Talk:Equation of time

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Notice the difference between this:

(with no spaces between digits and letters) and this:

That's part of what my recent edits did.

(On most browsers) 28° shows a superscript circle indicating "degrees"; I changed 28 deg to that. Similarly, in TeX, I changed this:

to this:

"Displayed" TeX should normally be indented; thus this

differs from this:

Michael Hardy 23:34, 5 Sep 2004 (UTC)

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The external link to the article by Brian Tung (currently dated and copyrighted 2002), is I think, important, because at the end it contains a link to a C program for the analemma or equation of time, which uses a more accurate formula than many use. Particularly, it is more complex than simply working out the Equation of Time due to the two components (eccentricity and obliquity) on their own, independantly of each other, then adding the two results. The movement of the Sun eastwards among the stars due to the orbital motion of the Earth, is itself uneven due to eccentricity of the orbit; this needs to be taken into account when working out the component of the Equation of Time due to the obliquity. As Brian Tung states in the last paragraph of his page, the formula which many use (working out the two effects independantly then adding them linearly) works reasonably well for small eccentricities and obliquities, but becomes noteably inaccurate for extreme orbits and inclinations.

Roo60 13:54, 2 Apr 2005 (UTC)

[edit] Almagest

I have no doubt that Ptolemaios already knew about the irregular motion of the sun. It is clearly evidenced in the duration of the seasons for example. But whether he fully appreciated that it also affects the length of the day, I am not so sure about that. How could he (or rather not he himself but any astronomer before him) measure it without a regular timekeeper such as a mechanical clock? Anybody with a copy of the Almagest who can look that up? Until then I keep it to the statement that the concepts of the equation of time and the analemma as we know them nowadays were not introduced until accurate clocks became available in the 18th century. --Tauʻolunga 20:19, 20 March 2006 (UTC)

I own a copy of Ptolemy's Almagest by G. J. Toomer, which is a complete English translation of the Almagest. However, it lacks any detailed commentary because there are many discussions of its contents elsewhere, for example in A History of Ancient Mathematical Astronomy by Otto Neugebauer. I probably should include a section on Ptolemy's discussion, but I should review these other explanations first. But to assuage your doubts, I give here Ptolemy's general description of the "inequality of the solar day" (page 170):

This additional stretch of the equator [59/60 time-degrees], beyond the 360 time-degrees, which crosses [the horizon or meridian] cannot be a constant, for two reasons: firstly, because of the sun's apparent anomaly; and secondly, because equal sections of the ecliptic do not cross either the horizon or the meridian in equal times. Neither of these effects causes a perceptible difference between the mean and the anomalistic return for a single solar day, but the accumulated difference over a number of solar days is quite noticeable.

His "360 time-degrees" is the daily sidereal rotation of the celestial sphere around a motionless Earth. This leaves the daily motion of the Sun along the ecliptic of 59/60° towards the east. The two reasons he gives are the same as those given in this article. The last sentence is self explanatory, going to the heart of your doubts. Only the last phrase introduces our annual "equation of time"—the rest of the paragraph deals exclusively with the variation in the length of the solar day itself. The Greek word that Toomer translates as "solar day" is "nychthemeron" (night+day) which is a valid English word according to both the Oxford English Dictionary and Websters Third New International Dictionary. — Joe Kress 06:17, 21 March 2006 (UTC)

Thank you. Meanwhile I found additional information on http://www.phys.uu.nl/~vgent/astro/almagestephemeris.htm showing that P. only used it for the fast moving moon, considering it ignorable for anything else. (Yes, in the moon will show up, I did not think about it when talking about sundials!) But also states that the E.T. as we know it nowadays was not defined until the late 17th century. So in fact we were both right. I must have a closer look at P.'s E.T. graph, before I go to update the article, but it seems that the secular change is quite visible. --Tauʻolunga 06:38, 21 March 2006 (UTC)

Do I understand that correctly, that Ptolemy knew about the Equation of Time, not through comparison of the length of the day with clocks, of course, but through comparison of the position of the sun relative to the stars? If so, (a) How does one measure that accurately without resorting to clocks?, and (b) How did he know (or why did he assume) that the stars move regularly rather than the sun? Art Carlson 08:12, 21 March 2006 (UTC)

