Talk:Equation of time/Archive 1

Archive 1

Archive 2

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.staff.science.uu.nl/~gent0113/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)

External Link - Equation of Time Longcase Clock

I would like to add a clock we have in our collection to the equation of time page as an external link. Could you let me know if this is OK. It is purely for research, non commercial. The clock is not for sale.

• Equation of Time clock C.1720 by John Topping[1]

I think the clock is very relevant to the page. If you believe it is could you add it to the section.

user: Danielclements —Preceding unsigned comment added by Danielclements (talk • contribs) 11:06, 12 September 2007 (UTC)

WikiProject Time assessment rating comment

Want to help write or improve articles about Time? Join WikiProject Time or visit the Time Portal for a list of articles that need improving. -- Yamara 12:17, 11 January 2008 (UTC)

364 in formula for B?

Why is the divisor 364 and not 365 or even 365.254?

92.4.0.19 (talk) 20:37, 15 April 2008 (UTC)

or 365.2425 ? 82.163.24.100 (talk) 19:22, 25 April 2008 (UTC)

Specific question about correctness

In the practical use section, it says "At the spring and fall equinoxes, the cone degenerates into a plane and the hyperbola into a line." Is this correct everywhere, or just at the equator? It seems to me that this statement is only true at the equator. At the tropic of Capricorn, the gnomon would draw a line just once a year during the solstice, not during the equinoxes, and in a temperate region like England it would never trace a line at all

This makes me wonder whether the shape traced is actually a hyperbola at non-equatorial locations. I can clearly see that it is a hyperbola at the equator, but elsewhere the math is just too complicated for me.Fluoborate (talk) 18:04, 5 May 2008 (UTC)

The article describes the effect from a geocentric perspective. Since from that point of view the Sun always moves in a circle, just with a different declination depending on the time of year, the production of a hyperbole at any latitude is obvious (I thought). The other way to look at it is from an inertial reference frame. In that case, think of a plane containing a line parallel to the axis of the Earth, for example, a plane parallel to the surface of the Earth at the Equator at your longitude. As the Earth rotates, this plane will always be parallel to the corresponding horizontal plane at the Equator, so the path traced out by the shadow of a gnomon will be the same in both cases. Since one point and one plane produce a conic section, any point and any plane will do so. (An exercise for the reader: In which plane will the shadow follow a circular path?) The article is correct, and you don't even need to do any math to see it. --Art Carlson (talk) 19:46, 5 May 2008 (UTC)

I am sorry, but I entirely don't understand. What do you mean by "one point and one plane produce a conic section"? Do you really mean "parallel" every time you say "parallel", or did you mean "perpendicular" some of the time? Can you use the words "perpendicular" and "tangent" a few times? Can you please over-define the planes, points, and lines you are talking about - for instance, "the equator is a circle, a grand geodice around the Earth, defined by the intersection of the Earth's surface with a plane that is equidistant from the North and South poles. The plane containing the circle of the equator is perpendicular to all lines of longitude at the equator, perpendicular to the axis of Earth's daily rotation, and parallel to the surface of the ground at the poles." Thanks.Fluoborate (talk) 07:09, 6 May 2008 (UTC)

Before I wax voluminous, tell me whether you prefer to think in geocentric or heliocentric terms. --Art Carlson (talk) 07:55, 6 May 2008 (UTC)

Definitely geocentric. Also, if you want to see me waxing voluminous on astronomy just tonight, go look at Talk:Tropical year. Thanks.Fluoborate (talk) 09:23, 6 May 2008 (UTC)

Good. Let's you and me do an experiment on May 6th, me in Munich and you in Timbuktu. We each take a board and drive a big nail into it so that the head is 10 cm above the board. Then we tilt the board until it points to Sirius at 08:00 UTC. Then we each mark the path traced out on our own board by the shadow of the head of the nail during the course of the day, as long as it is daylight both where you are and where I am. We both fly to Cairo and compare the marks on our boards. They are the same! Are you with me so far? It's just a short step now to the conic sections, but I want to be sure you see that it doesn't matter where on Earth we set up our sundial, as long as we do not restrict ourselves to horizontal surfaces. --Art Carlson (talk) 12:41, 6 May 2008 (UTC)

Thank you. I understand perfectly now. The key to my understanding was this sentence alone: "Then we tilt the board until it points to Sirius at 08:00 UTC." That sentence made me realize for the first time ever that a sundial does not need to be parallel to the ground - I simply had never even conceived of tilting a sundial, even though it obviously doesn't cause any problems. This allowed me to "adjust another variable" in my mental picture, and the new picture fully convinced me that the shadow traces a hyperbola usually and a line at the equinox. I never thought this was untrue, I was just unconvinced and confused before.

I didn't understand your first response because the planes were defined a bit ambiguously, but now I can clearly determine what you are talking about by analogy to your sundial explanation. The answer to your exercise for the reader is this: When standing at the equator, hold the board with a nail in it straight up and down (vertical), so that the surface of the board is like a wall and the nail juts out horizontally. While keeping the board vertical, rotate it until the edges point directly East and West. On any day but the equinox, this configuration will trace out a circle on the board. Ellipses and parabolas are also possible, with other configurations, of course.Fluoborate (talk) 12:04, 7 May 2008 (UTC)

I'm glad I could help. This medium sometimes lends itself to misunderstandings. But surely you have seen sundials mounted on walls before?! Once you start thinking about that, you realize that the wall you want to use will not always be facing directly south. If you are mathematically minded, that will be enough to get you thinking about arbitrary orientations. Another place the concept turns up is if you move to another city and take your sundial, which was designed for your home town, with you. You can still use it at your new location if you tilt it to adjust for the difference in latitude and longitude. The various possibilities are discussed in the sundial article, e.g. Sundial#Reclining-declining dials. --Art Carlson (talk) 13:00, 7 May 2008 (UTC)

P.S. Is there a way to change the article to avoid misunderstandings like this? --Art Carlson (talk) 13:02, 7 May 2008 (UTC)

Source of Equation

Does anyone know where the approximation in the "More Details" section was taken from? I used it for some work, and although I plotted it up to check the shape and am sure it is accurate enough for my purposes, I would rather have a "real" reference for it. Thanks. 141.163.157.53 (talk) 10:10, 5 August 2008 (UTC)

More details

From one year to the next, the equation of time can vary by as much as 20 seconds, mainly due to leap years. - given that the Y-axis of the graph is in minutes, might "a third of a minute" be better ??? 82.163.24.100 (talk) 19:22, 25 April 2008 (UTC)

The equation of time also has a phase shift of about one day in 24.23 years. The equation as read from a table of 1683 lags 13 days behind the one of 1998. - could that be in part because 1683 was Julian and 1998 Gregorian ? 82.163.24.100 (talk) 19:22, 25 April 2008 (UTC)

The section starts with an incomplete equation, namely:

$\alpha -M+\psi ,\!\,$

Should it not be something like:

$E=\alpha -M+\psi ,\!\,$

or at least an equation that defines E (equation of time) as function of other parameters. I'm not an expert on this subject, so I don't know what the general equation of time is. —Preceding unsigned comment added by WJvandeBerg (talk • contribs) 12:45, 17 April 2009 (UTC)

Erroneous value for the rate of increase of length of day

AmouDaria (talk) 14:22, 4 October 2009 (UTC): I think there is an error at the end of the section "Eighteenth and early nineteenth centuries" where it says The tiny increase of the mean solar day itself due to the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is ... According to the Wikipage on Tidal acceleration, the Earth's rotation period is increasing by about 2ms per century. It's certainly much less than the 1 second per year which the articles states.

