Talk:Tropical year/Archive 3

What the mean tropical year is not
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.    D b f i r s   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): "365.242 189 6698 days &minus; 0.000  006  153  59 T &minus; 7.29 × 10&minus;10 T2 + 2.64 × 10&minus;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 156.61.250.250 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:

"365.242 189 623 days &minus; 0.000  006  152  2 T &minus; 6.09 × 10&minus;10 T2 + 2.6525 × 10&minus;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.    D b f i r s   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)

Standard method of verification
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. 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (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 Reference desk/Science. 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (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 Talk:Will.i.am.   On another occasion, a celebrity told him her name had been misspelt in Wikipedia and that was enough to get it corrected. 156.61.250.250 (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. 156.61.250.250 (talk) 15:12, 19 May 2015 (UTC)

Support reversion of purported information from 1983 Astronomical Almanac
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.   D b f i r s   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. 156.61.250.250 (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? 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (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? 156.61.250.250 (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.    D b f i r s   14:48, 15 May 2015 (UTC)


 * Please give the URL. 156.61.250.250 (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. 156.61.250.250 (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.    D b f i r s   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. 156.61.250.250 (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?    D b f i r s   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. 156.61.250.250 (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.    D b f i r s   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. 156.61.250.250 (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. 156.61.250.250 (talk) 14:13, 19 May 2015 (UTC)
 * The text that I removed didn't mention Clavius or any figures he might have had available.   D b f i r s   14:35, 19 May 2015 (UTC)
 * No, because this is a general discussion of theory.  Clavius' actions are discussed in the Gregorian calendar article. 156.61.250.250 (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. 156.61.250.250 (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.    D b f i r s   21:42, 20 May 2015 (UTC)


 * As I wrote in the 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. 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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 156.61.250.250 (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. 156.61.250.250 (talk) 15:21, 21 May 2015 (UTC)

Source updated
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.

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:" "365.242 189 6698 days &minus; 0.000  006  153  59 T &minus; 7.29 × 10&minus;10 T2 + 2.64 × 10&minus;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. 156.61.250.250 (talk) 18:25, 11 May 2015 (UTC)


 * Virtually the same, yes, but the formula gives an exact correspondence. 156.61.250.250 (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.    D b f i r s   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. 156.61.250.250 (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. 156.61.250.250 (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. 156.61.250.250 (talk) 15:22, 26 May 2015 (UTC)

Meeus's calculation of equinox and solstice tropical years
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 = $1,721,139.292$ + $365,242.137$ Y + $0.061$ Y2 + $0.001$ Y3 - $0.001$ Y4

June solstice

JDE = $1,721,233.254$ + $365,241.726$ Y - $0.053$ Y2 + $0.009$ Y3 + $0$ Y4

September equinox

JDE = $1,721,325.705$ + $365,242.496$ Y - $0.117$ Y2 - $0.003$ Y3 + $0.001$ Y4

December equinox

JDE = $1,721,414.4$ + $365,242.883$ Y - $0.008$ Y2 - $0.009$ Y3 - $0$ Y4

Years +1000 to +3000

Y = (year - 2000)/1000

March equinox

JDE = $2,451,623.81$ + $365,242.374$ Y + $0.052$ Y2 - $0.004$ Y3 - $0.001$ Y4

June solstice

JDE = $2,451,716.568$ + $365,241.626$ Y + $0.003$ Y2 + $0.009$ Y3 - $0$ 4

September equinox

JDE = $2,451,810.217$ + $365,242.018$ Y - $0.116$ Y2 + $0.003$ Y3 + $0.001$ Y4

December solstice

JDE = $2,451,900.06$ + $365,242.74$ Y - $0.062$ Y2 - $0.008$ Y3 + $0$ 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. 156.61.250.250 (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. 156.61.250.250 (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)

Linear interpolation produces a time of JDE $2,451,989.053$ 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):

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)

Precision
All measured or evaluated quantities should be stated in accordance with MOS:UNCERTAINTY. In particular, if a source gives a number to a certain precision, such as 365 solar days, 5 hours, 48 minutes, 45 seconds, and an editor decides to convert it to a different unit of measure, such as days with a decimal point, the precision of the converted quantity should be as close as possible to the original quantity. In the example, the time is expressed to the nearest second, which is 0.00001157 days, so five digits should be retained after the decimal point and the converted quantity should be expressed as 365.24219 days.

