Ancient sky watchers had noticed that the motion of the Sun and Moon across the sky has some variations. They did not realize that the Earth revolved around the Sun and the Moon around the Earth and that the Earth was also rotating on its axis, so they could not know WHY these variations occurred. We now know the reason for our seasons and the Phases of the Moon and many other such things.
But as a Theoretical Physicist, I find it annoying when really poor quality information is provided to the public! It was actually possible to create a calculator with even far greater accuracy than this calculator is set up to do, nearly in the millionth of a second range, but even Physicists recognize overkill! So it was decided that a reasonably reliable accuracy of around one one-hundredth of a second, valid for several thousand years either in the past or future was sufficient for this presentation. If you need greater accuracy, derive it yourself!
Prior to around 500 years ago, there were not yet any mechanical clocks and so daily life was generally based on the apparent motions and positions of the Sun and Moon in the sky. An interesting detail is that they decided to DEFINE an interval which we now call an HOUR as being 1/12 of the DAYLIGHT period on that day. The result was that an hour in the summer was MUCH LONGER than an hour in the winter! But intervals of time were not yet that important to anyone! Ancient Greeks had invented crude water clocks, which therefore obviously had some major problems of their own, but they also discovered that there were some other variations, such as where the Sun seemed to cross the Meridian Circle ahead of or behind where they might have expected it to have been at noon, over a range of what we would now call over half an hour. (about February 12 compared to November 3). When clocks were invented around 500 years ago, and then greatly improved in accuracy around 250 years ago, the design of those mechanisms were all based on processes that occurred at precisely accurate and repeatable intervals. And so it was natural that the length of a day or hour or year was DEFINED as a specific interval of time. This was a wonderful advance but it caused some minor errors when it was used to try to ACCURATELY calculate where the Sun or Moon might be at any moment, specifically at noon! In fact, around November 3 of each year, the two clocks are different by more than 16 minutes! Around February 12, they are different by about 14 minutes in the opposite direction! The fact that Mariners needed to accurately know the time in order to determine their Longitude and Latitude, made this a serious issue to deal with! Since that was then the only use of really accurate timekeeping, the number that we now call the Equation of Time was initially defined as always being in one direction, based on the operation of ships. Much later, when accurate time became useful for business and commerce, the current definition was adopted, where the True Sun is sometimes ahead of and sometimes behind clock time. We will call Mean Solar Time as being what we call clock time (with Mean essentially meaning Average), and we will call Apparent Solar Time as being the time which should be true based on the apparent location of the Sun at that instant. In a more technical way, we define the first as being the Hour Angle of the Mean Sun and the second as the Hour Angle of the True Sun. The Mean Sun is fictitious, and is defined as moving with uniform angular speed along the Celestial Equator. These can equally be defined as the Right Ascensions of the Mean Sun and the True Sun. The Equation of Time continuously varies throughout the year, from the combined effect of two primary causes. (There ARE many very tiny effects which also exist, mostly due to gravitational perturbation effects of other planets, and some interesting consequences of the Moon's motion, but they are generally very tiny effects and mostly considered to simply be perturbations of the two primary effects, which we will discuss later.) A couple centuries ago, the definition was opposite the sign of what is currently defined, which causes some confusion regarding the correct sign. According to modern definitions, if the Equation of Time is a negative number, that is an indication that the true Sun has moved AHEAD of the Mean Sun, and therefore a sundial would give a time that was FAST, and so the needed CORRECTION has to be positive to get to Clock time. The two primary causes of variation are the varying speed of the Earth in its orbit due to the eccentricity of that orbit (and Kepler and Newton's analysis of the speed variations which result); and due to the fact that the Earth's rotational axis is tilted to the plane of its orbit, which is called the Obliquity of the Ecliptic, which is essentially a geometric factor. We will consider them separately.
