Talk:Tropical year

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Maya versus Gregorian[edit]

I find it interesting that the northward equinox of March agrees with Mayan-Amizaduga figures of 1507 years (365.2422 = 1508x365 =1507 tropical) while the southward equinox of September supports the Gregorian 1600 years (and 400) figure of 365.2425 days showing two cultures can both be precise from their viewpoint rather than slam the Catholics as not having Mayan genius. — Preceding unsigned comment added by (talk) 01:42, 11 October 2012 (UTC)

Naming conventions[edit]

Karl, lets settle on a convention about Latin, Greek, English words. I think we should stick to whatever is current in English. If you start to Romanize or Hellenize English words, there is no end - anglosaxons completely screw up foreign words; e.g.:

equinox would be aequinox (equus = horse, aequus = equal (sic!))

Homer would be Homeros

Now the English word is perihelium, not perihelion; like the stuff is helium, not helion. -- Tompeters

This would be a good argument, except for the fact that it's "perihelion" in English. What dictionary are you using that says otherwise? --Zundark, 2001 Oct 25
OK, I screwed up -- Tompeters
Bad example: Helium ends in -ium because it was first found spectroscopically in the sun and they thought it was a metal: and metals get -ium or -um on the end e.g. Thorium, Hafnium, Aluminium, Neodymium, Molybdenum. If the naming convention for noble gases was followed strictly, Helium actually should be called Helion, though no-ones going to rename it at this late date - Malcolm Farmer
Thanx for pointing that out, I never noticed. Good to see someone writing Aluminium, americans usually say aluminum.


The current definition is ambiguous and on first reading it sounds like a sidereal year. A tropical year is the time between successive equinoxes (or solstices) - JGBell

I hope my ammendment deals with this well - Karl 20 June 2006 UT

I don't believe the ambiguity and potential for confusion is cleared up. The current definition still emphasizes the sun's relationship to the stars; how is this relevant for the tropical year? Hgilbert 13:20, 9 April 2007 (UTC)

NPOV violation[edit]

This article is written from a northern-hemisphere perspective and needs some revising to make it hemisphere-neutral. Terms like "vernal equinox" and "summer solstice" are deprecated because they cause excessive confusion, especially for people who live south of the Equator. Terms like "Spring equinox" and "Summer solstice" should only be used in contexts where the local equinox or solstice is important (like the timing of Pagan festivals); when referring to the equinox or solstice that occurs in a particular month regardless of season, the name of that month should be used to identify the equinox or solstice (March equinox, December solstice).

I suggest that "vernal equinox" be changed to "March equinox", along with other similar changes to remove references to specific seasons where they are inappropriate. It would also be helpful to clarify why the March Equinox is important (because our calendar is based on the old Roman calendar, and the old Roman calendar began at the March equinox). --B.d.mills 02:32, 28 Feb 2005 (UTC)

I cannot agree with your suggestion for two reasons. First, relative to the astronomical portion of the article, the International Astronomical Union, with the concurrence of all of its member countries in the southern hemisphere, defines the vernal equinox as the point where the Sun crosses the celestial equator on its way from south to north, applying that definition world-wide, including the southern hemisphere. Second, for the calendrical portion, almost all churches, both East and West, define 21 March to be the vernal equinox, in the Julian and Gregorian calendars respectively, with at least the Roman Catholic Church specifically stating that this definition applied to all lands recently discovered (before the 1582 promulgation of the Gregorian calendar) in the southern hemisphere. Consequently, all Christians in the southern hemisphere continue to celebrate Easter in the northern spring. In this case, calling it the March equinox would be repetitive. As a compromise, the present clarification "(northern)" in the first paragraph could be expanded into a full statement that all seasons are northern hemisphere seasons. — Joe Kress 03:39, Mar 2, 2005 (UTC)
I disagree with the assessment because it does not remove the seasonal ambiguity. A statement that "all seasons are northern hemisphere seasons" would require further research to discover what the seasons are in the southern hemisphere. Therefore, it is of limited geographic scope and is not suitable for an international audience. Wikipedia has a boilerplate tag "Limitedgeographicscope" that can be used for this. One may as well refer to the equinox as the March equinox throughout, and have a single blanket statement that the March equinox is commonly known as the vernal equinox. The only objective reason for preferring vernal equinox to March equinox is if the IAU mandates the use of that term exclusively; and I have not yet found any indications that that is the case. --B.d.mills 11:13, 23 Apr 2005 (UTC)
A small clarification: March 21st is actually the ecclesiastical equinox. The true equinox most commonly occurs on March 20th, but may also occur on March 19th or March 21st due to the inaccuracies of the Gregorian calendar. -- B.D.Mills  (T, C) 01:24, 17 October 2008 (UTC)

Ephemeris Time[edit]

The article now says:

The time scale is Terrestrial Time (formerly Ephemeris Time) which is based on atomic clocks

I don't understand this. One could understand this as "Terrestrial Time was formerly called Ephemeris Time". This is not the case. Ephemeris Time is different from Terrestrial Time; and is not based on atomic clocks. Terrestrial Time is now used a lot where Ephemeris Time used to be used. But wouldn't the formulae be different in the past if they used Ephemeris Time in the past? Until someone who really understands this, sorts this out, I will delete the text in parentheses "(formerly Ephemeris Time)" -- Adhemar

You have some valid points. Although Terrestrial Time (TT) is based on atomic clocks, it is a uniform time just like Ephemeris Time (ET), which was defined relative to the motion of solar system bodies, especially the Sun and Moon. TT (1991) is the new name for Terrestrial Dynamical Time (TDT, defined 1976). It was renamed because TDT was not dynamical, i.e., it was not based on the motion of the solar system. ET was the time base used in all national ephemerides until 1983. The offset of TT from International Atomic Time (TAI) was intentionally chosen to be 32.184 s so that it would equal ET, and thus ET can be directly substituted for TT in most astronomical equations. This is indeed done by Jean Meeus in his "Astronomical Algorithms". I am replacing the objectionable phrase by "(formerly, Ephemeris Time was used instead)". — Joe Kress 03:20, August 30, 2005 (UTC)

The 2000.0 value of the tropical year[edit]

Hi Joe, at 24 November 2005, I reput the attested value of the TY 2000.0, originate from P. Bretagnon.

The problem now is: Further in the article, there is an other, lightly contradictory value. We should clear that up. If your given value is from VSOP-87, you must know that Bretagnon himself was one the most decisive co-author of VSOP-87.

A new improuved theorie should exist, but I don't know sources, except the numbered Meeus mention. --Paul Martin 17:55, 5 December 2005 (UTC)

Yes, the contradiction needs to be cleared up. The value now under Current mean value was in the original article by Tom Peters, and was no doubt obtained from Astronomical algorithms (1991). I obtained the following newer value for the mean tropical year by applying the method described by Borkowski in [2] to the mean ecliptic longitude of date for the Earth given by Bretagnon et al. on page 678 of [3] (1994):
365.242190402112 – 0.000061525135τ – 0.000000060932τ2 + 0.000000265246τ3 + 0.000000002536τ4 – 0.000000000338τ5
where τ is the number of Julian millennia since J2000.0 (negative earlier) and the least significant digits match those of the mean ecliptic longitude given by Bretagnon. Thus I can support the new value obtained by Bretagnon in 2000, although it does seem to have fewer digits than the underlying mean values warrant. In addition, at least its linear term is needed. Unfortunately, I do not have access to More mathematical astronomy morsels. Does the article cite Bretagnon's original article, or even better yet, the article containing the new mean ecliptic longitude of date? Or did Meeus obtain it by personal communication with Bretagnon? — Joe Kress 20:49, 6 December 2005 (UTC)

Your latter presumption is the right one. At page 358 Meeus gives the VSOP-87 formula:
365.242 189 623 – 0.000 061 522τ – 0.000 000 0609τ2 + 0.000 000 265 25τ3
Then he gives the elements of your reference two, in his article with the reference number six.
With this VSOP-87 formula above, he refers to an endnote of chapter, Note 2 at page 365, where he wrotes:
The small difference, 0.067 second, is due to the fact that the elements by Simon e.a. (6) use a slightly different value of precession.
On 2000 March 1, P. Bretagnon told me that recently still another, improuved value for the length of the tropical year at epoch 2000.0 was obtained: 365.242190517 days. This value was derived from a new theorie, "ten times more accurate than VSOP87", due to Xavier Moisson who worked at the Bureau des Longitudes, Paris.
Until now, I dont know the exactly references for this quoted Moisson's theory. Within the next few days, I will either try to find his works or to attempt to contact himself. Let's stay in dialogue concerning this topo. -- Paul Martin 22:15, 6 December 2005 (UTC)  (P.S. "obtained by Bretagnon" in the article – by rereading – seems to be false.)
PS2: This should be the Analytical Planetary solution VSOP2000.
Thanks for the VSOP2000 ref. — Joe Kress 05:47, 7 December 2005 (UTC)
The different lengths of only the four listed tropical years average to near 365.242189670 days, not to near 365.242190517 days, so the former length for the mean tropical year must remain in the article until the four different kinds of tropical years can be recalculated. Thus we must distinguish between the two mean tropical years in some way. A more recent VSOP2000 reference is A. Fienga and J.-L. Simon, "Analytical and numerical studies of asteroid perturbations on solar system planet dynamics", Astronomy and Astrophysics 429 (2005) 361-367. Unfortunately, it does not provide any newer mean values. Moisson's earlier paper is "Solar system planetary motion to third order of the masses" Astronomy and Astrophysics 431 (1999) 318-327. — Joe Kress 09:18, 12 December 2005 (UTC)
I have determined to my satisfaction that the 'new' value for the mean tropical year given by Meeus contains a typographical error of 5 for 4. The printed value should have been 365.242190417 days. But an even later value for the J2000 mean tropical year can be derived. To determine these I used the various elliptic solutions by the Observatorie de Paris and took from them the mean mean motions or frequencies N (mean includes all powers of T whereas mean mean only includes the linear term), referred to the fixed equinox J2000 (not to the equinox of date), given in VSOP82 (280) (6.2830758491800 rad), Simon 1994 (675) (1295977422.83429"), which is identical to VSOP87 (310) (6.2830758499914 rad), and the solution of Moisson 1999 fitted to DE403 (321) (6.2830758508994 rad) and DE405 (324) (6.28307585085 rad). I then added four different values for general precession p to them, IAU 1976 (664 in Simon 1994) (50.290966"), Simon 1994 (664) (50.288200"), IAU 2000 (50.2879695"), and P03 (50.28796195"), the last two in "Expressions for IAU 2000 precession quantities" (685 KB pdf) Astronomy and Astrophysics 412 (2003) 567-586 (571, 581). Finally, I used the method I described earlier. The resulting values for the mean tropical year at J2000 (limited to the same number of significant digits as in the frequencies N) are:
          365.242 189 669 78 (VSOP82 & IAU 1976)
          365.242 189 622 61 (VSOP87 & IAU 1976)
          365.242 190 402 11 (VSOP87 & Simon 1994)
          365.242 190 467 07 (VSOP87 & IAU 2000)
          365.242 190 469 20 (VSOP87 & P03)
          365.242 190 414 29 (DE403  & IAU 2000)
          365.242 190 416 42 (DE403  & P03)
          365.242 190 417    (DE405  & IAU 2000)
          365.242 190 419    (DE405  & P03)
The first value, rounded to 12 significant digits, is that originally given in the article. The second value was cited by Meeus in "The history of the tropical year" Journal of the British Astronomical Association 102 (1992) 40-42, also rounded. The presence of both values in the literature assures me that my method is correct. When the difference between the second and third values is multiplied by 86,400 s/d, 0.06735 s results, so its confirmation by Meeus is also assuring. The penultimate value is apparently the actual value given to Meeus by Bretagnon, derived from Moisson's "ten times more accurate theory" (fitted to DE405) and a "slightly different value of precession" (IAU 2000). The last value is the latest available from the Observatoire de Paris (not explicitly stated, but implied). Of course, when they finally complete VSOP2000, VSOP2004, or whatever, the value will change again. Joe Kress 23:57, 22 December 2005 (UTC)
I thought that More mathematical astronomical morsels might give some more info, so had to wait until I found a copy of it, but it did not, so I should fold some of this info into the article. — Joe Kress 10:16, 25 January 2006 (UTC)

Tropical year history[edit]

Hi Joe, I didn't come back to this talk page since 7 December 2005. (I often omit to put the pages to my watchlist, in exchange I make the tour of articles currently in my interest.) So I didn't saw you adds, excuse. I'll read and study it within the next days.

The inducement why I would let you a message was: The German Tropical year article has been recently reworked in – it seems me – a very good way. Especially historic evolution of the cognition of the TY is accurately described and well documentated. From the ancient times, over the Alfonsinischen Tafeln (1252) with 365d 5h 49m 16s, the Prutenischen Tafeln (Erasmus Reinhold, 1551) with 365d 5h 55m 58s and the Rudolphinischen Tafeln (Johannes Kepler, 1627) with 365d 5h 48m 45s [our nowaday value!], to the modern scientific values.

I don't know if you understand the German language, but I can translate the most interesting parts, provided that you are ready to make a copyedit, before we'll add it on our English article.

What do you think of this proposal? Paul Martin 17:27, 23 January 2006 (UTC)

I used Google translation. I agree that the historic evolution of the tropical year would be a valuable addition to the English article. It should be noted that the Babylonian value was for the sidereal year. I remember another value for the Babylonian year mentioned by Otto Neugebauer in A history of mathematical astronomy. I was disappointed that Arabic values were not mentioned in the German article, but basically relegated to a see also. I also didn't see any mention of trepidation. — Joe Kress 10:16, 25 January 2006 (UTC)
The history has been condensed from "The history of the tropical year" by Meeus and Savoie; there is even an online version, linked to from the English 'Tropical Year' article. The Arabic astronomers are not mentioned by M&S, and the allusions to trepidation are too vague there to be very useful, so for both cases I'll need additional sources to be able to add something substantial. The Babylonian values have been taken from Neugebauer's HAMA, p. 528ff.
Joe, before I ask this on HASTRO-L, do you happen to know who was the first to implicitly or explicitly use the "360°" definition for the tropical year rather than the "equinox to equinox" definition? One of the French analysts or Newcomb, maybe? Some sources imply A. Danjon, but that seems too late. Thanks and Bye -- Tosch 22:26, 25 January 2006 (UTC) (= de:User:Sch)
Off hand I don't know. That would require some research, although I do have a lot of the literature nearby. — Joe Kress 04:27, 26 January 2006 (UTC)

Different duration by starting point[edit]

The article states that the duration of the tropical year depends on the chosen starting point. I do not understand at all the given explanation. It seems to me that the revolution speed just before reaching the same point again is not relevant. During the whole cycle all speeds are met. Can someone supply a more clear explanation? −Woodstone 21:41, 11 December 2005 (UTC)

The anomalistic year (perihelion-to-perihelion or aphelion-to-aphelion) is longer than any tropical year, so all tropical points (including the four equinoxes and solstices) move earlier relative to it each year—alternatively, the perihelion and aphelion move later each year. The approach to or recession from any tropical point by a perihelion or aphelion changes the length of that particular tropical year, making it longer or shorter. By definition, the mean tropical year is the average of all tropical years. The average of only the four quadrature tropical years (equinoxes and solstices) at J2000 is 365.242189557 days, which is quite close to the associated mean tropical year of 365.242189670 days. It would be much closer if the lengths of 360 tropical years, one for each degree of ecliptic longitude, were averaged. Of course, this view is complicated by the Gregorian year, which differs from the mean tropical year. — Joe Kress 09:18, 12 December 2005 (UTC)
I think I understand this, but I am not sure. Could someone who is very knowledgeable in this field please tell me if this statement is exactly and perfectly accurate: "The length of the tropical year depends on which equinox you choose as the starting point because the Earth's axis of rotation varies relative to its ecliptic plane due to precession and nutation." I think this statement is true if I understand everything correctly, but I could be wrong in multiple different ways. Is precession and nutation the only reason, or just the primary reason? Which is more important right now, precession or nutation?
If this is the correct explanation, then let me try to speculate further. If the phenomenon that makes a vernal equinox year different than a winter solstice tropical year is EXAGGERATED HORRIBLY, with all other variables held constant, then the following might occur: The vernal equinox one year occurs when the sun is on top of the constellation Pisces. The vernal equinox the next year occurs when the sun is on top of Aquarius. The vernal equinox in the year following that occurs when the sun is on top of Capricorn (sun during equinox is moving in JUST ONE DIRECTION), and it is NOT on top of Pisces (which would happen if the position of the sun relative to the celestial sphere during the equinox oscillated back and forth).
An alternative reason for the discrepancy between different starting points could conceivably be that "the location of the aphelion and perihelion, relative to the celestial sphere and inertial space, is changing." I am relatively sure this is not the reason. To what extent do the aphelion and perihelion move from year to year? Does the aphelion rotate around, causing the ellipse of Earth's orbit to spin like a hoola hoop around the sun? Thank you.Fluoborate (talk) 17:32, 5 May 2008 (UTC)
Fluoborate wrote "The length of the tropical year depends on which equinox you choose as the starting point because the Earth's axis of rotation varies relative to its ecliptic plane due to precession and nutation" and wanted to know if this is correct. Well, many things are going on, but I believe the largest contribution to different tropical years can be explained as follows:
  1. Pick a starting point, such as the vernal equinox. Note the position of the Sun as viewed from the center of the Earth against the background of the distant stars that have no perceptible proper motion (the "fixed" stars).
  2. Use the equations of motion of the solar system to determine when the Earth will next be in the same position in it's orbit, relative to the fixed stars. This is a sidereal year.
  3. Due mostly to precession, the Sun will appear to be in a different position after one sidereal year. The Earth will have to move a little further in it's orbit to make the stars behind the sun be the same as at the beginning of the year. Because the Earth moves at different speeds during various parts of its orbit, the additional time that will be required to make the Sun appear in the right place will vary, depending on what starting point was chosen. --Gerry Ashton (talk) 18:06, 5 May 2008 (UTC)
Nice try but your last paragraph confuses sidereal with tropical. Return of the apparent position of the Sun against the distant stars is the definition of a sidereal year (sidera=stars). —Tamfang (talk) 15:59, 20 May 2008 (UTC)

length, and how to name the equinoces[edit]