• The rate of change of the Earth's rotation is given in Equation of time#Eighteenth and early nineteenth centuries as "2 ms per day per century" and in Tidal_acceleration#Quantitative description of the Earth-Moon case as the "change in the length of the day" is "+1.7 ms/cy" (average over the last 2700 years which includes the effects of glacial rebound). No disagreement there.
• The current value of the Earth's rotation is given in Equation of time#Eighteenth and early nineteenth centuries as "about 1 second every year". Leap second#Reason for leap seconds gives this more accuratly as "0.6 seconds per year". These values also agree.
• 0.6 s/yr is 1.6 ms/dy. Given the rate of change, we would then expect the mean solar day to have been exactly 24 hours about a century ago. In fact, the second was originally defined with reference to the year 1900. (See Second#Modern measurements.)

It looks to me like everything holds together. Should we edit the article to make these points clearer? --Art Carlson (talk) 08:06, 5 October 2009 (UTC)

[From Terry0051] It's an attractive idea, but unfortunately such clarity would be illusory for a number of reasons.

The changes in earth rotation are irregular, and the values you just cited are estimates of long-term smoothed-out trends. Also, the (retrospective) definition of the second relating to epoch 1900 referred to the length of the _year_ as at 1900 Jan 0, not the length of the _day_. As seen in Ephemeris time - history, the sources cited there make two points clear in particular.

(a) The ephemeris second was not the same as the mean solar second at 1900, the ephemeris second was already shorter. More recent publications cited under 'Reason for leap seconds' confirm this mismatch between the second at 1900 and the ephemeris second, e.g. in "The Physical Basis of the Leap Second", by D D McCarthy, C Hackman and R A Nelson, in Astronomical Journal, vol.136 (2008), pages 1906-1908, where it is stated (page 1908), that "the SI second is equivalent to an older measure of the second of UT1, which was too small to start with and further, as the duration of the UT1 second increases, the discrepancy widens."

(b) When the ephemeris second was first proposed and adopted there were no thoughts that it would be used for anything other than an astronomical time scale, its adoption for civil purposes was thought up separately later (aagin, the sources under "Ephemeris time - history" show this).

References on the various 'time' pages show that an estimate of about 1820 has been made for the time when the (smoothed-out average) mean solar second was equal to the ephemeris second aka the modern SI second. But that is retrospective reconstruction, the main historical basis for choosing the ephemeris second at the length it now has in its redefined form, the SI second, is also described under "Ephemeris time - history" and "Ephemeris time - redefinition of the second": it was to make the existing solar tables match the astronomical reality by redefining the second instead of changing the tabular coefficients. The main objective in that was to free the definition of the second for purposes of accurate astronomical timekeeping from any dependence on the irregular rotation of the Earth. So things unfortunately don't hold together as cleanly as they have sometimes been made to appear. Terry0051 (talk) 10:11, 5 October 2009 (UTC)

[From Terry0051] A postscript:

The current discussion of the 'Equation of Time' in the main article here really relates to a rather early historical stage in the development of timekeeping -- from about the 1650s/1670s to about 1750. It was only during that historical period, that

(a) consensus had been reached about the roles of the obliquity of the ecliptic and the ellipticity of the earth's orbit around the sun in producing the equation of time, as described in the main article, and

(b) it was still supposed that these were the only two components of the equation of time.

To be sure, the remaining components of the equation of time are very much smaller indeed; but since the second half of the 18th century it has been also known, not only that the nutation and aberration and the planetary perturbations of the Earth's orbit around the Sun have their contributions to the equation of time, but also approximately how much.

To the coarse approximation that the equation of time is treated as the same from one year to the next, these things are negligible. But that coarse approximation is rough even by early 18th-century standards, where one often finds the equation of time plotted or tabulated against longitude of the Sun rather than against calendar date, which avoids the error involved in treating common years and leap years alike. Terry0051 (talk) 10:45, 5 October 2009 (UTC)

Rearrangements/clarifications/corrections and 'dubious' tag

The historical sections are a little rearranged, with a few more sources/citations and a few consequential corrections.

The asserted fact that Huygens followed earlier practice in using an offset has been labeled 'dubious' (because I have seen some earlier tables that used no offset, and none that did) -- sources/citations needed if this statement is to be supported.

Ptolemy's discussion of the equation of time in the Almagest (referred to elsewhere on the talk page) implied an offset (he did not actually supply a table but his computational procedure implies this) AstroLynx (talk) 08:44, 20 July 2009 (UTC).

So, would there be a source for that, and for its amounting to an 'earlier practice' and for Huygens following that earlier practice, rather than the offsets being coincidental? Terry0051 (talk) 09:08, 20 July 2009 (UTC)

The offsets are slightly different. Ptolemy defined the equation of time (EoT) to be zero at the epoch of his tables (1 Thoth, 1 Nabonassar or 26 February, 747 BCE) and for solar longitudes between 301.02º and 330.75º his EoT is slightly negative in value. Huygens calculated his EoT table in such a way that its minimum value is zero (i.e. all values are positive). There is a detailed discussion of Huygens's EoT table and its use in regulating mechanical clocks in the Oeuvres complètes of Christiaan Huygens (online at the GALLICA website), volumes IV (pp. 53-57), XV (pp. 523-525), XVII (pp. 191-237) and XVIII (pp. 112-113) AstroLynx (talk) 13:26, 20 July 2009 (UTC).

About the historical and present-day sign conventions for the equation of time: Flamsteed's tables of 1672/3, like others of his time and before, did not give signed numbers; they tabulated unsigned numbers, and then separately instructed when to add and when to subtract them. The concept of signed numbers was not yet current. Kepler's tables instructed both addition and subtraction and led to a kind of mean, but it was a substantially wrong mean because he misapplied the annual effects of the earth's orbital eccentricity for reasons to do with the moon's annual equation. Huygens was first to make the annual variations correct, but added an offset, and Flamsteed both gave the correct annual behavior and his tables led to the mean.

Terry0051 (talk) 23:33, 17 July 2009 (UTC)

I've removed the statement about Huygens following earlier practice, as it is both dubious and not particularly relevant. Looking at the EoT table in the Rudolphine Tables (Kepler, 1627), I agree with Terry0051 that Kepler's values would give a mean: however, the same page implies that Tycho himself used an equation with an offset. Cassini made an interesting comment sometime between 1688 and 1693 when he published his revised ephemeris of the Galilean moons:

"I didn't speak at all in my first tables [from 1668] about the astronomical equation of time, it being a subject on which modern astronomers disagree and so I left each one the freedom to use his own method…" [my translation from the original French]

Cassini did include an EoT correction in his revised ephemeris. Physchim62 (talk) 08:49, 11 November 2009 (UTC)

Improved formulas

The formula must be congruent for N=1 and N=366, as these specify the same day (Jan 1st in non-leap years). But, sin(2.pi.(N-81)/364) gives different values for N=1 and N=366. The formula is wrong.

In http://www.ips.gov.au/Category/Educational/The%20Sun%20and%20Solar%20Activity/General%20Info/EquationOfTime.pdf the divisor for B is 365, not 364.

In http://eco-energy.info/upload/dnload/S/Solar_energy_chapter_6.pdf the divisor for B is once again 365!

http://www.mail-archive.com/sundial@uni-koeln.de/msg11178.html, Mr. Roger Sinnot of Sky & Telescope gives a much better expression for B as

  T = ( # of days since Jan 1st 1900 ) / 36525

Notice that in the same thread Mr. Ron Anthony provides a more exact equation, again using a divisor of 36525 and number of days since Jan 1st 1900; then Prof Paolo Gregorio uses a similar formula with a divisor of 36525, but counting the # of days since Jan 1st 2000.

