Science and the garden. Clockwork compared to organic growth.
The equation of time is the difference, over the course of a year, between the time as read from a sundial and a clock.
How Do They Compare?
I'm sure some of you know all this--the science grads learned astronomy, I think. The basics at least. It seems (to moi) that geometry, trigonometry, all the maths sciences must come from astronomy.
Or, taking the organic approach:
Stand up and look around. Which way is north? Which way is east?
Face the east; the direction in which you find the sun in the morning.
Make your first bow: you are moving....
The sundial can be ahead (fast) by as much as 16 min 33 s (around November 3) or fall behind by as much as 14 min 6 s (around February 12).
It's about regularity, exactitude of one action as it reacts--with exactitude--to others. Action?
It results from an apparent irregular movement of the Sun caused by a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit.
"irregular movement of the Sun"--compared to the movement of a model sun, a mechanical few-dimensional model in which many observable effects are absent.
Observation
The equation of time is visually illustrated by
an analemma.
Analemmas
Due to the earth's tilt on its axis (23.45°) and its elliptical orbit around the sun, the relative location of the sun above the horizon is not constant from day to day when observed at the same time on each day. Depending on one's geographical latitude, this loop will be inclined at different angles.
Plotting the analemma with the width exaggerated shows that it is slightly asymmetrical due to the misalignment of apsides and solstices.
See equation of time for an in-depth description of the horizontal characteristics of the analemma.
http://en.wikipedia.org/wiki/Analemma
The equation of time describes the horizontal characteristics of an analemma.
Apparent Time versus mean time
The irregular daily movement of the Sun was known by the Babylonians, and Ptolemy has a whole chapter in the Almagest devoted to its calculation (Book III, chapter 9). However he did not consider the effect relevant for most calculations as the correction was negligible for the slow-moving luminaries. He only applied it for the fastest-moving luminary, the moon.
Until the invention of the pendulum and the development of reliable clocks towards the end of the 17th century, the equation of time as defined by Ptolemy remained a curiosity, not important to normal people except astronomers. Only when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, did the difference between clock time and solar time become an issue. Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial, while mean solar time is the average as indicated by clocks.
http://en.wikipedia.org/wiki/Equation_of_time
So...the world was running on "apparent" time until the clocks became accurate enough, in Europe, in the period 1660-1700, to take over. Before then, the equation of time--as a problem--was only seen by astronomers.
But here comes the clock. Tick tock.
No more sundials. We're on mechanical time now. Mean time.
(A vision of the well-tempered harpsichord...and the evenly tempered pianoforte.)
Until 1833, the equation of time was mean minus apparent solar time in the British Nautical Almanac and Astronomical Ephemeris. Earlier, all times in the almanac were in apparent solar time because time aboard ship was determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. Since 1834, all times have been in mean solar time because by then the time aboard most ships was determined by chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of adding the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign.
(You're with me...we're hanging in there...)
No one needed mean solar time. Correction: at least someone needed, wanted, or both needed and wanted mean solar time. And it came, with ships' clocks. The clock was the time source; they could now begin to forget about the sky--unless it interested them. It was no longer necessary, and because of flips here and twists there, the sky can sometimes be wrong, compared to clock time.
As the daily movement of the sun is one revolution per day, that is 360° every 24 hours or 1° every 4 minutes, and the sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, clearly the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summertime, if any.
The sun dial is waaaaay out! Well, by up to fifteen minutes either way. Only at certain times, though. Over the course of the year it hits two points where it is spot on--to within a minute.
How a Sundial Works
Equatorial or Equinoctial sundial
The simplest sundial is a disk mounted on a bar. The bar must be parallel to the Earth's axis of rotation. The disk forms a plane parallel to the plane of the Earth's equator. The disk is marked so that one edge of the shadow of the bar shows the time as the Earth rotates. Usually noon will be at the bottom of the disk, 6AM on the western edge, and 6PM on the eastern edge. In the winter, the north side of the disk will be shaded, and hard to read. In the summer, the south side will be shaded.
http://en.wikipedia.org/wiki/Sundial#Garden_sundial
Garden Sundial
The classic garden sundial uses the same principle, except the lines of the disk are projected, using trigonometry, onto a face that is parallel to the ground. The advantage of the garden sundial is that it keeps time all year, and its face is never completely shaded in the daytime (as vertical sundials are). For use in a public area, this sundial can be made visible by placing it in a square, or making the face of frosted glass, elevated high in the air, and visible from underneath. The top edge of the gnomon is parallel with the axis of the Earth's rotation. The shadow will cross time markings on the face.The markings of each edge are aligned with the edge of the gnomon that produces the shadow. The angle of the face markings from the root of the gnomon (the substyle) are calculated from the formula:
face-angle = arctan(sin(latitude)*tan(hour-angle))
The angle of the style:
style height = latitude
(See Logo programming language for a sample program to draw a garden sundial)
Here come the tech guys and gals with their calculations. We can work this out!
But our problem wasn't that the sundial in the square wasn't accurate.
The high priests of where we've been, where we are, and where we're going. They need accurate measurements.
As though the siesta were a bad thing if its length varied by up to, what? Ah, yes. Up to thirty minutes over a given hour. (Spread over the year, and on either side.)
The Sun
That big ball of love that keeps radiating us with life-giving rays...
Someone calculated how long large mammals would survive if the sun were suddenly switched off. The measurements were in months; not days, not years, certainly not decades.
The sun! Stare at it and you'll go blind.
Whereas clocks. Tick tock. The sun. Wander wander. Too bright to observe directly with the naked eye. But there are ways....of observing it without the naked eye. Such as a sundial.
