Talk:Tropical year

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[edit] Naming conventions

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.

[edit] Definition

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)

[edit] NPOV violation

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)

[edit] Ephemeris Time

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)


[edit] The 2000.0 value of the tropical year

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 [1] to the mean ecliptic longitude of date for the Earth given by Bretagnon et al. on page 678 of [2] (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)

[edit] Tropical year history

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)

[edit] Different duration by starting point

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)

[edit] length, and how to name the equinoces

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)

[edit] Table formatting

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)

[edit] Solar year

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)

[edit] Undefined variable

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)

[edit] Iranian calendar

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 + 8⁄33 formerly used in the Iranian calendar is even better, and 365 + 8⁄33 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)

[edit] Dubious things

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)

[edit] Why the tropical year varies by starting point

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)