Also, I don't think that the article points out that the equation of time varies per year. In order to calculate a general equation of time, one must calculate all factors involved. A quite general formula, valid for several years, is given by US Naval Observatory:

    JD = Julian Date
    UT = Universal Time (hrs) 
    T <- (JD + UT/24 - 2451545.0)/36525.                                             Number of centuries from J2000.
    L <- 280.460 + 36000.770 * T                                                     Solar mean longitude, in degrees (remove multiples of 360 degrees from it.)
    G <- 357.528 + 35999.050 * T                                                     Mean anomaly, degrees
    M <- L + 1.915 * sin (G) + 0.020 * sin (2*G)                                     Ecliptic long., degrees
    e <- 23.4393 - 0.01300 * T                                                       Obliquity of ecliptic, degrees
    E <- -1.915 * sin (G) - 0.020 * sin(2*G) + 2.466*sin(2*M) - 0.053 * sin(4*M)     Equation of time, degrees (x15 for hours, or /4 for minutes)

Notice a factor on T of 36525 as the noted gentlemen proposed.. not 36400. I believe that we must correct the page!

Thanks! :^) —Preceding unsigned comment added by 76.182.26.210 (talk) 21:19, 26 November 2008 (UTC)

Done. Do you think this or that can be considered a WP:RS, so that we can add a reference? Of course, we could also consider adding one of the more accurate formulas. --Art Carlson (talk) 08:17, 27 November 2008 (UTC)

Appreciate the correction. I think that both docs fall within fair use for general public, but I'm not the owner. You may want to consider this equation too [2] ... it is from the NOAA, and is way easier than the algorithm above, but does not handle leap years well -- that can be of course corrected by using D <- 2*pi*frac(T*100), where T is per the formula above and frac(x) is x-trunc(x):

    D <- (2*pi/365) * ( N - 1 + (Hour-12)/24 )                                                              N = 1 for Jan 1st, etc..
    E <- 229.18 * (0.000075 + 0.001868 cos(D) - 0.032077 sin(D) - 0.014615 cos(2D) - 0.040849 sin(2D)       Equation of time, minutes

Thanks! :^)

The last equation of the US Naval Observatory set could probably be improved (shortenend) by rewriting it as

    E <- L - M + 2.466*sin(2*M) - 0.053 * sin(4*M)                                   Equation of time, degrees (/15 for hours, or *4 for minutes)

Note also the correction in the comment where /15 and *4 is written instead of *15 and /4 as was in the version above. Correction posted by Arif Zaman 19:24, 24 February 2009

Here's a good citation:

Hughes, David W.; Yallop, B. D.; Hohenkerk, C. Y. (1989), "The Equation of Time", Monthly Notices of the Royal Astronomical Society, 238: 1529–1535, retrieved 2010-08-10, An Equation is developed which gives the Equation of Time as a function of Universal Time. This enables it to be calculated for any epoch within 30 centuries of the present day, to a precision of about 3 s of time. We also give several expressions for the Equation of Ephemeris Time which ignores the distincion between the time-scale of the Ephemeris and Universal Time, and so may be compared with expresions given in old text books.

It gives a formula similar to the USNO one above, but with further refinements. I found it in the Sundial mailing list archives 71.41.210.146 (talk) 13:14, 10 August 2010 (UTC)

General Relativity

How big is the impact of General Relativity on the (classical) Equation of time, as Earth follows an elliptical orbit around a star? Hcobb (talk) 21:37, 25 August 2010 (UTC)

Calculating the precession of the earth's orbit due to general relativistic effects, I get 3.84 arc seconds per century. The precession of Mercury is about 43 arc seconds per century. I'm not sure if general relativity causes other effects, but my guess based on the above is that its not too much. PAR (talk) 18:35, 26 August 2010 (UTC)

Computer code to calculate Equation of Time and Solar Declination

See Talk:Declination.

DOwenWilliams (talk) 21:42, 30 August 2010 (UTC) David Williams

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)

Comment by Alex Vermeulen (a bit late) to Art Carlon's answer dated 9 April 2006.

Now I understand your formula! 81 days (the vernal equinox after New Year's day Number 1, N=1) minus 4 days (the perihelion Jan., 4), is 77 days. Approximatively you take that to be equal to 91 days (one quarter of a year). Of course, with one quarter of a year (approximately 90 degrees earth's orbit around the sun) you can switch from sine to cosine, which was the thing I thought was wrong. So, your formula is correct after all. I, for one, find your approximation a bit too rough, but that is a matter of taste. By introducing a sine term with an argument of N-4, you would have it made clear that it is the day of the perihelion that is the reference for the ellipticity of the orbit. And by introducing a sine term with argument 2(N-81)you would have made it clear that it is the day of the vernal equinox (March 21)that is the reference for the effect of the obliquity of the earth's rotational axis. The formula then gets even simpler than yours, inasmutch that it has one term less. That would also give a formula that is more commonly found in literature. Again, it is a matter of taste. Do you consider to change the text according to my suggestion? I am not going to do that myself, because as such your nicely presented text and figures are basically correct.

Alex Vermeulen, Zoetermeer, The Netherlands, Jan. 10,2009, corrected Jan 15.

The article states that the eccentricity of the Earth's orbit at the extremes increases or decreases the real solar day by 7.9 seconds, and that the obliquity of the ecliptic at the extremes increases or decreases the real solar day by 20.3 seconds. If that were correct, a real solar day could be no more than 28.2 seconds longer than 24 hours, even if the two factors combined perfectly. However, on December 22-23, 2008, according to a Table found at this webpage -- http://www.minasi.com/dolog.htm -- the solar day will be 29.77 seconds longer than 24 hours. Therefore, if the Table is correct, either or both of the eccentricity and obliquity factors are somewhat understated. Perhaps someone can recalculate those figures. Rodneysmall (talk) 20:16, 13 March 2008 (UTC)

OK consider the 9.873 sin 2M term in the series approximation, which is largely due to the axial tilt. Its time derivative is 9.873 min x 2 x 2pi / 365.25 d . -cos 2M, which has a range of plus or minus 20.4 seconds per day -- close enough to the 20.3 quoted. As for the -7.655 sin M term due to eccentricity, time derivative is -7.655 x 2pi / 365.25 d . sin M, for +-7.9 seconds per day... just as claimed. So these numbers are correct inasmuch as the two terms separate into axial tilt and eccentricity (which they don't do perfectly)!

I think the difference arises due to the fact that the series approximation is just that. I have had a fiddle and find the range of gradient in the approximation to be up to 28.1 seconds longer day length or down to 22.5 s shorter. (Note, they do in fact combine nearly perfectly as you suggest, because the December solstice and perihelion are only about 9 days apart.) For a more exact calculation of my own devising I get up to 29.9 s longer or 22.5 s shorter. As expected, they are larger -- the neglected higher frequency terms of the series would contribute relatively more strongly to the gradient (day length) than to the eqn of time itself.

In reality the two effects are not entirely separable so any independent statements of sizes of effects is always going to be somewhat approximate. --Russell E (talk) 08:18, 14 October 2010 (UTC)

Thanks, Russell. I note that the formula used by http://www.minasi.com/dolog.htm shows that the longest day this year is December 23, which is 29.7761 seconds too long. I e-mailed them, asking what formula they used, but did not receive a response. Just for my own edification: (1) If the earth's orbit were currently perfectly circular, what would be the maximum change to the length of day caused by the current obliquity?; and (2) If the earth's axis were not tilted relative to the plane of its orbit, what would would be the maximum change to the length of day caused by the current elliptical orbit? Rodneysmall (talk) 16:12, 24 November 2010 (UTC)

Both components of the length of day are discussed in Earth's rotation. — Joe Kress (talk) 04:28, 25 November 2010 (UTC)

Thanks, Joe. I'll check out that page. Rodneysmall (talk) 02:31, 27 November 2010 (UTC)

Obliquity of the ecliptic

Our explanation of the contribution from obliquity is correct, but I fear it is understandable only for near-experts. I'm not sure if it is possible to find a simple explanation, but I have to try. I thought it might be helpful to look at the cumulative rather than the differential effects. I submit the following alternative text for comments and consideration:

However, even if the Earth's orbit were circular, the motion of the Sun along the equator would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the Ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. To understand this effect, think of a globe with the Ecliptic drawn as a great circle that intersects the Equator at longitude 0 and 180. These points of intersection represent the equinoxes. "Clock time" is measured along the Equator, because the rotation of the Earth is uniform like a clock. "Sundial time" is measured along the Ecliptic, because it relies on shadows cast by the Sun. The solstices are represented by the points on the ecliptic halfway between the equinoxes, and these will be located exactly at longitude 90 E and 90 W (although on the Tropic of Cancer and the Tropic of Capricorn, rather than on the Equator). That means a clock can be set to agree with a sundial four times a year, at the equinoxes and the solstices. The points halfway bewteen an equinox and a solstice, however, will not be located at longitude 45 or 135 but slightly closer to the equinoxes. The deviation of the angle measured along the ecliptic from that measured along the Equator corresponds to the deviation between clock time and sundial time and is positive during spring and autumn and negative during winter and summer. This is responsible for the component of the equation of time with a six-month period.

