AW: Shadow Sharpener Again
Patrick Powers wrote ... The basic formula is actually f=(s^2)/(L), where f is the focal length, s is the radius of the (infinitely thin!) hole and L is the wavelength of the light. I would express this a bit differently, since a pinhole does not form an image in the sense that a lens does. Consider different pinholes imaging the sun on a plane at a fixed distance. The images are all the same size, but which one is sharpest? The images from the biggest pinholes are fuzzy by the size of the pinhole. But if you make the pinhole too small, then diffraction takes over and the images get blurry again. Patrick's formula tells you about what compromise you need to make to get the sharpest image. But remember that sharpness isn't everything. In particular, the smaller the pinhole the dimmer the image. If brightness is a problem, you might want to make the pinhole a few times bigger than this, particularly for large dials. If the pinhole size is fixed and you vary the the projection distance, the image gets fuzzy at short distances. At distances above that given in Patrick's formula, the sharpness doesn't improve much. near-perfect shadow sharpener should work when used on sundials. --Art Carlson -
difference between equinoxes and midsummer
Dear John, Let me try it this way. Take the Earth's orbit as it is and change the tilt from 23 degrees to 10 degrees, but still pointing in the same direction. Does this change affect where the Earth is at any particular moment? No. Does this change affect the positions on the orbit that correspond to the solstices and equinoxes? No. Therefore it does not change the time (measured not with a sundial but in seconds since the Big Bang) that the solstices and equinoxes occur. The answer are the same if we change to tilt to 1 degree, or 0.1 degree. The tilt is needed to define the seasons, but the amount of the tilt makes no difference at all in the lengths of the seasons. The tilt does affect the Equation of Time due to something that I like to think of as a coordinate transformation. The trick is that the coordinate systems for any degree of tilt happen to coincide at the solstices and equinoxes, which is why this part of the Equation of Time is zero on these four days. You wrote: I'm sorry but I have to disagree. BETWEEN the Vernal Equinox and the Summer solstice the correction due to the tilt is NOT zero. Every day EXCEPT at the equinox and solstice the day is a bit shorter (as the sun is early) due to this tilt contribution. Summing up these days (Solar days which the Civil calendar uses and not Sidereal days which astronomers use) leads to a shorter Spring than the summer where the days are now a bit longer. The sundial is fast compared to the clock for every day from April 16 to June 14, but that doesn't mean that the solar day is always less than 24 hours during this period. Take the beginning of June, for instance. Looking at the Equation of Time, we see that one each successive day, the sundial is about 9 seconds less fast, compared to a clock, than the day before. That means the solar day is 24 hr 0 min 9 sec long. (If you think I made a sign error, the length of the solar day around May 1 calculated this way is 23 hr 59 min 52 sec.) Servus, Art Carlson -
AW: AW: difference between equinoxes and midsummer
John Shepherd wrote: Now back to the original question: Why is the difference between the time between the Vernal equinox and the Summer Solstice different from the Summer Solstice and the Autumnal Equinox? This effect is approximately due to the tilt of the Earth's axis http://www.uwrf.edu/sundial/Eqntime.html ) on the Equation of Time (EoT), which can be approximated by a sine wave of a period of 6 months and amplitude of 10 minutes. The actual length of a day, as defined by solar noon to solar noon, is the Equation of Time minus the EoT. This is what must be integrated over the period involved. What I meant by averaging is that an integral over a period is equal to the average over that period TIMES the period. In this case the average of the half period of a sine wave is 10 mins*2/Pi or 6.37 mins. This is multiplied by 90 (or more accurately 92) days gives about 10 minutes. The solar time is less than the standard time by this and we get the same number but of opposite sign for the period after the solstice. So the difference is twice that or approximately 20 minutes. The elliptical orbital effect is very small on this difference essentially cancelling. We're talking about the same question now, but I beg to differ on the answer. The tilt of the Earth's axis cannot explain any difference in the length of the seasons. The only reason you need to bring the tilt of the Earth into the discussion at all is to define the equinoxes as the times when the Earth is on the line through the sun which is perpendicular to both the axis of the Earth's orbit and the axis of the Earth's rotation. The Equation of Time itself has nothing to do with the question, but if it did, the component with the 6 month period couldn't explain the difference because it is zero at the equinoxes and solstices. The eccentricity of the orbit, on the other hand, is on the order of 1%, and 1% of a year is a few days, so without doing a detailed calculation, the average difference ((spring+summer)-(fall+winter)) could be on the order of the 21 hours cited by Willy. The magnitude of (spring-summer), since the perihelion is near the winter solstice, must be much smaller. Up to five minutes ago, I was going to insist that the eccentricity of the orbit explains the effect. It is certainly true that that contributes a difference, but can it be that we still don't have the right answer, the one that explains the lion's share of the 21 hours? (Or else I still haven't understood John's answer. It happens.) --Art Carlson -
AW: difference between equinoxes and midsummer
John Shepherd wrote: 1. The equation of time gives the difference between the sun time and standard time. Your difference is cumulative or integral of the daily difference. The orbital effect has a maximum difference of about 8 minutes (this does not include the inclination effect). Averaging this approximately sinusoidal variation over 6 months is approximately 7 minutes per day. 7 times 180 days = 21 hours. Actually this point works the other way around. The difference between the length of any given day and the mean day is only handful of seconds. These snippets must be integrated to arrive at the Equation of Time. Integrating the Equation of Time doesn't produce anything meaningful. Actually, I don't think it is possible to directly deduce anything about the length of the seasons (Willy Leenders' question) from the Equation of Time. The answer to his question depends on the mismatch between the direction of the tilt of the Earth's axis (relative to the plane of the orbit) and the axis of the ellipse of the Earth's orbit. This is, however, related to the relative phase of the annual and semi-annual components of the Equation of Time. --Art Carlson -
AW: Polar ceiling sundial
Since a caustic is a very different animal from an image, is there any chance of getting around the 2 minute limit on sundial accuracy due to the sun's angular diameter? Does the caustic of an extended object form a line, or is it also smeared out? (I suspect there's no free lunch here, but I thought I could ask.) --Art -Ursprungliche Nachricht- Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Auftrag von Tim Yu Gesendet: Tuesday, January 08, 2002 3:22 AM An: Sundial List Cc: Tim Yu Betreff: RE: Polar ceiling sundial [David] What is a caustic curve? See the website: http://www.cacr.caltech.edu/~roy/Caustic/ A simple Java applet demonstrates how a caustic curve is formed by parallel light rays bouncing off a cylindrical, reflective surface. Tim
AW: Caustic and 2 minute limit.
Dear Bill, dear John, I realize that a shadow smeared over 2 minutes can be read to a fraction of that period (especially if it is symmetircal, as in John's dials), and that using images can give you a sundial with extreme accuracy. (What is the limit? Except with an azimuthal dial, I expect the first limit you hit would be the variation in atmospheric refraction.) The cost is comlexity (if focussing elements are used) or contrast/ease of reading (if pinholes are used). I did some experiments along the lines Bill suggests, although with pinholes, two years ago and convinced myself that I could determine a point in time under real-life conditions within 2 or 3 seconds. Making a complete sundial capable of this accuracy, however, looked like a difficult project. I was just curious if caustics could possibly give you the accuracy of an image in a way that is intuitive to read. That is, if you use a simple image, you have to tell the user whether to use the leading or trailing edge of the image. Bill's idea of using a double image solves this problem neatly and is probably more accurate anyway, due to its symmetry. --Art -Ursprungliche Nachricht- Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Auftrag von [EMAIL PROTECTED] Gesendet: Tuesday, January 08, 2002 3:09 PM An: [EMAIL PROTECTED]; sundial@rrz.uni-koeln.de Betreff: Re: Caustic and 2 minute limit. In a message dated 1/8/2002 4:19:42 AM Eastern Standard Time, [EMAIL PROTECTED] writes: Since a caustic is a very different animal from an image, is there any chance of getting around the 2 minute limit on sundial accuracy due to the sun's angular diameter? Art, I can't address the issue of caustics, but the 2 minute barrier can be broken by using two focused images of the sun, side-by-side, separated by a tiny amount of space. This space could be, say, 15 seconds of time, and would serve as the time indicator. If you have any doubt that this is feasible, I have a close up photo of my dial which operates using a single focused image of the sun, and although the image is 2 minutes wide, it is readable to better than 1 minute. The edges of the image are razor sharp, and it is easy to see that a design with two of these images side-by-side is achievable. Someday I may make one, but it is not high on my list of things to do. This JPEG is available to any who request it. Bill Gottesman Burlington, VT 44.4674N, 73.2027W
AW: Ceiling Sundial
You likely have a sheet of glass already clamped in place nearby -- the window. Couldn't you calculate a vertical dial for the right orientation, print it on a transparency, tape the transparancy to the window glass, and mark out the lines with a laser pointer or perhaps with a projector that casts shadows of the lines onto the ceiling? --Art -Ursprungliche Nachricht- Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Auftrag von Dave Bell Gesendet: Wednesday, January 02, 2002 8:28 PM An: J Lynes Cc: Mailing List Sundial Betreff: Re: Ceiling Sundial I like it!! Printing the dial artwork on transparency film should work well. Rub it down onto a thin sheet of glass supported in a frame, perhaps with a film of water or the like to keep it in place. The frame would need to be accurately leveled and oriented, but could easily be clamped in position, once that is determined. One hitch might come in, if the mirror is placed on the inside sill of a fixed picture window, making it hard to get the dial center over the mirror... Dave 37.29N 121.97W On Wed, 2 Jan 2002, J Lynes wrote: Here's a simpler proposal. Transfer the declination lines and hour lines of a horizontal sundial onto a transparent sheet. Mark a small circle on the centre of the mirror. Support the horizontal transparent sheet, rotated from north to south, with its nodus vertically above the centre of the circle, at a distance equal to the height of the transparent sundial's gnomon. Project a laser beam through the transparent sheet onto the centre of the circle. Make sure the beam passes through the sundial scale at a point corresponding to some chosen time and date. The reflected spot on the ceiling is the appropriate point on the ceiling sundial. Repeat for other dates and times. John Lynes
Trigon-Folding
Mystery solved. There are two different ways of carrying out the fold in the first part of your step F. Of course, I first did the one that doesn't work. --Art -Ursprungliche Nachricht- ... Actually, I wasn't able to follow your instructions, Edley. I get line 6 to be parallel to line 3 (45 degrees). I think there's a mistake, but I haven't figured it out yet. ...
AW: Trigon-Folding
Neat stuff. You can have it a bit easier, though, even if not quite so general. Take a rectangular piece of paper and lay it in front of you with one the the short sides near you. Fold it in half from left to right (the long way) and unfold it again. Now bring the lower left corner onto the crease from the first fold, and crease a second fold through the lower right corner. The second crease makes a 30 degree angle with the lower edge. Actually, I wasn't able to follow your instructions, Edley. I get line 6 to be parallel to line 3 (45 degrees). I think there's a mistake, but I haven't figured it out yet. --Art Carlson -Ursprungliche Nachricht- Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Auftrag von Edley Gesendet: Saturday, December 01, 2001 1:48 AM An: sundial@rrz.uni-koeln.de Betreff: Trigon-Folding Dear Membership, Here is more help for Emergency Sundial Makers. Trigon - Folding When you need 15, 30, 45, 60, 75 degree angles to lay out radially from a gnomon to create hour lines and don't even have a pencil, but do have something foldable; paper, foil, starched linen, etc., here is how to fold these angles. There turns out to be a number of ways to do this, but I'll describe only one. It involves trisecting an angle. I found the method on http://chasm.merrimack.edu/~thull/geoconst.html Starting from a scruffy piece of fom (foldable material) with no straight lines in it's shape. A. Fold ...