Is there any good reason not to say that the tropical year is 31,556,945 seconds long? Also, I agree with the radical practice of saying "March equinox" rather than "vernal equinox" and saying "September equinox" rather than "autumnal". I'm surprised anyone has to explain this. You'll have to excuse me, I'm in a peremptory mood. --arkuat (talk) 09:19, 15 April 2006 (UTC)

The fundamental unit of time in astronomy is the day, not the second. — Joe Kress 02:08, 16 April 2006 (UTC)

Yes, the nychthemeron is fundamental. However, the nychthemeron is wobbly and variable, and can't really serve (in the long run) as a unit of elapsed time. The second, defined as it is in the vibrations of a cesium atom of determined isotope, can. Therefore, lots of people who are not astronomers will continue to think of the day (not the true wobbly nychthemeron) as 86,400 seconds. --arkuat (talk) 02:33, 17 April 2006 (UTC)

I wasn't clear. The day of exactly 86,400 SI seconds is the fundamental unit of time in astronomy. The SI second is not fundamental in astronomy. All of the lengths in the article are given in terms of this constant 'SI day', not in terms of a variable nychthemeron, which itself is getting progressively longer. The tropical year is not 31,556,945 seconds. Its length in 1900, which is embodied in the definition of the ephemeris second, was 31,556,925.9747 seconds, which to the nearest whole second is 31,556,926 seconds. The number of transitions in the ground state of a cesium atom in the definition of the SI second was intentionally chosen to be the same length as the ephemeris second. A more current (2000) length of the tropical year already given in the article is 365.242190419 days or 31,556,925.252 seconds, precise to a millisecond. — Joe Kress 06:00, 18 April 2006 (UTC)

Table formatting[edit]

Hi Paul Martin. It is not primarily a matter of "pretty" in someone's opinion, but of uniformity across wikipedia. A similar look everywhere makes the whole more professional and attractive. For that reason a few templates have been developed and widely discussed to standardise table formats. Individual HTML to modify table formats is discouraged.

Apart fromt that you do not seem to realise that other users may have a narrower screen, where both tables do not fit. Also the standard for notation of time used in WP is with a colon between hour and minutes, not a capital H. −Woodstone 16:42, 8 June 2006 (UTC)

Solar year[edit]

How on Earth (pardon the pun) could the name Solar year not be mentioned? Who uses the term "tropical year" - people at the tropic/s? In decades of academic life I have never heard the solar year be re-Christened as the "tropical year" but then again Ipods are new too. So let's at least have the name "solar year" mentioned in the intro, and not be buried in the backround as a redirect. Seems to me this borders as an attempt to foist a near-neologism on the world (pardon the pun, again.) IZAK 08:30, 29 September 2006 (UTC)

The term tropical year is long standing practice in astronomy. It reflects the apparent movement of the Sun between the tropic of Capricorn and the tropic of Cancer. −Woodstone 09:09, 29 September 2006 (UTC)
The term tropical year is used to distinguish it from the sidereal year, the latter reflects the apparent movement of the sun relative to the stars rather than the tropics. The sidereal year is also a solar year. Karl 13:00, 29 Sept 2006 (UTC)

Undefined variable[edit]

The article contains four equations that contain the undefined variable Y. The first of these equations is:

365.242 374 04 + 0.000 000 103 38×Y days

--Gerry Ashton (talk) 20:57, 10 February 2008 (UTC)

An anonymous editor changed a to Y on February 9. However, the definition for a was not given until the next section. I'm moving it up to its first occurrence. — Joe Kress (talk) 08:54, 12 February 2008 (UTC)

Iranian calendar[edit]

Although it is close to the vernal equinox year (in line with the intention of the Gregorian calendar reform of 1582), it is slightly too long, and not an optimal approximation when considering the continued fractions listed below. Note that the approximation of 365 + 833 formerly used in the Iranian calendar is even better, and 365 + 833 was considered in Rome and England as an alternative for the Catholic Gregorian calendar reform of 1582.

This has several flaws, most notably that it disagrees with Omar Khayyam, which asserts that his calendar was entirely astronomical, and changed month when, and only when, the sun entered a new sign of the zodiac. This means that it has no regular pattern, and its average length is, by hypothesis, the true mean tropical year.

Since most calendars, including the several proposals to revise the Gregorian calendar, have periods comparable to the precession of the perihelion, the insistence on 365.2424, the temporary high value of the vernal equinoctial year, is also uncalled for.

The claim about "Rome and England" requires a source; the idea that England would replace the Julian calendar with a reformed calendar of its own is an extraordinary claim, and requires extraordinary evidence. Septentrionalis PMAnderson 13:44, 3 May 2008 (UTC)

I agree with removing this, for a different reason. Every claim about the length of the year should specify whether the unit of measure is solar days or SI days. Since the logical unit of measure for calendars is solar days, many of the claims that quibble about ten-thousandths of a day are meaningless because the length of solar days in the distant past or future can't be estimated to that precision. --Gerry Ashton (talk) 16:50, 3 May 2008 (UTC)

Dubious things[edit]

I put a "dubious-discuss" tag on "The time between successive passages of a specific point on the ecliptic... var[ies] (because the orbit is elliptical rather than circular)." I am essentially sure this is untrue or at least misleading. The time between two successive passages of the same single point on the ecliptic is a sidereal year, and that does not vary due to Earth's eccentricity, only much higher-order effects like other planets and moons could affect that. The sidereal year is essentially constant even with high eccentricity. That is a corollary's of Kepler's laws and kinematics. The second statement contained in that sentence is entirely true, though.

I notice that the variation in the tropical year by start point is about .001 days, while the sidereal year is about .01 days longer than the tropical year, and the anomalistic year is about .003 days longer than the sidereal year and almost .02 days longer than the tropical year.Fluoborate (talk) 08:05, 6 May 2008 (UTC)

Using the highly accurate SOLEX9.1 I calculated the three consecutive passages of the Sun at the heliocentric ecliptic longitude it had a May 6 2008 at 00:00 TDT. First gap, 365.2419 days. Second one, 365.2594 days. Saros136 (talk) 09:40, 6 May 2008 (UTC)

I just realized that ecliptic longitude is defined with the vernal equinox as the zero point. If I invented astronomy, I definitely would have fixed ecliptic longitude with respect to the celestial sphere, but I didn't invent it. So this sentence is entirely true. Edit: I would have deleted the "dubious" tag myself at 09:16, when I had these revelations, but I forgot.Fluoborate (talk) 12:52, 21 May 2008 (UTC)

Why the tropical year varies by starting point[edit]

I am fairly sure that I have figured out exactly why the tropical year varies by starting point, I will now try to explain:

First, ignore the sidereal year, it does not matter in any way to this calculation and/or thought experiment. We don't care at all about astrological sign, we care about climate. The anomalistic year is about .02 days longer than the tropical year. This means that if viewed from above/looking down on the Northern hemisphere, with the aphelion and perihelion taken as fixed points (so the celestial backdrop moves, who cares), then the equinoxes and solstices move gradually clockwise around Earth's orbital path.

Let us define the coordinate system a little better: the perihelion will be fixed at the top of the "clock face", at 12 o'clock, and the aphelion will be fixed at 6 o'clock. The vernal equinox starts at 12 o'clock, and it gradually sweeps through 1 and then 2 o'clock over many years. If you are interested, the celestial backdrop also gradually rotates in a clockwise direction, but it rotates much more slowly than the apsides.

Now, for simplicity's sake let's say that the equinoxes move (precess) exactly 1 degree on the clock face each year. The equinoxes move 1 degree clockwise. Let us also remember Kepler's laws, which state that the Earth is moving fastest around the clockface at 12 o'clock and slowest at 6 o'clock. We have defined position on the clock face in degrees, but how will we define time? Remember, the Earth is NOT the hand of a clock, it accelerates due to Kepler's laws. Plus, the Earth moves counterclockwise, which is also unlike a clock hand. The equinox is really much more like a clock hand, it moves slowly and constantly clockwise. We will define time as days and years, there are 360 days in a year and one year is an anomalistic year, the amount of time needed to go from 12 o'clock to 12 o'clock or 4 o'clock to 4 o'clock or any other full revolution. How long does it take to go from 3 to 9 (counterclockwise, we are talking about the Earth)? It would take 180 days for a circular orbit (180 degrees at constant speed equals 180 days), but because 12 o'clock, the perihelion, fell right in the middle, it will be LESS than 180 days from 3 to 9, because Earth was going extra fast.

So, how long is an tropical year starting from the vernal equinox on the year that the vernal equinox happens to fall at exactly 12 o'clock, 360 degrees? Well, if the orbit were circular then the tropical year would be 359 days - the Earth travels 359 degrees counterclockwise and the equinox travels 1 degree clockwise, and BAM, the Earth reached its equinox again. But the equation is much more complicated with an elliptical orbit - the sum might be 359.2 degrees plus 0.8 or 358.8 plus 1.2. What is the solution? Where will the Earth and the equinox, traveling in different directions around the clock, finally meet, and how much time will have elapsed?

This problem can be solved exactly using calculus. The speed of the Earth is a cosine-like wave, and you can take the integral of that wave to determine where the Earth is at every given time. The position of the equinox is simply a sloped straight line. The intersection of the integral of the Earth's speed with the equinox's speed is the place and time where they intersect, and that time minus the start time is the length of the vernal equinox tropical year. Now, you don't actually have to do any calculus to figure out the answer if you draw all the graphs well:

Let's redefine the coordinate axis so that the Earth travels in a positive direction as t increases. This means the starting point is now 0 degrees. 11 o'clock, which it passes early, is now 30 degrees, and the Earth ends an anomalistic year at 360 degrees. The speed of the Earth looks like a biased cosine wave (cos[t] + 5, or something), so the speed has a maximum at 0 degrees, a minimum at 180 degrees, and another maximum at 360 degrees. It's integral is therefore the area under this curve, and the area under this curve represents how far Earth traveled. The Earth will get a chance to travel approximately the first 359 days of this curve, but the last day was near a maximum, so cutting off the last day severely diminishes the area of the graph. This means the Earth didn't travel as far (maybe only 358.8 degrees), and the equinox had to travel farther, thus making the vernal equinox year seem long. The Earth's average speed was less than 1 degree per day because the last day, that got cut off by the approaching equinox, was going to be a fast day.

If the year had started on the autumnal equinox and the aphelion instead, the graph would have been a NEGATIVE cosine wave plus a bias factor. The day at the end which got cut off would have been a "slow day" for the Earth, meaning that the average speed of the Earth before then was faster than 1 degree per day.

If you are unconvinced by drawing pictures, do the calculus. I had to make some variables up, but my math even worked out when I tried to calculate the vernal equinox tropical year duration.Fluoborate (talk) 09:16, 6 May 2008 (UTC)

alternate measurements[edit]

This passage was added on June 3:

... when measuring the motion of the earth relative to the sun (via earth rotation studies or lunar rotation studies) astronomers find the tropical year describes an arc of 360 degrees and the sidereal year is 360 degrees and 50 arc seconds. Again the difference is due to precession, however the reason for the different measurements is the first uses a static solar system model (non moving solar system) and the later uses a dynamic solar system model (a moving solar system).

It looks as if it might make sense if its terms were made more explicit. What are "earth rotation studies" and how do they measure the motion of the earth relative to the sun? How is the rotating reference frame chosen? How is it that the "moving solar system" model, whatever that means, makes one planet's tropical year coincidentally come out to exactly 360 degrees? —Tamfang (talk) 17:59, 9 July 2008 (UTC)

I'm not sure what the editor means by a static versus a dynamic solar system. But a tropical year does subtend 360°, that is, a tropical year is the time needed for Earth to move from one vernal equinox to the next vernal equinox, arbitrarily defined as 360° or one revolution. For this discussion I assume that Earth's orbit is circular, thus I ignore the difference between the mean tropical year and the vernal equinox year. The view is from the barycenter of the Solar System or the center of mass of the Solar System, which is often outside the Sun in the direction of Jupiter. The vernal equinox is the intersection of the equatorial plane and the ecliptical plane where Earth ascends along the ecliptic from south of to north of the equatorial plane. Both planes move during that year, so the vernal equinox has moved due to precession during one tropical year relative to the fixed stars, even though the subtended angle is defined as 360° or one revolution. In general, the longitude of Earth at any time is the angle via the barycenter from Earth to the equinox of date, that is the moving or precessing equinox. This is standard when specifying the position or longitude of any Solar System body, including the Sun and all planets (including Earth). The Moon's position is also specified relative to the equinox of date, but now the view is from the center of Earth. Relative to the fixed stars, Earth must move an additional 50" along its orbit beyond 360°. — Joe Kress (talk) 06:39, 10 July 2008 (UTC)

Earth must move an additional 50" along its orbit?!

50" = Earth axis precession (per year), not orbit. —Preceding unsigned comment added by (talk) 15:30, 24 July 2008 (UTC)

Move jargon to separate article?[edit]

Perhaps coverage of the technical term, as used by professional astronomers, would be less confusing if the jargon was moved to a separate article.

The article says:

the vernal equinox year that begins and ends when the Sun is at the vernal equinox is not an astronomer's tropical year.

Okay, so that isn't an astronomer's "tropical year". Then whose "tropical year" would it be?

I read the referenced document, "The history of the tropical year" by Jean Meeus and Denis Savoie. It contains the statement:

the definition of the tropical year ... has nothing to do with the beginning of astronomical spring.

TV weather broadcasts often say that the vernal equinox is the "beginning of spring". Is "astronomical spring" different from the "beginning of spring" as it is commonly understood?

Both here and in the article on twilight, the astronomical definitions seem to be quite different from those understood by and useful to the general public.