--Art Carlson 11:54, 3 July 2007 (UTC)

I also had some trouble understanding the discussion of the contribution of obliquity, and felt it could benefit from a graphic, so I created one myself. I exported it here: http://mysite.verizon.net/ksbrace/obliquity.png I don't have authority to upload a file, but if anyone thinks this would be a useful addition and would like to grab this graphic and put it along the right margin, feel free. I was thinking of this caption:

This graphic illustrates the component of the Equation of Time due to the earth’s tilt. It shows the relative movement of the sun against background stars while the earth rotates once at the winter solstice (2a) and vernal equinox (2b). For a given absolute apparent angular motion of the sun in a day, the west-to-east component varies between the equinox and solstice, making the noon-to-noon period less than 24 hours near the equinox, and more than 24 hours near the solstice.

Let me know if you see any errors or suggested improvements, I'd be happy to correct them.

Karlbrace (talk) 12:57, 5 December 2010 (UTC)

Equation of time based on a model of the Sun's motion?

The "solar time" does not appear to be well defined. Observations of transit times for the prime meridian give a discrete function for the equation of time. Since the right ascension of the Sun is constantly changing there will be a different equation for each meridian. Comparing these would require an accurate determination of longitude or hour angle. A continuous function would require some model for the Sun's motion either for the time of transit for some meridian or the right ascension of the Sun. So one could have an "equation of time" for Ptolemy's epicycle for the Sun, Kepler's elliptical orbit or some other model used to compute an ephemeris. Defining the equation of time as the difference between solar time and mean solar time requires both "clocks" to be read simultaneously and both right ascensions would also have to be determined simultaneously. One could then compare an "observed" equation of time with that of a model to determine its accuracy. --Jbergquist (talk) 14:09, 30 January 2011 (UTC)

The Astronomical Almanac for the year 2011 states on page C2 that "apparent solar time is the timescale based on the diurnal motion of the true Sun" where "diurnal motion" is defined in the glossary as "the apparent daily motion, caused by the Earth's rotation, of celestial bodies across the sky from east to west." There are models for the position of the Sun and the Earth, and for the rotation of the Earth. I believe the weakest of these models is the model for the rotation of the Earth. There is also an expression to convert Earth Rotation Angle, observed with Very Long Baseline Interferometry, to mean solar time. This implicitly models a fictitious mean Sun, the position of which conceptually defines mean solar time. I have not seen any explicit equations for the position of the fictitious mean Sun. So certainly there are models available. Are you looking for an expression for solar time based on these models? If so there is an expression for 2011 on page C3 of the almanac, accurate to about 3 seconds of time. There is a calculation procedure valid form 1950 to 2050, to a precision of 0.1 minute of time, on page C5. Jc3s5h (talk) 16:16, 30 January 2011 (UTC)

Clocks in Ancient Times

An equation of time may have existed prior to that of Huygens' pendulum clock. We know the ancient Egyptians and Greeks used the clepsydra to measure the passage of time and the Antikythera mechanism indicates the existence of mechanical "clocks" at that time. --Jbergquist (talk) 14:50, 30 January 2011 (UTC)

The article already states that Ptolemy described the equation of time in the Almagest III.9 written c.140. However, he did not use it to determine mean time from apparent sundial time. Instead he used it to correct intervals of apparent time into mean time intervals, specifically for the interval between lunar eclipses originally given in apparent time. His description in the Almagest is cryptic, but he did provide a complete annual table in his Handy tables of which only fragments survive. Thus the earliest table appears in Theon's Handy tables written c.295. A graphical form of Ptolemy's equation of time is given by Robert H. van Gent in his Almagest Ephemeris Calculator. Also see A History of Ancient Mathematical Astronomy, Otto Neugebauer, 1975; and A survey of the Almagest, Olaf Pedersen, 1976. — Joe Kress (talk) 06:36, 1 February 2011 (UTC)

Let me get this straight. Ptolemy calculated an equation of time based on what we would call the obliquity of the ecliptic. And he applied it to the description of the motion of the Moon, i.e. The Moon can be found at such and such a position in the sky at each hour of the day, which will differ from the relative position to the Sun by up to a few degrees. Did that help him make sense out of the motion of the Moon? Was it a measurable effect, in that sense, or a mathematical curiosity? Who was the first person to compare the position of the Sun directly with the position of the fixed stars to measure the Equation of Time (or verify a theoretical formula for it)? What kind of time piece did he use, or did he find a clever way to do it directly? Who was the first one to suspect/measure that the motion of the Earth around the Sun is not regular (due to the eccentricity of the Earth's orbit)? Any way you cut it, these effects are on the order of (30 sec/dy)=0.03%, and can't be easy to measure without modern technology. The problem is compounded by the difficulty of observing the Sun and the stars at the same time. I have wondered about these things for years, but have neither read an answer nor figured one out for myself. --Art Carlson (talk) 13:33, 1 February 2011 (UTC)

East and west are the midpoints between the farthest north and south that the sun rises and sets. Except for some difficulty in determining the precise times of the solstices, when the sun does not move rapidly from day-to-day along the horizon, the number of days for each season was specified in 79 AD by Pliny the Elder, a non-mathematical Roman writer, to be 90 days 3 hours for winter, 94 days 12 hours for spring, 92 days 12 hours for summer, and 88 days 3 hours for autumn [3]. This only required the ability to count days and to average those counts over several years. This was observed long before any theory of the motion of the sun existed. Even though Pliny's total is a quarter day too long, it does show both that the sun does not move with uniform velocity along the ecliptic and the magnitude of that deviation from uniform motion.

The position of the sun among the stars is determined via an intermediary such as the moon (or Venus at its brightest). At full moon, the moon rises when the sun sets, so the sun's position among the stars must be halfway further along the ecliptic from the observed position of the moon after the sun sets.

To precisely determine both seasons and positions, a technical (geometric and mathematic) model of the motion of the sun and moon is needed. The Greeks, beginning with Eudoxus of Cnidus about 365 BC, developed a geometric model, while the mathematical details came from Babylonia and its several-century record of lunar and solar observations, especially lunar eclipses (the Babylonians did not have a geometric model). These were combined by Hipparchus about 140 BC and improved by Ptolemy about 140 AD.

Ptolemy's equation of time was a mathematical curiosity. Indeed, it was reconstructed by modern astronomers from Prolemy's own table of the Sun's anomaly in the Almagest. Immediately after the development of the pendulum clock by Christiaan Huygens, John Flamsteed put the equation of time into its modern form as the difference between mean and apparent time. Despite modern criticism, the Greek model of deferents and epicycles was more than adequate to predict the equation of time long before it could be directly measured.

