I mis-stated what the secant is. I said: 'The secant of the Sun's altitude is the *direct* distance from the tip of the object to the tip of its shadow, divided by the height of the object. Of course you probably don't have time to measure with measuring-tape, and you just estimate that ratio."
ii meant: Divide the direct distance between the tip of the object and the tip of its shadow, by the horizontal distance between the base of the object and the tip of the shadow. But, as a practical matter, instead of measuring and calculating, you just estimate by what percentage the direct tip-to-tip distance is greater than the base-to-shadow-tip distance. For example, say it looks as if the object tip to shadow tip is 25% greater than the base-to-shadow-tip distance. That sec Alt Sun. So multiply h by 1.25 And, as I said, optionally multiply a rough estimate of cos dec, which will always be somewhere between 1 and 1 - 1/12. ...1 at equinox, and 1 - 1/12 at either solstice. Michael Ossipoff On Wed, Oct 24, 2018 at 10:30 AM Michael Ossipoff <email9648...@gmail.com> wrote: > Dan-- > > It's as you said. The Watch Method works best (and is only really any > good) when the Sun is low. So, it's really only any good in winter, or very > late or early in the day. > > ...and a lot of people do most of their hiking in the summer. > > Its accuracy increases with latitude. > > But, as Favio pointed out, there's error due to EoT and longitude, if you > don't take those things into account. ...and, by definition, if you're > using the ordinary watch method, you aren' taking those things into account > > For a long time, the Watch method was the only solar direction-finding > method ever mentioned in outdoor books and articles. > > But, in 1962, a kid in this country suggested something simple, and > everyone wondered why it hadn't occurred to them: > > The Shadow-Tip method: > > 1. At the tip of the shadow of a twig, place a pebble, or make a mark in > the dirt. Of course you could place a stick vertically in the ground and > use its shadow. The stick needn't be straight. > > 2. After 5 or 10 minutes, again place a pebble or make a mark at the > shadow-tip's new position. > > 3. A line between those two marks will be roughly east-west. > > The Shadow-Tip method is both easier, and more accurate than the > Watch-Method. It's accuracy is greater at lower latitudes. Of course, an > additional advantage of Shadow-Tip is that it doesn't require any equipment > at all. ...and doesn't reqiuire EoT or longitude. > > But neither of those is the method that I use. For many years, I've been > routinely using an approximation to the Time-Altitude (TA) method. > > TA calculates the Sun's azimuth from the time and the Sun's altitude. Of > course you need EoT and longitude. You don't need latitude. It has been > used some by navigators and surveyors, but it isn't usually favorite. But > its formula is brief, and it lends itself to the practical and convenient > approximation that I use. > > The approximation of TA that I use, I call "the Altitude Watch Method > (AW). > > Instead of making it look complicated by first posting the TA formula, let > me first just describe AW: > > 1. Say you know the longitude, and the EoT for the current day. Adjust the > time accordingly. For longitude, that amounts to adding 4 minutes for each > longitude degree east of your timezone's central meridian, or subtracting 4 > minutes for each degree west of that meridian. > > 2.. Of course each hour moves the solar hour-angle 15 degrees, and each > additional 4 minutes moves it another degree. That's the Solar hour-angle, > from the meridian. > > 3. Multiply that h value by the secant of the Sun's altitude. That can be > estimated by observing the shadow of a post, tree, building, etc. > ...or of a pen held vertical against your forearm or the palm of your hand. > > The secant of the Sun's altitude is the *direct* distance from the tip of > the object to the tip of its shadow, divided by the height of the object. > Of course you probably don't have time to measure with measuring-tape, and > you just estimate that ratio. > > Multiplying h by sec Alt of the Sun greatly improves accuracy, and that > sec Alt is probably all you need to take into account to correct your h > estimate, for practical purposes. > > 4. If you don't know and take into account the Sun's declination, that's > ok, because it doesn't have much effect. But, on the other hand, we usually > have a rough idea of the Sun's declination. For example, right now, toward > the end of October, it's going to be somewhere between 0 and -23.44 degrees. > > So, optionally, mutltiply h * sec Alt Sun by the cos dec, the cosine of > the declination. Roughly estimating that is much easier than it sounds: > > The cosine varies between 1 and 0. For 0 degrees, the cosine is just 1. > For plus or minus 23.44, the cosine is about 1 minus 1/12. > > So, if it were the winter solstice, you'd subtract, from sec Alt Sun, 1/12 > of whatever sec Alt Sun is. Right now, we're between the equinox, when dec > Sun is zero, and the solstice, when dec sun is -23.44. So, the estimate > for the solar azimuth (measured from south) would be gotten by reducing > sec Alt Sun by something less than 1/12 of itself. Say half of 1/12? You > can guess about that, or just disregard it. > > I've nearly always gotten very good results with AW, though there are > combinations of time-of-year and time-of-day when it loses accuracy. > Midsummer and roughly mid afternoon or morning. > > The great advantage of AW of the Watch method (W) is that AW is much more > accurate. > > The main advantage of AW of the Shadow-Tip method (ST) is that AW doesn't > require you to stop walking. AW can be used in a car (where a magnetic > compass isn't accurate due to nearby steel. (...except for an installed > compensated compass). > > ST is what I recommend to people, because it's by far the easiest method, > and much more accurate than W. But I use AW, because it combines good > accuracy with great convenience. > > > > > > > > > > > > > > > > > > > > > On Mon, Oct 22, 2018 at 12:20 PM Dan-George Uza <cerculdest...