In light of this I have reverted this edit.

Of course this line of reasoning would not apply to quantities that are known exactly, such as the mean length of a Gregorian year expressed in calendar days. Jc3s5h (talk) 14:05, 17 February 2016 (UTC)

Ambiguity
In the formula containing $$\sin \omega T$$, is this meant to be parsed as $$\sin (\omega T)$$ or $$(\sin \omega) T$$? I'm guessing the former on the ground that if it were the latter the constant omega could be eliminated merely by simplifying the formula.

Apropos of that, if the formula without sin is good for plus or minus ten millennia, does the improvement make a significant difference over that range, or only over a wider range? And if the latter, what is that wider range? A hundred millennia? More? Vaughan Pratt (talk) 01:01, 1 March 2016 (UTC)


 * The formula which Vaughan Pratt questions was introduced in May 2015 in this edit. Since the passage lacks a citation to a reliable source as required by WP:V, I have removed it. Jc3s5h (talk) 01:29, 1 March 2016 (UTC)

Length of tropical year section
There is a problem with this as currently written. The first table gives example intervals between successive passages of the apparent sun across the celestial equator travelling north (the "vernal equinox year"). The average of them is 365d 5h 49m 07.8s. As expected, these correlate with the data in the second table (for example, the "vernal equinox year" quoted for AD 2000 is 365d 5h 49m 0.80s, converted to h/m/s from the decimals of a day given). However, elsewhere in the article the mean value is stated to be 365d 5h 48m 45s (or its decimal equivalent). The values in the second table are stated to be mean values, so either the calculation of them is wrong or they are misdescribed. 77.98.244.158 (talk) 15:13, 31 March 2016 (UTC)


 * I've double checked the values from the Meeus, J. & Savoie, D. (1992) paper, and they have been correctly described. You will have to give a more detailed explanation of what you consider is incorrect, and this time use decimals. Explicitly state which values you consider to be misdescribed. That is, what they are in the Wikipedia article compared to what they are in the reference. David.moreno72 (talk) 15:51, 31 March 2016 (UTC)


 * On page 42 of the paper the authors say

"It should be noted that the tropical year is not equal to the (mean) time interval between two successive spring equinoxes."

That's a fair enough comment. The earth is pulled by lunar, solar and planetary gravity so its progress is going to be inconsistent. They go on to say

"In present - day astronomy the tropical year is defined as the time interval needed for the mean tropical longitude of the Sun to increase by 360 degrees."

This is the astronomers' "mean tropical year", currently 365d 5h 48m 45s, which in decimals is 365.242190 days. In the past 2,000 years they say it has decreased from 365.24231 to 365.24219 days at AD 2000. They continue

"The (mean) time interval between two successive spring equinoxes is not, rigorously speaking, equal to the tropical year ..."

The problem here is that the time interval between two successive spring equinoxes has already been equated to the tropical year, and the mean time interval between two successive spring equinoxes has already been equated to the mean tropical year. So any calculation which gives a value for the mean time interval between two successive spring equinoxes other than 365.242190 days is by definition wrong. 77.98.244.158 (talk) 17:44, 31 March 2016 (UTC)


 * The astronomer's mean tropical year (currently about 365.24219 ephemeris days) is a theoretical calculation which is not equal to the mean time between equinoxes or solstices unless you average over all points of the orbit, or average over many thousands of years, by which time the mean tropical year will have changed as you correctly observe. Currently, the mean time between vernal equinoxes has been shown to be about 365.24237 ephemeris days (epoch 2000) by Meeus, J. & Savoie,and their theoretical calculation can be verified by averaging over a couple of millennia.  The fact that different values have been wrongly equated does not change the numbers.    D b f i r s   18:14, 31 March 2016 (UTC)