The Greek lunar theory was adequate for Hipparchus to discover the moon's first inequality (equation of the center due to eccentricity, modern 6.3°sin M' + 0.2°sin 2M', where M'=moon's mean anomaly) and for Ptolemy to discover the moon's second inequality (evection, modern 1.3°sin 2D−M' where D=moon's mean elongation from the sun). However, Ptolemy had to include a "crank" in his lunar theory in order to "save the phenomena" (make the positions predicted by his model agree with the moon's observed positions) even though that crank caused the predicted variation in the distance from the Earth to the Moon to drastically disagree with the observed variation. The third inequality was discovered by Tycho Brahe (variation, modern 0.7°sin 2D).— Joe Kress (talk) 01:20, 3 February 2011 (UTC)

Edits by Rpba

Rpba has made edits, such as this series, claiming "The equation of time varies over the course of a year, in a way that is almost exactly reproduced from one year to the next, except for a shift in phase of one day ahead in the calendar every 25 years" (new part underlined). The source given is a table by Tompion from 1683. This change is inappropriate for several reasons:

• There are many advances in astronomy since 1683 so such an old source is not a reliable source.
• There are many sources of change in the equation of time from one year to the next, so it is inappropriate to single out this source of change and imply there are no other sources of change.
• The level of detail in the statement is inappropriate for the lead of the article; there are already sections explaining year-to-year differences and such details, if they can be clearly explained with reliable sources, should go in the "Explanations for the major components of the equation of time" section. Jc3s5h (talk) 12:48, 14 March 2011 (UTC)

Mathematical details section

Should the mathematical details section be deleted or drastically trimmed? From the reader's point of view, it does not give numerical results unless the reader is prepared to write a computer program: "A numerical value can be obtained by infinite series, or numerical methods such as iteration." Also, no accuracy estimate is provided for the equations; why should a reader bother to understand and implement these equations without knowing first whether the accuracy will meet the reader's needs. From editor's point of view, it is hard to maintain because vandalism or errors could easily introduced, and it is not referenced point-by-point to reliable sources such that the equations in the section could be compared to the equations in the sources. Jc3s5h (talk) 15:13, 7 April 2011 (UTC)

I have implemented the method described in the Mathematical details section in a C# program. I evaluated it for each of 365 consecutive days, beginning at 23:51 UT1 January 2, 2008. This time was chosen because it is the time of perihelion for that year. I chose 2008 because that is the most recent full calendar year for which the Multiyear Computer Interactive Almanac has observed rather than estimated values of ΔT.

I found the maximum error, 35.2 seconds, occurred on December 17, 18, 19, and 20.

If others think this value is reasonable, perhaps the discussion of approximations should be trimmed accordingly. Jc3s5h (talk) 00:48, 5 May 2011 (UTC)

Reference section

I notice that the citations in the References section do not follow any consistent format. I also notice that the first edit to incorporate a citation does not follow any standard format, and thus could not be followed because we wouldn't know how to format anything other than a book, where only the author and title were important. So I conclude the citation format is up for grabs. Jc3s5h (talk) 20:16, 15 May 2011 (UTC)

Length of day chart

I am missing the "Length Of Day" difference chart on THIS page. Here is only described, when and how much is the Sun ahead or behind it's noon position at noon, but the "Length of Day" is not charted anywhere... It should probably be the first derivative of your "Equation of Time" chart... Only on "Solar Time" page itself is described, when and how much is the apparent Solar day longer or shorter, but the rather short note there is not sufficient, a chart here or there or at both pages would be preffered... (90.181.83.227 (talk) 15:50, 30 August 2010 (UTC) Semi)

The two online sources cited provide charts of the length-of-day. To place such a chart in a Wikipedia article it must not have a copyright. So a public domain chart is needed, either one drawn before 1923 or one drawn by a Wikipeida editor and released into the public domain. — Joe Kress (talk) 05:05, 31 August 2010 (UTC)

Which links? (None of the titles is obviously relevant to this specific point.) If we can't have a chart, can we have a verbal description of the extremes of the derivative, i.e., when the noon-noon intervals are longest and shortest and what they are? —Tamfang (talk) 20:49, 19 June 2011 (UTC)

the almanac is not part of the definition

“ Results of the equation of time are published in tabular format by authoritative sources (for example, The Astronomical Almanac) or derived from equations originally formulated by various mathematicians. Results of these methodologies generally are tied to Universal Time as kept at Greenwich, England, applicable world-wide at the same moment of Universal Time. ”

I question the value of putting any of this in the lead paragraph, before the definitions of apparent and mean solar time. The EOT is not defined by whether anyone computes it or publishes the computations. Objections to moving it to after the History section? —Tamfang (talk) 01:59, 24 July 2011 (UTC)

Not really- or maybe lowering to the end of the lead. My question is whether most of our readers would benefit from having it there- was it what they were searching for? :There is much to be done to clean up this important article. Two tasks on my list are to check and rewrite the references so they are consistent (that means Harvnb to me) check the Maths to see if they are using the internationally symbol conventions. BSS Time BSS Symbols give some starting points. Good hunting--ClemRutter (talk) 09:20, 24 July 2011 (UTC)

The first citation was added with this edit. Since the format does not follow any recognizable style guide, and thus it would be impossible to decide how to add the wide variety of sources in the article, I would say you are free to clean up the references however you like. Consider {{sfn}} which is similar to the bulk of the present format, but provides one-way links from the short footnote to the bibliography entry.

As for the math, check out my comment dated 5 May in the Mathematical details section above. I would be happy to send you the C# code; just go to my user page, expand the Toolbox section on the left (if necessary), and click Email this user. Jc3s5h (talk) 11:32, 24 July 2011 (UTC)

Definition of EOT

The Wiki page Equation of Time (EOT) states the EOT is the difference between apparent solar time and mean solar time, but then goes on to relate this difference to the longitude of place at the same instant of time.

I question whether relation to longitude of place at the same instant of time is helpful when defining the EOT.

The Astronomical Almanac makes no reference to longitude of place when defining the EOT. The Almanac provides two methods for determining EOT, one method uses a table of values for calendar dates and the other uses an algorithm, both methods determine EOT for an instant of Universal Time (UT), the time at Greenwich, England.

Once EOT is determined for an instant of UT, any observer on the earth can then determine when the EOT would apply to the observer’s time zone; for example, the Eastern Time zone is 5 hours earlier than UT.

While it is correct that the time of UT implies longitude zero degrees at Greenwich, England, I think the Wiki statement as now written suggests that an observer’s longitude anywhere on earth is a causative factor in determining EOT. In other words, as an example, at the same instant of time, an observer at longitude 123 degrees west would conclude EOT would differ from EOT for an observer at longitude 234 degrees west.

But the ‘same instant of time’ should refer to UT time, in which case both observers would determine their respective local times and realize the EOT at the same instant would be the same for both observers. — Preceding unsigned comment added by Onecomment (talk • contribs) 17:30, 11 July 2011 (UTC)

Have a go at rewording it. Our problem here is text written by mathmos tends to get changed over time, as the literati attempt to clarify the prose. --ClemRutter (talk) 17:54, 11 July 2011 (UTC)

The sentence that Onecomment changed said: The equation of time is the difference between apparent solar time and mean solar time, both taken at a given place (or at another place with the same geographical longitude) at the same real instant of time. This does not say the EOT depends on longitude; it says that the two versions of solar time, to be consistent, must be taken at the same longitude — any longitude will do, so long as the same one is used to measure both angles. —Tamfang (talk) 20:37, 22 July 2011 (UTC)

I inserted "at a given longitude" and ClemRutter changed it to "at a single point", with this summary: Not longitude dependent. Both of these are defined previously as local times, not the time of a reference meridian. Again, mentioning longitude does not assert that the EOT depends on longitude, and there was no mention of a reference meridian. Furthermore, if it's wrong to mention longitude then it's at least twice as wrong to specify "a single point" and thus both longitude and latitude. —Tamfang (talk) 22:34, 23 July 2011 (UTC)

It seems apparent that it is not ingrained in many reader's mind that both varieties of time are the same at a given longitude. Indeed, the most common time, civil time, is not necessarily the same at a given longitude due to political considerations in drawing time zone boundaries. So I suggest sticking with ClemRutter's formulation. If there is a need to point out that mean time is the same at a given longitude, and apparent time is substantially the same at a given longitude, that can be done in a different sentence with a suitable explanation. Jc3s5h (talk) 23:06, 23 July 2011 (UTC)

My wording is a helpful fudge. If you write the formula, in the form of a c language function or php (as I will in the next few weeks)., you will see that longitude is not in the input arguments, and it returns a value that will be time difference. The only input is day number, which traditionally is represented on the bottom axis of the graph.