Perhaps coverage of the more technical terms, as used by professional astronomers, would be less confusing if the jargon was moved to separate articles. -Ac44ck (talk) 05:53, 14 January 2009 (UTC)

The lay term is just "year". "Tropical year" is a technical term and should be fully described here. --Gerry Ashton (talk) 22:53, 14 January 2009 (UTC)
The first paragraph of the lead says:
A tropical year ... is the length of time ... from vernal equinox to vernal equinox
That is the definition I learned. It also seems to be important in some ecclesiastical calendars. The other forms of "tropical year", including those marked by solstices, etc., would seem to be of more interest to academics or professional astronomers.
The second paragraph of the lead says:
A tropical year can equivalently be defined as the time taken for the Sun's tropical longitude ... to increase by 360 degrees
What I glean from the article suggests that this second definition is not equivalent, as the time for "the Sun's tropical longitude ... to increase by 360 degrees" is different from a "vernal equinox year".
The Vernal equinox and mean tropical year section says:
The tropical year is not equal to the time interval between two successive spring equinoxes
It seems that the article tries to cover too much ground and presents conflicting information.
Why would a farmer want to know when the Sun's tropical longitude had increased by 360 degrees? Wouldn't they be more interested in planting 'x' days after the equinox? -Ac44ck (talk) 23:43, 14 January 2009 (UTC)
The farmer might be equally interested in planting x days before the autumnal equinox. --Gerry Ashton (talk) 18:45, 15 January 2009 (UTC)
In theory — the domain of academics and professional astronomers. It is more likely that a farmer is going to note the passage of the vernal equinox and count forward than to anticipate the autumnal equinox and count backwards.
For whom is it most convenient to define the tropical year as the time for the Sun's tropical longitude to increase by 360 degrees? The article may have POV issues in pushing the definition currently (but apparently not historically) preferred by astronomers.
Version tag in software is bad enough. Of what use is "Tropical Year 2.0" to the general public? The date of Easter is knowable in "Tropical Year 1.0". When was "New Years Day" for "Tropical Year 2.0"? The answer would seem to be, "It depends when one started counting." If everyone can define their own tropical year, for what purpose might the general public use "Tropical Year 2.0"?- Ac44ck (talk) 19:23, 15 January 2009 (UTC)

I agree with Gerry Ashton that "tropical year" is a technical term which must be described in a technical manner with suitable 'jargon'. The equivalent lay term is "solar year" to distinguish it from a "calendar year" or possibly from a "lunar year" or "sidereal year" if the general reader realizes that the latter two terms even exist. The conflicting information provided by the article is the essence of Wikipedia's neutral point of view, where all points of view, here in the form of definitions, must be presented in the same article. All definitions are the same if the cardinal points like "vernal equinox", "summer solstice", etc. are understood to be points on the celestial sphere, not points in time. For example, the vernal equinox is the northward intersection of the ecliptic with the celestial equator. The amount of time that the Sun takes to return to the same such point in space depends on the Sun's apparent speed, producing several kinds of tropical years. What the article calls the astronomer's tropical year (the mean tropical year) is the average of all possible speeds, specifically the speed of the ficticious mean sun. Nevertheless, even if these cardinal points are viewed as points in time, all definitions in the article are approximately the same, covering a span of time of only 2.5 minutes, which is of no concern to any member of the general public, certainly not a farmer. — Joe Kress (talk) 02:21, 17 January 2009 (UTC)

I disagree with both of you. It is not _a_ technical term. Rather, it is a catchall label for several technical terms. And the (latest) astronomical definition is given excessive weight. Actually, there are two astronomical definitions -- and they of uncertain relative priority. But those two are clearly given priority over others.
If a visitor lands on this page, it is for a reason; suggesting that they do, indeed, realize that a "sidereal year" exists and it is somehow different from what they need to find here. What they will find here at the moment is at least four versions of what "tropical year" means. They may leave lacking clarity about which one is important for their need. But if they were taught as I was, they won't lack clarity that their understanding was "wrong".
NPOV does not require presenting "conflicting information", it requires presentation without prejudice.
This is conflicting information:
  • A = B
  • A = 1
  • B = 2
And it sends the visitor away scratching their head; possibly writing off Wikipedia as unreliable.
Similar befuddlement in direct quotes from the article:
  • A tropical year ... is the length of time ... from vernal equinox to vernal equinox
  • The tropical year is not equal to the time interval between two successive spring equinoxes
A "year" is a unit of time. This is a statement about coordinates divorced from any useful context: "All definitions are the same if the cardinal points like 'vernal equinox', 'summer solstice', etc. are understood to be points on the celestial sphere, not points in time." Okay, a point at zero declination and some right ascension exists. So what? Why wouldn't I care about "points in time" in the context of defining a "year"?
This unqualified statement is not in keeping with NPOV:
The tropical year is not equal to the time interval between two successive spring equinoxes.
This is more accurate:
In astronomy, the tropical year is not equal to the time interval between two successive spring equinoxes.
This is one astronomical definition:
What the article calls the astronomer's tropical year (the mean tropical year) is the average of all possible speeds
Another is:
the time taken for the Sun's tropical longitude ... to increase by 360 degrees
I gather that this latter value may vary from year to year, and that "the average of all possible speeds" is not recalculated for each year. And what would it mean to recalculate it for "a year" if there is no preferred starting point in an astronomical tropical year?
The two astronomical definitions seem incompatible and of unclear relative priority.
To dismiss a 2.5 minute interval as "of no concern to any member of the general public" is not the role of a Wikipedia editor. And if a 2.5 minute interval is unimportant, then why state the length of a tropical year to the nearest ten-thousandth of a day (an interval of about 10 seconds)?
I detect at least four important definitions of a tropical year:
  1. The interval between instances of the vernal equinox (zero declination for the Sun in the northern hemisphere's spring)
  2. An interval between successive occurrences of a selected solstices or the autumnal equinox
  3. The interval for solar longitude to increase from some arbitrary point by exactly 360 degrees
  4. Mean tropical year -- a value divorced from any recurring, observable instant
None of those are necessarily the same.
And the article seems to have a dismissive view toward all but the last two because it is the astronomical definitions (du jour) that are the "important" ones.
The nebulous (either without a preferred starting point, or unrelated to something observable) astronomical definitions are not the ones that show up often in Google searches:
solar year: the time for the earth to make one revolution around the sun, measured between two vernal equinoxes
The most common definition of the tropical year is the interval between two successive passages of the Sun through the vernal equinox
How is the definition using a "fictitious mean sun" useful to astronomers? If there is no "New Years Day" in such a tropical year, how is it used? It seems comparable to a "fortnight" to me -- an archaic interval which is seldom-used by astronomers but for which a conversion factor was needed, so one was assigned. But when the actual "Sun's latitude increased by 360 degrees" value is needed, the average conversion factor may be ignored by astronomers. True?
If they haven't settled on one of them, then why bother the rest of us with the ambiguity? I think the nit picking that boils down to "we won't say when it starts, and you can't observe one anyway" belongs elsewhere.- Ac44ck (talk) 06:02, 17 January 2009 (UTC)

The key word that is missing from the lead is "mean" or "average". Jean Meeus in "The history of the tropical year" states that the modern definition of "the tropical year is the time needed for the Sun's mean longitude to increase by 360°." This is a slight but valid simplification of the official definition in the Explanatory Supplement to the Astronomical Almanac (p.80): "The tropical year was defined [by the 1955 IAU General Assembly in Dublin] as the interval during which the Sun's mean longitude, referred to the mean equinox of date, increased by 360°." "Mean" excludes all periodic variations from the mathematical description of the apparent motion of the Sun, including nutation, planetary perturbations, and the variation in the Sun's apparent speed throughout the year due to the eccentricity of Earth's elliptical orbit. Including the latter component results in the "vernal equinox year", the "summer solstice year", etc. When the remaining small periodic variations are included, nutation and planetary perturbations, none of which have a period which is a whole number of years, what might be called the true tropical year results, which varies from year to year and from point to point in the Sun's (Earth's) orbit by about two minutes. The addition of the missing word(s) to the (erroneous) definition you learned means that the lead should read "A tropical year ... is the average length of time ... from mean vernal equinox to mean vernal equinox". This is close to the official Explanatory Supplement defintion, but some important points are still glossed over. "Sun's mean longitude ... increased by 360°" means that any point during the seasonal year is a valid starting and ending point, not just the vernal equinox, as long as these mean longitudes are referred to the mean equinox, not the true equinox. Thus an increase in the mean longitude from 100° after the mean equinox to 460° after it is valid. "Mean" requires the exclusion of all periodic terms from the complete analytical expression for the tropical year. This is not the average over many years from one equinox to the same equinox, but is the average over many years and many different observations from any arbitrary point during the seasonal year to any other arbitrary point divided by the number of whole and fractional years between those points for each pair of observations. — Joe Kress (talk) 23:51, 17 January 2009 (UTC)

Thanks for the reply.
It is curious that the IAU made this decree 50+ years ago, and the common perception remains that it is synonymous with a vernal equinox year.
The astronomical definition of twilight seems similarly out of touch. I never knew of anyone to think of noon as happening during twilight (at other than polar latitudes) until I read the initially confusing definition in Wikipedia. In calculating periods of not-quite-daylight, I came to find convenience in the astronomical definition for the purpose of tabulating data. But the notion that it is twilight at noon remains nonsense in conversational speech.
A sidereal year is conceivably observable. An anomalistic year is conceivably observable. The mean tropical year seems to be observable only as a fluke or in a carefully contrived situation. I don't understand the utility of that definition, but it is what the IAU said it is -- in situations where the IAU's decree is important; until they come out with "Tropical Year 3.0". - Ac44ck (talk) 03:55, 18 January 2009 (UTC)

I do not believe the vernal equinox is close to the Earth's perihelion.[edit]

In Section Vernal equinox and mean tropical year It says

..and on average the vernal equinoxes come slightly further apart because the vernal equinox is close to the Earth's perihelion

But at.. it says..

Currently, perihelion occurs about 14 days after the northern hemisphere's winter solstice, making its winters milder than they would be otherwise, and southern hemisphere winters more extreme.Dave 2346 (talk) 14:50, 23 November 2009 (UTC)

If the perihelion is after the winter solstice, then it is less than 90° from spring equinox; that's not "close" but it's close enough to cause the effect mentioned. Perhaps a better word should be found. —Tamfang (talk) 05:09, 24 November 2009 (UTC)

Not 20 minutes[edit]

"Because of a phenomenon known as the precession of the equinoxes, the tropical year, which is based on the seasonal cycle, is slightly shorter than the sidereal year, which is the time it takes for the Sun to return to the same apparent position relative to the backdrop of stars. This difference was 20.400 minutes in AD 1900 and 20.409 minutes in AD 2000."

This is incorrect. I believe the incorrect equation is being used. The difference in the tropical and sidereal year is not 20 minutes. This is clearly not happening. Total precession around the zodiac would be only 72 years(because 20 minutes covers 5 full degrees of the sky) if this were happening. The difference is only 50 arcseconds per year, which in real time is only 3.3 seconds.

To say this: "One sidereal year is roughly equal to 1 + 1/26000 or 1.0000385 tropical years." is not correct. It's 1 year plus 1/2600 of the angle of the sky, which is 50 arcseconds of angle. We've been lumping in the precession measure with the measure for the spin of the earth in a year. They are independent movements. The stars do not shift 20 minutes(5 degrees) compared to the sun at vernal equinox every year.--Markblohm (talk) 18:47, 9 December 2009 (UTC)

Please keep discussion at one location, that is at Talk:Axial precession (astronomy)#20 minutes per year - is that right?. — Joe Kress (talk) 21:38, 9 December 2009 (UTC)
which unfortunately is a massive volume of misconceptions. More to the point, this issue is about 20 minutes of time, and about the apparent motion of the Sun (or Earth in its orbit) w.r.t. an inertial reference frame, not about the rotation of the Earth around its axis. The Sun moves 360 deg. per year, about 59'11" per day, 150" per hour, so about 50" (the annual precession of the aequinoxes) in about 20 minutes of time. The Earth rotates 360 deg. per (sidereal) day, so 15" per second, so about 50" in about 3.3s BUT THAT IS TOTALLY IRRELEVANT IN EXPLAINING WHY THE TROPICAL YEAR IS SHORTER THAN THE SIDEREAL YEAR AND BY HOW MUCH! Tom Peters (talk) 14:27, 10 January 2010 (UTC)
Yes, I think we've covered that at the place mentioned. —Tamfang (talk) 06:33, 13 January 2010 (UTC)

Citation format[edit]

In my sandbox, I am creating a new version of the article to address the {{Refimprove}} template. I am finding it necessary to cite different pages in the a few sources, which would be easier to do with parenthetical referencing. Is there any objection to this change? --Jc3s5h (talk) 02:58, 21 January 2010 (UTC)

A standard template used to cite different pages in a single reference at different locations in the article is {{rp}} (presumably for repeat page). It inserts the page or pages that are cited as a superscript immediately after the superscripted number of the reference, which has no page numbers itself. For example, after one phrase,<ref>Reference</ref>{{rp|491}} and after another phrase.<ref>Reference</ref>{{rp|302}} — Joe Kress (talk) 08:17, 21 January 2010 (UTC)

Reasons for tags[edit]

The primary citation used in the article, Meeus & Savoie, gives not just one definition but many different definitions for the tropical year.

Numerous RS's (and their comparison with the data-sources on which they rely) show that for astronomical standards purposes, the number quoted for a tropical year-length, at any epoch, is usually now obtained from the inverse of the linear coefficient of the elapsed time in the adopted polynomial expression for the sun's change of mean longitude with time relative to equinox and ecliptic of date, when expressed for that epoch.

In other words, if the linear coefficient in the expression for mean solar longitude w.r.t. equinox and ecliptic of date (starting from epoch X) is x degrees/day, then the tropical year length (at epoch X) is 360/x days. (Inverting instead the 1st derivative of the entire polynomial gives the variation of year-length with time.) The numbers given in the cited Meeus/Savoie source for Leverrier and Newcomb's estimates of the tropical year numerically match that, even though their verbal description is not so clear.

The reasons for the other tags arise from the same basis. It needs to be explained which definition is being used in which context. Terry0051 (talk) 20:15, 22 January 2010 (UTC)

I am working on this. You can see what I've don so far at User:Jc3s5h/sandbox3. I've gone through as far as the 18th and 19th century history. Let me know if you think I'm headed in the right direction. --Jc3s5h (talk) 20:55, 22 January 2010 (UTC)

Thanks. If it's ok with you I'll post on your sandbox3-talk page. Terry0051 (talk) 22:42, 22 January 2010 (UTC)

Agreed. --Jc3s5h (talk) 23:13, 22 January 2010 (UTC)
  • You probably have this all in hand, but apropos of the "dubious" tag against the opening definition, and the above discussion, we need to make sure that said definition remains intelligible to ordinary readers. For example, we cannot lead off with an explanation that involves "linear coefficients", "first derivatives of polynomials" and so on. 99.9% of readers will not have the faintest idea what it's talking about. (talk) 05:08, 23 January 2010 (UTC)
Terry0051: I find the "unreferenced" and "dubious" tags somewhat offensive. If you really look into it, there are infinitely many possible definitions of the "tropical year", as this article tries to explain. A conventional but somewhat naive modern astronomical definition is, the period from the polynomial expression of the Sun's mean longitude as measured from the northward equinox. However, historically a tropical year ran from one summer solstice to the next, or from northward aequinox to the next. Those periods are not equal and also not equal to the astronomer's "mean mean" tropical year, for rather arcane reasons that the article tries to explain. It is somewhat relevant because the Gregorian calendar year was defined to set and keep the vernal aequinox at 21 March; some calendar reformers claim that the calendar year could be improved by better matching it to the somewhat smaller "mean mean" tropical year, but that will make the date of the vernal aequinox drift and conflicts with the churches' Easter computus. Most astronomers don't know and don't care so you won't find a proper treatment in most astronomical text books. It appears that Meeus was the first to realize and publish about the distinction when he formulated empirical expressions for the moments of the aequinoxes and solstices, maybe only 20 years ago (in his Astronomical Algorithms; in his earlier Astronomical Formulae he used a single approximate formula for all seasons). I suggest you help improve the text rather than flag it. I think earlier versions of the article were better but the quality has eroded when people started trying to explain fragmentss they found obscure. Tom Peters (talk) 15:53, 23 January 2010 (UTC)
Tom, I am working to improve the article in my sandbox, and Terry has been helping. Any precise statement about a tropical year (T. Y.) would be dubious if it didn't identify which tropical year is being referred to, and what time scale is employed. Also, we do not rely on original research by Wikipedia editors. Taking an algorithm out of a book and using it to derive data about tropical years probably qualifies as original research, so there is a strong argument that any such results, if done by a Wikipedia editor, are unreferenced.
Meeus' could be called original research, but published on paper. The "derived" in the reference refers to taking the derivative of his polynomials to go from moments in time, to speed. This is a trivial operation for anyone who had calculus in secondary school. The "Moisson" value takes some more computation, as explained in the note; I suppose one could consider it "original research" in that the parameter that is of interest here is less obvious in the paper, but can be derived from the original data in a straightforward manner. The chances to get this computation published anywhere as "original research" are nil, so conversely I think Wikipedia should not disqualify such digestion of original data as "original research" unfit for inclusion: the values given can be verified from the sources. I think that in this form it is useful information adapted to the subject of the encyclopedic article. Tom Peters (talk) 22:56, 25 January 2010 (UTC)
It also occurs to me that the arguments about whether a calendar does a better job of tracking the mean T. Y. or the mean northern vernal equinox T. Y. have become moot because the unpredictable nature of the Earth's rotation, which has been revealed since the invention of atomic clocks, swamps the difference between these two varieties of T.Y. --Jc3s5h (talk) 17:27, 23 January 2010 (UTC)
Well, it's not entirely unpredictable, several mechanisms are well understood and can be modeled. Just because we measure time in SI seconds with atomic clocks, the rotation rate of the Earth becomes irrelevant, and we can compare a pure sidereal with a pure tropical year. And it so happens that when using a second based on the rotation of the Earth rather than the SI second (i.e. using UT rather that TDT), the date of the northward aequinox is stable over several thousands of years around present. So a case can be made to keep basing the calendar on the northward aequinox year in UT, rather than some mean mean tropical year in some ephemeris time scale. Tom Peters (talk) 22:56, 25 January 2010 (UTC)
My suggestion for the article: don't try to be exactly precise in the definition of the tropical year in the beginning of the article. The major point is the distinction between sidereal and tropical periods caused by precession. Then at closer look, the motion of the perigee also comes to play a role through the variable speed of the Earth in its orbit: but that is a second-order effect. Tom Peters (talk) 22:56, 25 January 2010 (UTC)
I concur with putting off overwhelming details until later in the article. I also suggest you read the WP:No original research article. Converting units or finding a density given a population and an area would qualify as trivial calculations; doing calculus or writing computer software would not. I grant you this does leave a gap between what can't be calculated by Wikipedia editors and what a journal would consider worthy of publication, but we have no way of telling the difference between a Wikipedia editor who is also a professional astronomer, versus one of those physics cranks who use to make the science related newsgroups nearly useless. --Jc3s5h (talk) 23:33, 25 January 2010 (UTC)

I agree with Jc3s5h's approach. I apologize if I've posted anything unduly long/complicated. I agree with keeping things as simple as the truth and sense will allow. The tags were not intended to be offensive, but I would respectfully defend their use. (Tom Peters' reaction came down to 'so fix it'. I would have started, though not immediately, but Jc3s5h immediately posted that he's working on it, it seems to me he's taking a good thoughtful approach.)