AW: Lunar ephemerids
Fernando wrote: Without intending to be so meticulous as we think Germans are, I'd like to do something similar (but much, much simpler), like observing if seeds sowed in the new moon do any better than seeds sowed in the waning moon, etc. I'm afraid you will have to be meticulous if you don't want to waste your time. (Leaving aside the question of whether the project is likely to be a waste of time regardless of how carefully it is done.) If you want to plant the seeds outdoors, you will need many (many!) years before you can get statistically significant results because you have to control not only for the season but also for the weather in each year. For example, you need to compare two sets of seeds, both planted at the equinox, but one set in a year where the moon was full at the equinox and the other in a year where the moon was new at the equinox. But that is not enough because you have to be sure that the temperature, cloudiness, and percipitation at the time of planting and several weeks before and after were similar. Your only hope to prove an effect would be to plant the seeds indoors and keep the temperature, humidity, and light at constant levels over several months. Several plantings would be necessary to be sure the seeds weren't drying out or something from one planting to the next. If you could manage to prove a small but consistent effect it would have no immediate application because the weather and other effects would certainly be more important in deciding when to plant in any given year. On the other hand, an incontrovertible positive result would be extremely interesting from a scientific point of view -- precisely because it would contradict so much of what we believe to understand about the world. Best regards, Art Carlson
AW: diameter of reflected sun image
The classical experiment using a mirror to detect minute rotations is not by Michelson and Morley, who used an interferometer, but by Cavendish, who measured the universal gravitaional constant in the lab. But the technique has been used often. --Art Carlson -Ursprüngliche Nachricht-Von: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]Im Auftrag von John CarmichaelGesendet: Tuesday, August 14, 2001 4:15 PMAn: [EMAIL PROTECTED]Cc: sundial@rrz.uni-koeln.deBetreff: Re: diameter of reflected sun image Hi Fritz Good to hear from you! What an interesting story. I seem to remember an experiment by Michaelson-Morley at the turn of the last century where they used mirrors to amplify the small movements in light. (I think they were trying to prove the the old theory that Einstein later disproved thatlight traveled through an "either" and that its speed changed).
RE: question on EoT
Dear fellow dialists, I am forwarding this inquiry I received privatly from Yaaqov Loewinger. It seems right up our alley. Regards, Art Carlson -Ursprüngliche Nachricht- From [EMAIL PROTECTED] Fri Mar 2 10:21 MET 2001 Date: Fri, 02 Mar 2001 10:33:16 +0200 From: Y. Loewinger [EMAIL PROTECTED] X-Accept-Language: en-US,hu,de-CH,fr-FR MIME-Version: 1.0 To: [EMAIL PROTECTED] Subject: Old equation of time:Equation d'Horloge This is a multi-part message in MIME format. --20215C5ACD79E9D29D96B5DD Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Dear Mr. Carlson, I read yr. excellent article on the Equation of Time. I posted following questions, in the History of Astronomy Discussion= Hastro group, and didn't get satisfactory answers. can you help me ? Thanks in advance! ( you can write me- if you like , in German). Subject: setting of clocks prior the introduction of modern mean time History of Astronomy Discussion Group [EMAIL PROTECTED] Dear List members, Prior the introduction of modern mean time into civil life (round 1780--1830), modern equation of time (= EoT) and modern distribution of time, civilian public clocks were set with the help of sundials. But as clock time is by definition a mean time, I wonder to which kind of mean time were the public clocks set generally, prior to, say 1780 ? I see here 3 possibilities: 1. public clocks were simply set always to sundial time , without using any equation of time tables. It seems that public clocks were set rather often, say once a week, as their quality was rather poor and as in such a short interval EoT changes only insignificantly, so it was simply neglected. 2. clocks were set to Nov. 3 mean time, as round this date modern equation of time is maximal (~ + 16 minutes). So, beside looking up sundial time, tabular values of equation of clocks running from 0 (on ~Nov 3) to ~31( on ~Febr 11) min, were added to sundial time to get clock time. This equation of clock = in French: equation de l'horloge (EdH) is the old Ptolemaic equation of nychtemeron(=day and night) of the Handy Tables. As this EdH was tabulated in the 18 th century French almanacs Connaissance des Temps, I guess they were widely used all over Europe (otherwise they would not have bothered printing it !). Beside this EdH the French almanac of, e.g. of 1751, indicated also the value of modern mean time at true noon, so modern mean time and modern EoT were also in use. Which one, EoT or EdH were used for setting civilian clocks ? Can we assume that the EdH table was really used everywhere in civil life in Europe, and can we assume in historical research that time indications in documents from the 18 th century mean Nov 3 mean time ? See an attached diagramm of Equation d'Horloge. 3. the third possibility to set clocks was, to set them to Febr 11 mean time. Round this date modern EoT is minimal, ~ 14 min. If a clock is set on that date to sundial time, so a Febr 11 equation of days, which runs from 0 ( on ~Febr 11) to 31(on~ Nov 3) min, had always to be *subtracted* from sundial time, to get clock time. Such equations of days can be found till the 17 th century in astronomy books ( e.g. the one of Huygens, near 1640), and it is also of Ptolemaic origin, standartly used in the middle ages. They seem to be less popular than EdH, perhaps because adding of EdH seemed to be easier for clock setters. To sum up my question: can we assume that a standard church clock, say in Central Europe, was set to November 3 mean time, still in the late 18 th century? Best regards Yaaqov Loewinger, dipl.ing. ETHZ -- Y. L o e w i n g e r mail: P.O.B. 16 229 ; 61 161 Tel Aviv / Israel tel.: 972-3- 604 61 79; ++ 523 98 33 fax : 972-3- 546 90 76 e-mail : [EMAIL PROTECTED] --20215C5ACD79E9D29D96B5DD Content-Type: image/gif; name=EdH-curve- with title.gif Content-Transfer-Encoding: base64 Content-Disposition: inline; filename=EdH-curve- with title.gif R0lGODdhgAI/AfcAAP8A
Re: Sun's Apparent Angular Diameter
Concerning Bill Gottesman's proposal of a method to measure the solar diameter: Well, it's basically a very good idea, but there are a number of traps to watch for. First, the slits need to be parallel. They also need to be aligned closely to North-South (or rather, perpendicular to the sun's motion at the time the measurement is made). Finally, what you are measuring is the time it takes the sun to change its longitude by one solar diameter, so there is a correction for the declination of the sun and another for the motion of the Earth around the sun. If all you are interested in is the ratio of the solar diameter at the two solstices, then the first correction cancels out, but not the second one. Also, it may be harder to determine a point in time accurately than a point in space. I would favor two parallel pin-slits casting images on a plane normal to the suns rays. The angular diameter (in the direction perpendicular to the slits, which should be oriented close to North-South and used near noon to eliminate any residual effects of atmospheric refraction) is inversely proportional to the distance between the slits and the plane when the images just touch. That is, the measurement consists of moving the plane until the images on it just touch and then measuring the distance from there to the slits. --Art Carlson
Re: Gnomon for Vertical Decliner
Mac Oglesby [EMAIL PROTECTED] writes: It does leave one surprised that apertures are quite commonly installed at an angle to the plane receiving the shadow. Is this irrational or are they just optimizing to some other feature? I mean, what's really so great about circular spots? What you really want is readability, which is a compromise between brightness and blurriness for any pinhole. Assuming you are interested in an accurate reading of the declination as well as the time, the best pinhole may be the usual choice of a circle in a plane perpendicular to the sun's rays (which is itself also a compromise). Extending this logic, a vertically elongated pinhole in a vertical plane might have some advantages over either of the other arrangements. Hmmm. Art
Re: Gnomon for Vertical Decliner
[EMAIL PROTECTED] writes: Oy vey! Maybe this will restart the Shadow Sharpener thread going again! Sounds like quite a project-good luck. I would like to suggest that if you use a pin-hole, that the aperture be parallel to the dial face. This may seem obvious, but it wasn't obvious to me 2 years ago. This way a round hole will always cast a round image, and will not spread into an ellipse as the angle of the sun changes. It is true that parallel rays shining through a circular hole will produce a circular image on a flat surface parallel to the hole. But an extended source shining through a pinhole will produce an elliptical image (unless the optical axis is normal to the surface). Consequently, the edges will be fuzzier in one direction than the other, whether or not you want to consider that to be an elliptical image. I suspect that a circular hole parallel to the surface is still the best compromise. Regards, Art Carlson
! message from owner-sundial !
Daniel Roth [EMAIL PROTECTED] writes: This message is sent for two reasons: 1st to remind how subscribing and unsubscribing works and 2nd to bring into discussion again the allowed length of a message including attachments. ... The length of a message is limited to 25 kB. Many subscribers still send messages, which are longer. This requires a manual intervention by the list owner. Please take into account that your attachment has to be downloaded by members, which may have only a 14.4 baud modem. Please vote for one of the following choices: I have a connection through a research institute, so I'm not bothered by any size message. Still I don't want to vote for no limit because I am also not particularly bothered by expediencies like requesting a file by email or by browser. My vote is for something very close to the lowest common denominator, i.e., if more than two or three members have a serious problem that can be solved without terrible inconvenience to the rest of us, they should be accommodated. --Regards to all, also to our brethren still living in the stone age, Art Carlson
Re: Bifilar dial in Genk Sundial Park
Chris Lusby Taylor [EMAIL PROTECTED] writes: Frans W. MAES wrote: I know one more case of an interesting bifilar dial. Using a pole style and a specially shaped curve in the equatorial plane, one may obtain a polar dial with straight, parallel E-W date lines, perpendicular to the hour lines. This principle was described in the Bulletin of the Dutch Sundial Society in 1979 by Th.J. de Vries. [...] http://www.biol.rug.nl/maes/zonwyzer/en/zwappi-e.htm This is an exciting sundial. Who would have guessed that you could achieve straight, parallel, date lines? Brilliant. Is the formula for the curve available, please? (Don't tell me - it's a catenary, right?) The principle is relatively straightforward. As the description says, the pole style and base plate together constitute a polar dial. (Since the second shadow is not needed to tell the time, I would hesitate to classify this as a bifilar dial at all.) At any given time of day, the shadow plane will always cut the edge of the yellow glass at the same point. For different dates/declinations, the shadow of this point will move up and down by the distance L*tan(D), where D is the declination and L is the distance from the edge of the gnomon to the edge of the shadow. At noon, L must equal the height of the style, H. The trick is to make L = H for every time of day. If x is the distance from the base of the style, measured in units of H, and y is the distance above the base plate in the same units, then the equation for the necessary curve is this: x = (1-y)*sqrt(1-y^2)/y Have fun proving this! --Art Carlson
Re: outdoor decor sundial question
Dave Bell [EMAIL PROTECTED] writes: I'd call it a fairly expensive joke! Note that a real dial should, roughly speaking, have the hours from 0600 to 1800 in a semicircle, running from East through North to West (in the northern hemisphere). This is a clock face, with only room for 12 hours in a day! The sundial at http://www.shopoutdoordecor.com/cgi-local/SoftCart.exe/online-store/scstore/p-AWS209S.html?L+scstore+wxsc3599ff367336+981445839 is certainly poorly (criminally?) designed in that it gets out of whack as the declination changes (by about +/- 1 hour, even if properly mounted). The simple fact that there are only 12 hours in a circle does not, however, make it totally useless. Since the foot of the gnomon is on the circle rather than in its center, the shadow falls at about the right spot near sunrise and sunset. In fact, if the circle were either perpendicular to the gnomon or elongated to an ellipse along the 6-12 axis, it could be turned into a perfectly fine sundial. In the Sundial Installation Instructions, the company states, These sundials are designed for ornamental use and give an approximation of time. As a very accurate sundial would require constant adjustment and less ornamentation, these models have been selected to give years of enjoyment without the aggravation of constant tuning. I find these words rather painful, knowing that sundials certainly can be accurate (limited in most cases by the Equation of Time), and having seen many examples of the beauty the artisans of this list can bestow on such an accurate dial. --Art Carlson
Re: Length of the year
Richard Mallett [EMAIL PROTECTED] writes: As for determining the length of the tropical year ... with a gnomon between successive solar solstices, I don't believe this is a good method. One can determine the exact date/time of an equinox much more accurately than that of a solstice (although the solstice is conceptually a bit easier to deal with). Can you elucidate please ? I would have thought that the solstices, representing the extremes of solar altitude (measured when the Sun crosses the meridian) would be easier to determine. Suppose you can measure the declination give or take one tenth of a solar diameter, i.e., to +/- 3'. Around the equinox, the declination changes by about 1' per hour, so your measurement would allow you to pin down the time of the equinox to +/- 3 hrs. At the solstice, the declination varies quadratically from its extreme value by about 0.22'/dy^2. In the worst case, you measure a value 3' below the maximum, so you might actually be right on the solstice, but you could also be at a date, either before or after the solstice, where the declination is 6' smaller than the extreme value. So the uncertainty in your measurement of the soltice can be as large as +/- sqrt( (6') / (0.22'/dy^2) ) = +/- 5 days. --Art Carlson