To expand further the value given will be applicable for all points on the planet at that time even if it is 12:00 noon on a watch in London-and 2 pm in Moscow when the value is taken. For the lead this is good enough. Later it may be pointed out that when Boris looks at his watch, he needs to convert his local time to Universal Standard Time before he estimates whether the day number- it he is not using integers, and realise his watch runs two hours fast so he needs to subtract 2/24th of a day - so the value when his watch reads 250.80 days that it should be 250.76.

Practically when using a sundial, you have to make three corrections when taking a reading. Daylight Saving correction + Longitude Correction + and equation of time. The equation of time is the only one here that is longitude independent. As I said earlier, the problem here is how to express the mathematics in terms that the intelligent layman will understand- I only aspire to that skill. --ClemRutter (talk) 09:04, 24 July 2011 (UTC)

I keep saying mean and actual solar time must be measured at the same longitude for the comparison to be meaningful and you keep responding that the equation of time is independent of longitude (and also dragging civil standard time into it irrelevantly) as if you think one assertion contradicts the other; this frustrates me, but convinces me of your point that mentioning longitude invites unnecessary confusion. —Tamfang (talk) 00:14, 3 August 2011 (UTC)

My views on the current definition of the Equation of Time (EOT)

Here is the existing sentence from the first paragraph:

--The equation of time is the difference between apparent solar time and mean solar time measured at a given instant at the same point on the Earth.

It seems to me the first segment of the sentence is all that is needed for a proper definition: (The equation of time is the difference between apparent solar time and mean solar time).

The remaining wording of that sentence is, in my view, not essential for understanding by the average reader, might be confusing to the average reader, and is problematic in its own right, as follows:

First, the use of the word ‘measured,’ the use of this word suggests someone (the user?) goes out to make measurements in order to determine the EOT. While this may be possible given the equipment and technology, etc, I don’t see it happening. I don’t see how the use of this words helps the average reader understand the definition of EOT.

Next, (measured) ‘at a given instant’; an instant is an element of time but is not defined here even though we are talking about kinds of time. Is the ‘given instant’ an element of apparent solar time or of mean solar time (or both) or possibly even of another system of time against which the apparent and mean systems are measured?

More importantly, I think the average reader understanding there are 2 time systems running, as it were, in parallel would realize intuitively the difference between them is merely that, a difference between the two time keeping systems at any point.

Even further, why is it necessary to state ‘at a given instant’ when it would not make much sense to find a difference otherwise. Suppose there are two clocks on a shelf and we know they usually give a different time. To compare the clocks, we would look at both and note the difference, say 30 seconds. The average person is going to do this intuitively and is not going to look at one of the clocks at say, 12 noon, and then wait some period of time to look at the other clock in order to learn the difference spanning the period of time waited.

Last, the wording ’at the same point on the earth,’ this implies there is a specificity per the EOT to the same point on the earth, but there is none as the EOT is everywhere the same on the earth. In fact, the existing paragraph goes on to say this (’that difference is the same everywhere’).

It seems to me this can only leave the average reader scratching his head as to why the EOT would imply a difference at the same point on the earth when the difference at every other point on the earth is the same difference.

Here’s an analogy. We say a man’s home is his castle, implying the man is King of his own home. Fine. Now suppose the man is the King of England. Would it make much sense to say the King of England is king of his own home?

No, it would not. The King of England would be king everywhere in England, not just his own home. All Englishmen will know this.

Just MHO. — Preceding unsigned comment added by Onecomment (talk • contribs) 02:54, 12 August 2011 (UTC)

These are all valid points, and I agree that the definition is unnecessarily complex. Furthermore, while in most contexts "EOT" refers to the Earth's EOT, it does apply to other planets as well. I will make some changes to the opening section. If anyone has issues with what I've done, feel free to discuss and/or revert.

BTW, I don't understand your king analogy or how it applies here. Anyway, thanks for your input. --Lasunncty (talk) 07:02, 12 August 2011 (UTC)

Thank you for making those changes. I withdraw the analogy as not quite a propos. — Preceding unsigned comment added by Onecomment (talk • contribs) 20:07, 14 August 2011 (UTC)

Alternative Approximation

My mistake, the equation is correct. Them darn computers like to use radians and us humans like to think in degrees. Forget my discussion below, all is well with the world. This is a great page. — Preceding unsigned comment added by 24.117.221.82 (talk) 04:41, 6 December 2011 (UTC) Under this heading it states... EoT = 720 * (C − nint(C)) where EoT is the equation of time in minutes... I believe in this equation EoT will be units of seconds. The discussion mentions C ranging from 0, 1, or 2. If this is the case, I believe EoT will be units of Seconds. — Preceding unsigned comment added by Pnplibi (talk • contribs) 04:05, 2 December 2011 (UTC)

No. At the end of the calculation, the Equation of Time is presented in minutes. However, the number C is an angle, in "half turns", i.e. multiples of 180 degrees. The Arctan operation theoretically has multiple values, spaced 180 degrees apart. For example, any calculator will tell you that the arctan of 1 is 45 degrees. But the tan of 225 degrees (45 + 180) is also 1, as is tan(405 degrees), and so on. So why doesn't the calculator give you these other angles when you ask for arctan(1)? Just custom. But for the Equation of Time calculation, these other values are sometimes needed, which means that an integer number of half turns has to be added or subtracted from the value of C that the calculator has generated. In fact, 0, 1, or 2 must be subtracted from C, at different times of the year. The final number of half turns, after this subtraction is done, is multiplied by 720. This is because the earth takes 720 minutes (12 hours) to rotate one half turn, so the result of this multiplication is the Equation of Time, in minutes.

Try it! Any published table of the Equation of Time will tell you that on Oct 28, which is day 300 of the year, counting from zero on Jan 1, the Equation of Time is almost exactly 16 minutes. I actually did the calculation a few days ago, using the method we are discussing, and the result came out to 15.97533 minutes, which is only about 1.5 seconds less than 16 minutes. I think that's pretty good agreement.

DOwenWilliams (talk) 03:10, 7 December 2011 (UTC)

Mathematical detail section again.

Some of the terminology is quaint!

INT(C+0.5) is described as a computer function- does anyone use FORTRAN anymore, or perhaps it was athe original 1964 version of BASIC? Referring the to half-turns for π/2---hardly.

We also have the question of 2.44 or 2.45- both are used at different points.

Keep up the good work.--ClemRutter (talk) 10:23, 23 November 2011 (UTC)

You probably meant 23.44 and 23.45, and π not π/2. Quaint. DOwenWilliams (talk) 15:25, 23 November 2011 (UTC)

Thats why I don't dare to edit the article- :) --ClemRutter (talk) 16:57, 23 November 2011 (UTC)

Yes. Perfection is hard to achieve. It's impossible to make everyone happy. DOwenWilliams (talk) 17:05, 23 November 2011 (UTC)

I was looking at the typographical conventions used and they vary throughout the article.