 * This definition seems to be something you have made up.  You claim that if you average over "many thousands of years" you will get one value, but if you average over "a couple of millennia" you will get another.   You can of course slice the sine curve any way you wish, but you, Meeus, Savoie, Laskar or someone else may slice it in many different ways.   This is hardly the way to arrive at a scientifically defined mean.   To show the inadequacy of this approach, your argument is meaningless because to the British the phrase "a couple" means two, but to an American it simply means "several".
 * There is an internationally agreed definition which is used in the Astronomical Almanac.  The 2015 edition, page C3, says

... L is the mean longitude of the Sun, corrected for aberration, given by:


 * L = 279.344 [degrees] + 0.985 647 d

and where d is the interval in days from 2015 January 0 at 0h UT, given by:


 * d = day of year (from B4 - B5) + fraction of day from 0h UT.

Now take a look at the 2016 edition. The information is the same except that "2016" replaces "2015" and the value of L is 279.105. The mean vernal equinox is the instant when the longitude reaches 0 degrees. The "time interval between successive vernal equinoxes" is the gap between the equinox moment in 2015 and the equinox moment in 2016. Calculate them and let me know what that value is. 77.98.244.158 (talk) 10:42, 1 April 2016 (UTC)


 * If you average over two thousand years centred on 2000, you get the value calculated by Meeus and Savoie from a modification of Laskar's formula. If you average over 23,000 years or so, you will, in theory, get the Laskar value, though that is slowly changing, so the averaging method is only theoretical.  We've had this argument before when you were using your London IP address.    D b f i r s   10:10, 2 April 2016 (UTC)


 * You don't do any averaging.  You get the longitude directly from the table.   Did you do the calculation? 77.98.244.158 (talk) 16:56, 2 April 2016 (UTC)


 * Yes, I know about the mean tropical year calculation, but that's not what I was referring to because it's a theoretical value. You got the average wrong when you did it under a different IP.    D b f i r s   18:57, 4 April 2016 (UTC)


 * The mean synodic month is a "theoretical value" but that didn't stop Rambam making it the basis of the Hebrew calendar.  What's your argument? 77.98.244.158 (talk) 19:29, 4 April 2016 (UTC)


 * There's nothing wrong with a theoretical value, but one needs to consider how it relates to reality. Sorry, I shouldn't be arguing with you at all.  I'll stop now.  We've been over all this before.    D b f i r s   19:35, 4 April 2016 (UTC)


 * I'm glad you've seen the light.  Let me just remind you that the tables in the Astronomical Almanac are constructed on the same principles as the ones utilised by Clavius and his astronomers in devising the calendar.   You can't get more related to reality than that. 77.98.244.158 (talk) 20:18, 4 April 2016 (UTC)
 * I've seen the light about who you really are. I've no argument with the tables in the Astronomical Almanac, just with what they represent.  Clavius had numerous calculations available.  Why do you dislike Jean Meeus so much?  Why don't you publish your own paper to refute his work?  Cites from "willbell.com: "In the field of celestial calculations, Jean Meeus has enjoyed wide acclaim and  respect ..." and "Jean Meeus explores the frequency of blue moons, planetary groupings, and a  great deal more, as only this master of astronomical calculations could."   D b f i r s   20:25, 4 April 2016 (UTC)
 * Come again?

I've no argument with the tables in the Astronomical Almanac, just with what they represent.

What on earth is that supposed to mean? 77.98.244.158 (talk) 20:35, 4 April 2016 (UTC)


 * Exactly, mean.   D b f i r s   20:38, 4 April 2016 (UTC)


 * Are you so out of touch that you have to start talking in riddles? 77.98.244.158 (talk) 20:44, 4 April 2016 (UTC)


 * Don't you remember our previous discussion when you were editing from a selection of London IP addresses?   D b f i r s   20:47, 4 April 2016 (UTC)

"Length of tropical year" section
The section includes the sentences: " Using the oversimplified definition, there was an equinox on March 20, 2009, 11:44:43.6 TT. The 2010 March equinox was March 20, 17:33:18.1 TT, which gives a duration of&#8239; . " [Added reddish space to mark where the sentence become incomplete.] In the article's edit history, it looks like there was some kind of edit war, which I do not feel like jumping into. Someone should correct this sentence. The correct value that should be in this place is " 365 d 5 h 48 m 34.5 s ". — al-Shimoni  (talk) 19:51, 6 August 2016 (UTC)


 * Corrected, along with removal of other changes by sock of banned user. Jc3s5h (talk) 20:27, 6 August 2016 (UTC)

--

Which year?
The Gregorian average year-length is not designed to approximate mean tropical year.

