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    Corrections for speed and bearing
    From: Robert Eno
    Date: 2005 Jun 5, 10:32 -0400
    I am generally humbled by the erudite discussions that I see on this site. While I have been a practicing astro-navigator for quite a few years, I continue to learn new things from the list members.
    In response to Frank Reed's excellent discourse, I am reminded of tables 1a and 1b -- "correction for motion of observer" & correction for intermediate time" -- in publication A.P. 3270 (Sight Reduction Tables for Air Navigation). These tables essentially solve the problem described by Frank, however the lowest entry argument for speed is 90 knots. Other entry arguments include the relative bearing of the object being sighted. I believe that some time ago, I posed a question to this list on whether or not there is a similar set of tables for the more sedate speeds as experienced by the sea-going navigator.
    Several years ago, I had an opportunity to take some sun shots from a Hercules transport aircraft en-route from Thule, Greenland to Frobisher Bay. I had never before taken a sight from such a fast-moving platform so everything was new and scary. For one thing, it took me a long while to clue into the fact that as we were rocketing south at over 250 knots, the sun's apparent altitude was also climbing at an appreciable rate; a fact which at the time, baffled me, until the switch in my brain finally flipped on. In this case, it is clear to see why the above-mentioned tables were required for air navigators. By the way, I gained a much higher respect for air navigators from this experience. It is not an easy thing to pinch an observation under such conditions. I cannot imagine how difficult it must be to take star and planet sights.
    But this also begs the question: does this correction really matter for the surface navigator who is not likely to be moving faster than 7 - 10 knots? Having myself, taken numerous sights at sea, including noon sights (just in case George questions my experience!) it seems to me that this factor is not terribly significant. Or is it? 
    Frank, you'd been doing me a big favour if you could provide me (us) with a practical example of technique which you have described, using some real numbers and speeds. I know, I know, I could possibly sit down and figure out myself but I am feeling lazy these days. I haven't picked up my sextant for several months and only now thinking of getting in a few rounds of practice observations.
    ----- Original Message -----
    From: Frank Reed
    Sent: Saturday, June 04, 2005 7:54 PM
    Subject: Latitude and Longitude by "Noon Sun"

    First things first: I've put the phrase "Noon Sun" in quotes here because the set of sights required for this system goes a little beyond the standard procedure for shooting the Noon Sun for latitude only.
    This short method of celestial navigation will get you latitude and longitude to about +/-2 miles and +/-5 miles respectively --more than adequate for any conceivable modern practical purpose. You can cross oceans safely and reliably for years on end using this technique if it suits you to do so. Its enormous advantage is simplicity. It's easy to teach, easy to demonstrate, easy to learn, and also easy to re-learn if necessary. I mention this because most people who are learning celestial navigation today will quickly forget it. What's the point of learning something if you can't reconstruct your knowledge of it quickly when and if the need actually arises to use it? It's tough to resurrect an understanding of the tools of standard celestial navigation on short notice, but easy with this lat/lon at noon method. Additionally, this method does not require learning all the details of using a Nautical Almanac (you don't need one at all --only a short table of declination and equation of time, possibly graphed as an "analemma") and it needs no cumbersome sight reduction tables.