To be more specific, the first 'dubious' tag maybe could have been better placed, I defend it because it relates to the combination of the 1st and 3rd lead paragraphs together. There must be something wrong there (with all due respect to the arcane explanations), because, if it really is true that the mean tropical year-length is different between the solstices, than between the equinoxes, it would have to follow that at some time the solstice and equinox would coincide, which is physically impossible.

I suggest the existing first paragraph would become good if 'vernal equinox' is qualified (both times) by 'mean', and if the complicating reference to the solstice is left till later.

I've re-read an article by Jan Meeus about the various intervals between equinoxes and solstices. It is clear he started out by referring primarily to equinoxes and solstices based on true/apparent positions (which show effects of several kinds of periodical perturbation). He has even said this expressly, see Jean Meeus, 'Mathematical Astronomy Morsels' (1997), p.347, reprinting in translation his articles originally published many years before. (The very title of the book 'Mathematical Astronomy Morsels' conveys that there is intentionally an element of entertainment in his treatment of this subject, rather than of standard-setting. The subjects were treated as a jeu d'esprit, though in a learned and instructive way. This is meant respectfully to the author Jean Meeus, it was clearly his intent, he has been a great educator and mediator in bringing mathematical astronomy to a wide audience; and in the days when calculating/computing power was becoming available, but still expensive, he created highly ingenious intermediate approximations for astronomical theories and calculations, intermediate between on the one hand the full theories, which were still unfeasible then for the computing resources of most users, and on the other hand the more commonly-quoted crude approximations, which are/were of unusefully low precision.)

All of the considerations based on true, perturbed, positions of equinoxes and solstices seem to be at some distance away from the big first encyclopedic point about the tropical year length. This is already present in effect in the 1st paragraph (arguably confused by other material). It is the year length that is locked in sync with the seasons. That is a physical reality, and even those who are content with equinoxes defined to the nearest day can be interested in a good precise long-term measure of that reality. It results from the relation between the sun and the reference-frame defined by the equinox, and although both of these move with irregularities, a mean can be found that represents the long-term physical reality. (If there are competing measures or definitions, all of them that are valid will converge to represent that reality.)

(Also, if there is any connection between on the one hand Jan Meeus' jeux d'esprit about equinoxes and solstices, and on the other hand the medieval/renaissance debates that took place about year-length before the 1582 calendar reform, it seems a reliable source has not yet been cited to show that. Among other points, the exact times of solstices and equinoxes were difficult to observe and to measure. There is much good information in "Astronomical aspects of the calendar reform" (J Dobrzycki), though there may also be other and even controversial sides to the questions treated.) With good wishes. Terry0051 (talk) 14:18, 26 January 2010 (UTC)

Dubiosity tags[edit]

1. Def of tropical year: the tropical year is defined as the time passing of Sun's passage throught the point of vernal equinox, until the next such passage, as seen from earth. That is the tropical year, and that is irrespective of eccentricity of Earth's orbit. A mean tropical year, could however be said to compensate for the eccentricity. There's no meaning with looking it up in Meeus, because he doesn't provide definitions, and is not using mathematical definitions derivations as much as he should. The tropical year is never defined by any other point than by vernal equinox. In older times certain years might have been defined by choosing other points, but tropical year is an astronomical term. Rursus dixit. (mbork3!) 17:03, 27 January 2010 (UTC)

The term tropical year is not a name for a year determined by any reference point, but those who make that mistake do so by using it as the vernal equinox year. Actually using Meeus's Astronomical Algorithms is very informative here. Differentiation of the formulas for the mean equinoxes or solstices gives formulas for the mean vernal equinox, September equinox, June solstice, and December solstice years. Adding those formulas and dividing by four gives a formula that calculates the tropical year value given on page 408, and that matches the value gotten through differentiating the mean longitude formula given on page 183. Saros136 (talk) 07:12, 20 September 2010 (UTC)
Quite incorrect. You should try finding a book defining tropical year. My book Astronomi och Astrofysik, by Gunnar-Larsson Leander is in Swedish but it says "the time that sun needs to move anticlockwise along the ecliptic from the vernal equinox and back to that again ... one calls tropical year" (literal translation –> weird syntax), i.e. the same as "Tropical year". My objection stand. Rursus dixit. (mbork3!) 07:06, 21 September 2010 (UTC)
I found the ultimate source. I'll read it and then answer. However: there are enough examples defining it as the full turn backward eccliptic movement from vernal equinox to vernal equinox, to use that definition as a starting point in the text. Either one can provide the very generalized, intransparent definition first, or one can start with the vernal equinox tropical year and expand to a generalized notion. Either way, multiple sources refers to the vernal equinox year as being equal to the tropical year, that's no mistake. Rursus dixit. (mbork3!) 07:25, 21 September 2010 (UTC)
That was Meeus and another guy named Savioe, making a very confused appearance. They claim that there's no coherent definition. Then they claim that the tropical year of antiquity was defined as the turn of sun from one tropos (any equinox or ecliptical longitude) to the same one 360° on, and that modern authors claims so too. After that, they claim that it's easy to see that the tropical year is not the same as that turn along the ecliptic because of nutation and planetary perturbations. There's no order in their text! They are making the objection, not the various authors. They're not distinguishing unnutated and unperturbed ecliptic from the perturbed one, they're not introducing deviations after the main definition, they're starting with the deviations and then trying to define everything at once in a confused verbal mess that explains nothing. I now understand why Meeus never provides derivation in his books: it is because he cannot write explaining discourses.
The best thing is to stick to the original defining sources of antiquity: the tropical year is the time of the anticlockwise movement of sun along the ecliptic from one tropos to the same one next time, (in modern times usually vernal equinox) and then add complications such as nutation and planetary perturbations afterwards as is customary in scientific pedagogy, not bash with all complications at once like Meeus, and then deny that there is a definition. Rursus dixit. (mbork3!) 07:47, 21 September 2010 (UTC)
Well, I wasn't using Meeus and Savioe for my post. A book by Meeus. The Meeus and Savioe article says the definition has changed. I thought it wasa fairly good although somewhat messy. They do not claim there is no coherent definition, now. It has changed over time. The definition is very simple-it is a way of expressing the rate of change in the sun's mean longitude at an instant. But it didn't seem confusing to me because I was familiar with the subject, having first read about the history and distinctions in Marking Time, by the astronomer Duncan Steel, who covers it quite thoroughly in his appendix, and is correct.
It's true that most who write consider the tropical year to by synonymous with the vernal equinox year. But what's interesting is that everyone agrees about the length of the tropical year, even though the different definitions lead to different lengths. If you take the vernal equinoxes times of, say, 1900 and 2100 (true equinox) and divide the difference by 200, you'd get a figure about a second different from that figured using modern value for the vernal equinox. I got 365d,5h,49m,2.34s, using the dates generated by Solex 11.0. The figure given by Meeus and Savioe, for J2000 vernal equinox year ends with 1.11s (although they didn't use the min-sec form). Other sources come very close if not the same. But the figure everyone agrees that the tropical year equals ends with 45 m45.2s, which is what one would gets from the equation given by Meeus&Savioe, and is the average of the four equinox or solstice years. So there is a contradiction for those who say the tropical year is the VE year, yet give the modern value for the tropical year (which is 16 s shorter)
As far as which is correct...the atomic second was set to be equal to ephemeris second, which was officially defined as the fraction 1/31 556 925.9747 of the tropical year for 1900 January 0d 12h ephemeris time. The y-d-m-s form ends with 45.9747 seconds, the modern one with 45.7 just a fraction of a second different. Because they used the same definition. Saros136 (talk) 10:19, 22 September 2010 (UTC)
The lead of the article should present the current definition. It doesn't really matter if you don't like the way Meeus and Savioe wrote their article, because others agree with them as far as the current definition is concerned. For example, see "year, tropical" in the glossary of the Astronomical Almanac Online. —Preceding unsigned comment added by Jc3s5h (talk (talkcontribs) 12:35, 21 September 2010
The "tropos" in "tropical year" means turn, and refers to the turning back of the Sun in its motion at the solstices. So the tropical year need not be measured from the northward aequinox by definition. And the length of the period from the tropic of Cancer to the next event, is on average different from the period of the aequinox to the next. That difference is the second-order effect due to the long-term motion of the perigee with respect to the aequinox.
But again, the major distinction should be with the sidereal year. For either one, all short-periodic perturbations like elliptic terms, planetary, and nutation, can and should be ignored: only the long-period precession matters. Tom Peters (talk) 22:16, 1 February 2010 (UTC)
Yes, right! Tropical year is generally contrasted with sidereal year, and that difference neatly connects to precession. Rursus dixit. (mbork3!) 07:11, 21 September 2010 (UTC)

2. Alternate def in second paragraph is perfect only if the sun starts at degree 0 and runs 360° degrees forth.

3. The third paragraph might be explained if preceeded by: "because of the eccentricity of earth's orbit around the sun, and the displacement of sun due to gravitational influence from Jupiter, and less so by other major planets, the time from one passage of the vernal equinox to next such passage, might vary from year to year". Don't use the argument of choosing any other point than vernal equinox, first it has nothing to do with the tropical year, secondly it confuses.

4. I think an unqualified "tropical year" can be said to refer to the mean tropical year, but citation is needed, as asked for.

IMHO. Rursus dixit. (mbork3!) 17:03, 27 January 2010 (UTC)

I'm afraid the 'definition' offered by Rursus has multiple ambiguities. First, the true or apparent sun can have up to about 1" latitude north or south, which makes the passages across the zero lines of ecliptic longitude, right ascension and declination happen at three different times. Second, the inequalities of the true/apparent sun in ecliptic longitude are not only due to the orbital eccentricity, but also to the lunar and planetary perturbations. Third, the inequalities of the true equinox (the nutation) are different from year to year. In short, and in addition to the multiple ambiguity, the period referred to is never the same from one 'year' to the next. This seems hardly a definition. "The vernal equinox" is ambiguous until one has specified either mean or true equinox of date, the difference at any time being defined by the value of the nutation either in longitude (for the ecliptic longitude) or in right ascension (for measurements parallel to the equator).
In view of all these factual considerations, it seems that the matters put forward in my previous post still hold. Terry0051 (talk) 22:27, 28 January 2010 (UTC)
Nothing to be afraid of ;-), but you're right. 0° to 360° refers to ecliptic longitude, nothing else. Then it is perfectly unambiguous. Rursus dixit. (mbork3!) 06:52, 21 September 2010 (UTC)
The definition of the ephemeris second in The explanatory supplement to the Astronomical Almanac (1992/2006) p.80 clarifies "The tropical year was defined as the interval during which the Sun's mean longitude, referred to the mean equinox of date, increased by 360°." Earlier in the paragraph "the tropical year was understood to be the mean tropical year". This definition is loaded with hidden meaning. "mean" means that all periodic terms are ignored (deleted), including terms defining the orbital ellipse and its movement, hence the periodic terms (the equation of the center) needed for the vernal equinox year are not included. However, polynomial terms are retained, hence Newcomb's linear decrease in the length of the mean tropical year is retained, so the [instantaneous] tropical year at noon Greenwich mean time on 1900 January 1 differs from the [instantaneous] tropical year on 1900 February 1, etc. "of date" means that general precession is included (no periodic terms are needed) but nutation is ignored (periodic terms would have been required).
Even though "year, tropical" in the glossary of the Astronomical Almanac correctly includes the phrase "the tropical year comprises a complete cycle of seasons", the hyperlinks incorrectly lead, via "ecliptic longitude", "dynamical equinox", and "true equator" to the "true equator and equinox", which is affected by nutation, so it is not the "mean equinox". Confirming that the Astronomical Almanac uses the mean tropical year we find "tropical year is 360^\circ/\dot{\lambda}" on page L8, where λ is defined on page C1 as 279°.319067 + 0.98564736 d, hence the tropical year is 365.242190 days when limited to the eight significant digits in the coefficient of d, which is the same value given for the "tropical year" on page C2. — Joe Kress (talk) 02:26, 28 September 2010 (UTC)

Calendar year section conflicts.[edit]

The Gregorian calendar, as used for civil purposes, is an international standard. It is a solar calendar (it is designed to maintain synchrony with the tropical year). It has a cycle of 400 years (146,097 days). Each cycle repeats the months, dates, and weekdays. The average year length is 146,097/400 = 365+97/400 = 365.2425 days per year, a close approximation to the tropical year. (Seidelmann, 1982, pp. 586–81)

The citation has typos (should be 1992, and pages 576-81), but that's not the big problem. The book is Explanatory supplement to the Astronomical almanac. by Seidelmann, but the calendar article is by Doggett who considers the topical year to be the mean interval between vernal equinoxes , unlike Seidelmann himself or this article.

The third paragraph does consider the tropical year, not the vernal equinox year, to be the standard. Blackburn, B. & Holford-Strevens Saros136 (talk) 05:37, 12 August 2010 (UTC)

fixed the cite. Saros136 (talk) 06:21, 12 August 2010 (UTC)

References and notes[edit]

Please use numbered references with <ref> as all other Wikipedia articles. Notes can use letters. (talk) 17:15, 14 March 2011 (UTC)

Please see Wikipedia:Citing sources, especially the statement "Editors are free to use any method for inline citations; no method is recommended over any other." Jc3s5h (talk) 18:17, 14 March 2011 (UTC)
Jc3s5h, you seem to support this method of referencing sources. I just want to point out that the vast majority of the readers of this article do not care from which page of which book the information he is reading comes from. I have no idea why you would support cluttering the article with that information. Could you please explain your thinking? Dave3457 (talk) 23:00, 9 September 2011 (UTC)
The method is effective when different parts of the same source will be cited many times. It is also easy to understand, so future editors will not have trouble figuring out how to add new citations. Also, readers who are familiar with the literature will recognize the sources immediately, without having to click to go to the footnote. These reasons are sufficient to cause a significant number of scholarly publications to use parenthetical citations. Jc3s5h (talk) 00:20, 10 September 2011 (UTC)
I can definitely appreciate why scholarly publications whose readers are familiar with the literature might use this form, and I can also appreciate that this is somewhat of a "specialized" subject. But Wikipedia is targeted at the general public not academics. I can't help but feel that we should be making Wikipedia as accessible as possible and the citations add clutter and a bit of confusion. I don't know what the "rules" are concerning this method, but may I suggest a compromise between the two points of view where we have the font of the references be at least reduced in size. It would make things clearer and less distracting for the general reader. Below is an example using the "small font" option at the top of the editing window.
... early astronomers did so by noting the time required between the appearance of the Sun in one of the tropics to the next appearance in the same tropic. (Meeus & Savoie, 1992, p. 40)
I personally think that it is a substantive improvement. Dave3457 (talk) 02:05, 11 September 2011 (UTC)
Irrespective of whether consensus ended up being reached regarding the referencing style, this is not a proper use of an editnotice (they go on the edit page, by reference to a special subpage). They're not supposed to be used as cleanup tags, and we're not supposed to add permanent disclaimers. I'm excising the editnotice, without implying an opinion on the style of referencing currently used here. TheFeds 06:57, 5 September 2012 (UTC)
The template does indeed seem to not be working. I don't know if a template can provide an edit notice. But the template is not being used as a cleanup tag. If it were working, it would provide a reminder to editors who might be more accustomed to seeing other citation styles. It seems to me this issue should be dealt with at the template, not in individual articles. If the goal is beyond the capabilities of templates, delete the template. If it is never appropriate to remind editors that an article uses a style that they might not be used to, delete the template. If it could work, and if reminders are appropriate, find someone to fix it. Jc3s5h (talk) 13:35, 5 September 2012 (UTC)
After further investigation I see the template won't work when it is added to the text of the article, it would have to be added as an editnotice by an administrator or account creator. Since I am neither, I've just restored TheFeds' deletion. As an aside, I don't agree with TheFeds that the edit notice is a disclaimer. The disclaimers page is all about providing warnings or labels regarding article content; that page does not address providing technical hints for editors on how to edit the article. Jc3s5h (talk) 14:30, 5 September 2012 (UTC)

Please see the referencing method of Mayer–Vietoris sequence for perhaps a compromise that all parties will be satisfied with. I can add an editnotice to Template:Editnotices/Page/Tropical year if required. Just use {{editprotected}} to attract the attention of an admin. — Martin (MSGJ · talk) 15:40, 5 September 2012 (UTC)

It would be easy to overlook the year in the last post that questioned the citation style: 11 September 2011. That edit suggested reducing the size of the inline citations. I would oppose that because I think the default font size in articles is too small, and I have my browser set to zoom in on all Wikipedia articles. So obviously I oppose shrinking anything but the most useless text. Jc3s5h (talk) 15:55, 5 September 2012 (UTC)
I'll add that I think the present form of citation is suitable for this article because there is a lot of misinformation about the tropical year. It's literally a religious conflict. Many Orthodox churches follow the Julian calendar, which tracks the tropical year more loosely than the Gregorian calendar or several proposals to reform the calendar. Much of the calendar reform literature was written before the invention of atomic clocks. So this is a topic where who made a claim and when they made it is likely to be on the mind of readers, and providing a short summary of this information inline is helpful. Jc3s5h (talk) 16:11, 5 September 2012 (UTC)

Proposal of new sub-section[edit]

I would propose to add the following table as a new sub-section after the sub-section "Mean tropical year current value"

Any protests? Any questions? Other comments?