The Gregorian Calendar's leapyear-system was designed to keep the calendar as stable as possible with respect to the March equinox. ...so that the March equinox's calendar-date would vary as little as possible.

Twice, the article says that the Gregorian Leapyear system was designed so that the calendar's mean year would approximate the mean tropical year. That's incorrect.

Michael Ossipoff--MichaelOssipoff (talk) 02:38, 11 January 2017 (UTC)


 * I agree that the aim of the reform was to maintain the position of the March equinox in the calendar. What is not clear is whether the reformers were aware that the vernal equinox year and the mean tropical year are not currently the same.  There are editors (one of them banned for sockpuppetry) who refuse to accept the concept of a mean time between vernal equinoxes.  It would be useful if we could find more sources (other than Jean Meeus) that discuss this difference.    D b f i r s   06:49, 11 January 2017 (UTC)


 * If you read the cited reference (page 123 in the Gregorian Reform of the Calendar: Proceedings of the Vatican Conference to Commemorate its 400th Anniversary) it is apparent that the 16th century decision makers were considering the mean tropical year (I believe that at that time they would not have had the computational or theoretical wherewithal to define it as we do today, so would have to rely on observations over many years, or preferably, centuries). The extent to which they could distinguish this from the mean vernal equinox tropical year is not clear. The way I interpret this, until presented with even better sources, is that the overall goal was to follow the established religious convention that the calculation of the date of Easter called for the vernal equinox to fall as nearly as possible on the calendar date March 21, and the means to this end was to make the mean length of the Gregorian calendar year very close to the length of the mean tropical year. Jc3s5h (talk) 10:35, 11 January 2017 (UTC)

Change in citation style
With this edit ShelleyAdams changed the citation style, from APA style to Citation Style 1, and continued the use of parenthetical referencing in a different form. WP:CITEVAR calls for such changes to be done after obtaining consensus on the talk page. Since consensus was not obtained in advance, do the editors active on this page wish to ratify the change? Jc3s5h (talk) 15:13, 6 November 2017 (UTC)

Change of length
Last year I added, after the cubic equation for the length of a tropical year, the following: The following expression is a slightly better approximation for the time period (maximum error about 1 second); it is easily verified that it has the same first three derivatives, and extrapolates better since it is not a cubic function:

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

with ω=0.0160440243. This expression has a long-term average 0.49 seconds less than the value in 2000. Ten thousand years ago the tropical year was 34 seconds longer than now, and ten thousand years from now it will be 33 seconds shorter, but then it will start to get longer again.

deleted this saying that it appears to be "original research". I disagree. It is clear that the value and first and second derivatives are the same as the cubic formula given in our article, and it's easy to show that the third derivative is also correct. It doesn't take "research" to determine that. The information about the maximum error is from a simple comparison with the data in the reference. The information about the behavior over the period is also derived from the reference, and simple calculation.

Eric Kvaalen (talk) 12:23, 19 February 2018 (UTC)


 * Curve fitting is original research. Exceptions are described within that policy at "Routine calculations". In addition, neither the Wikipedia article nor the paper cited give the detailed results of the 10,000 year numerical integration that the cubic expression is fit to. For you to make any statement about how well your expression fits, or whether it extrapolates better than the cubic expression, you would have to reproduce the 10,000 year numerical integration, which is far beyond a "routine calculation". Jc3s5h (talk) 14:56, 19 February 2018 (UTC)

Emoji’s
Why are emojis being used to display the symbols in the ‘length of a tropical year’ section? Ayil676 (talk) 13:58, 9 March 2018 (UTC)


 * There are no emojis. The symbols are accepted astronomical symbols for the first points of Aires and Libra. See http://www.oxfordreference.com/view/10.1093/oi/authority.20110803095820164 and http://www.oxfordreference.com/search?q=first+point+of+Libra&searchBtn=Search&isQuickSearch=true


 * Jc3s5h (talk) 14:10, 9 March 2018 (UTC)


 * I'll elaborate a bit by pointing out that "first point of Libra" and "first point of Aries" are rather wordy. Alternatives are "northern vernal equinox" and "northern autumnal equinox", but those are both wordy and ambiguous; they can refer either to a direction or to a point in time. The symbols are short and always refer to direction, not time.