Re: A Sundial as a Prize
... A photo of a dial similar to the one made for Patrick Moore can be seen on the internet at http://www.lindisun.demon.co.uk/smallest.htm I have a question for Tony Moss about the dial pictured. Unless there is another scale on the back we can't see or the dial plate can be turned over, this dial can only be used in summer. That's OK, but then why do you include the Equation of Time for the whole year? --Art Carlson
Re: Length of the year
Gordon Uber [EMAIL PROTECTED] writes: The length of the tropical year was determined with a gnomen between successive solar solstices. The length of the sidereal year was determined from successive heliacal risings. From Time in History by G. J. Whitrow. I have long wondered how to make accurate observations of the sun relative to the stars (as John Sheperd put it). Given the key word heliacal rising, I have been able to find the definition and some discussions on the Net. I find it surprising that this could be, as John Sheperd said, pinned down to a single day. Wouldn't this depend on the brightness of the star and the viewing conditions and God knows what? On the other hand, the position of a given star at sunrise will change by 1 degree from one day to the next, which seems like a manageable distance. And I suppose what counts (for present purposes) is not what the actual relationship between the sun and the star is, but just the reproducibility of the phenomenon. Still, you would need to take years where the meteorological conditions were comparable. As for determining the length of the tropical year ... with a gnomon between successive solar solstices, I don't believe this is a good method. One can determine the exact date/time of an equinox much more accurately than that of a solstice (although the solstice is conceptually a bit easier to deal with). --Art Carlson
Re: Length of the year
Allan Pratt [EMAIL PROTECTED] writes: According to a source I read, Hipparchus, a 2nd C BC astronomer calculated the length of the year to within six minutes of accuracy. Considering that at best he had a sundial and a water clock, how did he do this? I hope a historian will answer this, but I am willing to speculate. H's minutes were surely defined not with respect to a cesium clock but as a fraction of a day. The year is defined by the seasons, i.e., the declination of the sun. The declination is most sensitive to the date around the equinoxes. Since the equinox is one of the most fundamental and easily observed astronomical events, it is plausible that the equinox had been determined and recorded, at least to the nearest day, for hundreds if not thousands of years before Hipparchus. If he had available an uninterrupted calendar and a record of an equinox 240 years earlier, then, by counting the number of days between that and a contemporaneous observation of an equinox and dividing by the number of years, he could calculate the length of the year to a the claimed accuracy: (1 dy) / (240 yrs X 365 dys/yr) = 1/87,600 = (6 min / 1 yr) / (60 min/hr X 24 hrs/dy X 365 dys/yr). Alternatively, if he knew what he was about, he could by careful naked-eye observation determine the time of the equinox to within a fraction of a day. If his observations had an accuracy of 0.1 day, then he would only need observations 24 years apart, easily within a professional lifetime even in those days. The observation must not necessarily be of the equinox. One could use solar eclipses in a similar way, or simply the date in spring on which the sun first becomes visible in a notch between two mountains. Note that you don't even need a sundial or a water clock for any of this! --Art
Re: Off topic, but not too much
SÈrgio Garcia Doret [EMAIL PROTECTED] writes: 1 - Assume the hours equals exactly 1/24th of the earth revolution time and suppose a disguster lover choose to retire into a cave, where daylight is entirelly shut off for a period of six months to the minute. ... What adjustment does his watch need? As pointed out by others, the assumption does not even come close to the actual definition of an hour, but what the heck. The watch owner has more important things on his mind. I see two answers, depending on the type of watch: 1) If it the usual stupid kind, no adjustment will be necessary. 12 o'clock is 12 o'clock, and the watch can't distinguish 12 noon from 12 midnight. 2) If the watch has a date display, then it must be adjusted by 12 hours, and it makes a difference whether you set it forward or set it back because you wind up on a different day. The correct procedure is to set it back to give the rotation about the axis time to make up for the revolution about the sun. --Art Carlson
Re: steriographic projection
Patrick Kessler [EMAIL PROTECTED] writes: Can anyone recommend an essay on steriographic projection? In particular I am searching for a proof that circles on the sphere are mapped onto the equatorial plane as circles. http://www.geom.umn.edu/docs/doyle/mpls/handouts/node33.html outline[s] two proofs of the fact that stereographic projection preserves circles, one algebraic and one geometric. You should also be aware of http://www.astrolabes.org/. Finally, if you are using search engines, or even a card catalog, you'd better spell stereographic with an eo. Have fun. Art Carlson
Re: Shadow Sharpener
I wrote: Nevertheless, I have a feeling that it may not be possible to improve on a simple pinhole. Let me reconsider that. Consider an aperture a distance L from a surface, so that the image of the sun through an infinitesimal pinhole would have the diameter D = L*(0.5 degree). With a circular pinhole of diameter dD, the brightness in the center of the image compared to that on an unobstructed surface is (d/D)^2. At the edge of the image, the brightness drops to zero over the distance d. With an annular slit of inner diameter D and thickness d, the brightness in the exact center of the image is zero. The brightness rises rapidly moving away from the center to very nearly (d*D) / ((pi/4)*D^2) at a distance of d, and increases more slowly to about (pi/2) times that value after that. (Mathematics available on request, at least in principle.) Considering only the initial rise over the distance D, the change in brightness with the annular slit is (4/pi)*(D/d) greater than with the circular pinhole. This factor is by design greater than one and can be made much greater. As an example, if you can work with a brightness 1/10 that of unobstructed sunlight, then a circular pinhole allows you to increase the accuracy by a factor of 3, but an annular aperture allows you can gain a factor of ten. If you can work with dimmer light, the improvement is even more dramatic. I don't know if it really works this way. Maybe all that bright light around blinds you so you can't see the small dark spot. On the other hand, your visual acuity may be increased by the fact that your pupil contracts. Experiments are needed. This analysis does suggest to me that significant gains might be obtainable for some geometries, e.g., noon marks, where the angle of the incoming light is always about the same, and it gives some guidance in choosing aperture dimensions. (Wouldn't it be great if we can come up with a useful sundial feature that the ancients didn't know about?) Cheers, Art
Re: Shadow Sharpener
It is easy to read a sundial with an accuracy a bit better than the solar diameter, even if the shadow is from a simple edge. The worthy goal of a shadow sharpener is to significantly improve on that accuracy. Since we still want to make the reading with a human eye, the best system will be determined to a large extent by psychophysics. Human vision is so complex that it is not obvious just what we are looking for, so the final judge will be experiments. Nevertheless, I have a feeling that it may not be possible to improve on a simple pinhole. The image produced by any of the systems discussed (a simple pinhole, an annular pinhole, or a classical shadow sharpener which is a pinhole downstream of a conventional gnomon) will be sharper if the holes are smaller, at the expense of brightness. It may be hard to find with arbitrary accuracy the center of the image produced by a simple pinhole even if it is perfectly sharp, but one should be able to locate the edge of the image as accurately as desired. I would thus suggest to the experimentalists that they always compare the clever designs with simple pinholes, where the pinhole diameter should be varied to find the optimum, and where both forms of reading, from the center of the image and from an edge of the image, are compared. A simple pinhole may also be less sensitive to variations in the distance to the scale and gross variations in the position of the sun during the course of the day. On the other hand, reading from the edge of an image may be less intuitive for the casual user of a dial. Cheers, Art Carlson
Telling Directions from the Sun and the Moon
One of the things that got me going on sundials was an article in the magazine of the German Alpine Club on telling directions from the moon. I found the procedure impossibly complicated and spent much time trying to understand celestial mechanics in order to think of alternatives. At long last, I have put my thoughts into words, which may be found at http://www.ipp.mpg.de/~Arthur.Carlson/sun-compass.html This may be of interest to some of you, but even if it is not, I would appreciate any feedback on its accuracy and pedagogical value. The audience is intended to be hikers more than dialists, and I would like to eventually publish the essay in said magazine of the DAV. --Art Carlson
Re: equation of time
Willy Leenders [EMAIL PROTECTED] writes: The equation of time has two causes. The first is that the orbit of the earth around the sun is an ellipse and not a circle. The second is that the plane of the earth's equator is inclined tot the plane of the earth's orbit. Please can anyone explain me the second cause so that I can conceive it. I am not a astronomer! I have given this question a lot of thought, but I realized when I was asked about it a few days ago that I am still not satisfied with my answer. I have tried to explain it in detail on my page http://www.ipp.mpg.de/~Arthur.Carlson/sundial.html;, but that isn't the intuitively obvious answer we would all like to have in order to claim that we understand the effect. If I had to answer in one sentence, I might say that the effect arises because the sun moves against the stars (in the Ptolemaic sense) on a circle (the ecliptic) that differs from the coordinate system we use to define time (the equatorial plane). You can see that it is a mathematical effect, as opposed to the physical effect of the eccentricity, by considering a planet that does not rotate, so you can place the poles anywhere you want. The hour angle of the sun during the course of the year, except at the solstices and equinoxes, will depend on your choice. You can do it in Dutch (for preference), in French, in German or in English. I can offer you German, if you have trouble understanding the English. Art Carlson
Re: Coming equinox
[EMAIL PROTECTED] (John Carmichael) writes: It's an interesting thought to use the moon's shadow at sunrise and sunset on the equinox to locate your east-west points. Although this can be done with the sun, you would have errors using the moon, unless there is an eclipse on the equinox also. If the moon is not in the same plane as the sun, it will not act like the sun. Since the moon moves about two minutes/hour eastward in the sky, the only way you can do this with precision is to use precisely calculated times and lunar coordinates such as those sent to you by Jim Cobb. First you would find the meridian with the moon, using Jim's data, then find east/west. Of course, the moon doesn't cast shadows during an eclipse, so we are really talking about taking a sight on the moon. Even during a total eclipse, without additional information, the errors will be several times larger than using the sun. If you have additional information, i.e., the coordinates of the heavenly bodies at various times, then there is no need to wait for the solstice or an eclipse, you can take a sight on anything at any time and deduce the cardinal directions from the result. I think the purist's way to find directions (a purist being a Druid with the knowledge and technology of Stonehenge), is to mark the directions of sunrise and sunset for a few days near the solstice. By interpolation, you can get the hypothetical direction to the rising sun for each hour of the period in question, and likewise for the setting sun. For one of these hours, the directions will be exactly opposed to each other. This is East/West, and the hour is the exact time of the equinox. Magic. Have fun out there with the coyotes! Art
Viviani's pendulum
Looking up Foucault's pendulum experiment in Meyers Grosses Taschenlexicon, I read the claim that Vincenzo Viviani in 1661 was the first to do the experiment, 189 years before Foucault! Browsing through the Web for more details, I was only able to find two further references: In http://www.newadvent.org/cathen/15183a.htm Foucault's pendulum experiment was materially forestalled [sic] by Viviani at Florence (1661) and Poleni at Padua (1742), but was not formally understood. and in http://www.physik.uni-greifswald.de/~sterne/Observatory/events.html Already in the year 1661 Vincenzo Viviani discovered this phenomenon. It was rediscovered by Leon Foucault in 1850. I'm hoping some of the erudite contributors to this group can give me a few more details. It seems like the experiment, while requiring some care, should have been within the range of 17th century technology. Did Viviani really look for rotation of the plane of swing of a pendulum? Did he know it would provide the proof of the Earth's motion that eluded his mentor Galileo? Did he get a positive result? Why was the experiment forgotten for almost two centuries? Thanks and best regards, Art Carlson P.S. I come to this question because I am reading Galileo's Daughter by Dava Sobel. I thought that the interest shown in this forum for her book on Longitude was justification enough for asking my question here. In addition, there are some connections with sundials through the time-keeping aspects of pendulua and through Galileo's attempts to solve the longitude problem using the moons of Jupiter. Apropos Sobel's new book, I'm a third of the way through. Up to now it's a remarkably straightforward biography of Galileo. It certainly won't have the fascination for this list that Longitude did.