Variations of the individual tropical years from the mean value above

Because of the gravitational attraction to the other planets of the Solar system the orbit of the Earth around the Sun will vary somewhat between different years depending on the constellation of these planets. The difference between the times from solstice to solstice for an individual year will therefore deviate with a few minutes from the mean value 365.2421897 given above.

Using the JPL planetary ephemeris and a standard model for the precession of the Earth axis the following deviations were found:

Start of year (Atomic time) Offset from 365.2421897 SI days (minutes)
1990/12/22 2:59: 2.1 -10.89
1991/12/22 9: 0:24.5 12.62
1992/12/21 14:35:52.6 -13.29
1993/12/21 20:28: 7.8 3.50
1994/12/22 2:19: 4.9 2.20
1995/12/22 8: 8:58.0 1.13
1996/12/21 14: 2:58.4 5.25
1997/12/21 19:51: 7.3 -0.61
1998/12/22 1:51:31.3 11.65
1999/12/22 7:25: 0.6 -15.26
2000/12/21 13:26: 7.0 12.35
2001/12/21 19: 7:58.9 -6.89
2002/12/22 0:55:50.2 -0.90
2003/12/22 6:57:44.1 13.15
2004/12/21 12:23:55.9 -22.56
2005/12/21 18:31:27.1 18.77
2006/12/22 0:12: 6.5 -8.10
2007/12/22 6: 4: 4.2 3.21
2008/12/21 12: 3:19.7 10.51
2009/12/21 17:40:13.2 -11.86
2010/12/21 23:44:53.2 15.91
2011/12/22 5:21:41.8 -11.94
2012/12/21 11:14: 1.9 3.58
2013/12/21 17: 5:38.3 2.85
2014/12/21 22:55:15.2 0.86
2015/12/22 4:44:29.3 0.48
2016/12/21 10:28:12.6 -5.03
2017/12/21 16:22:18.6 5.35
2018/12/21 22: 2:52.6 -8.19
2019/12/22 4: 8:18.1 16.67
2020/12/21 9:47:46.7 -9.28
2021/12/21 15:42: 6.9 5.58
2022/12/21 21:42:35.6 11.73
2023/12/22 3:10:37.1 -20.73
2024/12/21 9:18:53.8 19.53
2025/12/21 14:53:32.3 -14.11
2026/12/21 20:47:37.8 5.34
2027/12/22 2:41:37.0 5.23
2028/12/21 8:13:52.5 -16.50
2029/12/21 14:19:45.1 17.12
2030/12/21 20: 0:48.9 -7.69
2031/12/22 1:58:26.5 8.87
2032/12/21 7:49:15.7 2.07
2033/12/21 13:38:28.5 0.46
2034/12/21 19:29: 5.7 1.87
2035/12/22 1:13:54.4 -3.94
2036/12/21 7: 6:22.3 3.71
2037/12/21 12:47:12.9 -7.91
2038/12/21 18:51:44.3 15.77
2039/12/22 0:25:56.5 -14.55
2040/12/21 6:17:35.0 2.89
2041/12/21 12:12:42.6 6.37
2042/12/21 17:47:45.5 -13.70
2043/12/22 0: 0:44.2 24.23
2044/12/21 5:34: 0.7 -15.48
2045/12/21 11:33:48.3 11.04
2046/12/21 17:28: 8.2 5.58
2047/12/21 23: 2:31.1 -14.37
2048/12/21 5: 7:41.4 16.42
2049/12/21 10:43:23.7 -13.05

The mean of the durations of these 60 tropical years was 0.62 minutes longer than the theoretical value 365.2421897 SI days specified above

Stamcose (talk) 15:35, 13 June 2011 (UTC)

It would be more appetising to present this information in a bar diagram. And modify in the last sentence, the presumptuous "was" longer. −Woodstone (talk) 11:12, 14 June 2011 (UTC)
Are we to understand that you, Stamcose, derived these numbers from the JPL ephemeris? Such a synthesis is against the rules, I believe. — And I agree with Woodstone that it's inappropriately bulky. I'd make it a graph of cumulative offsets; a graph of year-lengths would (I imagine) look like white noise. —Tamfang (talk) 18:33, 14 June 2011 (UTC)

Why is this table important?

The section Mean tropical year current value says:

The mean tropical year, as of January 1, 2000 was 365.2421897 or 365 days, 5 hours, 48 minutes, 45.19 seconds. This changes slowly; an expression suitable for calculating the length in days for the distant past is
365.24218966986.15359×10^−6T7.29×10^−10T2 + 2.64×10^−10T3

The "mean value for January 1, 2000", how this now should be understood, is given with an accuracy of 1/100 seconds!

The formula for the "mean value" is even given as a function of time with an accuracy of about 1/100000 second!

With normal concepts one would talk about a mean value for 60 consequtive years (what I do) or possibly for 1 million consequtive years if a sensible analysis for this at all is possible! Sure, there is a theoretical model where "mean" for one single specified year is defined (in some special meaning). But instead of entering into complicated discussions/explanations about this it is more useful to generate such a tabel that explicitly shows how the durations of the individual years vary with many minutes! It is all a matter of many minutes, values with accuracy of 1/100 seconds or even 1/100000 seconds give a completly wrong impression of the accuracies involved. But with the table this is all clear and the many decimals make no harm anymore and could stay. But by preference be explained better!

Stamcose (talk) 22:33, 14 June 2011 (UTC)

Year "0"?[edit]

What is this year 0 referred to in two of the tables? Our calendar runs like this Dec.31st in year 1 BCE , and the next day is Jan 1st, in year 1 CE (supposed year of the birth of Christ). Of course, this standard was introduced som 600 years later (rulers first year of reign was used in,before and after the time of birth of Christ), and most continued to change years in March, even though Gaivs Ivlivs Cæsar decreed Jan as new years day from the year 44 BCE.

However, there has not been a year 0, not even in calculation, so why use this as anything? — Preceding unsigned comment added by (talk) 04:30, 19 February 2013 (UTC)

See Astronomical year numbering. Also, this usage is consistent with the sources that the tables were based on. Further, astronomical year numbering is more convenient for use with the equations that are included in the article. Jc3s5h (talk) 12:46, 19 February 2013 (UTC)

Calendar Year Fixing[edit]

If math people and astronomers would just think a little, just a little, they would have get the right answer for the Julian year calendar:
The reason for the day off every 128 years - not to be believed - is adding an extra day every 128 years!

365.242189... * 128 = 46,751.000192... days.
365.25... * 128 = 46,752 days.

As I said, extra day all along.
And here I read about ideas of disadd a Febuary 29 every year which is divided by 3200 - 775 leap years instead of 776. No need!

775/3200 + 365 = 365.2421875

The idea of this long period is a bummer and... unnecessary. why?
Because - if you divide both numbers of the 775/3200 fragment, you get: :31/128 = 365.2421875. This means 31 leap years (instead of 32) in a 128 year period. That's all! (יהודה שמחה ולדמן (talk) 19:00, 27 February 2014 (UTC))

The purpose of this article is to discuss improvement to the article. I can't figure out what the post of יהודה שמחה ולדמן is about, but it does not seem to be about improving this article. Jc3s5h (talk) 15:42, 28 February 2014 (UTC)
Do you have any compliments to say? Did you read anything here? Focus... יהודה שמחה ולדמן (talk) 23:27, 1 March 2014 (UTC)
יהודה שמחה ולדמן is arguing for a revision of the Gregorian calendar based on the mean tropical year. This was never the purpose of most calendars: they aim to match the vernal equinox year of 365.2424 days. No-one yet knows whether an adjustment will be necessary during the next 3200 years. See John Herschel's correction. Dbfirs 08:32, 16 January 2015 (UTC)

Not designed for tropical year[edit]

I don't agree that the Gregorian calendar was that it is designed to maintain synchrony with the tropical year, as the article says. The church didn't make a statement that (in modern terminology) they wanted to match the tropical year, or any average. It is just the modern way of thinking that judges it by secular standards. What is clear is that the Catholic Church's concerns were religious, and especially concerned Easter. The main issue was that the vernal equinox had been coming earlier in the year, and they wanted to keep it within a small range of days. How long the calendar does so depends on the vernal equinox year more than any other. And the Gregorian calendar article doesn't claim that it was designed for either the tropical or vernal year—it makes comparisons for both. Saros136 (talk) 06:51, 23 June 2014 (UTC)

Saros136 is quite right that the Gregorian calendar was designed to keep the astronomical vernal equinox close to March 21. The time between vernal equinoxes is one form of tropical year, although not the form used by modern astronomers. So the Gregorian calendar was designed to maintain synchrony with one form of the tropical year. Jc3s5h (talk) 12:13, 23 June 2014 (UTC)
The sentence I question referred to the tropical year, not a tropical year. The difference is crucial. The first normally means the mean tropical year. The writer or speaker has in mind a standard with a definite length, the one currently 365d5h48m45.1s long. I was judging the default meaning of the words, and using it myself.
The VE year is actually calculated by formula now, but it still represents closely a real interval, as opposed to an average. It is not just another kind of tropical year, it's more like the solstice years or the September equinox one.
Saros136 (talk) 03:23, 24 June 2014 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────The Alfonsine tables, first printed in 1483, were in general use in Europe at the time of the calendar reform. Dobrzycki critisizes the geometrical structure implicit in these tables as being "incoherent". A competeting hypothesis, published in the midst of the reform, was Copernican heliocentrism, and Dobrzycki considers an important improvement in the Copernican hypothesis to be "unequivocal definition of all the elements in the system of celestial coordinates". (p. 123) A new set of astronomical tables, based on the Copernican hypothesis, was the Prutenic Tables, which were used by the spokesman for the Gregorian reform, Christoph Clavius to investigate the actual and mean tropical years based on the vernal equinox, and decided the mean tropical year would be satisfactory.

So the design of the Gregorian calendar was based on the mean tropical year between vernal equinoxes, rather than the modern mean tropical year which is based on the mean elements of the Earth's orbit, and using calculations based on those elements, the time it would take for the Earth's ecliptic longitude to increase 360 degrees.

Work cited:

Dobrzycki, J. "Astronomical Aspects of the Calendar Reform." In G.V. Coyne, M. A. Hoskin, and O. Pedersen. Gregorian Reform of the Calendar:Proceedings of the Vatican Conference to Commemorate Its 400th Anniversary 1582&ndash 1982. (Vatican Observatory, 1983) p. 117–125. Jc3s5h (talk) 12:45, 24 June 2014 (UTC)

I have added a footnote to the article to indicate the exact type of tropical year used in the design of the Gregorian calendar. Jc3s5h (talk) 13:10, 24 June 2014 (UTC)

I'd recommend not using that guy for a source. He, like everyone else at the conference, thinks the vernal equinox year is the mean tropical year. Tom Peters :wrote about it on Calndr-L that "I found it striking that all of the authors of the conference equate (sometimes explicitly) the vernal equinox year with a (mean) mean tropical year of 365.2422 days. Apparently this distinction has not become generally known before Jean Meeus published expressions for the times of aequinoxes and solstices in his Astronomical Algorithms in 1991 (Chapter 26). However anyone looking for regularity or predicting of these events could have known." Saros136 (talk) 16:20, 26 June 2014 (UTC)

There's still usage issues here. Writing that the Gregorian calendar was designed for the tropical year, given common usage, is taken to mean the mean tropical year. Your footnote will be overlooked by most.
I still don't think we should say what the calendar was designed for. There were other factors, but we will never know all. They did not pick the most accurate they knew of.
What would be better would be a mention of the complications her in judging the calendars.
  • Most people think the mean tropical year should is that standard (and believe that 365d5h48m45s value represents that average gap between vernal equinoxes).
  • For those who know the modern language many just compare it to the MTY standard. Duncan Steel, in Marking Time: the Epic Quest for the Perfect Calendar is an astronomer who argues emphatically the in the spirit of the Catholic Church's main goal it should be compared to the VE year. (like most, he does not add tropical qualifier.) But probably many more others who may will continue to judge it by the MTY.
  • Because the tropical year with no other qualifier is not the VE year, those of us referring to it need to make it clear right away that we're writing about the latter. Most (like Tom Peters above) do not add that's it is a tropical year, probably because it's unnecessary.

Saros136 (talk) 16:42, 26 June 2014 (UTC)

(edit conflict) The archive of calndr-l you provided does not contain the words "found", "striking", or "Peters", so I please provide a locatable position in that document so I can see what you're writing about.
Dobrzycki makes it clear the year used by Christoph Clavius was the mean vernal equinox year, and that he consulted both the Alfonsine tables and the Prutenic Tables. In the same book North on p. 79 states "Finally, for future reference, I note that Clavius claimed that for the Gregorian reform the mean of Copernicus' maximum and minimum values for the tropical year, namely 365d 5 h 49m 16.4s, the mean of 365d 5h 55m 37.7s and 365d 5h 42m 55.1s. I think it should be obvious that this is a method of finding a mean vernal equinox tropical year, and since the reader knows the Gregorian calendar was designed in the 16th century, the reader will understand the method of calculation will not be the same as today. Is this article really the place to go into the detailed history of the design of a calendar that is over 400 years old? Jc3s5h (talk) 16:46, 26 June 2014 (UTC)
We're using words differently. The four years defined by one of the equinoxes or solstices all have different lengths. The tropical year, or mean tropical year, is the average of those four, although it is not defined as such. The writers at that conference did not recognize the difference, and this is why what they wrote does not make it, as you say, obvious that this is a method of finding a mean vernal equinox tropical year , or that Copernicus was printing data with that in mind. Clavius and the others then did know that the measurements were not simply intervals of the vernal equinox. Copernicus had already decided that the tropical year should be measured by the sidereal year and precession, according to a source in the article. And I gave the wrong link last time. Now it's right.
This messyness is why we shouldn't make such definite statements about the design.
At any rate Clavius should not be taken at face value. He was under attack, and by his own account, they judged what the most accurate numbers were and almost perfectly matched it. The most accurate proposal made (eight of 33 years a leap one), known of long before the reform, was rejected. Because of the smaller cycle of 33 rather than 400 years, it doesn't wander as much from its own target.
The problem with the article is still that it says the tropical year, which always refers to the one that is the average of the equinox and solstice years. The one that ends with either 45s of 46s in a time citation. The one that's the subject of equations in the article. If you want to have that footnote, either change the text to a tropical year, or even better put the VE reference in the text. Saros136 (talk) 19:53, 7 July 2014 (UTC)

Introduction of changes by[edit]

I will make one final attempt to explain the nature of the errors being introduced by My next step will be dispute resolution.

First, this is the main article about the tropical year. It is an appropriate place to discuss historical definitions of the tropical year, or alternate definitions used for particular purposes. In other articles, such as "Leap year", it would be wrong to write as if any of these historical or alternate definitions are the main value of the leap year adopted by the scientific community. It was wrong to remove an alternate definition that is useful for some purposes, such as understanding why the design goal of the Gregorian calendar is slightly different than approximating the tropical year (as defined by 21st century astronomers).

Also, if one reads page L8 in the "Notes and References" section of the Astronomical Almanac for the Year 2011 it explains how the tropical year value printed in the almanac (on page C2) is calculated. Note that the heading for the table of year lengths on page C2 is "The lengths of the principal years at 2011.0 as derived from the Sun's mean motion are:". This makes it clear the interval is not between the equinox crossing (or some kind of "mean equinox crossing" however that might be defined) some time in 2011 and the next equinox crossing some time in 2012. No, the duration of the tropical year is calculated at 2011.0.