Effects of EccentricityThis is clearest if we define an imaginary Intermediary Sun, as moving in the ECLIPTIC with the MEAN speed of the True Sun. We further define this as coinciding with the exact position of the True Sun when the Earth is at perihelion (around January 3).The apparent motion of the True Sun is in accordance to the Earth moving in an ellipse and in therefore obeying with Kepler's law of equal areas. The effect is that the apparent True Sun moves in an equivalent ellipse, with the Earth located at one of the foci of that ellipse. Near perihelion, the True Sun has GREATER angular velocity than the Intermediary Sun (due to that law of equal areas), and the True Sun therefore quickly passes and moves ahead of the Intermediary Sun. This makes the Equation of Time a positive quantity at that time, per the reasoning above. The difference between the angular positions of these two Suns continues to increase, until the point where the Sun's actual apparent speed has dropped to its mean value (again, the law of areas, now where the radius of the orbit has become its mean value). This occurs when the Earth is near the end of the minor axis of its orbit, around the first few days of April each year. At this time, the contribution of Eccentricity to the Equation of Time is at its maximum negative value. After this date, the True Sun continues to slow up (again in accordance to the law of equal areas) until when the Earth is at Aphelion, it has slowed to the point of exactly then again being at the same point in the sky as the Intermediary Sun. This obviously means that the Eccentricity contribution to the Equation of Time is exactly zero then, about on July 4 of each year, at Aphelion. The second half of the orbit has the True Sun LAGGING the Intermediary Sun in essentially similar ways. The maximum lag occurs around October 3 of each year, where the Eccentricity contribution to the Equation of Time is most positive. For an orbit that has great eccentricity, this mathematical problem is actually immensely difficult! In fact, it is referred to as Kepler's Problem, and no one has yet solved it with precise accuracy! There have been several solutions to Kepler's Problem presented, such as by LaGrange in around 1770. In 1816, Bessel discovered the exact form of the solution (as an infinite series). A number of computer-calculated solutions, generally based on Bessel's analysis, are pretty accurate now. The fact that the Earth has a rather circular orbit allows many simplifications in getting an APPROXIMATE solution for our purposes. Putting that another way, Bessel's infinite series converges very quickly because of the nearly circular orbit. Keep in mind that the Eccentricity and Obliquity are constantly changing and oscillating (as per some graphs shown below), so for long term calculations, lots of new complications enter the picture! As a first approximation, we know that when at Perihelion, the Earth is closer to the Sun by the eccentricity of 0.017, in other words, 0.983 its average distance. Given Kepler's Law of equal areas, we can easily determine the maximum velocity of the Earth when at Perihelion. At Perihelion, the very narrow triangle is SHORTER by that factor and so it is therefore WIDER by the same factor, to have the same area. That means that the velocity of the Earth is then 1.017 (actually, 1/0.983) times its average velocity. So during about the first day after Perihelion, when the Intermediary Earth proceeds exactly one (Mean) degree in its orbit, the True Earth must advance 1.01729 degree, or about 1°1'2". But this angular distance is along an instantaneous circle of a slightly smaller radius, so in just the first day after Perihelion, the True Sun advances a distance which represents about 7.9 seconds of time FARTHER than the Intermediary Sun does. This effect obviously reduces to zero by about 91 days later, and we might estimate that the path was close enough to being circular to allow some simplifications, such as assuming that this dependence is exactly sinusoidal. That is not technically true, but it is pretty close, and this results in a MAXIMUM amount of the Eccentricity contribution at about seven and a half minutes of time. General form of Bessel's solution;
E is the Eccentric Anomaly and M is the Mean Anomaly. It is also commonly referred to as a Kapteyn series. Bessel's method can be used to obtain a more accurate value, and that gives a maximum contribution of the (current) Eccentricity as being 7^{min}38.3^{sec}. (Keep in mind that the eccentricity of the Earth's orbit is constantly slowly changing, and so this value also therefore changes over centuries. It is interesting to see the precise value of the Eccentricity change even over a period of days, as is seen in the Calculator below. Keep in mind also that the other planets cause perturbations which cause our orbit to slowly be rotating around, and so our Line of Apsides, which defines the location of our Perihelion also moves around slowly [advancing], so the day and hour we might assign to Perihelion passage gradually change. These comments are made to remind the reader that for the Equation of Time to be accurately applied to distant times in the past or future, then corrections to those orbital parameters must be made.) This effect is fairly close to being sinusoidally shaped, so the Eccentricity contribution is sometimes (approximately) given by the sine of the (day number minus three [to identify where in the orbit we are as compared to the location of our Perihelion, assuming that the Perihelion occurred on January 3 ] divided by 365.25 times 360), times the maximum time contribution we just derived. (We will see this factor in the CRUDE formula for the Equation of Time which seems to get used these days, and which we placed at the very end of this presentation!)
Long-Term Changes in the Eccentricity of the Earth's OrbitHere is a graph which gives accurate values of the Earth's orbital eccentricity for much longer time scale than anything else discussed here, a million years before and after today.