• item seems to describe a constant

and all formulae seem to start with an indent

declination seems to spelled with a lower case d

I think we must keep the conventions consistent with related articles such as Declination. But I haven't checked. --ClemRutter (talk) 21:39, 14 December 2011 (UTC)

I think we use the lower-case "d" except where we are referring to the title of the Wikipedia article, which definitely starts with a "D".

I'll check about starting formulae with an indent.

I don't understand your first point.

Of course, there is bound to be some variation in conventions because those of us who have contributed to it come from varying backgrounds. My own British-Canadian background makes me write "centre", and put the comma a few words back outside the quotation marks. Other contributors have used U.S. conventions. I don't think we should force each other to adopt conventions that make us uncomfortable. Surely we can accept some variety. So long as clarity is not compromised, obsessive conformity achieves nothing.

DOwenWilliams (talk) 23:12, 14 December 2011 (UTC)

P.S. Ok. I've put indents ahead of formulae. I guess that's an improvement.

Looking further up the article, "points" (coded with asterisks) seem to be used when lists of similar things appear, in order to distinguish them. It has nothing to do with constants. Since we have no such lists in the "Alternative calculation" section, we shouldn't use points, and haven't done so.

DOwenWilliams (talk) 23:41, 14 December 2011 (UTC)

Question on consistency of entry statements

I raise a question with respect to the consistency of the following statements from the current 2nd and 5th paragraphs from the page entry for the Equation of Time:

Second paragraph: ‘Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time average to zero (with zero net gain or loss over the year).’

Fifth paragraph: ‘…its (the equation of time‘s) change from one year to the next is slight.’

The fifth paragraph appears to contradict the 2nd paragraph because how could a slight change from one year to the next result from differences which ‘average to zero’?

In almanac data, I found slight changes in the EOT one year to the next verifying the 5th paragraph, therefore, I question whether the phrase ‘…average to zero (with zero net gain or loss over the year)’ is correct. — Preceding unsigned comment added by Onecomment (talk • contribs) 17:36, 21 July 2012 (UTC)

Strictly speaking the 2nd paragraph is both ambiguous and wrong. It's ambiguous, because it doesn't state which year should have an average of 0; there is the calendar year, the tropical year, the sidereal year, etc. It's wrong because at the most precise level, UT1, which is the current version of mean solar time, is defined in terms of the earth's rotation angle (ERA) with respect to quasars and other radio sources outside the Milky Way, as determined by Very long baseline interferometry. The official definition was created with the position of the sun in the background, but the position of the sun does not appear in the formula. Jc3s5h (talk) 18:50, 21 July 2012 (UTC)

It is completely possible for the EoT to change year to year yet maintain an average of zero each (tropical) year. These statements do not contradict. However, I think there is a better way to describe MST than the statement in ¶2. --Lasunncty (talk) 04:52, 22 July 2012 (UTC)

Discrepacy in text and figure concerning λ_p

The figure 'The celestial sphere and the Sun's elliptical orbit ..." contains a λ_p bow arrow that is appr. 180° smaller than 282.9381° as given in the text. This is confusing and not healed by the hint 'For the sake of clarity the drawings are not to scale'. I suggest the author(s) should coordinate this - including a reversal of the Vernal Equinox arrow.

I think the author(s) are still around so that it would be undue on my side if I twiddled with their wording.

By the way: It would be helpful to introduce arrows into the figure indicating the direction of movement of the apparent and mean Suns. Modalanalytiker (talk) 15:16, 18 November 2012 (UTC)

The Equation of Time - Early Days

This article by Kevin Karney can be used to fill out detail : NASS Journal "Compendium" Vol 16 4 pp23- Dec 2009. On compilation disc (paywall)-- Clem Rutter (talk) 09:30, 29 April 2015 (UTC)

Mediaeval / Medieval

I inadvertently did not include an explanation with this change. An earlier change from medieval to mediaeval gave the reason that it "restored British spelling. 'mediaeval.'" A check with the British corpus on Google Ngram Viewer indicates that medieval has been the dominant British spelling since 1930. A similar result is found for the phrase "medieval Latin." --SteveMcCluskey (talk) 23:07, 2 December 2015 (UTC)

Spellings drift to and fro across the Atlantic. A more significant question is, what spelling was used by the original author of the article? If it was any spelling that is accepted in "correct" English, as listed in dictionaries, then it should be left in that form, without arrogant editing. DOwenWilliams (talk) 00:41, 3 December 2015 (UTC)

I disagree; spelling should reflect currently dominant usage, not the taste of a Wikipedia editor.The Oxford English Dictionary has an entry for medieval, not for mediaeval, although listing mediaeval as a historically attested form. --SteveMcCluskey (talk) 01:20, 3 December 2015 (UTC)

I have a copy of the Concise Oxford Dictionary here. It includes both mediaeval and medieval. However, they are both alphabetically placed among the other entries, which means they are not adjacent to each other. There are about 20 words between them. It is therefore not obvious that they are just two spellings of the same word. DOwenWilliams (talk) 03:48, 3 December 2015 (UTC)

I was referring to the online edition of the Oxford English Dictionary which lists both forms of the word under the single heading medieval. I take the OED as the authority on the language. FWIW, I'm a medievalist and have seldom seen the term mediaeval since the Medieval Academy of America changed its name in 1980. As the Google data indicates, mediaeval is an obsolete usage. --SteveMcCluskey (talk) 04:47, 4 December 2015 (UTC)

How about encyclopaedia (I know one editor who writes "Wikipaedia"), or aesthetics, or aeon, or haemophilia? DOwenWilliams (talk) 16:29, 4 December 2015 (UTC)

What about them? What do such uses have to do with the dominant use of medieval?

I looked at a few categories in Google Search and found the following results:

Source	Medieval	Mediaeval	Ratio
site:.ac.uk	478,000	43,600	11.0
Google Scholar	2,140,000	130,000	16.5
Google	154,000,000	5,860,000	26.3
site:.uk	31,000,000	291,000	106.5
site:.edu	15,900,000	106,000	150.0

Even among British academics, medieval is favoured by 11:1 while American academics favor it by 150:1. Unless you have clear evidence that mediaeval is the preferred use in British English, I will revert to the universally preferred use in both British and American English of medieval.

There is also a useful discussion of the historical decline of æ and œ forms at Wikipedia_talk:Manual_of_Style/Spelling. The point that "there has been a gradual trend towards simplification of æ to e throughout history" seems particularly relevant. --SteveMcCluskey (talk) 22:08, 4 December 2015 (UTC)

Maybe we should spell it "medieeval". That more accurately shows how it is pronounced than either of the common forms. DOwenWilliams (talk) 22:42, 4 December 2015 (UTC)

OK, I changed the two occurrences of mediaeval to medieval, which matches the four instances of medieval further down in the article. --SteveMcCluskey (talk) 02:27, 5 December 2015 (UTC)

Length of day

Someone was arguing with me that the equation of time gives us days of 23 h 45 min at some point, and of 24 h 15 min at another. I know he’s wrong, but we didn’t have much time to discuss, so I couldn’t prove/demonstrate to him that he’s wrong. Can someone help me, for the next time I see him? (He was also playing on the subtlely of “journée” v. “jour” French terms [we’re both francophones], which made no sense to me.) CielProfond (talk) 12:33, 11 February 2016 (UTC)

The accumulated difference between mean time and apparent time reaches about +/- 15 minutes, but there is never that much difference in the duration of a single day. Jc3s5h (talk) 15:12, 11 February 2016 (UTC)

That is what I’ve told him, but he maintains that there is a “journée” of 23:45 and another of 24:15, although he does admit that the “jour” is only a few seconds different from 24:00… I’ll have to talk with him to understand what he means by “journée”! CielProfond (talk) 11:19, 12 February 2016 (UTC)

The longest and shortest solar days, from one solar noon to the next, are about 15 seconds, not minutes, longer or shorter than 24 hours. I would guess that your friend mistook 24:00:15 and 23:59:45 for 24:15 and 23:45. DOwenWilliams (talk) 21:03, 12 February 2016 (UTC)

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Alternative calculation - Typo?

