega T/2)^2/\omega^2$$

with ω = 0.0160440243. Here the long-term average is 0.49 seconds less than the value in 2000. From these formulas we find that ten thousand years ago the tropical year was 34 seconds longer than now, and ten thousand years from now it will be 33 seconds shorter, but then it will start to get longer again. All of this was deleted by Jc3s5h with the instructions not to add "complex equations" without a reliable source. First of all I object to the deletion of everything just because he doesn't like the equation. The last two sentences are certainly true whether one uses the equation I added or the other -- it suffices to make a graph. We don't need a "reliable source" for such a simple thing. Routine calculations are not considered original research by the way.

Secondly, I do not agree that this is a complex equation. It's simply replacing the cubic with a sine series, and it is easy to verify that it has the same derivatives as I wrote. (Note that since the length of the tropical year oscillates with time it will be approximated by a sine wave much better than by a cubic! The cubic starts giving horrible results outside of the stated range.) I actually compared with the model of Laskar and found that this equation is better, but I dare not say that in our Wikipedia article because then someone would surely accuse me of doing original research! (Some people have the attitude that you have to stick with only what "reliable sources" say, no matter how obvious or easy some slight extension might be. I once had a long discussion with someone who insisted on writing an equation the same way as it was in some book, even though it gave incorrect results before Julian day zero just because of a question of how one defines a truncation!)

Eric Kvaalen (talk) 15:55, 21 August 2018 (UTC)


 * I haven't checked that your formula gives the same results, but length of the tropical year is not predictable by a simple periodic function. We've had a lot of dubious claims added to this article in the past, so it is best to follow Wikipedia policy of using only WP:Reliable sources.   Dbfirs  16:03, 21 August 2018 (UTC)


 * But we absolutely know that the cubic equation (from the "reliable source") is horrible if we go a bit outside of the range 8000 BC to AD 12000! Eric Kvaalen (talk) 12:51, 25 July 2019 (UTC)


 * ... but have we a reliable source that states that a periodic function gives better predictions?  Db<i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  17:53, 25 July 2019 (UTC)

Dispute possible original research
This footnote was added: "A more suitable sine wave expression can be found having the same first, second, and third derivatives."

I reverted and Eric Kvaalen reinstated the edit.

I dispute this statement for the following reasons:

If there is to be any statement about approximating the mean sun with an expression involving the sine function, this should be backed by a reliable source.
 * I would have to dust off my math books to be sure, but I believe a "sine wave expression" (which I take to mean Fourier series) would have to have an infinite number of terms to exactly match the first, second, and third derivatives of a cubic equation.
 * Deciding what a "suitable" expression is would require mature mathematical judgement, taking into account how the expression was to be used (which is not stated).
 * The statement goes far beyond what is allowed by WP:CALC.

I am mentioning this discussion at No original research/Noticeboard. Jc3s5h (talk) 18:43, 19 February 2021 (UTC)

Actual Length of Tropical Year
This article repeatedly mentions the 'mean' tropical year. Including the paragraph about 'Mean tropical year current value'. In other words, the actual length varies from year to year and the mean is taken over some interval of years. So, is there a formula that would show the actual length of any particular year and how this length varies over time? Assuming some definition of the tropical year such as the interval between March equinoxes. — Preceding unsigned comment added by ‎ 2600:8800:ff10:200:9460:8721:3133:4a76 (talk • contribs) 15:20, 18 May 2021 UTC (UTC)
 * I have never seen such an equation. The considerations are complex. I suppose someone could come up with an equation that was accurate within certain limits, during a certain span of years. Jc3s5h (talk) 02:11, 19 May 2021 (UTC)
 * A table appears here but I don't see what their source is. Like Jc3s5h, I don't see a simple formula to be likely. The numbers are computed using complex programs that track all the sources of variation. Zerotalk 07:35, 19 May 2021 (UTC)

Changes near 1 January 2022
I reverted changes by John Maynard Friedman with this edit summary: "Revert section to version by Nsae comp at 02:23, 1 January 2022‎. Revisions by John Maynard Friedman described Earth Rotation Angle, which is nearly sidereal time, not mean solar time."

I intended to only revert the changes in the Time scales and calendar section, but I accidentally reverted the changes to the lead too. After I read the changes to the lead, I decided to let the reversion stand.

The changes to the lead deserved to be reverted because instead of portraying the duration between successive vernal equinoxes as an example of a possible tropical year definition, it portrayed those durations as the one and only current definition of the tropical year. Of course, this claim is contradicted in other parts of the article. Jc3s5h (talk) 18:03, 1 January 2022 (UTC)