Re: OFF TOPIC -- OFF, OFF TOPIC
Fernando Cabral [EMAIL PROTECTED] writes: I've heard the French Assembly has approved a Resolution 495 which determines (so I heard) that every public organization in France has to replace Microsoft Windows by Linux. Even if it's not true it's a great rumor, so I have been working to spread it. My wife (a journalist) wasn't able to dig up anything, but a colleague found this in an article someone posted to the scientology group: Windows 2000 starts out against the wind Government investigations, bug reports tales of horror and strong competition: Linux Stuttgart, Germany February 22, 2000 Stuttgarter Zeitung [...] Swiss authorities believe Windows 2000 is too expensive and they are reviewing Microsoft's pricing politics. EU commercial competition commissioner Maria Monti is investigating complaints that Microsoft has arranged network functions in Windows 2000 so that they will work only with software which comes from the House of Microsoft. A French importer is also suing in an EU court over competition obstacles. The corporation bought a French language Microsoft program in Canada because it was cheaper there. Microsoft's French branch company prohibited the sale of this import. Last week, an EU court stated the importer's complaint was justified, thereby forcing the EU Commission to take the case. The French are happy over the decision: one initiative in the French Senate aims to have only software with free source code installed in all government agencies by the year 2002; the Culture Ministry is already converting to Linux. I hope everybody will excuse me for abusing this list's patience. An occasional off-topic post is no problem among friends. Art Carlson
Re: Declination Table
Daniel Lee Wenger [EMAIL PROTECTED] writes: The reading of standard time via a sundial may be accomplisted by mearly reading the declination of the sun and using an analemma, determining standard time. At no point is the current date needed to do this. Way, way back I explained why I was not totally satisfied with this method, essentially because there are (almost always) two values of the EoT for each value of declination. At the solstices there are even an infinite number of values (in some technical sense). Consequently, if you are interested in relating the sundial reading to clock time, you always need some knowledge of the current date. Art Carlson
Re: Design challenge
John Davis [EMAIL PROTECTED] writes: I have a question/challenge to all you sundial designers: what is the most accurate design for a Standard Time dial? ... As a starter, the Singleton dial recently discussed here would seem to be a reasonable candidate. It's main limitation, common to all dials which incorporate an EoT correction, is that it is drawn for a some MEAN EoT curve, and no allowance is made for the leap year cycle and the other minor variations. Is there some geometry of dial plate and style which minimises the time error caused by small year-to-year variations in the mean daily declination? If this is achieved, then the small change in the EoT over a single day may be allowed for. The maximum rate of change of the EoT is about 30 sec/day toward the end of December. Averaging over leap years can be done to make the chart wrong by at most half a day, or 15 sec. The diameter of the sun is 0.5 degree, or 120 sec of time. Before you worry about the leap year problem, you first need to find a way to locate the center of the shadow edge 8 times more precisely than the degree to which it is smeared. We (e.g., John Carmichael and I) have discussed here some designs which might be capable of this accuracy, but they tend to be a bit hard to use. If you insist, one possibility is a camera obscura with a slit (ideally oriented parallel to the Earth's axis). This gives a sharp line image of the sun, which can be used to read the time from a series of date lines like we have been discussing. If you're really worried about leap years, you can pile four years' worth of dates on top of each other. The other approach advocated by some, namely determining the EoT directly from the declination, rather than the date, will always suffer near the equinoxes. For example, if you determine that the declination is 23 deg 11 arcmin +/- 15 arcmin, the EoT can vary over a range of 11 minutes! --Art Carlson
Re: Azimutha Sundial (once more)
fer j. de vries [EMAIL PROTECTED] writes: Back to the bifilar dial : A bifilar dial can be constructed in such a way that the hourlines ( for local suntime ) are equi-angular spaced. Than it is also possible to correct for EoT and/or longitude by rotating the hourscale. So we have at least 2 possibilities to correct for EoT with bifilar sundials. Is there a resource on the Web with the theory of bifilar sundials, or at least a picture or some info on constructing them? I spent many idle hour trying to come up with a sundial that would allow an easy mechanical correction for the equation of time. The best I could devise was using a gnomon tilted halfway between the Earth's axis and either the vertical or the meridian. This allows the hour marks to be placed evenly around the circumference of a circle, so that the dial can easily be set forward or back by rotating the circle. The catch is that the center of the circle has to be moved to match the declination. (I assume that also this invention of mine is old hat and has a name that someone will kindly tell me about.) It sounds like the bifilar dial is the solution I was not clever enough to find myself. Given time, I would be able to work out the theory on my own, but I'm also willing to forgo some of the fun on this one and just read about it. Art
Re: Singleton's azimuthal
[EMAIL PROTECTED] (John Carmichael) writes: ... Why not follow John Singleton's notion (p. 51, BSS Journal for Feb 2000) and use your normal taut wire pole style? Have I missed something in the discussion? Maybe we all have. I think John Singleton's azimuthal will not work (except at noon, sunrise and sunset). I know this is a rather bold statement to make, but I think there is a general misconception that azimuthal dials can work with either a vertical gnomon or a polar axis gnomon as was originally suggested in an earlier discussion. This has always bothered me because it seemed impossible. If a polar axis works, then it would certainly solve the gnomon height problem. Rather than speculate, I did a simple experiment. Using a Spin drawing of an azimuthal for my location and an icepick for the gnomon, I quickly found out that the dial worked correctly when the icepick was vertical and became progressively worse as I tilted it towards the celestial pole. A dial with date rings (neither azimuthal nor Dali is quite the right name) can be designed for any gnomon, in particular for either a straight vertical gnomon or a straight polar gnomon, but any given dial plate will only work with its own gnomon. Your mistake was trying to use a vertical drawing with an polar gnomon. --Art
Re: thumbs down on azimuthals
John Carmichael listed the pros and cons of azimuthal dials and concluded that it is NOT an appropriate design for me to build. Of his pro arguments: 1. It looks different, original and pretty (especially if you like the Batman logo!) 2. It can be made to tell Standard Time 3. It requires a simple vertical gnomon 4. It can be designed by Fer's Spin program 5. It is horizontal (usually), and horizontals are very commercial 6. It tells time from sunup to sundown I have always placed great weight on number 2. Perhaps because I'm a physicist, I hate to see a machine that is an order of magnitude less accurate than its inherent possibilities (+/-15 min EoT compared to +/-1 min (of time) solar radius). An EoT chart is an awkward remedy. I got onto azimuthal dials (before I knew what they were called) as a way to build sundials that accurately show clock time, but also saw great possibilities for different, original and pretty designs (pro argument 1), and I offered my Arizona dial as an example. Now look at John's cons: 1. It requires an absurdly tall gnomon at middle and lower latitudes which would make the sundial look odd and would have severe shadow fuzziness problems in the summer. 2. To avoid using a tall gnomon, the shadow must be artificially extended by visual guesstimation or by a string shadow extender, both of which would make the dial less precise. Also, changing the date ring order complicates calculations and makes the dial even harder to read. 3. It is inherently hard to read even with just one hour time lines, especially for the novice, without instructions. 4. It is very difficult to make this dial precise with small time line divisions.(For fun, try Spin using five minute time increments (step hour=5/60=.0833, and you'll see what I mean) 5. Small time increments make the dial even harder to read. 6. There is severe time line compression on the inner date rings, making engraving and reading difficult. 6. If the geniuses on the Sundial List have a hard time understanding it, I doubt my customers ever will! These revolve around the short shadows of vertical objects at some times and places and the difficulty of reading the wildly curving lines. I think it is still possible to have the best of both worlds (except pro 3), specifically by using a polar gnomon. (Some other contributors are already playing with designs with concentric dates rings and non-vertical gnomons.) This would immediately eliminate cons 1 and 2 (too tall gnomon). It would be much easier to read, understand, calculate, and manufacture (the remaining cons, except perhaps the first of the sixes) because the would look nearly like a conventional sundial. The hour lines would be nearly straight since they only have to accommodate the EoT, not the declination. You can tell at a glance about what time it is (as with an uncorrected dial), or you can look for the date ring and tell the time within a minute or two. The flexibility of choosing the shape and location of the date rings remains (pro 1), so an Arizona dial, for example, is still possible. (Words, words, words! Will one of you that has been posting azimuthal dial plate designs please plug in a polar gnomon for me?) Are you interested in such a compromise, John? Regards, Art
Re: Metric v's Imperial.
Gordon Uber [EMAIL PROTECTED] writes: Let's face it: The Babylonians got it right when they developed the base-60 system. It was applied to the sixth of a circle (one sixtieth of this being a degree) and the hour, of which we still use the first and second minutes. Third minutes (sixtieths of second minutes) are not in common use, although I would note that the third minute of an hour is the period of U.S. power main standard 60 Hz alternating current. Coincidence? Is this the origin of our (English, at least) names for units of time? Seconds because it result from dividing an hour by 60 twice? (Min'-ute, I assume, is related to mi-nute' and mini.) Is it known whether the Babylonians, when they chose 360 degrees to a circle, were more concerned with the convenience of numbers divisible by 2's and 3's or with the fact that there are 360 days in a year (within a percent or two)? --Art
Re: metric
Peter Tandy [EMAIL PROTECTED] writes: ... Of course, for some specialised work, metric measurements are no better and no worse; atronomers for instance do better with the numbers they need to measure huge distances, when in a metric form, and physicists with the numbers they need to measure minute atomic distances. But neither of these is a measurement that us ordinary folk use on a day-to-day basis - and for those, Imperial with its greater number of divisors is far better. The way to cut the Gordian knot is to throw out everything and start over with a base 12 numeral system. Then the scientific calculations and the everyday divisions by 2, 3, 4, and 6 are *both* easy. (Time measurements with base 12 is another kettle of fish. 12 months in a year is good, but the 7 day week is still a killer. 24 hours in a day is close, but there's that pesky divisibility by 5 when splitting hours into minutes or minutes into seconds.) --Art
Re: Diverging Light Rays
Andrew James [EMAIL PROTECTED] writes: My idea is this: is it possible to combine the two points made? Arrange, say, two sets each of four posts with three 0.4 mm gaps between, one set having slightly wider posts but with the same gap, so as to make three light rays the outer two of which diverge by the same small amount - say 0.2 degrees - in each direction from the inner. Then balancing the appearance of the outer rays should give a rather more accurate estimation of the angle of the centre of the solar disc.Any takers? I'll buy it. I did a lot of thinking and some experimentation last summer. I used a slit and two pinholes and tried to balance the intensity of light on the two sides. I found I could judge the moment of symmetry within a second or two of time, which corresponds to one arc minute or better of angle, which I found very respectable. The principles are these: (1) your eye can judge symmetry better than just about anything else, and (2) the light passing through lenses/pinholes/slits varies most sensitively if the apparatus is aligned with the limb of the sun. My slit produced a line image of the sun. Both the diameter and the separation of my pinholes were about equal to the width of this line image. --Art
Re: optical resolution tables
I just wrote: ...You will find that you can make a beam anywhere within a few tens of a degree. (To be precise, 0.5 deg at sunrise and sunset, closer to 0.3 deg near noon.) I got that backwards. The sun subtends a larger azimuth when it is higher in the sky, so the beam can be formed to point in any direction in a range of something like 0.7 degrees near noon (at mid-latitudes). --Art
Re: drawing hour lines using gnomon
Arthur Carlson [EMAIL PROTECTED] writes: [EMAIL PROTECTED] (John Carmichael) writes: Let's say ... ... Will this technique produce the same shape hour lines at any time of the year? Yes. The hour lines will always have the same shape. This is even true if the gnomon is not aligned with the axis of the Earth, as long as it is straight. Some other respondents have touched upon the question of the orientation of the gnomon. I stand by my answer to the question as stated: John's technique will produce the same shape hour lines at any time of the year, for any straight gnomon. If you want to label these lines for clock time, the labels will have to change during the year. Or you can put labels on them that are valid for some day of the year and read corrections to these labels from a table. The advantage of a polar gnomon is that these corrections are just a function of the day of the year (the familiar Equation of Time). With, say, a vertical gnomon, the correction will depend both on the day of the year and also the time of day. (Obviously, I've been thinking too much about my Dali dial. I better go cool off my brain.) Art
Re: drawing hour lines using gnomon
[EMAIL PROTECTED] (John Carmichael) writes: Let's say you want to build a large sundial using the ground as the dial face. The ground is somewhat irregular and not quite horizontal. You decide to draw the hour lines, not by calculation, but by using the technique of building the gnomon (style) first and then marking the position of the shadow on the ground at selected time intervals using clock time and correcting for EOT and longitiude. You draw the hour lines from the edge of the dial face to the dial center, tracing the shadow. Since the ground is irregular and not flat, you notice that the shadow line is not straight, but irregular also, depending on the terrain. This produces hour lines that are not straight. Because the sun's declination changes during the year, changing the angle that the sun strikes the style, will this technique produce the same shape hour lines if it is done at any time of the year or is the declination irrelevant? Will this technique produce the same shape hour lines at any time of the year? I think it should. but I'm not sure. Yes. The hour lines will always have the same shape. This is even true if the gnomon is not aligned with the axis of the Earth, as long as it is straight. If the shadow passes through a point P, then it will pass through all the points of the plane containing P and the gnomon. If it does not pass through P, then it will pass through none of the other points in the plane (except those of the gnomon). The intersection of this plane with the ground (the shadow line) is just a subset of the points of the plane, so the same holds there. Art