(Jacques Laskar 1986, 1999 and 2004) We are currently headed for a minimum eccentricity (nearly circular) which should occur around 29,000 AD, when the eccentricity is then around 0.0023, around 1/7 of the current eccentricity.
Effect of the Obliquity of the EclipticThis is an effect that is essentially geometric rather than astronomical.The above discussion shows that the Intermediary Sun does NOT move at a uniform rate around the Ecliptic, but for the moment, ignore those Eccentricity effects and assume that it does. The Mean Sun, upon which our clock system is based, moves at constant speed along the Celestial Equator. We will additionally assume that these two (both imaginary) Suns would be in the exact same spot at the instant of the Vernal Equinox (around March 21). Given that we intend to focus on the projection of both on the Celestial Equator, and the Mean Sun is already there, it is clear that the AVERAGE speed of the Intermediary Sun is necessarily FASTER than that of the Mean Sun. (The total distance it has to cover, along the Ecliptic, is longer than the projected distance along the Equator, by a simple geometric effect.) This fact has some importance. Imagine the situation a brief time (here just over one day) later. The Intermediary Sun will have moved 1° at its constant speed along the SLANTED Ecliptic, which would have a Right Ascension change which is slightly less (due to the cosine of the angle of the Ecliptic to the Celestial Equator.) It is the PROJECTION of the movement along the Ecliptic ON the Equator. The Mean Sun, even though it is actually moving slightly more slowly, will have moved 1° ahead along the Equator. This results in the Mean Sun going ahead of the Intermediary Sun. This causes a NEGATIVE contribution to the Equation of time at this moment. During that first degree of travel, approximately one day, the total effect of these TWO factors (the difference in the average speed of the two and the projection losses of the Mean Sun due to the angle of the Ecliptic) is about 20.3 seconds of equivalent time. Consider about three months later, at the Summer Solstice. BOTH have necessarily advanced by exactly 90°, one along the Celestial Equator and the other along the Ecliptic. They are now moving in exactly parallel paths, parallel to the Equator, and so they have gotten to the exact same Right Ascension. They are NOT now moving at the same speed, with the Intermediary Sun necessarily moving somewhat faster (gaining around 20.3 seconds of equivalent time each day) (for having to travel the larger total path length by this analysis). After this moment, the Intermediary Sun starts moving back downward along the angle of the Ecliptic, and so the Mean Sun starts gaining on it, which causes a POSITIVE contribution to the Equation of Time, which continues for about three months, until the Autumnal Equinox. Their Right Ascensions are again identical then, and in fact they would be in the same spot in the sky. The second half of the year repeats this pattern. It results in FOUR times when the Obliquity of the Ecliptic does not result in any contribution to the Equation of Time (at the instants of the Equinoxes and the instants of the Solstices), and two periods of positive contributions and two periods of negative contributions. ROUGHLY, the first and third quarters of the year have negative and the other two quarters have positive contributions. This curve shape is also fairly close to being sinusoidal, with the acknowledgement that there are many tiny effects that we have ignored in this simplified analysis. Therefore, in CRUDE formulas for the Equation of Time, a fairly simple single term can be calculated, by simply determining the entire annual path length and dividing by 365.25 to determine the AVERAGE Mean Sun speed and the AVERAGE Intermediary Sun speed, and then calculating the geometric factor to know the projected length along the Celestial Equator of the motion of the Intermediary Sun along the tilted Ecliptic. (This term is also seen in the CRUDE formula given at the end of this presentation!). This all results in a greatest contribution to the Equation of Time from the Obliquity component as being (currently) around 9^{min}52.5^{sec}.
Long-Term Changes in the Obliquity of the Ecliptic(Jacques Laskar 1986 and 1999) Laskar has calculated that there was a maximum of the Obliquity of 24°14'07" in about the year 7530 BC and that there will be a minimum of the Obliquity of 22°36'41" in about the year 12,030 AD. These values are in close agreement with what this calculator produces.
Geographic LongitudeThis is actually not an astronomical or geometric effect, but one due to the fact that people over a large area want to use clocks which agree with each other! And so Time Zones were developed, where the official civil time is generally that of the Meridian which is divisible by 15 degrees. For example, Central Standard Time in the USA is determined for a location at exactly 90 degrees West Longitude.If your sundial is located at exactly 92 degrees West Longitude, then it is exactly eight minutes later that the Sun crosses your Meridian, which means that an 8^{min}0.0^{sec} adjustment must always be made for that location. In the Summer, an hour correction may also be necessary if the Civil Time uses Daylight Savings Time.