@gmail.com> > wrote: > >> Dear John and others, >> >> Thank you for your insights. Although I haven't been able to track the >> analysis I was looking for eventually I did find something similar. Google >> "On the method of direction finding by Sun and Watch by Norman Pye". The >> author makes an analysis for true azimuths and watch hour angles, dealing >> with values projected onto the horizontal plane. The directional error is >> due to the Sun moving in a different plane from the horizontal. From the >> table I attach below it seems that the watch method works best in winter >> because then the Sun stays close to the horizon and doesn't have a great >> spread in azimuth. >> >> Hope this helps, >> >> Dan >> >> >> On Mon, Oct 22, 2018 at 1:45 AM <john.pick...@bigpond.com> wrote: >> >>> Hi Dan, >>> >>> Sorry for the delay in replying. >>> >>> From a PRACTICAL point-of-view, as we all know, analogue watches >>> replaced sundials, digital watches replaced analogue watches, and smart >>> phones have replaced watches. Digital compasses replaced analogue >>> compasses, and now smart phones have replaced compasses. I confess to >>> having a mobile phone, and no longer wearing a watch. Also when I go >>> bushwalking, now I carry a GPS with real-time tracking on appropriate-scale >>> topographic maps. I still have a digital compass, but it was pretty fiddly >>> to use, so it now sits somewhere at home. I only use a magnetic compass >>> when doing serious field work, and I need to know the orientation of some >>> feature I am measuring. >>> >>> But what has practicality to do with anything related to sundials???? >>> >>> Several years ago I was also intrigued about the accuracy of using a >>> watch as a compass, and I decided to investigate it the empirical way. So I >>> made up a little “tool” and every weekend when I went bushwalking, I would >>> set it up and compare compass north with watch north at regular intervals. >>> >>> I ended up with quite a few measurements before life got in the way of >>> plans, and the project petered out. I still have the results but I have >>> never analysed them. In part because even then it was obvious to me that >>> the question could be investigated using standard equations. But my feeling >>> at the time was that the whole watch / north method was getting close to an >>> urban myth. I had collected several variations on instructions, but I seem >>> to have lost them in one of my several moves. But I do remember that >>> depending on which you used, the error could be 30o or more. None of the >>> methods said anything about the difference between true and magnetic north, >>> but that may be irrelevant anyway unless you are somewhere like Antarctica >>> where the difference can be 70o. More important would be DST which could be >>> a major trap for the unwary. >>> >>> At one stage the watch / north method was called the “Boy Scout”method. >>> During my travels on public transport when kids are going to school, I >>> haven’t seen too many wearing analogue watches. Or watches of any form. >>> They are completely welded to their smart phones, so it’s pretty obvious >>> that the method has gone the way of sundials. >>> >>> When you find the analysis of the watch / north idea, can you post it on >>> the List please? >>> >>> Cheers, John >>> >>> John Pickard >>> john.pick...@bigpond.com >>> >>> >>> *From:* Dan-George Uza <cerculdest...@gmail.com> >>> *Sent:* Saturday, September 29, 2018 4:57 AM >>> *To:* Sundial List <sundial@uni-koeln.de> >>> *Subject:* Accuracy of wristwatch as compass >>> >>> Hello! >>> >>> I'm sure you know the method of pointing the analogue wristwach hour >>> hand towards the Sun and then bisecting the angle to 12 o'clock in order to >>> find south (or north, if you live down in the south). Actually I guess what >>> you should be doing is bisect the angle to your noon time and not >>> necessarily 12 o'clock, but anyway. A few years ago I read an interesting >>> seasonal accuracy analysis of this method. I also vaguely remember the >>> demonstration involved Vitruvius' analemma and I'm pretty sure it was all >>> in a book. Thing is - I can't remember where! Can you help? >>> >>> Dan Uza >>> >>> ------------------------------ >>> --------------------------------------------------- >>> https://lists.uni-koeln.de/mailman/listinfo/sundial >>> >>> --------------------------------------------------- >>> https://lists.uni-koeln.de/mailman/listinfo/sundial >>> >>> --------------------------------------------------- >> https://lists.uni-koeln.de/mailman/listinfo/sundial >> >>
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