Re: gnononistically challenged
[EMAIL PROTECTED] (John Carmichael) writes: Thanks for taking the time to explain the Dali dial to us gnomonistically challenged dialists. I think I'm beginning to understand it, but will have to think about it some more. What threw me off was that I was thinking that a Dali dial would be draped over the edge of a table! It could be, hence the thread title. Perhaps it's best to start by thinking about a normal sundial with a horizontal plate and a paraxial gnomon. Design such a dial to read clock time without any EoT correction. Then design another dial to read clock time plus 5 minutes, another to read clock time plus 10 minutes, and so on for +15 min, -5 min, -10 min, and -15 min (always without any EoT correction). Now, to read the time from such a dial you don't need the whole plate. Any ring will do. So cut a ring out of each of the 7 dials you just made, but of different radii so that the rings do not overlap. Properly position all the rings around a single gnomon. Now if you want a +5 min correction, you just read the time from the +5 min ring. For a -10 min correction, use the -10 min ring. Label each ring with the dates that its correction is valid, and you have a version of my sundial. At any given hour, the shadow will always point in about the same direction, but will be shifted right or left a bit due to the EoT. If you move away from a paraxial gnomon, the concept becomes a bit trickier because a plate with radial hour lines will only be accurate for a particular declination of the sun, that is, a particular date. But, look! You are only using each ring for a particular date anyway, so it doesn't matter. Someplace around here we realize that most of what we have learned about sundials doesn't matter any more either. The gnomon could be bent. The plate could be warped. You just have to make sure that the shadow of the gnomon always crosses the date line at exactly one point, and that this point is never in the shadow of something else (like a fold in the plate). I used this extreme flexibility to propose a sundial with date lines in the shape of Arizona, but only after I checked a map to see that Tucson was conveniently located (somewhat south of the middle) and that no line radiating from Tucson intersects the border more than once. So. I think you have caught on to the concept by now. One thing I like about some dial designs is their inevitability. The plate *must* be horizontal if it is to always be illuminated when the sun is above the horizon. The gnomon *must* be paraxial to minimize the difference to clock time over the course of the seasons. This concept, in contrast, has very few constraints, although it is in a sense more accurate than the conventional form. How can this freedom best be used? Can we require the dial to do something new, that other dials can't? Is there an especially esthetic form? Have fun with it. Art
Re: Dali dials
[EMAIL PROTECTED] (John Carmichael) writes: I'm trying to understand your letter. Your design sounds very intrigueing. In fact, I've often thought of carving a map of the state of Arizona onto the dial face, with Tucson at the center of the dial. All the hour lines would radiate out from Tucson. With my vertical pointer in the center and the correctly oriented map on the face, you could point to any place within the state, like a boyscout does with his map and compass. I could even put latitude and longitude lines on the map. But I was going to use my cable coaxial gnomon, with a sphere on the cable to serve as the nodus for date readings instead of a vertical style with nodus sphere at tip. Your design would place Tucson north of center, and the hour lines would radiate from a point south of Tucson (probably in Mexico!). So with a vertical gnomon you would lose the ability to use the poiner to take a compass bearing. A neat idea to use the hours lines to show the azimuth, but that won't work with my idea. My hour lines wind up being wavy like the Equation of Time, and maybe bowed as well, so you can't use them to point anywhere. The main misunderstanding is that my design does not have a nodus. Methods which use the declination of the sun, either by using a specially shaped gnomon or by observing the shadow of a nodus, rather than an edge, are perhaps more esthetic, but they are inherently ambiguous at the solstices and double-valued the rest of the time. What do you mean by inherently ambiguous at the solstices and double-valued the rest of the time? The locus of the shadow cast by a nodus at, say, noon through the course of the year is the figure-eight-shaped analemma. When it is noon on any day, the shadow will fall on the analemma. But on most days, it will cross over the analemma twice, and you have to know which of these two crossings to use to tell time. You have to know whether the current date is in the first half or the second half of the year (which shouldn't be a big problem, even for absent-minded types like myself) and on that basis decide which branch is currently valid (which always is confusing, even for the mathematically and astronomically inclined). That's the double-valued part. The ambiguous part comes in because at the solstices, the shadow traverses the ends of the analemma loops tangentially, so it is hard to decide exactly when the crossing occurs. You don't have to choose between two distinct but well-defined candidates as during the rest of the year, but your one candidate becomes rather fuzzy, extending over several minutes. The only dates that are free of these problems are April 13 and August 31, the crossing point of the figure-eight. If we make the user do this work instead of the nodus, the figure-eight can be unfolded and made unambiguous. Are you talking about a moveable nodus? No. No nodus. None. Like with an analemmic dial, the time is read by looking at the intersection of a shadow line and a time line. In the present case, there are a multiplicity of time lines, one for each day of the year (or as many as you have room for). In contrast, in a conventional dial, the shadow line coincides with the line marking the time. With a nodus, the position of the shadow point tells the time. And from here on I'm completely lost! I can't imagine what the face might look like, or the gnomon, let alone how you would calculate such a dial. Wish I could see a picture! Here, here! Maybe someone can whip up a picture with Fer's program to give us something to point at while we're talking. (My PC is in the shop just now.) My idea is basically the azimuthal dial he mentions. My only contribution is pointing out that such a dial will still work even if the time lines are not circles and the gnomon is not vertical (or even necessarily straight). If you design it, I'll build it (if it works!) It will look rather bizarre, but should be accurate and easy to read. I'm hoping you or Fer will be intrigued and do the calculations. (Unless you are willing to do a lot of dot-to-dot drawing, the first thing you will need is a mathematical representation of the shape of Arizona!) If not, I'll put it on my pile and get to it manana. Regards, Art
Dali dials
A normal sundial has the gnomon coaxial with the Earth. This is done to keep the errors with respect to clock time to a minimum during the course of the year. If we have the ambition to make our sundial read clock time to better than +/- 15 minutes, then we have to correct for the Equation of Time. There have been many public discussions here and private ones in my head about the best way to do this. Simply reading a table or graph is inelegant and subject to errors of sign. Methods which use the declination of the sun, either by using a specially shaped gnomon or by observing the shadow of a nodus, rather than an edge, are perhaps more esthetic, but they are inherently ambiguous at the solstices and double-valued the rest of the time. One way of thinking about the nodus methods (which has come up here in discussions of the EoT with respect to leap years) is that the declination tells you what the date is, and the figure-eight-analemma allows you to find (with the restrictions mentioned above) the EoT for that date. It seems reasonable to suppose that everybody has a pretty good idea of the date already, so we are making the sundial do unnecessary work. If we make the user do this work instead of the nodus, the figure-eight can be unfolded and made unambiguous. (I am sure I have seem such a dial design somewhere, but I can't remember where.) For example, the date-lines can be made concentric circles, from Jan 1 innermost to Dec 31 outermost, and the EoT for each hour marked as a nearly radial wavy line. We could even trivially accommodate the change between standard-time and daylight-saving-time. As a practical matter, I think it would be easier to read clock time (corrected for EoT) from such a dial than from any alternative. The freedom opened up by this arrangement is astounding: The date-lines can have (nearly) any shape, and they could all have different shapes (as long as they don't cross). The gnomon need no longer be parallel to the Earth's axis; it doesn't even have to be straight! I could envision a Salvador Dali sundial, but maybe I should start with something for John Carmichael: Draw the outline of Arizona many times at different scales and put them inside of one another, but so that all the Tucsons overlap, and of course properly oriented with respect to the compass. Put an obelisk at the location of the Tucsons. Label each outline with a date and calculate (the hard part!) where the shadow of the obelisk will fall across that outline for that date and each hour of the day. For each hour, connect the points for all the dates and label the resulting wavy line with the hour. Voila! I'd love to do the design myself, but realistically I know I won't find the time any time soon, so I'd rather through the idea out to the world. Is the description clear enough? (The idea is probably between 500 and 2000 years old anyway.) Regards, Art Carlson
Re: Twisted band sundial
I can think of three ways to incorporate the Equation of Time into a twisted band dial: (1) A correction can be made in the hardware by simply turning the band around its axis. Since it is hung up on a polar support, this is easier to accomplish than with some other designs like horizontal plates. The drawback, of course, in addition to mechanical complexity, is that somebody has to constantly readjust the dial. (2) When the sun is at different declinations, the line of the shadow will be at an angle across the strip. Multiple lines can be put on the strip and you only have to read from the one with the angle matching the shadow. The drawback is the inherently bivalent nature of the analemma, i.e., for most dates there will be two lines that match and you still have to figure out which one to read. Furthermore, the accuracy is not great near the equinoxes because the EoT is changing but the declination is not. (3) The shadow is a line, but only a point is needed to tell the time. This opens up the possibility of adding date lines down the length of the strip and making the time lines the same wiggly shape as the Equation of Time graph (plus, ideally, an additional correction for the changing slope of the shadow mentioned above). To use the dial, you find the intersection of the shadow with the line for the current date and read the corrected time from the wiggly line passing through that point. I like this solution. It is easy to manufacture because all the time curves have the same shape and the same separation, and it is easy and accurate to use. Regards, Art Carlson
Re: FAQ commentary
Jim_Cobb [EMAIL PROTECTED] writes: I've thought of another tip for spotting worthless horizontal sundials (such as is sold in garden shops, etc)--if the shadow of the gnomon crosses the hour lines it's no good. This test requires only horizontal positioning, not polar alignment, and a lot will fail this test because the gnomon for cheap dials often does not intersect the dial plate at the convergence point for the hour lines. Actually it doesn't require horizontal positioning either, or even a shadow. For each hour line, you should be able to find a position for your eye such that the edge of the gnomon is superimposed on the hour line. If they ever cross, i.e., if you can ever see part of the hour line above the gnomon but not all of it, then the gnomon will not intersect that line in the dial plate, and the dial is worthless. --Art Carlson
RE: Heliograph
Tony Moss wrote: In my impecunious searches of WWII 'surplus' stores back in the 1950s I came across a Portable Heliograph Set' in a pouch. It was simply a mirror about four inches across with a sighting hole in the middle. A length of cord attached it to a short rod with a bead on top. In use the mirror was held in one hand near to the operator's eye. The cord was then stretched tight and the 'bead' used to 'sight' the target. If the mirror was then rotated until a sunray coincided with the bead above the other outstretched hand a flash of sunlight would be directed at the target. I learned a different method in Boy Scouts: While looking through the hole at the target, you will also see an image of your face in the back side of the mirror. There will be a spot of light on your face where the sun shines on it through the hole in the mirror. If you tilt the mirror until the image of the spot coincides with the hole in the mirror, the sun will be reflected toward the target. This method might be considered less intuitive than the stick-string-and-bead method, but I actually find it simpler. I am fairly certain it is also more accurate. It also takes less equipment, so it can be carried out without preparation with any two-sided reflecting surface. And while we're on the subject, I would be interested in learning how the heliograph in Peter Mayer's jpeg is aimed. It's not as easy as aiming a laser or a search light because information on the position of the sun as well as that of the target is needed. --Art
RE: Heliograph
Bob Haselby and Tony Moss dialoged: This sounds like a signal mirror ... It uses double internal reflection in the hole to give a virtual image of the sun Any chance of a diagram or somesuch to show how this works Bob? It could work like this: Set up two sheets of glass and a mirror so they are all perpendicular to one another. There will be a faint image of the sun reflected in each sheet of glass, but also a still fainter image due to a reflection from both sheets. The direction of this third image is also the direction the sunlight will be reflected from the mirror, so if you tilt the assembly until the faintest image is superimposed on your target, they will see the light. This does not yet sound like a practical piece of emergency equipment, but maybe it will give somebody enough of an idea to figure out how real signal mirrors work. --Art Carlson
RE: Easter ( a bit off topic)
Any rule for calculating the celebration of Easter depends on whether you are interested in the Western or Orthodox holiday. Furthermore, any calculations for the future will become wrong if the rules are changed. See, for example, http://www.smart.net/~mmontes/pr.wcc.19970324.html --Art Carlson
RE: Urgent request.
For the benefit of Tony Moss, a search on http://bible.gospelcom.net/cgi-bin/bible in KJV for every thing beautiful yielded: He hath made every thing beautiful in his time: also he hath set the world in their heart, so that no man can find out the work that God maketh from the beginning to the end. ... Of course, the authorized version is always what the client wants. Whether the Bible is infallible or not, the customer certainly is. --Art Carlson
RE: Sundial with a Second Hand
Bill Walton wrote: To get the desired accuracy the pin-holes' themselves must be very accurately aligned (not true if the free pin-hole technique is used and the hole moved back and forth until the shadow of the gnomon is centered, and on the hour mark, at the same time) They would not have to be more accurately aligned than the hour line itself. In fact, you could drill a hole in the center of each hour line and use that as your pinhole. You're on the money when the image of the gnomon bisects the image of the sun, regardless of where those images are projected. --Art Carlson Who knows what evil lurks in the hearts of men? The Shadow knows!