Leap YearsThe fact that the Earth rotates about 365.25 times each year, means that the calendar has an automatic shift of one-fourth day each year. This does not affect the size or shape of the sinusoidal curve, but it DOES cause it to be shifted 1/4 day each year, and then the leap year day fixes that and gets it back to the original. So every four years, the pattern is very close to the same, but in the intervening three years, the Equation of Time curve gets shifted by part of one day, which can therefore cause a difference of up to about 15 seconds or so.The reality is that this is actually irrelevant! The SIZE of the image of the Sun is large enough to take about two minutes to pass any point in the sky, so the shadow cast by a (normal) sundial CANNOT be very sharply defined! It is always fuzzy-edged, with a resulting sundial accuracy limit of at best about half of one minute either way. Of course, it is possible to use a lens in a much larger sundial structure, and then only pay attention to the leading edge of the Sun's image, and for such a device, these smaller corrections can become important.
Moment of PerihelionI decided that it was potentially appropriate to include a precise calculation for the moment of perihelion each year, but concluded that there are so many variables involved that it is probably not worth the trouble! But that very fact can be instructional.We know that perihelion occurs early in January each year, defining the instant when the Earth is closest to the Sun in its orbit. The date shifts back and forth several days due to the Leap Year effect, which is actually due to a quirk in our calendar system. A useful formula for determining the approximate date of perihelion for the Earth is: JDE = 2451547.507 + 365.2596358 * k - 0.0000000156 * k^{2} where k is an integer. If k is less than 0 then the perihelion found will be prior to the year 2000 AD. If k is 0 or greater, then the perihelion is after the beginning of 2000 AD. For example, if a value of k = -10 is chosen, the JDE from this formula is 2447894.911 which corresponds to January 3.41, 1990. This is moderately accurate for the perihelion of the BARYCENTER of the Earth-Moon system, but in that year, the positions of the Earth and Moon were such that the correct instant FOR THE EARTH was actually around 31 hours later! There ARE corrections which can be made based on this orbital motion of the Moon, which can involve as few as five extra terms, and which then provide an answer which is only one hour off for 1990 and generally around three hours error on average. Those corrections can be made far more involved, to also correct for the eccentricity of the Moon's orbit, the Regression of the Nodes of the Moon's orbit and the advance of the line of apsides of the Moon's orbit, to calculate values which are within around one minute, but there are many hundreds of terms which must be calculated in those corrections. However, there are MANY other (smaller) effects which also affect the precise accuracy of ALL of the numbers discussed in this presentation. An interesting example is that both Jupiter and Saturn have enough mass to affect the exact position of the Sun during their orbits, by the effects of each of their separate barycenters with the Sun. Specifically, there are years when Saturn and Jupiter are in Conjunction, where they are both on the same side of the Sun and therefore their contributions to their mutual barycenter cause the position of the Sun to have to shift a maximum amount the other way. There are other times when they are in Opposition where their effects on their mutual barycenter compete and the position of the Sun is nearest the actual barycenter of the Solar System. This sort of effect takes around 20 Earth years to cycle, and it occurs at different times of the Earth's orbital motion, so there is a longer-term, somewhat complex dance of an effect on the Earth's exact perihelion time! Neptune has an even more strange situation regarding this effect, where the barycenter effects due to Jupiter and Saturn often cause Neptune to have TWO perihelions per orbit, and they can be around ten years apart, both being around five YEARS off from the simplified calculation of the perihelion! This last happened for Neptune around 1880. Additionally, these equations are somewhat simplified in assuming that the planets have unperturbed elliptical orbits, that they are not each all affecting all the other planets' orbits. If the effects of all the thousands of different perturbations are included for the Earth, the calculations become extremely complex. Even relatively simple corrections require many thousands of terms in the calculations. Just listing the values of the multipliers for the terms takes around 60 published pages! For the purposes of this discussion, that did not seem necessary or appropriate here. And finally, there are various other interesting effects! For example, in 1246 AD, the Earth's perihelion coincided with the Winter Solstice, and the Equation of Time curve was exactly symmetrical! Equation of Time CalculatorThe (modern) Gregorian Calendar system is used here. For dates before 1582 AD, that calendar did not exist! It is necessary to use the Julian Day Number value given, and convert that to the calendar system which existed when and where your interest lies. For example, the displayed (Gregorian) dates for the Vernal Equinoxes around the time of Christ are therefore incorrect, but the Julian Day Numbers displayed are correct. This program calculates the Obliquity of the Ecliptic accurate to within 0.01 second of arc for 1000 AD to 3000 AD, and within a few seconds of arc for 8,000 BC to 12,000 AD. It calculates the Eccentricity of the Earth's orbit accurate to 9 places in near times (1000 AD to 3000 AD) and to 8 places within 10,000 years either way (8,000 BC to 12,000 AD). This calculator is based on the work of Jean Meeus and Jacques Laskar. Corrections are made for leap years and leap days, and also for many effects of Nutation in Obliquity. The Equation of Time, at Noon on the day chosen, should be accurate to about 0.01 second, within a thousand years and fairly accurately beyond that. Correction still needs to be made for the Geographic Longitude of a sundial and for Daylight Savings Time. This calculator automatically sets for today's date. Click on the year, month or date to then change it for any other date and then click Calculate. Use a negative year number to indicate a BCE date.