This calculation is found by using the mean orbital elements of the earth (specifically the mean geometric longitude λ) and calculating the angular velocity of the Earth, that is dλ/dt. Then the time to cover 360° is found by calculating 360°/(dλ/dt). So the length of the tropical year is the length of time it would take for the mean geometric longitude to increase 360° if the rate of change of that remained constant. For practical purposes it doesn't matter over a few years; the rate of change in both the 2001 and 2011 almanacs (p. C1) was 0.98564736 degrees per day. But we should not misstate the current definition just because our definition is "close enough" to the correct definition. Jc3s5h (talk) 15:02, 9 March 2015 (UTC)

I agree that we should emphasise the current scientific definition, but we should also retain the calculation on which the Gregorian calendar was based (as we do). Both values are important. I don't understand why removed the table with the separate calculations. Dbfirs 20:08, 9 March 2015 (UTC)
I have further examined the need to introduce a passage from Borkowski. I made up a spread sheet based on equation 3 in his paper, and found that to the 9 significant figures given in the Astronomical Almanac, Borkowski's equation gives the same length for a tropical year (365.242190 days of Terrestrial Time for any year between 2000 and 2003). Since the values agree, and the Astronomical Almanac is more prominent, and intended for a more general audience, than Borkowski's paper, I see no need to include the passage from Borkowski's paper. Jc3s5h (talk) 03:51, 10 March 2015 (UTC)
I supported your recent restoration, but I see that Woodstone doesn't like it. I assume his argument is that the difference was not perceived at the time the calendar was devised, which is a fair viewpoint, even though it was the vernal equinox that was being matched. I'm happy to leave things as they are, rather than revert, but I hope no-one starts removing further information. Dbfirs 17:51, 15 March 2015 (UTC)
Once again Jc3s5h emphasises we should be using official definitions:

The prestige of the United States and United Kingdom Nautical Almanac Offices is infinitely superior to an anonymous internet editor


The official terminology is "interval between mean equinoxes". (talk) 16:58, 10 April 2015 (UTC)

I'm unhappy with your continuing campaign to remove the calculations of Jean Meeus from all Wikipedia articles. I agree that it would be useful to know the exact details of the calculations, but one has been verified by your own method, and the others could be similarly verified. Wikipedia does not require that all sources provide full details of all calculations. Meeus cites the exact data used in his calculations, and the values are cited elsewhere. It is not fair to remove referenced data just because you don't like it. Dbfirs 14:27, 7 May 2015 (UTC)
I too would prefer to keep the Jean Meeus references. His calculations are cited in many publications and he is always careful to provide reliable references to the astronomical literature. AstroLynx (talk) 15:10, 7 May 2015 (UTC)
On Talk:Year I see that an average length of the vernal equinox year over two millennia 1000 - 3000 is 365.242 360 days. Meeus and Savoie gave a "mean time interval between two northward equinoxes for the year 2000" of 365.242 374 days. If they can't be bothered to explain how they get that figure why should we quote them? "Providing reliable references to the astronomical literature" is a joke: Wikipedia requires author, title and page number. If Meeus cites the numbers he actually crunched perhaps you could pass the information on so that we can see if he did the calculation right. (talk) 14:20, 8 May 2015 (UTC)
If you had taken the trouble to actually read the Meeus-Savoie paper, online here, you can see that the numbers were derived from the planetary theory VSOP87 of P. Bretagnon and G. Francou. The small difference with the value you cited from Talk:Year is probably not significant as the latter computation neglects higher order terms (T^3, T^4, etc.) in the polynomials representing the planetary motions. AstroLynx (talk) 14:53, 8 May 2015 (UTC)
That calculation of the average over 2000 years just verified the accuracy claimed by Jean Meeus for his year 2000 estimate. He didn't calculate his figure that way, of course, but used a refinement of the formula for mean tropical year, using ephemeris data from "Numerical Expressions for precession formulae and mean elements for the Moon and planets" - a paper by J L Simon, Pierre Bretagnon, J Chapront, M Chapront-Touze, G Francou and Jacques Laskar (whose formula you use). I agree that we would like to know exactly how he modified Laskar's formula for the separate equinoxes and solstices. If we can't find the detail, perhaps we should write to him, or perhaps Jacques Laskar can explain if you know him personally. See our article VSOP (planets) for the data used by Jean Meeus. Dbfirs 15:35, 8 May 2015 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────Meeus has published a few editions of a book, Astronomical Algorithms, and I have seen hints in internet searches this book includes information on year length. If someone has a nearby library with this book, perhaps you could take a look and see if it gives more details about his method. Jc3s5h (talk) 15:25, 8 May 2015 (UTC)

The formulae used for the four separate calculations are given in chapter 26 of Jean Meeus' "Astronomical Algorithms" (1991) according to one account, but I don't have access to a copy of the book. Dbfirs 16:25, 8 May 2015 (UTC)
The full details on how this is computed (again from VSOP87) is indeed found in Jean Meeus, Astronomical Algorithms: Second Edition (Richmond: Willmann-Bell, 1998), chapter 27 ("Equinoxes and Solstices"). See also Jean Meeus, More Mathematical Astronomy Morsels (Richmond: Willmann-Bell, 2002), chapter 63 ("The Gregorian Calendar and the Tropical Year") for even more detail. AstroLynx (talk) 16:39, 8 May 2015 (UTC)

(ec)Bouasse, cited by Meeus and Savoie, doesn't appear to know much. He thinks "the true tropical year is the time interval between two successive passages of the Sun through the vernal equinox" and "the mean tropical year is the mean of a large number of true tropical years". Yes, about 25,000 of them (using his definition) but does he know that? I see that VSOP 87 provides the mean tropical year. On page 41 Meeus and Savoie appear to have overlooked the fact that "the vernal point" is also subject to the equation of the centre. The claim that "it was the length of the tropical year as derived from" the Alphonsine Tables is wrong. The astronomers averaged three sets of tables, because they all gave different longitudes for the sun and there was no reason to trust one more than the others. [ - please sign your contribs correctly!]

On page 42 there is another blooper: "It should be noted that the tropical year is not equal to the (mean) time interval between two successive spring equinoxes". They then say that the equation of the centre affects the length of the mean tropical year. That's like saying the equation of the centre affects the length of the mean sidereal month, so that the sidereal month is not 27.32166 days at all, but variable depending whether you start from the ascending node, or the descending node, or the perigee, or the apogee, or some other point. [ - please sign your contribs correctly!]

So if the mean time interval between two successive equinoxes or solstices (same season) is found to be different from the mean tropical year we know they calculated wrong. We don't know how Meeus deduced the figures from VSOP 87, and until we do we can't use them. (talk) 17:33, 8 May 2015 (UTC)

All I can track down is [5]. They appear to be playing around with formulae for the elements of the planetary orbits. The only thing you can derive from that is the instant of any given equinox or solstice. Unless they can say which equinox or solstice they used we can't use their values. Presumably these are formulae which we have to put the arguments into to get a value, and they won't say which arguments they put in and what the result was. (talk) 17:58, 8 May 2015 (UTC)

This series of edits introduces numerous errors so I have reverted them. I'll mention a few of the errors:

"Since antiquity, astronomers have progressively refined the definition of the tropical year. Meeus and Savoie, 1992 p. 40 define it as the time required for the mean Sun's tropical longitude (longitudinal position along the ecliptic relative to its position at the vernal equinox) to increase by 360 degrees". Meeus and Savoie use numerous definitions, historical as well as current. In the lead it is essential to inform the reader of the current meaning, rather than obscuring it with names of authors.

"The mean tropical year (averaged over equinoxes and solstices) on 1 January 2000 was 365.242189 days according to the calculation of Laskar (1986), each day lasting 86,400 SI seconds." No, it is the time required for the mean ecliptic longitude of the Sun to increase by 360°.

"Values of mean time intervals between equinoxes and solstices (the mean tropical year) were provided by Meeus and Savoie (1992, p. 42) for the years 0 and 2000." The current meaning of mean tropical year (the word mean being optional) is the time required for the mean ecliptic longitude of the Sun to increase by 360°. The mean interval between a given equinox or solstice is different.

This was inserted into the lead:

Ancient tables provided the sun's mean longitude.[1][2] Christopher Clavius, the architect of the Gregorian calendar, noted that the tables agreed neither on the time when the sun passed through the vernal equinox nor on the length of the mean tropical year. Tycho Brahe also noticed discrepancies.[3][4] The Gregorian leap year rule (97 leap years in 400 years) was put forward by Petrus Pitatus of Verona in 1560. He noted that it is consistent with the tropical year of the Alfonsine tables and with the mean tropical year of Copernicus (De revolutionibus) and Reinhold (Prutenic tables). The three mean tropical years in Babylonian sexagesimals as the excess over 365 days (the way they would have been extracted from the tables of mean longitude) were 14,33,9,57 (Alphonsine), 14,33,11,12 (Copernicus) and 14,33,9,24 (Reinhold). All values are the same to two places (14:33) and this is also the mean length of the Gregorian year. Thus Pitatus' solution would have commended itself to the astronomers.

This is utterly unsuitable for the lead of the article. Indeed, it is unsuitable for this article; that sort of discussion, if it belongs in an encyclopedia at all, would belong in the "Gregorian calendar" article. But there are no citations, and I recall reading several different points of view about why the reformers adopted the value they did.

There is one change I see that, while not required, may be helpful to some readers: "The Gregorian calendar, as used for civil purposes, is an international standard. It is a solar calendar that is designed to maintain synchrony with the mean tropical year. I will add that change. Jc3s5h (talk) 23:37, 8 May 2015 (UTC)

The current meaning of mean tropical year (the word mean being optional) is the time required for the mean ecliptic longitude of the Sun to increase by 360 degrees. The mean interval between a given equinox or solstice is different.

How so? The interval between two mean occurrences of a given equinox or solstice is a direct function of the increase in the mean ecliptic longitude. (talk) 11:38, 9 May 2015 (UTC)

No, as the times of the equinoxes and the solstices also depend on the eccentricity and the line of apsides of the apparent solar orbit which slowly vary. Only over very long periods will the various tropical years (mean, vernal-equinoctial, summer-solstitial, autumn-equinoctial & winter-solstitial) be equal. AstroLynx (talk) 11:51, 9 May 2015 (UTC)
If you're adding a function which corrects for the equation of the centre you are applying a term to the mean value. So your mean value becomes actual. It is no longer mean. (talk) 12:03, 9 May 2015 (UTC)
To get the actual values of the equinoxes & solstices you also have to add the numerous planetary perturbations, the correction for the earth-moon barycentre and nutation. In the various equinoctial and solstitial tropical years discussed by Meeus these are neglected (i.e. "averaged-out"). AstroLynx (talk) 12:23, 9 May 2015 (UTC)
You're supposed to be a professor of mathematics but you're making a mistake that a secondary school mathematics student wouldn't make. As I understand it, you start with the mean then you concoct a suitable sine curve function to get the first approximation, then the second and so on. It's best to start with the correction that best matches the actual value then refine with successive functions. Some changes are believed to be very long periodic functions but it's not possible to separate the periodic from the secular. In the theory, correcting for the equation of the centre is done first, to get as quickly as possible very near to the actual value - these other corrections are just icing on the cake. The initial correction is the most significant adjustment to the mean - if you told a twelve - year old mathematics student that after doing that you still had a mean value she wouldn't believe you. (talk) 12:45, 9 May 2015 (UTC)
Your response tells me that you have no clue how these phenomena are routinely calculated. Perhaps you should first read the books of Jean Meeus and do some of the computations yourself before you criticize his methods. You are very quick in criticizing the works of others but what have you ever published on this topic?
I should furthermore correct you: although I do a lot of computing on a daily basis, I never claimed nor am I a professor of mathematics. AstroLynx (talk) 13:11, 9 May 2015 (UTC)
The word mean is meaningless (sorry) unless you specify over what values the average is being taken. Usually it is obvious, so there is little need to specify, but here we have various possibilities. In the case of the mean tropical year, the average is taken over perturbations in the orbit and is calculated for a particular point in time. There is no logical reason why one particular perturbation cannot be taken into account when calculating the mean time between vernal equinoxes on a specified date. The fact that the calculation agrees almost exactly with the mean over time (over both 2000 years and 200 years) suggests that the method has merit. Dbfirs 13:07, 9 May 2015 (UTC)
(Edit conflict). What AstroLynx said. "Mean" has many meanings in astronomy.'s reference to the equation of the center is apt; the mean anomaly is similar to mean longitude (although I'm still checking some subtleties like the distinction between the Earth and the Earth-Moon barycenter). Periodic terms such as the annual variation in angular speed due to the elliptical orbit have been removed; it is almost equivalent to treating the orbit as circular with the same period as the elliptical orbit.
We don't know the exact procedure Meeus and Savoie used, but they may very well have used the mean orbital elements; the orbit is treated as an ellipse. Periodic disturbances have been smoothed out, but the orbit is an ellipse and the changes in angular velocity due to the position of the Earth in its orbit will cause different lengths of the tropical year. Mean orbital elements are available in papers by J. Laskar ("Secular terms of classical planetary theories using the results of general theory", Astronomy and Astrophysics, v. 157 pp. 59-70, 1986) and J. L. Simon et al. ("Numerical expressions for precessional formulae and mean elements for the Moon and planets", Astronomy and Astrophysics, v. 282 pp. 663-683, 1984). Simon's results are used for the tropical year length printed in the Astronomical Almanac.
Another meaning for "mean" is the position of a celestial object (or plane) taking into account precession, but not nutation. So the true equinox would be the intersection of the true celestial equator of date and the true ecliptic of date. To have a mean equinox, one must be using a theory of planetary motion and Earth orientation that separates precession and nutation; the mean equinox is the intersection of the mean celestial equator of date and the mean ecliptic of date. For this meaning, the only difference is nutation; the Earth would have its true position in its orbit, affected by perturbations due to gravitational attraction by other planets. Such perturbations are smoothed out in the mean elements computed by Laskar or Simon et al, so the meanings of "mean" are different. Jc3s5h (talk) 13:23, 9 May 2015 (UTC) Added corrections & clarifications 20:21 UT.
The flaw in Jc3s5h's argument is that since the equinox precesses the "mean equinox" must be related to that movement. The technical term is "mean equinox of date". The "mean tropical year" is likewise the mean tropical year referred to a specific epoch (e.g. J2000). Astronomers specifically exclude terms relating to the equation of the centre in their definition of the mean tropical year. In the same way, effects due to obliquity and eccentricity are excluded from the definition of Greenwich Mean Time. (talk) 13:57, 9 May 2015 (UTC)
I apologize for not remembering that while the instantaneous equator is affected by nutation, the instantaneous ecliptic is not, so I should not have applied the terms "true" or "mean" to the ecliptic, and should have applied the term "of date" since I was referring to the instantaneous planes. I agree with what wrote at 13:57, 9 May 2015 (UTC). However, the mean vernal equinox tropical year does involve the equation of the center, and so is different from the mean tropical year. The mean vernal equinox tropical year is not normally used by modern astronomers (but Meeus and Savoie decided to discuss it their paper, and others, such as Richards ("Calendars" in Explanatory Supplement to the Astronomical Almanac, 2013, pp. 586) and McCarthy & Seidelman 2009 (full citation in article), p.18, have cited this paper.


  1. ^ See, for example,Tabule illustrissimi principis regis alfonsii, Prague 1401 -4 (Latin). A full set of Alphonsine Tables (including tables for mean motions, conjunctions of sun and moon, equation of time, spherical astronomy, longitudes and latitudes of cities, star tables, eclipse tables).
  2. ^ For an example of the information provided see Jacques Cassini, Tables astronomiques du soleil, de la lune, des planetes, des etoiles fixes, et des satellites de Jupiter et de Saturne, Paris 1740, available at [1] (go forward ten pages to Table III on p. 10).
  3. ^ Dreyer, J L E (2014). Tycho Brahe. Cambridge. p. 52. ISBN 978-1-108-06871-0. He remarks that both the Alphonsine and the Prutenic Tables are several hours wrong with regard to the time of the equinoxes and solstices. 
  4. ^ North, J (1989). The Universal frame: historical essays in astronomy, natural philosophy and scientific method. London. p. 29. ISBN 0-907628-95-8. "He noted on one occasion that the Alphonsine tables differed from the Prutenic by nineteen hours as to the time of the vernal equinox of 1588. 

What the mean tropical year is not[edit]

I reverted a change which indicated the current definition of the mean tropical year is the tropical year averaged over all points of the orbit. This is not how the calculation is done. I will post more details on how the calculation is actually done, but it will take a little time to prepare the details. Jc3s5h (talk) 11:00, 10 May 2015 (UTC)

Yes, I know that there is no averaging actually done in the formula, but the effect is to take the average over all points in the orbit (otherwise what does "mean" mean? -- it's not a time average, though it agrees surprisingly well with observed time averages). I'd certainly be interested to see the details, and also of the Meeus adjustments for the actual equinoxes and solstices if anyone can find them. Dbfirs 11:06, 10 May 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── The details of how to calculate the length of the mean tropical year (or just "tropical year" in the terminology of the Astronomical Almanac) is given on page L8 of the almanac for 2011:

The lengths of the principal years are computed using the rates of the orbital elements. Tropical year is 360°/[dλ/dt].