Ptolemy's knowledge of the equation of time was basically deduced from theory. The variation in the lengths of the four seasons by several days shows that the Sun does not have a uniform motion—it has an anomalistic motion. Ptolemy states in Book III chapter 4 that spring was 94 1/2 days, summer was 92 1/2 days, autumn was 88 1/8 days, and winter was 90 1/8 days. The modern values, which differ because of the movement of the apsides (perihelion and aphelion) relative to the equinoxes and solstices over two millennia are 92.76, 93.65, 89.84, and 88.99 days. In addition, it is obvious that the Sun does not travel along the celestial equator as indicated by the annual variation of its altitude at noon—the projection of its ecliptic motion onto the equator causes another nonuniform motion (the Greeks were masters of geometry). Ptolemy realized that these two annual effects cause the length of the solar day itself to vary. — Joe Kress 05:17, 22 March 2006 (UTC)

If you think about it in this way: What must P. have been a clever person to dare to make such bold statements about the motion of the sun which could not be experimentally verified in his time. The length of the seasons was about the only readily observable. And with some effort the moon. As the moon moves its own diameter in about 1 hour, a change of a half hour due to ET results in a well measurable position shift. --Tauʻolunga 06:10, 22 March 2006 (UTC)

Very impressive. Any idea how he was able to determine the length of the seasons to 1/8 day accuracy? The equinoxes I can imagine, but the solstices must be extremely hard to pin down. And one more thing, I may be thick, but I don't follow the comments on the Moon. Are you using the position of the Sun relative to the Moon and the Moon relative to the stars to determine the EoT? --Art Carlson 08:35, 22 March 2006 (UTC)

I was not present when the ancient Greeks made their measurements, neither have I access to their records, but I conjecture the following. Although it is hard to measure the day of the solstice directly, it is relatively easy with a big sundial to find that the sun's declination has changed already 10 days after and over the same distance 10 days before. The solstice then must be in the middle. Also a favourite trick was to look at the full moon, which is diametrically opposite the sun. And do not forget the power of repetition. Some people are confused and do not understand why astronomers cannot determine most star distances better than a couple of digits, while the length of the solar year, for example, is known to 10 digits. The answer is that the year is repeated again and again, and an average can be taken over a long time. P. had centuries of records to his disposal. Likewise the position of the moon could be tackled. The motion of the moon is too irregular to derive from limited measurements. But taking proper averages over repeated periods will quickly show up systematic discrepancies. If P. found that the moon was always, on the average, half of its diameter off (half hour of monthly motion) in a particular month, while some other value between zero and that in other months, then clearly that was a yearly effect, attributable to the sun. Would be an interesting topic for a graduating student to reproduce that. --Tauʻolunga 06:21, 23 March 2006 (UTC)

[edit] Formula correct?

The formula for the Equation of time as presented under en.wikipedia.org/wikw/Equation_of_time contains an error in the term denoted as -7.53cos(2B). That term shoud instead read +7.53sin(N-4) its argument reaching zero on N=4 or January 4. This date(plus or minus one day, depending mostly on the Julian cycle)is the perihelion. The EoT values presented in the accompanying graph are nevertheless fairly accrate. user: Alex Vermeulen, Zoetermeer, Netherlands Aril 9, 2006.

It looks OK to me. I suppose you want the argument of your corrected term to be 2pi(N-4)/365, rather than simply (N-4). I also suppose you are referring to the -7.53cos(B) term since there is no cos(2B) term and you mention the perihelion specifically. Note that B has an offset of 81 days, almost pi/2, which changes the cos(B) to something close to sin(2pi(N-4)/365), the remaining phase shift being presumably taken up by the sin(B) term.

There remains a question of the sign. Note from the graph that the maximum slope is negative and occurs near the beginning of the year, so both the annual and the semiannual periodicities must be decreasing near N=0, i.e. when B is about -pi/2. This requires the term with the B argument to be a cosine with a negative coefficient and the 2B term to be a sine with a positive coefficient. And so it is.

--Art Carlson 19:03, 9 April 2006 (UTC)