The calculated times of the beginnings of the seasons, corrected for Nutation and other effects, should be accurate to within a few seconds for most historical times. There are a LOT of effects which can cause the calculation of the apparent longitude of the Sun to be affected, and even one arc-second difference in that longitude results in a time difference of around 24 seconds! The Julian Day Number is given such that a more accurate value could be expressed if needed. HOWEVER, the calculated results here, involving the position of the Earth in its orbit regarding the start of seasons, have actually calculated the precise location of the BARYCENTER of the Earth-Moon system and not the actual location of the center of the Earth (or of any specific location on the surface of the Earth). For this reason, the precision of the timing of season beginnings, that is the Solstices and Equinoxes, can be affected by this factor. Therefore, depending on the Phase of the Moon (in other words, the relative positions of the Earth and Moon to our Barycenter), the values given above for the Season starts can be slightly off. Due to this effect of the Earth-Moon system, the (center of the) Earth can be as much as 2900 miles in front of or behind the barycenter. And then, depending on what time of day, a person on the surface of the Earth can be another 3950 miles ahead of or behind the Center of the Earth. At an Earth average orbital velocity of around 18.5 mi/sec, this can cause the Center of the Earth to have a variation of more than two minutes! MOST of the greater components of this effect are included in these calculations, but the Moon has a rather elliptic orbit which is also continuously having its perigee moving along! Some simple additional corrections regarding the Moon Phase and perigee changes should be pretty easy to add if better accuracy is required here. And if severe accuracy is needed, the VSOP87 program has several thousand mathematical components in that correction. The Naval Observatory calculated values for the moment of the beginning of the seasons seems to often be about a minute different from the values calculated here. I think that the reason for this is a choice made by the Naval Observatory. I think they may calculate everything FOR THE LOCATION OF THE NAVAL OBSERVATORY, and that is NOT identical to the location of the Barycenter of the Earth-Moon system (deep inside the Earth) which we calculate for. We see no reason to correct values for the position of the Naval Observatory, and we think this approximate one minute effect of Time Zones and Latitude might be the cause of this difference.
The following sort of defeats the central point of this presentation of extreme accuracy, but it is clear that there are people who may want to have a CRUDE formula to be able to calculate an APPROXIMATE value of the Equation of Time, rather than having to use this very sophisticated set of equations. The following simple formula can generally give a value that is within about half a minute of being accurate (where the above Calculator determines a value that should be accurate to around 0.01 second). E = 9.87 * sin(2*B) - 7.53 * cos(B) - 1.5 * sin(B) where B = (360/365) * (N - 81) where N is the day number in the year, with January 1 meaning N = 1. The formula above has to use radians and not degrees for the angles of this calculation. This gives E, the Equation of Time, APPROXIMATELY, in decimal minutes. The first term here is due to the Obliquity factor and the other two terms are due to the Eccentricity factor. But those are only APPROXIMATE calculations, which do not include any of the many effects due to the Moon's complex orbit or various other Nutation effects. NOTE: For a while, the equations for this true calculator had a peculiar error for the Equation of Time for the specific dates of March 21 and March 22 of each year! It actually had calculated it correctly, but it then displayed it as 'one day off' or, in other words, with an error of 1440 minutes! I am pretty sure I finally fixed that bug in the program, and it should now always give the exact right Equation of Time! Sorry for the inconvenience!
This presentation was first placed on the Internet in January 2009.
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago |