Talk:Equation of time

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[edit] Discussion

Notice the difference between this:

{3 \rm{hr} 56 \rm{min} \over 8}\times 2

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

{3\ \rm{hr}\ 56\ \rm{min} \over 8}\times 2

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:

28\ \rm{deg}

to this:

28^\circ

"Displayed" TeX should normally be indented; thus this

3\ \rm{hr}\ 56\ \rm{min}

differs from this:

3\ \rm{hr}\ 56\ \rm{min}

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

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)

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)