The text actually shows λ with a dot over it, but I don't know how to enter that in Wikipedia, so I substituted other notation for the derivative. The text refers to the 1994 paper by Simon et al., but other suitable expressions for λ (mean longitude) can be used. The passage containing the reverted edit cited Laskar (1986). That paper has an expression for λ on page 64. Richards in Urban and Seidelmann's Explanatory Supplement to the Astronomical Almanac 3rd ed. p. 586 provides the result of differentiating Laskar's expression and dividing the derivative into 360° (along with all necessary unit conversions and expressed to suitable precision):

365.2421896698 days − 0.00000615359 T − 7.29 × 10−10 T2 + 2.64 × 10−10 T3

...In these expressions, T is the number of Julian centuries (of 36525 days) measured from 2000 January 1 in Terrestrial Time (TT). It is given by T = (J - 245 1545.0 / 36525) where J is the Julian date

If we implement Laskar's expression for λ and plug in the dates and times copied by from the 1983 Astronomical Almanac (and convert from UT to TT) we get the following mean longitudes:

March equinox: 180.24°
June solstice: 270.23°
September equinox: 0.23°
December solstice: 90.23°

The results are about 180° different from what one might expect, because Laskar's expression is for the mean longitude of the Earth, while discussions of equinox and solstice often refer to the mean longitude of the Sun. Also, the quarter degree difference may be due to the Astronomical Almanac using a different expression for λ; the almanac was published in 1982 and Laskar published in 1985. Jc3s5h (talk) 12:47, 10 May 2015 (UTC)

Thanks for those details. I suppose it gets called the mean tropical year because it is derived from the theoretical "mean longitude" (which turns an ellipse into a circle). The true and mean longitudes are the same only at periapsis and apoapsis. In this sense, it is averaged over all points, but perhaps that's not the best way to describe it.
It's also interesting to compare the Laskar formula with that of Meeus & Savoie:

365.242189623 days − 0.0000061522 T − 6.09 × 10−10 T2 + 2.6525 × 10−10 T3

I suppose the exact formula depends on the exact values of the ephemerides used (VSOP82 for Laskar's formula and VSOP87 for Meeus & Savoie's).
The values for the equinox and solstice years were presumably calculated by the same method, but using the true longitude instead of the mean longitude. It would be useful to be able to confirm this. Dbfirs 15:07, 10 May 2015 (UTC)
Some sources describe the λ provided by sources such as Laskar or Simon et al. as the mean mean longitude. This is because term "mean longitude" has been used to mean the longitude for a circular orbit that has the same period of rotation as the elliptical orbit. But the sources we're discussing take the averaging process a step further, smoothing out any periodic variations with a period of less than around 10,000 years. Jc3s5h (talk) 15:18, 10 May 2015 (UTC)

Source updated[edit]

An edit today introduced this source:

  • P Kenneth Seidelmann, Explanatory Supplement to the Astronomical Almanac, Chapter 12, "Calendars" by L E Doggett, Washington 2006, ISBN 1-891389-45-9, p. 576, available at [6].

The source is a paperback reprint of a 1992 book; it is the second edition of the Explanatory Supplement. It was used to support the contention that the current definition of tropical year as the mean time between vernal equinoxes. The exact passage is

The tropical year is defined as the mean time 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 days − 0.00000615359 T − 7.29 × 10−10 T2 + 2.64 × 10−10 T3

This is the same expression given in the preceding section. Evaluating it for any date within a given year gives a negligible difference; the result is virtually the same for the vernal equinox, autumnal equinox, spring solstice, winter solstice, or any other date. For example when evaluated for the solstices and equinoxes in 2015, the maximum is for the vernal equinox (365.24218873 days) and the minimum is for the winter solstice (365.24218869), a difference of only 4 milliseconds.

The third edition of the same book, in the "Calendars" chapter by E. G. Richards, p. 586, (full citation in article "References" section) makes a different statement

The tropical year is today defined as the time needed for the Sun's mean longitude to increase by 360° (Danjon 1959; Meeus and Savoie 1992). This varies from year to year by several minutes, but it may be averaged over several years to give the mean tropical year. It may be noted that this definition differs from the traditional definition, which is the mean period between two vernal equinoxes.

The intervals between any particular pair of equinoxes or solstices are not equal to one another or to the tropical year; they are also subject to variations from year to year but may be averaged over a number of years. The arithmetic mean of the four average intervals based on the two equinoxes and the two solstices is equivalent to the value of the mean tropical year. These matters are discussed by Steel (2000).

The following approximate expression, based on the orbital elements of Laskar (1986), may be used to calculate the length of the mean tropical year in the distant past. Note, however, that The Astronomical Almanac has not used these equations, nor does it use the orbital elements from Laskar, but starting from the 2004 edition, it uses the orbital elements of Simon et al. (1994): [the same expression as above is given].

So whether Doggett was unaware that the definition of mean tropical year had changed, or went overboard in trying to make the text accessible to non-technical readers, the next edition of the same book took care to provide more precise wording and specifically reject the idea that the current definition of mean tropical year is the mean time between vernal equinoxes. Jc3s5h (talk) 17:50, 11 May 2015 (UTC)

The traditional definition is the mean period between two vernal equinoxes. A formula for calculating it (Laskar) is given. Laskar was 1986. The current method of finding the length of the mean tropical year uses the same formula but the orbital elements of Simon, 1994. So there is no change. This discussion is as sterile as the one about "natural born citizen" v "citizen at birth". The judges (the people who matter) have decided both phrases mean the same thing. Here, the people who matter (the astronomers) have decided that both phrases mean the same thing. End of story. (talk) 18:25, 11 May 2015 (UTC)
Virtually the same, yes, but the formula gives an exact correspondence. (talk) 18:44, 11 May 2015 (UTC)
What little change there is is caused by the the steady decrease of the length of the mean tropical year, given as 0.53 second per century by McCarthy and Seidelman (2009, p. 18, full source details in article). Since the vernal equinox and winter solstice are 276 days apart, you would expect a decrease of 4.0 milliseconds, which is just what you get from the formula. The formula does not give any special treatment to equinoxes or solstices; they're just dates like any other date. Jc3s5h (talk) 18:59, 11 May 2015 (UTC)
Astronomers are interested in facts. If they use the traditional definition, believing it to be the same as the modern one, then they need to be aware (as I'm sure some of them are) that the word "mean" is being used in a slightly different way. Wikipedia is written for the general reader, so an explanation of some of the subtleties would be appropriate. Dbfirs 09:27, 12 May 2015 (UTC)
Jc3s5h, you appear to have an agenda, as Dbfirs would put it. On 9 May you changed the direct quote "the time required for the mean sun's tropical longitude to increase" by replacing the word "tropical" with "mean". Then you removed the quotation marks because it was no longer the official definition without making it clear to readers what you had done. I think you have some explaining to do. (talk) 10:42, 12 May 2015 (UTC)
The statement before my edit was 'the time required for the mean Sun's tropical longitude (longitudinal position along the ecliptic relative to its position at the vernal equinox) to increase by 360 degrees (that is, to complete one full seasonal circuit)". ("Astronomical Almanac Online Glossary" 2015, s.v. year, tropical; Meeus & Savoie 1992, p. 40).'
The definition in the Astronomical Almanac Online Glossary is

year, tropical:

the period of time for the ecliptic longitude of the Sun to increase 360 degrees. Since the Sun's ecliptic longitude is measure with respect to the equinox, the tropical year comprises a complete cycle of seasons, and its length is approximated in the long term by the civil (Gregorian) calendar. The mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds.

Page 40 of Meeus and Savoie does not contain the exact wording between the double quote marks. Perhaps the best way to sum up Meeus and Savoie's paper is the first sentence: "When we look at different books on fundamental astronomy, we are surprised to find that the definition of the tropical year varies from one author to another." Jc3s5h (talk) 17:29, 12 May 2015 (UTC)

Note that elsewhere in the 1992 edition of the Explanatory Supplement to the Astronomical Almanac, online here and here referred to as ESAA2, the tropical year is defined as "the period of one complete revolution of the mean longitude of the sun with respect to the dynamical equinox" (p. 738; see also p. 80).

This disagrees with Doggett's definition (ESAA2, p. 576) but few astronomers were then aware of the fact that these definitions are not equivalent. The Meeus & Savoie paper only appeared in 1992 and before that I only know of one paper by J.J.M.A. de Kort, online here, who appears to have been aware of the distinction between the tropical year (in its 'mean' or modern definition) and the four seasonal variants.

By the way, the first edition of ESAA (1961) is online here. AstroLynx (talk) 14:50, 12 May 2015 (UTC)

Ha. So you deliberately suppressed (Jc3s5h, 17:29) the information that the mean sun's position is measured relative to the equinox. (talk) 18:11, 12 May 2015 (UTC)
The definition of the mean tropical year (aka the mean interval between vernal equinoxes) has not changed. There is a totally separate formula for the interval which Meeus is talking about. It is

365.242 374 8 + 10.34 10-5 T - 12.43 10-6 T2 - 22.63 10-7 T3 + 1.31 10-7 T 4

where T is a period of 365250 ephemeris days measured from J2000.0. This formula is valid from about 500BC to AD 4500. Provided Meeus' values agree with the formula we can use them, but we must make clear that this is not a mean and explain why - we have to tell the readers it does not take into account the fluctuations in the length of the year caused by various periodic gravitational perturbations and also tell them what it does take into account. (talk) 15:22, 26 May 2015 (UTC)

Support reversion of purported information from 1983 Astronomical Almanac[edit]

I support this reversion by User:AstroLynx of a paragraph supposedly supported by the Astronomical Almanac for the Year 1983. The cited page, C1, does not directly state any equonix or solstice dates. It does provide a formula for the mean longitude of the Sun, but that formula gives equinox and solstice times around two hours later than what is stated in the reverted paragraph times when the mean longitude of the Sun according to Newcomb's Tables of the Sun are multiples of 90°. Jc3s5h (talk) 19:04, 13 May 2015 (UTC) correction 15:16, 17 May 2015 (UT)

I support it too. Dbfirs 19:08, 13 May 2015 (UTC)
These times are technically Ephemeris Time, but the offset is only a few seconds. I put my trust more in the Astronomical Almanac than in anonymous internet editors (even though their real life identities are known) who can't be bothered to display their calculations. (talk) 16:24, 14 May 2015 (UTC)
Which times? I have the 1983 edition of The Astronomical Almanac on my desk and page C1 nowhere lists the dates and times which you cited for the equinoxes and solstices for 1983. Regarding your last remark, you are probably unfamiliar with the expression The pot calling the kettle black? AstroLynx (talk) 08:10, 15 May 2015 (UTC)
According to Jc3s5h's post "the formula" gives times of around 03:48 23 March, 11:15 22 June, 18:42 21 September and 02:10 22 December. In the penultimate section he says the sun is advanced 0.23 degree at the times the Astronomical Almanac locates the phenomena. Now the sun moves about one degree per day, so this implies an offset of about six hours giving times of around 0h 23 March, 7h 22 June, 15h 21 September and 22h 21 December. Is he working in Eastern Standard Time or does he

simply not know what he is talking about? (talk) 08:53, 15 May 2015 (UTC)

To which posting of Jc3s5h are you referring? I do not see these dates and times listed in his posting of 13 May. AstroLynx (talk) 09:48, 15 May 2015 (UTC)
12:47, 10 May 2015. (talk) 11:08, 15 May 2015 (UTC)
I see the 0.23 degree shift in this posting by Jc3s5h but not the dates and times you cite - did you compute this? AstroLynx (talk) 11:35, 15 May 2015 (UTC)
Jc3s5h computed the mean longitude of the sun at the time given in the article. He reported that he found this to be 0.24 degrees in advance of the longitude it would have at the equinox or solstice, i.e. his computed longitude was 0 degrees, 90 degrees, 180 degrees and 270 degrees plus this small increment. I then computed that since the sun moves about 1 degree per day he was saying that the equinoctial or solstitial times on his reckoning were approximately 24 x 24/100 hours earlier. (talk) 12:31, 15 May 2015 (UTC)
Both of you appear to have been misled by Laskar's 1986 formula for the mean longitude of the Sun. If you look up the original paper and read carefully you will see that the polynomial series (Table 5) are given relative to the epoch and equinox of J2000. To obtain the solar longitude relative to the mean equinox you have to add a similar expression for the precession - from 1983 to 2000 this correction is indeed about 0.23 degree. AstroLynx (talk) 13:23, 15 May 2015 (UTC)
Then surely the time I put in the article is right. (talk) 13:27, 15 May 2015 (UTC)
The times which you put in the article are correct but are, as they are not listed in the source which you cite, based on original research which is a no-no in WP. If you can provide a publication listing these times then it should be OK. AstroLynx (talk) 13:51, 15 May 2015 (UTC)
You keep putting into the article values that nobody can verify (the Meeus & Savoie figures). How do you justify that when you removed these figures which you yourself verified using the Astronomical Almanac which you have sitting on your desk? (talk) 14:36, 15 May 2015 (UTC)
The Meeus & Savoie figures are from a paper available on-line. They are also given in one or more books written by Jean Meeus, and are also quoted in several other publications. Dbfirs 14:48, 15 May 2015 (UTC)
Please give the URL. (talk) 15:02, 15 May 2015 (UTC)
See the ref list in the article. AstroLynx (talk) 15:22, 15 May 2015 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── This page gives a list of some papers that have cited Meeus and Savoie. ESAA3 (to use AstroLynx's convenient abbreviation) is not available online, but you can purchase it from your favorite bookseller. Jc3s5h (talk) 15:28, 15 May 2015 (UTC)

The paper refers to "the mean time interval between two successive March equinoxes". There is only one interval between two successive March equinoxes so it can't be a mean of anything. (talk) 15:33, 15 May 2015 (UTC)
There is only one interval between two particular successive March equinoxes, but I think Jean Meeus uses "mean" in the same sense that you (and the Astronomical Almanac) use the word to describe the tropical year. I agree that it would be useful to know exactly which perturbations have been averaged out. I think I can guess, but I'd like to see it spelt out for all versions of the tropical year, then the casual reader can understand the differences. Dbfirs 15:59, 16 May 2015 (UTC)
Then in that case the result of the calculation should be the value which the Astronomical Almanac gives, and which I agree. I think we are spending too much time on Meeus and Savoie - any other set of figures which could not be validated would have been removed without discussion long ago.
I don't think the reader wants to know about all versions of the tropical year - I've referenced the main ones in the article with pointers to sources if she wants to delve deeper. (talk) 10:04, 18 May 2015 (UTC)
On the contrary, I think most readers will be very interested in the various versions. Why do you wish to suppress them? Dbfirs 17:34, 18 May 2015 (UTC)
OK, if you want to provide complete coverage, I'll work on that basis. There are two tropical years involved, the mean and the actual. I've noticed that on more than one occasion you have altered a sourced definition of the tropical year to your own version and Jc3s5h has reverted you. These unsourced changes are original research - they are unverified and have no place in Wikipedia. Jc3s5h has explained in detail that unverified figures also cannot be included for the same reason. You put some in today - if you can justify your edit please do so before I or another editor takes them out. (talk) 17:46, 18 May 2015 (UTC)
I simply restored the sourced figures that you alone insist on deleting. I've no argument about the definitions. Dbfirs 21:06, 18 May 2015 (UTC)
What do you mean "sourced"? A nursery rhyme book may say that the moon is made of green cheese - we can't put that statement in citing "Mary's little book of nursery rhymes" because it's not been verified. (talk) 08:35, 19 May 2015 (UTC)
Confusing diversion? If so, then everything in Wikipedia about Clavius and the criteria applied in the production of the Gregorian calendar should go as well.
We mention the Gregorian calendar in the lead and we treat the subject in detail in the section "Calendar year". We're only giving half the story if we don't give the actual figures Clavius worked with when putting his calendar together. (talk) 14:13, 19 May 2015 (UTC)
The text that I removed didn't mention Clavius or any figures he might have had available. Dbfirs 14:35, 19 May 2015 (UTC)
No, because this is a general discussion of theory. Clavius' actions are discussed in the Gregorian calendar article. (talk) 15:03, 19 May 2015 (UTC)
What is a confusing diversion is the inclusion of values for the mean interval between equinoxes which bear no relation to reality. The Laskar value is backed up by pages of formulae and explanation - Meeus' values are backed by zilch. I know you are attached to the Meeus values, but there is no sane reason for this. In what other field of science would you accept values which have not been rigorously proved? The mean orbital period of every other body in the solar system is calculated after the equation of the centre has been smoothed - there is no reason for Earth to be the exception. Meeus has "deduced" his values. I could deduce from an analysis of soil brought back by Apollo astronauts that the moon is made of green cheese, but without supporting argument my claims would be inadmissible on Wikipedia. (talk) 09:42, 20 May 2015 (UTC)
As I wrote before, it would be useful to see the details of Jean Meeus's calculations, but they seem to be valid since they closely match reality. I test theories against observations, and I see a close match here. Other authors have cited Meeus, and have criticised astronomers who ignore reality in favour of a formula that gives only an average over all points of the orbit, or over many thousands of years. I don't doubt the validity of Laskar's formula, even though I haven't seen details of his derivation, but, as an average, it does rather too much smoothing for what I would like to see in the article. The updated version of Laskar's formula provides the astronomers' mean tropical year and this value should be, and is, given prominence in the article, but I see no reason why Meeus cannot take that formula and apply a variation to take into account the fact that Earth's obit is not circular. Jean Meeus is a respected astronomer, I trust him more than I trust anonymous editors of Wikipedia, and I object to your attempts to suppress his work. Dbfirs 21:42, 20 May 2015 (UTC)
As I wrote in the #Source updated section, Richards made sure to give a more careful definition of the tropical year than Doggett. In his 2000 book Marking Time (New York: Wiley) in the preface Duncan Steel criticizes Doggett's writing: "I will be highlighting various errors made over the years in calendrical matters, and not all of them occurred long ago. A recent egregious example is contained in the Explanatory Supplement to the Astronomical Almanac" and goes on to describe the problem discussed on this page. He continues "This is not merely inconsequential pendanticism; the Persian calendar may well be altered shortly based upon the erroneous belief of Iranian clerics that the definition given by the U.S. and U.K. government astronomers is correct." The distinction between the mean tropical year and the mean interval between vernal equinoxes is discussed in appendix B, pp. 380–381.
I think the different kinds of tropical year have received enough attention outside Wikipedia that the different kinds should be presented in this article.Jc3s5h (talk) 22:35, 20 May 2015 (UTC)
I think I can safely conclude from the above remarks that both you and Steel don't really know what you are talking about. There is zero chance of the Persian (you presumably mean Jalali) calendar being altered. The precise moment of the vernal equinox is a significant marker and has been so for thousands of years. This is Zoroastrian philosophy which remains deeply embedded in the Persian culture although the country has been nominally Islamic for 1400 years. (talk) 09:15, 21 May 2015 (UTC)
Nobody is suppressing anything. Meeus' work will be available as before. You say that his calculations "seem to be valid since they closely match reality". What was the experiment you set up, the results of which enabled you to make that statement, and what were the values you obtained which you matched against Meeus' in the course of your research? As for ignoring reality how are the times which Dbfirs has been removing unreal? The tables of the sun do "apply a variation to take into account the fact that Earth's orbit is not circular". We have adequate theory - why is it necessary to mess with it in unspecified ways? — Preceding unsigned comment added by (talk) 12:42, 21 May 2015 (UTC)
I've just had a look at Steel's comments and the errors are glaring. He says