    Here's how it's done:
    Start 20 or 30 minutes before estimated local noon. Shoot the Sun's altitude with your sextant every five or ten minutes (or more often if you're so inclined) and record the altitudes and times by your watch (true GMT). Continue shooting until 20 or 30 minutes after local noon. [note the difference from a noon latitude sight --we're recording sights leading up to and following noon-- usually these are thrown away]
    Next you need to correct for your speed towards or away from the Sun. For example, if we're sailing south and the Sun is to the south of us, then each altitude that we have measured will be a little higher as we get closer to the latitude where the Sun is straight up. We need to 'back out' this effect so that the data can be used to get a fix at a specific point and time. This isn't hard. First, we need the fraction of our speed that is in the north-south direction. If I'm sailing SW at 10 knots, then the portion southbound (in the Sun's direction) is about 7.1 knots. You can get this fraction by simple plotting or an easy calculation. Next we need the Sun's speed. The position where the Sun is straight overhead is moving north in  spring, stops around June 21, then heads south in fall, bottoming out around December 21 (season names are northern hemisphere biased here). It is sufficient for the purposes of this method to say that the Sun's speed is 1 knot northbound in late winter through mid spring, 1 knot southbound from late summer through mid autumn, and 0 for a month or two around both solstices (it's easy to prepare a monthly table if you want a little more accuracy). Add these speeds up to find out how much you're moving towards or away from the Sun. If you're moving towards the Sun, then for every six minutes away from noon, add 0.1 minutes of arc for every knot of speed to the altitudes before noon and subtract 0.1 minutes of arc for every knot of speed to the altitudes after noon. Reverse the rules if you're moving away from the Sun. Spelled out verbally like this, this speed correction can sound tedious but the concept is really very simple and it's very easy to do. [Incidentally, George Huxtable deserves credit for emphasizing the importance of dealing with this issue (although I don't think he ever spelled out how to do it)]
    Now graph the altitudes (use proper graph paper here if at all possible): Sun's altitude on the y-axis versus GMT on the x-axis. The size of the graph should be roughly square, maybe 6 inches by 6 inches so that you can clearly see the rise and fall of altitude. For longitude, you will need to determine the axis of symmetry of the parabolic arch of points that you've plotted. There is a simple way to do this: make an eyeball estimate of the center and lightly fold the graph paper in half along this vertical (don't "hard crease" the fold yet). Now hold it up to the light. You can see the data points preceding noon superimposed over the data points following noon which are visible through the paper. Slide the paper back and forth until all of the points, before and after, make the best possible smooth arch (half a parabola). Now crease the paper. Unfold and the crease line will mark the center of symmetry of the measured points with considerable accuracy. Reading down along this crease to the x-axis, you can now read off the GMT of Local Apparent Noon. Reading back up the crease to the data, you can pick off the Sun's maximum noon altitude (which is probably already recorded but if you missed the exact moment of LAN you can get it this way).
    Next we need two pieces of almanac data: the Sun's declination for this approximate GMT on this date and the Equation of Time for the same date and time. You do NOT need a current Nautical Almanac for this. The exact value of declination and Equation of Time varies in a four-year cycle depending on whether this year is a leap year or the first, second, or third year after. So we don't need an almanac for this. A simple table will do (where to get one? Today, they're very easy to generate on-the-fly... or you could use an old Nautical Almanac... or you could also use an analemma drawn on a sufficiently large scale).
    Apply the Equation of Time to the GMT of Local Apparent Noon that you found above. You now have the Local Mean Time at LAN, and you already know the Greenwich Mean Time. The difference between those two times is your longitude. Convert this to degrees at the rate of 1 degree of longitude for every four minutes of time difference. Done. We've got our longitude.
    Now for latitude. Notice that we didn't correct any of our altitudes for index correction or dip or refraction or the Sun's semi-diameter. These corrections are totally unnecessary for the longitude determination. But we need them for latitude. Take the Sun's altitude at the time of LAN (read off the "crease" or actually observed by watching the Sun "hang" at the moment of LAN). Correct it for index correction, dip, refraction and semi-diameter as usual. This gives you the Sun's corrected observed altitude. Subtract from 90 degrees. This "noon zenith distance" tells us how many degrees and minutes we are away from the latitude where the Sun is straight up. The latitude where the Sun is straight is, by definition, the "declination" that we have looked up previously from our tables. So if the Sun is north of us at noon, then we are south of the Sun's declination (latitude) by exactly the number of degrees and minutes in the noon zenith distance. If the Sun is south of us at noon, then we are north of the Sun's declination by the same amount. A simple addition or subtraction yields the required latitude. Done.
    We've spent about ten minutes making and recording observations of the Sun's altitude over the course of 45 minutes to an hour, and reduced those observations to get our latitude and longitude at noon with about five minutes of paperwork. Not bad!
    Again, the overwhelming advantage of this "short celestial" is that it can be taught easily, learned quickly, and RE-learned quickly on the spot if necessary. An additional advantage is that it requires an absolute minimum of materials. You need a sextant (metal if at all possible, but plastic will do), a decent, cheap watch or small clock, tables of refraction and dip (one sheet of paper), a four-year revolving almanac of the Sun's declination and equation of time (another sheet or two of paper), and some graph paper and a pencil. You could even print out these (or equivalent) instructions and throw everything in the case with your sextant.
    As for disadvantages, they really depend on the student and his or her expectations. What is it that we want to do with celestial navigation? Why study any method? And for a thousand students, you will get a thousand answers. The days are gone when celestial navigation was essential and fixed curricula could be dictated for students to either take in their entirety or leave. This field has moved on to the stage of "a la carte" learning. It can be a pain in the neck for instructors accustomed to doing things the same way year after year but it's a real liberation for students and possibly also for more creative teachers and "information publishers".
    42.0N 87.7W, or 41.4N 72.1W.
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