Talk:Tropical year

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Naming conventions[edit]

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

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

Homer would be Homeros

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

This would be a good argument, except for the fact that it's "perihelion" in English. What dictionary are you using that says otherwise? --Zundark, 2001 Oct 25

— Preceding unsigned comment added by Zundark (talkcontribs) 11:43, 25 October 2001 UTC

OK, I screwed up -- Tompeters — Preceding unsigned comment added by Conversion script (talkcontribs) Revision as of 15:51, 25 February 2002 UTC
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

— Preceding unsigned comment added by Malcom Farmer (talkcontribs) 12:10, 25 October 2001

Thanx for pointing that out, I never noticed. Good to see someone writing Aluminium, americans usually say aluminum.— Preceding unsigned comment added by Conversion script (talkcontribs) Revision as of 15:51, 25 February 2002 UTC

Ephemeris Time[edit]

The article now says:

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

I don't understand this. One could understand this as "Terrestrial Time was formerly called Ephemeris Time". This is not the case. Ephemeris Time is different from Terrestrial Time; and is not based on atomic clocks. Terrestrial Time is now used a lot where Ephemeris Time used to be used. But wouldn't the formulae be different in the past if they used Ephemeris Time in the past? Until someone who really understands this, sorts this out, I will delete the text in parentheses "(formerly Ephemeris Time)" -- Adhemar — Preceding unsigned comment added by (talkcontribs) 09:34, 29 August 2005

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)

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

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

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

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

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

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

Expressions for "mean" equinox or solstice

Years -1000 to +1000

Y = year/1000

March equinox

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

June solstice

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

September equinox

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

December equinox

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

Years +1000 to +3000

Y = (year - 2000)/1000

March equinox

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

June solstice

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

September equinox

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

December solstice

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

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

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

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

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

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

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

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

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

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

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