The actual human activities of a year, at least in terms of agriculture and other matters of import to ancient peoples, began in the spring after the winter hiatus. Thus the spring, or vernal, equinox is the appropriate marker point to use in defining the length of the year.

I have observed that here trees may blossom as early as January and as late as April. What has that to do with the vernal equinox? The year counted from the vernal equinox doesn't need a special name. To give it one only confuses people south of the Equator whose spring occurs in September. — Preceding unsigned comment added by (talk) 12:58, 21 May 2015 (UTC)

Steel gives a value "right now" claimed to be accurate to 1/10 second. When is "right now" and where are his calculations? He gives values of 1/10 second, 20 seconds and half a day and then says

It is necessary to know the length of the year with precision of that order (about ten seconds).

As Astrolynx would say, "the cited number(s) come(s) from nowhere". Then we get to the nub of it:

Nevertheless, it is possible to calculate mean (averaged over many orbits) values for these different "years". One can quote values to six decimal places, although in reality only at best the fifth is meaningful. In the present epoch (close to the year 2000) these values are ...

So the key information which is missing is

  • how many orbits were averaged and
  • what is the epoch of the averaging procedure.

Steel then boldly claims that it is wrong to define the tropical year as the mean interval between vernal equinoxes. That is the definition. To say that is wrong is the same as saying that the definition of the second as 9 192 631 770 cycles is also wrong. Then comes the gobbledegook:

Two sets of averaging have been applied to arrive at the result, corresponding to the two uses of "mean" in another description of the tropical year elsewhere in the Explanatory Supplement. The tropical year was defined [in 1900] as the interval during which the Sun's mean longitude, referred to the mean equinox of date, increased by 360 degrees.

Steel makes unjustified criticisms of the procedure used to determine when the calendar will need correction as a result of confusing ephemeris time with universal time. He correctly states that the aim of the Gregorian calendar is to keep the vernal equinox steady, goes on to say that the figures indicate an error of one day in eight thousand years and then spoils it by claiming that figure is wrong. In fact, it's right. Here's the analysis: — Preceding unsigned comment added by (talk) 13:35, 21 May 2015 (UTC)

Because of tracking errors the frequency of centennial leap years will have to be reduced if the dates of the equinoxes and solstices are to be maintained. Tidal friction causes a progressive increase in the length of the day, the retardation in clock time compared to about 1820 being known as delta T. — Preceding unsigned comment added by (talk) 13:45, 21 May 2015 (UTC)

If the calendar is left unaltered the dates of the equinoxes and solstices will continue to move backwards as they have done since it was first introduced in 1582. The calendar could be reconfigured so that the mean vernal equinox never falls later than 1 PM (GMT) on 19 March. The significance of this is that the astronomical equinox in turn falls no later than noon GMT on 21 March. This prevents Easter Sunday falling on the same day as the astronomical equinox anywhere in the world. — Preceding unsigned comment added by (talk)

The trigger for the introduction of the Revised Gregorian calendar would be when the mean vernal equinox in a year giving remainder three on division by 400 was calculated to fall for the first time earlier than 1 PM (GMT) on 18 March. The preceding leap year would be cancelled. Thereafter all centennial years would normally be common, until the third year following was calculated to have a mean vernal equinox later than 1 PM (GMT) on 19 March, in which case the preceding leap year would be reinstated. — Preceding unsigned comment added by (talk)

Extrapolating delta T forward, based on the average rate of increase over the past 27 centuries, the tipping point will be reached in 8403, when the mean vernal equinox is calculated to fall at 3 AM (GMT) on 18 March, conveniently very close to the year (AD 8599) when the Easter table in the Book of Common Prayer of the Church of England expires. AD 8400 would be common, with the next two centennial leap years in AD 8800 and AD 9700.

These dates are only provisional, since the future rate of increase of delta T cannot be predicted with complete certainty. Looking further ahead, when the mean solar year drops below 365.24 days the minimum four - year interval between leap years will have to be extended. — Preceding unsigned comment added by (talk) 15:12, 21 May 2015 (UTC)

Looking ahead millions of years, if there are still people around then, when the solar year falls below 365 days, August would lose a day. Below 364 days, December would lose a day. Below 363 days, January would lose a day. Below 362 days, August would lose another day. Below 361 days, December would lose another day. Below 360 days, June would lose a day. Below 359 days, April would lose a day. Below 358 days, September would lose a day. Below 357 days, November would lose a day. Below 356 days, January would lose another day, thus restoring the lengths of the months to those of the Roman Republican calendar, which was replaced by the Julian, itself replaced by the Gregorian. (talk) 15:21, 21 May 2015 (UTC)

Standard method of verification[edit]

It is standard in academic life to identify sources in one of two ways depending on how the information is received:

Personal (or private) correspondence

Relates to writing received from the informant (e.g. a personal letter)

Personal communication

Denotes that the information was transmitted by some other means. (talk) 12:32, 16 May 2015 (UTC)

In academic papers you cannot refer to anonymous sources like you are now trying to do on WP. Anyway, I share the general opinion of other editors that this is original research and does not belong here. AstroLynx (talk) 13:13, 16 May 2015 (UTC)
What you're saying is that if someone does some experiments and somebody writes that up in a journal Wikipedia cannot quote that journal. That is totally and utterly false. You are not an anonymous source. Your identity is known.
A request to an administrator is not a public declaration. Revert this again and you will be reported for attempting to reveal the identity of an editor. I notice that since March 2015 (only two months ago) you have been blocked three times. Do you really want to risk a fourth block? You do realize that each new block lasts longer than the sum of all previous blocks? You will not be warned a third time. AstroLynx (talk) 07:35, 19 May 2015 (UTC)

Plugging that information into Google leads directly to your real name and all your personal information. (talk) 14:13, 16 May 2015 (UTC)

Be careful about revealing an editor's identity - this is called harassment and can lead to an immediate block. AstroLynx (talk) 14:24, 16 May 2015 (UTC)
On the contrary, there are many cases where an editor complained of being "outed" and it was discovered she had provided the information herself. (talk) 15:04, 16 May 2015 (UTC)
You mention "the pot calling the kettle black". That's rich coming from someone who uses the facilities of publicly - funded institutions to propagate his false claims. I suppose your philosophy is that you can get away with misbehaviour if no one knows who you are. See Wikipedia:Reference desk/Science#Journal publication and peer - reviewer anonymity. (talk) 12:45, 17 May 2015 (UTC)
As you can read yourself it is very uncommon (and certainly not standard practice) for scientific papers to be published anonymously. Anyway, even if I would give you permission to cite my verification of your "original research" you still would not be able to refer to a published source. And you cannot refer to this TalkPage as Wikipedia is not a reliable source. AstroLynx (talk) 08:31, 18 May 2015 (UTC)
Wrong again. Anything you say on Wikipedia can be edited and re - used by anyone at will. (talk) 11:43, 18 May 2015 (UTC)
Exactly, so what is your point? AstroLynx (talk) 07:35, 19 May 2015 (UTC)
How ridiculous can you get? What is posted on Wikipedia is not like a card posted in the window of a corner shop. It's directly accessible (no password required) from every computer, tablet and smartphone on the planet. And on your last comment, my point is that you don't have the faintest idea what you are talking about. (talk) 08:09, 19 May 2015 (UTC)
Developing the theme of my last sentence, blocks are not punishment. Their purpose is to prevent disruption. If you get too many points on your driving licence you are disqualified (blocked in wikipedia parlance). If you rape someone you may go to jail for seven years. If when you come out you steal a flowerpot from someone's garden you don't go back to jail (get blocked) for a minimum of seven years - you might get one month if the judge was being particularly harsh. (talk) 08:15, 19 May 2015 (UTC)
Wikipedia has something called WP:IAR designed to prevent the transmission of knowledge being impeded by Luddites. Jimbo has many times explained how sourcing works. See his post of 05:24 on 27 September 2012 at On another occasion, a celebrity told him her name had been misspelt in Wikipedia and that was enough to get it corrected. (talk) 08:55, 19 May 2015 (UTC)
For the avoidance of doubt, what you write in Wikipedia can be used by me to make the point that the material has been peer - reviewed by a Ph. D. in Astronomy. That flows not only from the fact that you signed away the copyright in your words but also from the fact that copyright does not prevent someone from reporting, in their own words, what somebody else said. (talk) 15:12, 19 May 2015 (UTC)

Meeus's calculation of equinox and solstice tropical years[edit]

Some other editors asked about how Jean Meeus calculated the mean time intervals between March equinoxes, June solstices, September equinoxes and December solstices that appear in Meeus & Savoie's 1992 paper on page 42.

I obtained a copy of the Meeus book Astronomical Algorithms, 2nd ed. (1998, corrected printing of August 10, 2009). As the author says in the "Introduction" (p. 1) "This book is not a general textbook on astronomy. The reader will find no theoretical derivations."

Chapter 27 (pp. 177–182) is devoted to calculating equinoxes and solstices. In general, and in Meeus, an equnox or solstice is when the apparent longitude of the Sun is a multiple of 90 degrees (apparent means geocentric, referred to the true equinox and ecliptic of date).

One suggested procedure is to begin by calculating "the instant of the 'mean' equnox or solstice, using the relevant expression in Table 27", then applying corrections provided in the chapter. This brings up the question of what "mean" means. One possible meaning is the instant when the geometric longitude of the Sun, referred to the mean equinox and ecliptic of date, and where mean equinox means the equinox direction that results from the intersection of the ecliptic with the mean celestial equator. But evaluating the expression for 2001 and comparing it to the Astronomical Almanac for the year 2001 gives a difference of about 13 minutes 103 seconds, and greater amounts for some other cases shown below, so it is unlikely this meaning is intended.

Looking at the expressions below, they are evidently the result fitting a polynomial to a list of Julian dates of solstices and equinoxes for the listed year ranges; they contain secular, not periodic, terms. In the expressions, JDE stands for Julian Ephemeris Day, in other words, Julian day using the Terrestrial Time timescale.

Expressions for "mean" equinox or solstice

Years -1000 to +1000

Y = year/1000

March equinox

JDE = 1721139.29189 + 365242.1374 Y + 0.06134 Y2 + 0.00111 Y3 - 0.00071 Y4

June solstice

JDE = 1721233.25401 + 365241.72562 Y - 0.05323 Y2 + 0.00907 Y3 + 0.00025 Y4

September equinox

JDE = 1721325.70455 + 365242.49558 Y - 0.11677 Y2 - 0.00297 Y3 + 0.00074 Y4

December equinox

JDE = 1721414.39987 + 365242.88257 Y - 0.00769 Y2 - 0.00933 Y3 - 0.00006 Y4

Years +1000 to +3000

Y = (year - 2000)/1000

March equinox

JDE = 2451623.80984 + 365242.37404 Y + 0.05169 Y2 - 0.00411 Y3 - 0.00057 Y4

June solstice

JDE = 2451716.56767 + 365241.62603 Y + 0.00325 Y2 + 0.00888 Y3 - 0.0003 4

September equinox

JDE = 2451810.21715 + 365242.01767 Y - 0.11575 Y2 + 0.00337 Y3 + 0.00078 Y4

December solstice

JDE = 2451900.05952 + 365242.74049 Y - 0.06223 Y2 - 0.00823 Y3 + 0.00032 Y4

Evaluating these expressions for the years -1, 0, 1, 1999, 2000, and 2001, then subtracting the JDEs of adjacent years to find the year length, agrees with the values given in Meeus and Savoie (1992, p. 42).

Jc3s5h (talk) 15:43, 27 May 2015 (UTC), corrected 14:45, 28 May 2015 (UT).

If you apply the formula to three successive years, you will get three successive JDEs. Is there any difference between the lengths (JDE2 - JDE1) and (JDE3 - JDE2)? I would have thought that the parameters change so slowly that there would be no discernible difference at all. (talk) 18:07, 27 May 2015 (UTC)
True. When I evaluated them for -1, 0, 1, 1999, 2000, and 2001, the difference of the earlier year pair was the same as the later year pair, to 5 decimal places, which is the accuracy to which the coefficients are given in the book. Jc3s5h (talk) 18:19, 27 May 2015 (UTC)
You did a calculation before and you got a completely misleading answer because you didn't take account of precession. Thirteen minutes is not a lot when you consider that the mean equinox is nearly two days' travel along the ecliptic from the true equinox. Can you please provide the full calculation by which you ascertained the mean vernal equinox moment using the 2001 Astronomical Almanac alongside the full calculation using the Table 27 formula in Meeus so that we can see how the 13 - minute discrepancy arose. (talk) 11:19, 28 May 2015 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────The calculation of what Meeus describes as "the instant of the 'mean' equinox" for March 2001 using table 27.B is simply a matter of substituting 2001 into the formulas given above and yields JDE = 2451989.052214. Using the US Naval Observatory's Multi-Year Computer Almanac this converts to 20 Mar 2001 13:15:11.3. The Astronomical Almanac for the Year 2001" p. C6 gives the following data (showing only the relevant columns)

Date Julian Date Ecliptic Long. for Mean Equinox of Date
Mar. 20 2451988.5 359°27′00.34″
Mar. 21 2451989.5 0°26′37.55″

Linear interpolation produces a time of JDE 2451989.05341 which is 13:16:55 Mar. 20. In my previous interpolation of the table in the Astronomical Almanac I made an error in converting degrees, minutes, and seconds to decimal degrees.

Meeus on p. 179 gives lists of errors for the true equinox and solstice times computed using his tables versus more accurate methods for the years 1951–2050. That would be 400 instants; 395 instants have errors less than 40 s, and the largest error is 51 s.

Comparing the passage of the Sun through geometric longitudes of 0°, 90°, 180°, and 270° in issues of the Astronomical Almanac that are at hand vs. the values from Meeus's expressions as above gives the following differences (Table 27.B minus almanac, seconds):

Year Mar. Jun. Sept. Dec.
2001 -103.24 134.08 669.26 455.37
2003 408.12 250.05 258.32 162.14
2011 293.34 503.10 121.70 79.83

The magnitude of these differences seem too great for Meeus's table 27.A and 27.B to be intended as an approximation to the geometric longitude, mean equinox and ecliptic of date. Jc3s5h (talk) 16:17, 28 May 2015 (UTC)