In the alternative calculation section, it says

$C={\frac {A-\arctan {\frac {\tan B}{\cos 23.44^{\circ }}}}{180^{\circ }}}$.

Shouldn't the cos be multiplied instead? Like this:

$C={\frac {A-\arctan(\tan B\cdot \cos 23.44^{\circ })}{180^{\circ }}}$,

as it seems to corresponds to the equation "α = tan⁻¹(cos ε tan λ)" (longitude to right ascension conversion) above. --Cebus (talk) 20:05, 21 July 2017 (UTC)

Calculation section is way too complicated

There is no need to go on for three pages when this could easily be described in a paragraph. The basic equations are buried among tons of unnecessary information that make it very hard to follow, even for someone who knows how to do the calculation. --Lasunncty (talk) 03:20, 9 August 2017 (UTC)

Semi-protected edit request on 19 September 2017

The text "the path of the Sun seems to take in the ..." is grammatically incorrect. I believe the text was meant to say "the path the Sun seems to take in the ..."

Darc-gray (talk) 15:54, 19 September 2017 (UTC) Darc-gray (talk) 15:54, 19 September 2017 (UTC)

Done Primefac (talk) 15:57, 19 September 2017 (UTC)

Semi-protected edit request on 9 October 2017

In the section "Remark on the continuity of the equation of time" the formula to get the arctangent nearest a specific value is incorrect.

The x inside the round function has to be changed to arctan x:

arctan_η x = arctan x + π round (η − arctan x/π)

See the German version for comparison, including a graph: https://de.wikipedia.org/wiki/Arkustangens_und_Arkuskotangens#Arkustangens_mit_Lageparameter 217.6.20.163 (talk) 13:28, 9 October 2017 (UTC)

Done Primefac (talk) 13:53, 9 October 2017 (UTC)

Refactor of the Code Snippet

Cleaned up the code and made it more readable:

• changed the Double objects to use the double type instead
• made the variable names more meaningful
• made the coding style more uniform

BEFORE:

/*
    Equation of Time calculation
    *** No guarantees are implied. Use at your own risk ***

    Written by Del Smith, 2016-11-29

    Based on "Equation of time" WikiPedia article as of 2016-11-28
    (which describes angles in a bewildering mixture of degrees and radians)

    It appears to give a good result, but I make no claims for accuracy.
*/
// constants  (all angles as radians)
private static final Double pi = atan(1.0)*4;             // tan(pi/4) = 1  (45 degrees)
private static final Double lambda = 23.44 * pi / 180;    // Earth's inclination in radians
private static final Double omega = 2*pi/365.25;          // angular velocity of annual revolution (radians/day)

// calculate the Equation of Time from 'day of year'
private static Double EqnOfTime(int day_number) {
    // the result is in seconds
    // add to the Mean Solar Time  (UTC + Longitude*240 sec.) to get Apparent Solar Time
    // D                                                          day of the year
    // A = W*(D+10)                                               angle of revolution (circular)
    // B = A + 0.0333*sin(W*(D-2))                                angle of revolution (elliptical)
    // C = ( A - atan[  tan(B) / cos (inclination) ] ) / 'pi'     angular correction
    // EOT = 43200 × (C − int(C+0.5))                             Equation of Time in seconds
    Double delta = Integer.valueOf(day_number).doubleValue();
    Double alpha = omega*((delta+10)%365);                              // angle in (mean) circular orbit, solar year starts 21. Dec
    Double beta = alpha + 0.0333*sin(omega*((delta-2)%365));            // angle in elliptical orbit, from perigee 3. Jan (radians)
    Double gamma = ( alpha - atan( tan(beta) / cos(lambda))) / pi;      // angular correction
    return (43200. * (gamma - Double.valueOf(gamma+0.5).intValue()));   // equation of time in seconds
}

AFTER:

/*
    Equation of Time calculation
    *** No guarantees are implied. Use at your own risk ***

    Originally written by Del Smith, 2016-11-29
    Rewritten by xxc3nsoredxx for improved readability, 2017-08-06

    Based on "Equation of time" Wikipedia article as of 2017-08-6

    It appears to give a good result, but I make no claims for accuracy.
*/
// Use the constant for pi defined in java.lang.Math
private static final double meanVelocity = 2 * Math.PI / 365.24;    // Mean angular orbital velocity of Earth in radians/day
private static final double obliquity = 23.44 * Math.PI / 180;      // Obliquity of the Earth (tilt) on it's axis;

// Calculate the Equation of Time from 'day of year'
// day represents the day of the year with 0 being January 1
// Returned is the value for the Equation of Time in seconds
public static double equationOfTime(int day) {
    // The returned result is in seconds
    // Add to the Mean Solar Time  (UTC + Longitude*240 sec.) to get Apparent Solar Time

    double approxAngle = meanVelocity * ((day + 10) % 365);                                       // Angle the Earth would move on it's orbit at mean velocity from the December
                                                                                                  //     solstice until the day specified
                                                                                                  // 10 comes from the approximate number of days from December solstice to January 1
    double correctedAngle = approxAngle + 0.0334 * Math.sin(meanVelocity * ((day - 2) % 365));    // Angle the Earth would move from solstice to specified day, with a correction
                                                                                                  //     for Earth's orbital eccentricity
                                                                                                  // 2 is the number of days from January 1 to Earth's perihelion
                                                                                                  // 0.0334 is the angle provided in the article (1.914 deg) in radians
    double angleDifference = Math.atan(Math.tan(correctedAngle) / Math.cos(obliquity));           // Difference between the angle moved at mean velocity and at the corrected velocity,
                                                                                                  //     projected onto equatorial plane
    angleDifference = approxAngle - angleDifference;                                              // Split into multiple lines for better readability
    angleDifference = angleDifference / Math.PI;                                                  // Divided by pi to get the value in half-turns
    return 43200 * (angleDifference - (int)(angleDifference + 0.5));                              // Return the Equation of Time in seconds
                                                                                                  // Corrected for any excess integer count half turns
                                                                                                  // 43200 comes from the number of seconds it takes for Earth to complete a half turn
                                                                                                  //     (12h * 60m * 60s)
}

Xxc3nsoredxx (talk) 06:48, 7 August 2017 (UTC)

Done Though I included in the edit summary requesting a revert if any other editor did not see the refactored code as an improvement. Thanks for improving. jd22292 (Jalen D. Folf) (talk) 16:58, 8 August 2017 (UTC)

I've removed this code entirely from the article -- the wiki is a particularly bad place to put code, for a number of reasons. From a purely aesthetic perspective, it displays poorly. Then there is the fact that, unlike prose, it is not comprehensible to most readers which means that errors can be introduced easily and persist for lengthy periods of time. It also quite frequently invites partisans of other programming languages to post their own versions, which can result in large parts of articles devolving into competing implementations of algorithms -- this is a very frequent problem on technical pages.

Sites like github or even wikisource are better suited to this sort of thing, and we can certainly reference them in the external links. But WP really isn't and shouldn't become a repository of source code. In articles (such as those detailing algorithms) that merit an explicit implementation, this typically done (as it is in the literature) in pseudocode rather than in a particular programming language. This choice future-proofs the article somewhat as the popularity of programming languages tends to wane over time, and it also shields the implementation from the vagaries of a syntax and formatting that are necessary for a computer but which do little to help a human understand what is being done. In the case of our article I don't really see how even a pseudocode implementation would significantly add to the technical explanation already in the article. Eniagrom (talk) 11:58, 20 December 2017 (UTC)