The sun does not move along the celestial equator but rather along the ecliptic. At the equinoxes part of the yearly movement of the sun appears as a component in the change in declination, leaving less for the component in right ascension. The sun slows down by up to 20.3 seconds every day. At the solstices, on the other hand, all yearly movement is in right ascension only, but at this declination, 23.4°, the meridians are closer together, which speeds up the sun by the same amount. The inclination of the ecliptic results in the contribution of another sine wave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points are reached on the equinoxes and solstices, while the maxima are at the beginning of February and August (negative) and the beginning of May and November (positive).
The maths can be done to approximate this living system, by which I mean it is happening now and we're part of it, and we're alive, yeah!
But the maths of clockwork is easier, and therefore...
...I wanted to say cheaper.
More manageable?
What terrible things would happen if we all worked on solar time?
Well, everyone on the planet would have a different time...there would be no common time.
No easy means of communicating the time, or...clocks create an easier method of communicating time and can be calibrated more accurately, from which fact more accurate observations of the world around us can be made.
The clock separates time from the sun. Time is local, but "time"...the tick tock...the tick and the tock are universal constants?
That tock follows tick.
The exact shape of the equation of time curve and the associated analemma slowly changes over the centuries due to secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but in reality they vary up and down over a timescale of hundreds of thousands of years. When the eccentricity, now 0.0167, reaches 0.047, the eccentricity effect may in some circumstances overshadow the obliquity effect, leaving the equation of time curve with only one maximum and minimum per year.
Yes, time will outlive us. But without us: who cares about mechanical time? Not the sun. Not our galaxy.
It is a human invention to replace, for various practical reasons, solar time as measured on sundials, to mark out the passing of a day.
Months are funky as you like. Twenty eight days, thirty days, thirty one days. Thirty days hath September; April, June, and November. All the rest hath thirty one, except for February alone, which has twenty eight.
I'm remembeing that badly, I'm sure.
Back To The Equation of Time
Practical use
If the gnomon (the shadow casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will (usually) be the conic section of the hyperbola, since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an "analemma". By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined.
You want to know the time? Buy a watch!
Or a mobile phone.
Sundials are too clumsy to fine tune. Or: their tune is too fine and it is too clumsy to turn them into clocks.
(He types, looking at the clock. I'm sure it's accurate to within five minutes or so, but more or less it reads: "Evening".)
And, with a flurry of maths signs, the article finishes in a list of links to helpful sources.
Water Clocks
Around 1400 B.C. (about 3,400 years ago), water clocks were invented in Egypt. The name for a water clock is clepsydra (pronounced KLEP-suh-druh). A water clock was made of two containers of water, one higher than the other. Water traveled from the higher container to the lower container through a tube connecting the containers. The containers had marks showing the water level, and the marks told the time.
Water clocks were very popular in Greece, where they were improved many times over the years. Look at the picture below. Water drips from the higher container to the lower container. As the water level rises in the lower container, it raises the float on the surface of the water. The float is connected to a stick with notches, and as the stick rises, the notches turn a gear, which moves the hand that points to the time.
http://www.arcytech.org/java/clock/clock_history.html
Water clocks worked better than sundials because they told the time at night as well as during the day. They were also more accurate than sundials.
Joost Bürgi, or Jobst Bürgi (February 28, 1552, Lichtensteig, Switzerland - January 31, 1632, Kassel, Hesse-Kassel) was a Swiss clockmaker and mathematician. He invented logarithms independently of John Napier, since his method is distinct from Napier's. Napier published his discovery in 1614, and this publication was widely disseminated in Europe by the time Bürgi published in 1620, at the behest of Johannes Kepler.
There is evidence[1] that Bürgi arrived at his invention as early as 1588, six years before Napier began work on the same idea. By delaying the publication of his work to 1620, Bürgi lost his claim for priority in historic discovery.[2]
Bürgi was also a major contributor to prosthaphaeresis, a technique for computing products quickly using trigonometric identities, which predated logarithms.
The lunar crater Byrgius is named in his honor.
http://en.wikipedia.org/wiki/Joost_B%C3%BCrgi
He made the first clock to have a minute hand. It was not very precise.
And they were off! Inventing clocks!
Here's an atomic clock.
Katori uses six laser beams to create a pattern of standing electromagnetic waves. This creates a series of energy wells, each of which supports one strontium atom, in much the same way as each dimple in an egg box holds an egg (see Diagram). This prevents the electromagnetic fields of individual atoms interfering with those of their neighbours, and allows the oscillating signals of many atoms to be measured at once.
Previous attempts to make clocks this way failed because the trapping lasers themselves interfered with the atoms' oscillation frequency. Katori's group has got round this by tuning the frequencies of the lasers so they alter the upper and lower transition energy levels of strontium by exactly the same amount, so the oscillation frequency remains unaltered. Katori claims that this "optical lattice clock" will keep time with an accuracy of 1 part in 10^18.
(Ed: That means it is probably out at any one moment by no more than 1 part in 1,000,000,000,000,000,000)
http://www.newscientist.com/article.ns?id=dn7397
I feel I have wandered into someone's subject!
As Gaianne wrote:
The problem is not with the science as science, but with how science supports a particular philosophical/economic/political agenda. When the goal of harmony is replaced by the goal of progress, the sustainable is explicitly abandoned in favor of exploitation.
http://www.eurotrib.com/comments/2007/4/16/18483/2608/23#23
Yet there be cycles in them thar subatomic devils.
Hope you enjoyed the trip!