FW: Shadow Sharpener
Roger Bailey wrote: I tried your Shadow Sharpener test today and was amazed at the result. Me, too! It was easy, just using the shadows falling on my desk. My pinhole was made by sticking a paper clip through a Post-It, which I stuck to the edge of a clip board on my window sill. The gnomon was first a seam down the middle of the window, then I changed to another Post-It stuck to the window so I could adjust the width of the shadow. With the image of the gnomon a bit smaller than the image of the sun, I watched the brightness of the spots on each side where the sun was shining through. There was only about a two second interval when they looked balanced. That means even a quick and dirty set-up can yield an accuracy approaching +/- 1 sec! (I am in San Diego near midday, so the shadows may be running a bit faster than usual.) --Art Carlson
RE: a peculiar sharpener
John Carmichael wrote: The design which worked the best was a 1/8 inch spherical bead, suspended by thin brass crosswires, in the exact center of a 1/4 inch round hole. (The style was about 24 inches from the analemma). A very curious thing happens with this type of style. The bead alone, by itself, casts a shadow that was twice as big as the bead; but when the 1/8th in. bead is in the center of a 1/4 hole, with a space of 1/16th of an inch between the bead's edge and the hole edge, the bead's shadow miraculously sharpens into a tight, dark shadow that is only 1/16th of an inch in diameter, smaller than the bead itself The wires which keep the bead suspended in the middle of the hole are so thin that they don't cast a visible shadow onto the analemma. And Richard M. Koolish calculated: The linear diameter of the diffraction spot (Airy disk) produced by a pinhole of a given diameter is: spot = (2.44 * wavelength * focal_length) / diameter The optimal size is where spot = diameter, so: diameter * diameter = (2.44 * wavelength * focal_length) diameter = sqrt (2.44 * wavelength * focal_length) An example of a pinhole for a distance of 100 mm and a wavelength of 550 nm is: diameter = sqrt (2.44 * .000550 * 100) = sqrt (.01342) = .366 mm Using a distance of 24 inches = 610 mm, this becomes 0.9 mm = 1/32 inch, still several times smaller than John's hole. I think the explanation lies in simple geometrical optics. Imagine putting your eye where the shadow is being cast and looking back toward the style and the sun. I would like to suppose that the distance to the style was something closer to 14 (subject to objection and correction from John), so that the image of the sun would be just eclipsed by the 1/4 inch bead, giving a black shadow at the center. Just a little off-center, an arc of the sun would show around the bead, so the brightness would grow, but only until the disk of the sun runs into the edge of the hole. Thereafter the brightness would decrease slowly until the sun is entirely outside the hole. This would lead to a shadow with a diameter-at-half-brightness of about 1/16 inch, within a diffuse bright field with diameter on the order of 1/4 inch. The size of the shadow is reduced at the cost of reducing the contrast with the surrounding lighted area. The principle is much the same as a sundial that images the sun through a pinhole: a sharper image is a dimmer image. --Art Carlson
RE: A GIANT PRECISION SUNDIAL
John Carmichael wrote: Does this mean that there is no upper limit for the size of a sundial? * Seems obvious to me. The limitation in most configurations is the fuzziness of the shadow, which also implies that size doesn't improve precision. If this is true, then one second time line markings could be placed on the dial face, couldn't they? I haven't done the math, but if the one second lines at high noon ,when they are closest, were spaced at an easy to read distance of about a 1/2 inch apart on a giant horizontal sundial, then the height of the style and the diameter of the face could be determined. It would be a large sundial indeed! The Earth rotates at 360 degrees/day = 0.073 mrad/sec. Divide this into 1 cm/sec and you get the scale of the sundial, 140 m. Monumental, but doable. It has long been my dream to design and construct such a sundial, maybe not with one second markings, but with 30, 20, or 10 second time lines. (What are the time divisions on the large sundial in Japur India, does anyone know?) I'd like to use the same basic design that I use for my horizontal string sundials (see website). The sundial face could be located in a park and people could walk on it. The cable style would reach way up to a pulley attached to a building roof edge or southern wall. A very heavy counterweight suspended from the cable would apply tension, making the cable straight. The diameter of available stranded metal cable may be the limiting size factor here because if the sundial were too large and the cable too narrow then the shadow would completely disappear (like telephone lines do on the ground). The diameter of the sun is 8.7 mrad, so the style would have to be at least 140 m X 8.7 mrad = 1.2 m thick to provide an umbra. Consider using a thinner cable with a 1.2 m ball attached, so that the date can also be read with great accuracy. You should be able to determine the exact day of the solstice with that precision, and the equinox within 15 minutes! I think you will never be able to locate the position of the shadow this accurately, however, without imaging optics. The most convenient lens to use for dialing purposes is that in your eye. You can get a very accurate reading if you look for the place or time where the image of the sun disappears behind an appropriately sized object. The biggest drawback of this approach, particularly in a public park, are the hundreds of people who will be blinded by looking too long into the sun. One solution that would be appropriate for a park would be a shallow, arc-shaped pond, preferably with the bottom black except for the dial markings. The visitors would walk along the pond until the reflection of the sun is blocked by the reflection of the style. This also solves another problem of a configuration where the style is viewed directly, namely the error due to different eye levels. This would be a sundial the builders of Stonehenge could be proud of! --Art Carlson
RE: accurate vs. precise
Speaking of barleycorns reminds me that one can have a lot of fun with units. My favorite combination has components atmosphere = 101,325 newton/m^2 yard = 0.9144 m barn = 1 x 10^(-28) m^2 Combining these we get the barn yard atmosphere = 9.265158 x 10^(-24) joule a unit of energy. Just to relate this to our everyday experience, I would like to point out that the barn yard atmosphere is also a convenient unit of temperature, lying just between the Fahrenheit and Celsius degrees. I once heard that the mass of the electron in pounds is exactly 2.00 X 10^-30, but I don't know whose pound you need to use to get this. (When the Germans say pfund, they mean half a kilo.) --Art Carlson
Re: eleven days
Martin [EMAIL PROTECTED] wrote: Regarding Franks mention of simple folks cry of give us back our 11 days Well I would be pretty riled too if the rent was due 11 days early as I'm sure evil land lords would have used the change in the calender as a good excuse to ring money from the masses. I bet they didnt get paid earlier!!! Actually, they were just having to pay the rent for the extra leap days that had been incorrectly added to the calendar century after century. They can be glad they didn't have to pay interest on the back rent. I suppose a progressive pope might have decreed that the eleven days stuck should include the day rent was due, letting everyone live some three weeks rent free. It is interesting to note that the days of the week were not skipped, so that the day following Thursday, October 4, 1582 became (in Catholic countries) Friday, October 15, 1582. Otherwise there would have been additional difficulties determining a week's wages. All in all, it would have been a lot easier, from a practical point of view, to not drop the 10 or 11 days all at once. The same effect could have been achieved by just declaring that none of the next 40 or 44 years would be leap years. -- Art Carlson
Re: Internet Time
John Carmichael writes: Hey, did anyone see the CNN story last night about the watch company ,Swatch that is now selling timepieces which tell Internet Time? I can't remember exactly, but they said one minute of normal time=about 1 1/2 minutes Internet Time, and that the idea behind it is to facilitate timekeeping around the world for internet users. Everybody everywhere (even on Mars?) will be using on the same time! Arthur C. Clarke believes that the current timezone system will be abandoned and everyone will use Universal Time in the future. I agree with Arthur. Or am I wrong, will we all be using Internet Time instead? You're both wrong. In the future everyone will use local solar time. This is certainly the most natural time for any living thing. The need to physically transport time to find the longitude and the need to provide timekeeping at night and on overcast days led to the rise of mechanical clocks. Unfortunately, these were not sophisticated enough to tell the true time, but had to be satisfied with an unnatural uniform scale. It is, of course, trivial for any microprocessor to convert its clock pulses to solar time and also to convert the time at any other place to the local solar time. If I am in Germany and want to arrange a meeting with someone on Japan, I would say, After lunch would be good for me, say 2 PM?. My email or voice mail would not only be translated to Japanese, the time would also be translated to the local time of my correspondent, something like 11 PM. The airlines use this principle already, in that the arrival and departure times on the ticket are always the local times. The radio transmitters used now to synchronize clocks will in the near future be ubiquitous and short range, so that you will never have to adjust your watch while traveling. In a similar way, the attempt to change timekeeping to a base ten system is an anachronism. The decimal system is a lifesaver if you have to do complicated (scientific) calculations in your head or on paper. For simple (everyday) calculations, a system based on 12 (or 24 or 60) or possibly 16 is much more convenient. The processors which are taking over all but the simplest calculations for us have no trouble with 12 inches in a foot and 24 hours in a day. The implications this has on the demand generated and respect tendered for the skills preserved in this mailing list are obvious. -- Art Carlson
Re: lunar eclipse
Jim_Cobb [EMAIL PROTECTED] writes: I noticed that this time disagrees with the time given in the almanac, so I thought I should provide more information so as not to impugn the reputation of the excellent xephem program. The 16:08:17 UT time is what xephem computes as the time of the full moon. I do not know how to get it to reveal the maximum eclipse time. Well, that's interesting. I would have defined full moon as the time when the moon is most nearly opposite the sun, which would be the same as the time of maximum ecclipse. How else can it be defined? There must be something like a projection into the ecliptic. Art Carlson
Re: Analemmatics on a Gradient
[EMAIL PROTECTED] (Mr. D. Hunt) writes: In relation to the recent question/replies, regarding detecting/correcting 'errors' in the setting of sundials - is there any feasible way of varying the layout of an Analemmatic dial, to cope with it being on a GRADIENT ? My own thinking is that this is just NOT possible, if the dial has to tell 'correct' time (disregarding EOT effects) at all times of DAY, plus at all seasons of the YEAR - but will welcome any comments/confirmation, on this. The gnomon, whether vertical or not, together with the direction to the sun, defines a plane. The intersection of this shadow plane with the ground plane, whether horizontal or not, defines a line. If you think of the celestial sphere as being a finite size and centered on the base of the gnomon, then the position of the sun projected along the direction of the gnomon onto the ground plane will lie on the shadow line. The orbit of the sun during the course of a day is a circle, generally not centered on the base of the gnomon. The projection of the orbit on any day will be an ellipse, though the center of the ellipse will move from day to day. An analemmic sundial is designed by rescaling all the ellipses to the same size, then translating them to lie on top of each other, which implies that the gnomon must also be translated to a particular position for the projection to be accurate on that date. The upshot is, an analemmic sundial properly designed for sloping ground will be just as accurate as one on the level.