 * As for being emoji, according to Time, emoji were invented in 1999. The symbol for the first point of Aries may be found on page 7 of J.M.A. Danby's Fundamentals of Celestial Mechanics, published by MacMillan in 1962. I suspect the symbols are hundreds, if not thousands, of years old, but don't have a publication at hand to verify that. Jc3s5h (talk) 14:42, 9 March 2018 (UTC)

ΔT
Referring to the 16th century reform of the Julian calendar, the article says "Participants in that reform were only partially aware of the non-uniform rotation of the Earth". No source is given for those participants having any awareness at all, and it is not mentioned at ΔT. If there was such awareness it would improve both articles to provide details and a source. Otherwise it should be removed. Zerotalk 10:14, 4 September 2018 (UTC)
 * Around the time "partially" was introduced there was an edit war by a banned user going on. I'm unable to sort through all the edits to figure out the reasoning behind this change, but I've corrected it. Jc3s5h (talk) 15:44, 4 September 2018 (UTC)
 * I see that the "partial" wording was added in this edit by Dbfirs. Jc3s5h (talk) 16:28, 4 September 2018 (UTC)
 * Yes, I was thinking of the orbital rotation, but I'm happy to see the "partially" removed. It was some sort of compromise, but I can't remember the details now.   Dbfirs  19:45, 4 September 2018 (UTC)
 * The only reason I could imagine to use "partially" is that the last few countries to adopt the Gregorian calendar did so after the invention of the Shortt–Synchronome clock in 1921, which was capable of measuring irregularities of the rotation of the Earth. But I doubt there was time for astronomers to assimilate measurements using this clock in time for the last few conversions from Julian to Gregorian calendar. Jc3s5h (talk) 20:24, 4 September 2018 (UTC)
 * Since ΔT is only about 3 hours in 2000 years, I'm not sure why it is even relevant. Calendar reform is not concerned with such tiny fractions of a day. Wouldn't it be better to just remove the mention? Zerotalk 01:17, 5 September 2018 (UTC)
 * In the future, the effect will accumulate more rapidly. Those who wish to evaluate various calendars in the distant future will have to pay more and more attention to ΔT.
 * Also ΔT is closely related to the length of day. Obviously, if the day is longer, the number of days in a year (expressed to quite a few decimal places) is less. The last sentence in the lead shows this already affects the 5th decimal place, and the affect will become greater. Jc3s5h (talk) 01:30, 5 September 2018 (UTC)

Formula in Calendar year made no sense
The formula in the Calendar year section contained the formula $365 97/400$ = 365.2425, which doesn't make any sense as written. I assume the original author intended to write 365 + $97/400$ (which is mathematically correct), and so have made the correction. Ross Fraser (talk)


 * They are both the same. Why make the change?   Db<i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  13:07, 8 October 2018 (UTC)


 * $365 97/400$ is the normal way to write the number. It is not a "formula", it is a mixed number, which is taught in elementary school. 365 + $97/400$ is an abnormal way to write the number. Jc3s5h (talk) 17:07, 8 October 2018 (UTC)


 * The problem isn't the mixed fraction, it's the formatting. It reads as 365 to the power of 97 divided by 400.  Ross Fraser (talk)


 * I agree that the typesetting could be clearer. I suppose you could take this up with the authors of the "frac" template.  There will be many other places in Wikipedia where a mixed number is shown in this way. Would $$ 365 \tfrac{97}{400} $$  be clearer? <i style="color: blue;">D</i><i style="color: #0cf;">b</i><i style="color: #4fc;">f</i><i style="color: #6f6;">i</i><i style="color: #4e4;">r</i><i style="color: #4a4">s</i>  11:54, 9 October 2018 (UTC)


 * Yes, very much so! This would eliminate 97 looking like an exponent.  Ross Fraser (talk)