Re: speed of light
John Carmichael writes: We could make this question even more complimented if we consider the speed of light. When we see the sun's center on the horizon we are seeing light that left the sun about 8 minutes earlier. The sun really has already set. (of course this has no practical effect on sundial time, but is fun to think about!) What does that mean, The sun really has already set.? I would say, By the time the light now leaving the sun gets here, I will have moved behind the edge of the Earth. But the sun really is located in the direction from which the light I see is coming. (There's a teeny tiny shift in the direction if I am moving perpendicular to the line of sight, but that is not the case at sunset.) Art Carlson
Re: sundial setting
[EMAIL PROTECTED] (Philip P. Pappas, II) writes: Thank you for your thoughtful comments. I make the statement that the time method is the prefered method for setting a sundial if and only if the sundial is properly designed, constructed and leveled (correcting for the EOT and longitude of course). I would say that it is the preferred method *especially* if you suspect you have a poorly made dial. If you set it up by the time method, then at least you know it is accurate at one time for two days of the year. This is not guaranteed to, but is likely to reduce the errors on average. 4. This has just occurred to me and is probably not relevant but it has got my mind wondering. As we know, the earth is a flattened sphere. Gravity, from which we derive a vertical (and subsequent horizontal) reference comes from the centre of the earth's mass. This is presumably right in its centre, assuming that differences in local density do not move it by much. But as we move towards the flattened poles the angle to the centre of gravity will no longer be a true vertical. But even so, it is this centre of gravity which is the true reference point for the earth in its orbit around the sun. Then there is the centrifugal force due to its rotation. Will this effect a true vertical? At the equator - no, but imagine a point at 45 degrees latitude, where the centrifugal force must have some effect on any plumb line/spirit level. I guess that all of these effects are so tiny as to be irrelevant, but I would like to know how much they modify the results. These effects are one and the same. The Earth is flattened at the poles *because* centrifugal force pulls it out around the equator. At the Equator and at the poles the vertical passes through the center of the Earth, inbetween it doesn't, but that doesn't affect the accuracy of a properly designed dial. Just for fun, the radius of the Earth is 6,378 km and the difference between the the equatorial and the polar semiaxis is 21.4 km. This makes the maximum discrepancy in the angle about (2*21.4/6378) = 0.4 degree. Art Carlson
Re: sundial setting
An analemmic dial would be insensitive to refraction effects, wouldn't it? Art Carlson
Re: Help needed with unusual sundial
Dear Bob, Fun problem. 1. If I were setting the thing up, I would turn the existing disk so that the local longitude pointed up, not that of Greenwich. That way the observer can see at a glance where in the world he is, as well as the approximate time anyplace else in the world. There is a blemish on the longitude disk in your last photo; it looks as though it could be a locking screw. 2. To trace out the path of the sun, the hole would have to rotate around the axis, so I would want the hour angle arc to be similar to the one you have drawn, but rotatable. The hole would be in the slide, which moves up and down the arc at the middle +/- 23.5 degrees. 3. Either the hour scale must be attached to the hour angle arc and move under a fixed pointer or the pointer must be attached to the arc and the scale fixed. I like the idea of having the scale on the globe, since it reduces the number of parts. If the globe moved with the hole, however, it would be easier and more accurate to align the sun spot up with a line on the globe, rather than judging when the spot is round. On the other hand, if the globe is fixed, then it would be simpler and more elegant to read the time directly with the spot of light, rather than with an extra pointer. Furthermore, the globe can then sensibly be an actual globe, with a map of the world on it, rather than just a sphere. (But then why bother with a separate longitude disk?) Good luck, and lots of fun. Art Carlson
Re: Invention to tame moon monsters
Roger Bailey [EMAIL PROTECTED] writes: I was experimenting with the shareware program Astronomy Lab. One calculation that this program plots is the Moon Angular Speed in degrees per day. This is the lunar equation of time we have been looking for. In minutes rather than degrees, the variation is up to 14 minutes on top of the 48 minute average daily correction that we have been quoting. The moon's equation of time is the variation on that average angular speed. The graph shows this well as the sum of two periodic cycles. The major cycle is the monthly lunar cycle. The moon speeds up when it is closest to the earth (perigee) and slows down when it is most distant (apogee). The cycle ranges from about 11.8 degrees (47 min) to 14.2 degrees (57 min). A yearly cycle is added to that giving maximum peaks of 15.4 degrees (61min) when the full or new moon (lunation) is in phase with the lunar orbital cycle. Arthur C. noted the connection between the lunar and solar (year) cycles. I don't understand why the position of the sun should have an effect on the angular velocity of the moon. Does the yearly cycle superimpose another oscillation (like making the moon run generally slower in summer than in winter) or does it modulate the amplitude of the monthly cycle (like making the moon run at a more nearly constant speed in summer than in winter)? Art Carlson
Re: Best angle to catch sun light - off topic
Fernando Cabral [EMAIL PROTECTED] writes: Now I am planning to build a house for a small farm I have. I've been thinking on how to take the best advantage of the solar power. This includes where to have a garder with a nice sundial and where to place the solar panels for water heating as well as (perhaps) electricity (at least in Brazil solar panels for electricity are very expensive). At 19 37' 57 S, it is clear that the panel should be facing North. But what is the best angle with the horizon. And, if I can have several panels, is there a practical to calculate the best angle of each so as I can guarantee the highest possible insolation level? Say, if I have three panels, is it best to place them side by side, with the same inclinatation and declination? Perhas if one is a inclined towards the East with a certain angle and the other to the West with a proper angle I can capture more light? Goods questions, to which I don't have the answers. I would even question the basic assumption that the panels should face north. You may want to have fresh hot water as soon as you can in the morning, especially if you shower then, in which case the panel (or one of them) should face east. At midday and in the evening you can use the heat that has been collecting all day. By the same token, it is probably better to point the panel low to the horizon (angle between the normal and the vertical equal to 23.5 degrees minus the latitude, since you are in the tropics) because you will want to produce more hot water with less sunshine in the winter. (You may not have enough of a winter that that matters, but there may be similar considerations for rainy/dry season, afternoon thundershowers, etc.) If you determine the optimum angle simply by integration of the sunlight over some period, then that angle will be the same for every panel. If your use patterns are different and the storage characteristics poor, then you might want to do something like point one panel to the east for morning hot water and one to the west for evening hat water. Do you need some inclination to drive convection through the collectors? Or to prevent rain water from collecting on the panel? Since you say there is a great variety of orientations of panels in the city, can you get answers to some of your questions by interviewing residents with different orientations? Art Carlson
Re: moon monsters
Dear John, Your explanations sound like about the right level for a users' manual. Maybe because I'm a scientist, I think it is important to at least mention the major sources of error. In my opinion, the biggest problem is determining the exact phase of the moon by looking at it. (Of course, you can get pretty accurate by looking it up in the newspaper.) I would guess a misjudgment of the phase by up to one day is common without a lot a practice. That will result in up to 48 minutes of error. The EOT might be considered small compared to that, but I think I would mention that it should be used. Some of your users may get a kick out of an accurate measurement. The other errors, like those due to the tilt (5 degrees) or eccentricity (0.0549) of the Moon's orbit, I expect to be on the order of the square of the parameters, which is under 1%. But 1% of what? 1% of a day is 14 minutes, so I would need to give this some more thought. As far as your eclipse observation goes, I suspect the eclipse either took place when the correction was small, or you misjudged when the center of the eclipse was. Since a lunar eclipse takes a fairly long time, the moon time at the start and the end can differ by several minutes. Art Carlson
Re: moonlight readings
John Carmichael writes: I have a section which tells how to tell time by using moonlight and a sundial. I provide a table of corrections from which the time can be estimated if one knows the age (the phase) of the moon. One question though: Is it nessary to correct moontime with the Equation of Time ? Since the Equation of time is due to the eccentricty of the earth's orbit around the sun and the tilt of the earth's axis, it seems to me that this has nothing to do with the moon and should not be considered in the corrections. Am I right? Roger Bailey [EMAIL PROTECTED] writes: Hello John, My advice is Don't go there. There be monsters! * Good advice. The motion of the moon is quite complicated and the equation of time shortcut will not work. You were right is concluding that the solar equation of time does not apply, and the eccentricity and obliquity of the ecliptic were the determinants of the equation of time. I wouldn't agree the Equation of Time does not apply, just that other corrections are much larger. John does, after all, want to correct for the phase of the moon, so the position of the sun is relevant. The major problem with the moon is the time between new moons (lunation) is 29.53 days, different from the orbital period of 27.32 days. This means the declination cycle, connected with the orbital period, is out of phase with the lunation cycle. This makes it sound like these are two separate orbital parameters. They are simply connected by the length of the year: 1/27.32 - 1/29.53 = 1/365 In fact, the time between any two particular adjacent lunations will have a correction closely related to the Equation of Time. For night time checks, I use a nocturnal and determine the time based on the rotation of the big dipper around Polaris. The date / sidereal time correction is easier to build into the instrument. Even easier than correcting a sundial for the Equation of Time. I have been interested for some time in the related problem of finding directions from the moon, possibly given watch time. I haven't formulated the mathematics yet. To quantify the error of various methods I will need some more information on the distribution of the relative positions of the sun and moon. This is certainly known. Is it also readily available in a comprehensible form? Art Carlson
Re: Definition of Time?
Paul Murphy [EMAIL PROTECTED] writes: September 11-24 , 1752 Unfortunately, Warren, even this depends where you were at that time! Had you been in a place where the Gregorian Calendar had been accepted in 1582, quite a lot might have happened. On the other hand had you been in Russia, you would have to wait until 1917 to find the lost days!! I wonder something every time I hear about idiot savants who can tell the day of the week of any calendar date. Do they ever make the switch from the Julian to the Gregorian calendar? If so, when? I suspect the psychologists examining them don't know enough about the calendar to realize there is an issue. It's like claiming they can recognize any prime number instantly without asking, say, if the product of two particular ten digits primes is prime. Art Carlson
Re: What's sum of series of increasing powers?
Tad Dunne [EMAIL PROTECTED] writes: I'm working on an Excel spreadsheet and need a formula or function that will give, for an input A and B, the sum of all the powers of A for integers from 1 to B. Example: 1.05 + 1.05 squared + 1.05 cubed ... S = A + A^2 + A^3 + ... + A^(B-1) + A^B = A + A*( A + A^2 + A^3 + ... + A^(B-1) + A^B ) - A^(B+1) = A + A*S - A^(B+1) S*(1-A) = A - A^(B+1) 1 - A^B S = A * --- 1 - A Cheers, Art Carlson
Re: Milennium Clock
Tom Mchugh [EMAIL PROTECTED] writes: One thing which doesn't seem to have surfaced in the discussion yet, is the imponderable effect of plate tectonics upon the accuracy of any type of sundial over a period of 10,000 years, which effect would cause both a latitude and longitude change in the position of any fixed dial. Since different plates move in different directions and at different speeds, one would have to compute a correction table based upon the particular plate upon which the dial is located. If you only use the sun through a N-S slit to synchronize the clock, then latitude shouldn't matter. The longitude correction will be small, but perhaps not negligible for some locations. Another possible source of error is that therre is a good likelyhood that many parts of the Northern hemisphere will be under a mile or two of ice in 10,000 years. There is substantial geological opinion to the effect that we are now enjoying the balmy climate of an interglacial period. And, if one builds a dial in an area not likely to be crushed under a glacier, there is still the problem of changes in rotation rate due to the shifting of thousands of cubic miles of water from the deep ocean basins to northerly land glaciers. I don't know whether anyone hascome up with an accurate model of the effect of glaciation on the rate of change of earth's rotation and nutation c. Danny Hillis is thinking of a desert. Everything lasts longer in a dry climate (see the pyramids). You are right that no potential for glaciers should be a site criterion. I wouldn't expect the change in rotation rate due to ice to be a problem for the same reason that the secular slowing shouldn't be. The error cannot accumulate because you are constantly synchronizing the clock to the sun. I do worry about keeping the clock synchronized if the sun disappears for several months due to a very bad turn of weather, a nuclear winter, or a meteorite impact. Art Carlson
Re: Milennium Clock
fer j. de vries [EMAIL PROTECTED] writes: On this list many is said about the equation of time, the precession and so on in relation to the milennium clock. And in the quoted mail is said the clock should be accurate to the minute in 10,000 years. Is this possible at all? Think of the decrease of the earth rotation. This affect, or at least a part of it, isn't predictable. To synchronize the atomic clocks to the civil time, which still is based on suntime, leap seconds have to be added. These leap seconds can't be predicted precisely. So at this time it is unknown how many will be needed in the coming 10,000 years. It will be many more then 60 I think. And this correction will be needed to synchronize the milennium clock. I assume that leap seconds will be added as needed so that in Greenwich averaged over a year noon clock time agrees with noon sun time. The Millennium Clock will be synchronized to mean sun time. In this case there will be no drift. A problem doesn't develop until the day is so much out of synch with the clock mechanism that the error can accumulate during a long period of cloudy weather to a half-swing of the pendulum. I suspect this will take much longer than 10,000 years (though I have been proven wrong in my suspicions on other topics in this list). Future generations can fine tune the clock for this eventuality by lengthening the pendulum a tiny bit. Art Carlson
Re: Precession / EoT
Luke Coletti [EMAIL PROTECTED] writes: Arthur, Please investigate for us Touche. I think some of these questions will move to the back burner. To significantly improve my understanding of celestial mechanics, I need to do some systematic reading. This thread started with the Millennium Clock. The plan is to have the clock tell clock time, but to synchronize it using the sun. I would want it to still be accurate to the minute 10,000 years from now. We have established that in that case it will have to take the major changes in orbital parameters into account. Additionally I was wondering if we should anticipate another calendrical reform in the next 10,000 years. Although there would be ways to keep the months better aligned with the seasons, say by making all millennium years leap years, not just those divisible by 400, on a 10,000+ year time scale things change so much that no rule will remain satisfactory for long. Whether future generations let the seasons drift, introduce intercalary days ad hoc every few thousand years, or establish a completely different calendar is impossible to predict. In conclusion, I would base the primary display of the Millennium Clock on the Gregorian calendar, with separate displays for other calendars (Chinese, Moslem, Jewish, Mayan) and astronomical data (precession of the equinoxes, phases of the moon). Art Carlson
Re: Precession / EoT
Luke Coletti [EMAIL PROTECTED] writes: Below are some data that may help you, the calculation date is Jan 1 Noon UT, EoT values are in the form TA-TM. The discussion to date has been more about the variation of our orbit and Earth's alignment within, however all these events need to be related to a calendar and since there is not a even multiple of days in our orbital period I think you can see how the EoT becomes unsynced. Note that after twenty years, which falls on a four year boundary from the start, the delta is 1.3 secs. ... Column 1: day of year, Column 2: year, Column 3: days in year Column 4: days from J2000, Column 5: Solar Day Length, secs Column 6: EoT, secs, Column 7: EoT delta, secs 1 2000 366+0.0 -28.5750 -198.0059 +0.00 1 2001 365 +366.0 -28.3493 -219.3062 -21.300274 1 2002 365 +731.0 -28.4230 -212.3139 -14.307972 1 2003 365 +1096.0 -28.4950 -205.3031 -7.297152 1 2004 366 +1461.0 -28.5650 -198.2742 -0.268264 1 2005 365 +1827.0 -28.3388 -219.5665 -21.560536 ... We see a -28.3 sec jump when the calendar is changed by one day. What's left over is -0.268 seconds, which accumulates. It would seem that the difference would reach -28.3 sec after 422 years, at which time we would want to leap over a leap year. Why does the Gregorian calendar skip a leap year every 133 years (on average)? Whatever the ratio between the length of the day and the length of the year, one can find calendrical rules which approximate the ratio. Over what time scale do irregularities in the perturbations of the planets or slowing down of the Earth's rotation rate through tidal drag cange the ratio enough to make simple leap year type rules invalid? Art Carlson
Re: Precession / EoT
Luke Coletti [EMAIL PROTECTED] writes: You still appear to be asserting that in calculating the Longitude of Perihelion (over a 10,000 year period), only Precession need be considered and the shifting of Perihelion due to perturbation and other smaller combined effects can be ignored. I'm sorry to appear picky, but this assertion is most definitely incorrect. You have convinced me that Danny Hillis will indeed have to put a lot of (slowly turning) gears into the Millennium Clock. I am surprised at the size of the change in Eccentricity, but maybe I shouldn't be. The perturbations from the other planets probably affect it in much the same way as they affect the Longitude of Perihelion. Even the change in Obliquity amounts to nearly a minute, and it would be a shame if the people in the year 12000 could not set their watches by the Clock. One thing that threw me is that you quoted 12 arc-sec/yr as the rate of shift of the Perihelion during the period 1980-2020. The numbers you give above for the period 2000 to 12000 amount to 66 arc-sec/yr. Can you explain why the shift is so irregular? Can you explain why the shift exists at all? These questions are getting very deep into celestial mechanics, and I am willing to defer discussion until I have at least read Meeus. Regards, Art Carlson
Re: Precession / EoT
I wrote: ... I think the processes which change the eccentricity and obliquity of the Earth's orbit work on a much slower time scale than the precession of the equinoxes, so that we can still use the same Equation of Time 13,000 years from now. Does anybody know for sure about this? Luke Coletti [EMAIL PROTECTED] replied: You are right regarding the relative small changes in the values of Obliquity and Eccentricity over the period of a precession cycle. However, it is the phase relationship of the two effects and not their values that play the dominant role in the variation of the Equation of Time over this period. The Analemma will indeed look considerably different 13,000 years from now. My mistake. The magnitude of the obliquity and eccentricity change very, very slowly, but the phase is determined by the precession of the perihelion (also very, very slow) and the precession of the equinoxes, with the 25,860 year period we are discussing here. Putting it another way, dialists (not only those using astrolabes) do care about the stars because they care about the perihelion, which is fixed relative to the stars. The Millennium Clock will either need to calculate the changing Equation of Time, or else average its sighting of the sun over several years, which I presume to be more difficult. I would anyway rather see the clock designed for the astronomically significant cycle of 25,860 years than for a numerological cycle of 10,000 years. If the mechanism of the clock inherently contains the precession period, that is one more reason to make the display correspond. The 10,000 time frame was chosen, among other reasons, because that is the length of time since the development of agriculture and technology. On the other hand, there were some damn good painters active 25,860 years ago. Art Carlson
Re: Precession
Sonderegger [EMAIL PROTECTED] writes: I think in the northern hemisphere summer is always in July, because the beginning of spring is here always when then sun crosses declination of 0 degree from south to north (= crossing the ecliptic). The places of the stars on then sky will change in this 13000 years. Right. The question is not whether summer will be in July, but whether July will be in summer. That only depends on how you choose your calendar system. The Gregorian calendar reform was an effort to keep July in the summer. The precession is a physical process, which does not depend on our calendars. What it does is change the constellations that are visible at night in July. Since we don't use the stars for sundials, it won't make any difference. What will matter is the Equation of Time, as mentioned by Luke Coletti. I think the processes which change the eccentricity and obliquity of the Earth's orbit work on a much slower time scale than the precession of the equinoxes, so that we can still use the same Equation of Time 13,000 years from now. Does anybody know for sure about this? Art Carlson
Re: refraction
Thanks for the graph, Luke. If I take +/- 20 sec as the accuracy of a very good sundial, then I see that I have to start correcting for refraction around 10 deg altitude, i.e., the first hour after sunrise and the last hour before sunset. Since Pete Swanstrom's earliest observation is at 7:10 am on May 6, when the sun in Boise is 16.5 deg above the horizon, it is not surprising that he saw no refraction effect. If you don't feel like a stroll in the park at 5:30 am, Pete, how about taking a series of measurements in the last hour before sunset, between 7:30 and 8:30 pm? Art Carlson
Re: Sundial in Pretoria
Anton Reynecke [EMAIL PROTECTED] writes: When I was a young boy it amazed me that it was actually possible, but now realise it is just a form of sundial. It is situated in Pretoria and can only be seen in action less than an hour every year (annum), and is is a special feature of the Voortrekker Monument (roughly translated as Pioneers Monument) from South-Africa's controversial past. A spot of sunlight (about a foot in diameter) shining through a hole in the roof, onto the middle of an epitaph in the centre of this monument, at exactly 12 'o clock (Standard time) on December 16, every year, to commemorate a certain event in history. On the other days of the year, the sunlight does not enter the monument at all. This is not possible. In the first place, the sun is in exactly the same place in the sky at 11:55 on Dec 26 as it is at 12:00 on Dec 16, so these times cannot be distinguished in principle. The finite size of the sun's image makes this much worse, so some sunlight is bound to get through the hole on at least 10 days no matter how you arrange it. If the date you want to commemorate happens to be an equinox, then you could in principle create a monument which lets in a ray of light at a particular time on that day, but you can only use a sliver of about 0.0002 the area of the sun's image. Did primitive peoples determine the solstices from direct observation near the solstices or by halving the days between the equinoxes, which can be determined accurately? Art Carlson
Re: Help with Trig Problem
[EMAIL PROTECTED] writes: Could someone help me solve for declination of the sun or latitude from the equation for altitude: sin(Alt)=sin(dec)*sin(lat) + cos(dec)*cos(lat)*cos(local hour angle) I would like to know how this is solved as much as just knowing the answer. You want to add a sine wave and and a cosine wave with different amplitudes. The result will be a sine wave with a phase shift. Use the trig identity: sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b) My a will be your dec. My b can be found from the ratio of the coefficients: coefficient of cos(a) cos(lat)*cos(local hour angle) tan(b) = - = -- coefficient of sin(a) sin(lat) We have to multiply both sides of your equation by 2 2 -1/2 c = [ ( sin(lat) ) + ( cos(lat)*cos(local hour angle) ) ] so that the sum of the coefficients, cos(b)^2 + sin(b)^2, is unity. Then we have: sin(a+b) = c*sin(Alt) Written out (nearly) in full: dec = -arctan( cot(lat)*cos(local hour angle) ) + arcsin( c*sin(Alt) ) [Disclaimer: Don't believe it or use it until you or someone else has checked it.] --Art Carlson-- [EMAIL PROTECTED] http://www.ipp.mpg.de/~Arthur.Carlson/home.html
Re: lat long
Clem asked: ... Anyone know of a web site that would have the lat long of cities around the country? Maybe a database one could query? I just happened across a site giving lat, long, and time zone of cities around the world: http://www.quantum.de/zahlen/koord-int.html Don't let the German language interface phase you. Art -- To study, to finish, to publish. -- Benjamin Franklin Dr. Arthur Carlson Max Planck Institute for Plasma Physics Garching, Germany [EMAIL PROTECTED] http://www.rzg.mpg.de/~awc/home.html
Re: Octaval hours
I am familiar with temporal hours for which the period of time between sunrise and sunset is divided into 12 hours. I have recently come across octaval hours for which the period of time between sunrise and sunset is divided into 8 hours. ... This eight hours were called also tides in ancient Saxon language, and what I know is that that word doesn't mean tide like today, but simply something like space of time. There is obviously a close and practical relationship between time and tides. In German, time is Zeit, and tides is Gezeiten. As for dividing a period of time into eighths, that is a natural result of repeated halving. Like we divide inches into fractions, pounds and pints into ounces, the compass into points, or a dollar into bits (like shave and a haircut...). It was the astrology-loving Babylonians that liked to divide things into twelve parts or 5X12=60 parts, which gave us our system of measuring time and angles. On rainy days I ponder the question, whether it would have been better if evolution had given us 4 fingers, so we would count everything by halves, or 6 fingers, so we would always use base 12, which makes it easy to find the 1/2, 1/3, 1/4, or 1/6 part of anything. I'm sure 5 fingers was a bad choice since it led to this muddle of the bases 8, 10 and 12. Art -- To study, to finish, to publish. -- Benjamin Franklin Dr. Arthur Carlson Max Planck Institute for Plasma Physics Garching, Germany [EMAIL PROTECTED] http://www.rzg.mpg.de/~awc/home.html
Re: Thanks for the responce (wedding time)
Luke Coletti wrote: Define sunrise as the time when the apparent altitude (H) of the upper limb of the Sun will be -50 arc minutes (34' for refraction + 16' for semidiameter). I thought this might give Bart a way out, but it goes in the wrong direction. If you define equinox the way any good witchdoctor would, as the date when day and night are equally long, or as the date when sunrise and sunset are 180 degrees apart, atmospheric refraction gives you a systematic discrepancy relative to astronomical data, shifting the equinox a good day farther into the winter. Unfortunately, Bart wants to get married a day farther into the summer. How about getting married in a valley, and counting the time the sun rises/sets over the mountain tops? Art -- To study, to finish, to publish. -- Benjamin Franklin Dr. Arthur Carlson Max Planck Institute for Plasma Physics Garching, Germany [EMAIL PROTECTED] http://www.rzg.mpg.de/~awc/home.html
Re: exact time of equinox fall '97
Bart wrote:
I am a sundial enthusiast an have been an on-looker of this list for somew
time. Now I need some help from all of you. I am planning on getting
married in september of 97. I hope to get married on the equinox
(autumnal) I would like to know the exact time of the equinox given in US
Central time zone. Also, I wonder what is the tolerence for the equinox.
i.e. can the whole day be considered as the equinox or even three days,
one before and one after?
My point is that if the equinox happens on a tuesday and we want to get
married on a sunday can we call it the equinox?
First off, congratulations! May you have love, joy, prosperity, and
many happy years together.
Which brings us to the question, How long do you plan to stay married?
At your golden anniversary, you will have been married 18,262 days, so
one day more or less will be at the 50 ppm level, generally not
considered significant in human affairs.
On an astronomical level, the equinox is the *instant* when the sun lies
in the equatorial plane of the earth. A reasonable level of exactness
(for a sundial enthusiast) would be to insist that at least some *part*
of the sun lies in this plane. Using this criterion I come up with ...
0.25 deg X 365.24 dy
= 0.637 dy
sin(23.45 deg) X 360 deg
... plus or minus 15 hrs 18 min, which doesn't leave you much leeway.
I've tried to find a different definition which could push the limits
out, but I don't think I can even make two full days out of it without
cheating.
Why do you want to get married on the equinox? The equinox is the
symbol of duality. It divides the year into summer and winter. At the
equinox the day and night are equally long. Sunrise and sunset divide
the horizon into two equal parts. Thus it is indeed an appropriate
symbol for marriage, for complementarity, for two natures which are
different but equal, and indeed are only recognizable next to each other.
This is a good philosophical basis for a life together. But the union
will lose its magic if you try to keep too close account of whether
you both are profiting equally from it. In this spirit, I would
suggest that you plight your troth near the equinox, but don't
calculate the time too pedanticly.
With best wishes,
Art Carlson
--
To study, to finish, to publish. -- Benjamin Franklin
Dr. Arthur Carlson
Max Planck Institute for Plasma Physics
Garching, Germany
[EMAIL PROTECTED]
http://www.rzg.mpg.de/~awc/home.html
Re: Millenium Sundials
Ian Elliott writes: Please note also that the 2nd Millennium will not be complete until the END of the year 2000. The origin of the Gregorian calendar is 1 Jan AD 1. Add 2000 to get the start of the 3rd Millennium, i.e. 1 Jan 2001. That's true, and people like us find it interesting, but if you stay home the night before 1 Jan 2000, you're going to miss one hell of a party. Since the turn of the millennium doesn't correspond to any astronomical event, and not even to the 2000th anniversary of a historical event, I decided what we're celebrating is the fact that the way we write the year will change in 4 digits. Pure numerology. Like watching your odometer turn over. And that happens when 1999 changes to 2000. Hope to see you then, Art -- To study, to finish, to publish. -- Benjamin Franklin Dr. Arthur Carlson Max Planck Institute for Plasma Physics Garching, Germany [EMAIL PROTECTED] http://www.rzg.mpg.de/~awc/home.html
