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    Whole horizon mirror, left/right error
    From: Frank Reed
    Date: 2011 Feb 7, 05:25 -0800

    Some background:
    Friday before last, January 28, just before midnight, I happened to click by accident on a list of boat shows online, and I thought to myself that I should check to see when Ken Gebhart would next be in Chicago since I had never made it to see one of his presentations during these events. Well, I clicked around a bit and discovered that he was in town at that very moment and would be doing a talk on Basic Celestial Navigation at 10:30 the next morning. You can't fight that kind of coincidence, so I made my way down to Navy Pier, an event and convention center on the lake shore in Chicago, and wandered around until I spotted a booth full of sextants. I surprised Ken and also John Karl who was there for the event. We chatted for a bit, and I looked over the collection of books he had for sale at his booth. As Ken noted, who else sells a hard-copy Chauvenet in the world today? He had just about every navigation book you might imagine, including of course a couple by John Karl, as well as texts in positional astronomy like that wonderful old paperback "Astronomical Formulae for Calculators" by Jean Meeus that so many of us cut our teeth on. He was even selling a recent edition of Lecky's "Wrinkles" (new printing, old book!). And of course, there were the sextants, which always intrigue mariners even in these electronic days.

    Ken's presentation was excellent, as expected. He gives a great, quick overview of finding one's position by the Sun using an analemma, the simplest shadow observations for altitudes, and crossing circles of position on a globe. It's definitely something that they can take home and try out. Towards the end, he spent just a little time talking about sextants, which of course he's there to sell, and he described the difference between the "traditional" horizon mirror and the "whole" horizon mirror. For those who haven't seen them, most sextant horizon mirrors are split, clear on the left side to see the horizon and mirrored on the right side to catch the reflected image of the Sun from the index mirror. That's the "traditional" horizon mirror, and in many cases, it gives a fairly narrow "window" near the center of the field of view where you can get the Sun to sit on the horizon. The whole horizon mirror, by contrast, is partially reflective all the way across so that it's possible to see the Sun's reflected image (stars, less so) as well as the horizon all the way across the field of view, which many people find convenient. But this raises a small question of error. How much error do you get if you bring the Sun to the horizon well to the left or right of center in the sextant's field of view?

    Quick answer:
    The error is almost certainly negligible if you're using a telescope for your sights since then the field of view is rather small anyway.

    Details:
    This is a type of collimation error. If your sextant's telescope is tilted with respect to the frame of the instrument, you will get an error that increases with increasing observed angle. It also increase quadratically with the tilt (the actual tilt of the telescope relative to the frame). While it's usually described in terms of telescope alignment, this collimation error also applies to the line of sight itself. If I look through a sight peep or a sighting tube, my line of sight is constrained to be nearly parallel to the plane of the sextant. But if I then bring the Sun to the horizon so that it is near the far edge, left or right, of the field of view, the line of sight is no longer parallel to the frame. The error can be calculated geometrically. It's found in older textbooks on observing instruments, and it's given by (tilt)^2*tan(alt/2) where tilt is the angle relative to the plane of the instrument and alt is the measured altitude (or other angle). The result is the error in the observation as a pure angle (the angle in "radians") and assumes also that the tilt is a pure angle. If the tilt is minutes of arc and we want the result in minutes of arc, divide the above by 3438. By the way, you may run across versions of this equation with "sin(1")" in the formula. This is an obsolete 19th-century style of writing trigonometric equations, and if you see that sine of one arcsecond, just drop it. For modern users, either work in pure angles (in radians) or work in some specific common angular measure like minutes of arc. Now it turns out that this error formula can be approximated very nicely by a rather simpler equation:
    error in minutes = (tilt in degrees)^2*(altitude in degrees)/90.
    It's good enough for quick estimates. For example, if we bring the Sun to the horizon ONE degree to the left or right of the center of the field of view, and the altitude we're measuring is 75 degrees (a high noon Sun perhaps), then the error is about 0.8 minutes of arc. That's not much, but it's at the level where we start to pay attention. And if we make contact TWO degrees away from the center of the field, then the error is four times greater, or 3.2 minutes of arc, which is definitely worth avoiding. Two degrees off-center means that the center of the Sun's image would be four full Sun diameters away from the center of the field of view. That's a long way, and with a telescope, you would probably never see that. That's why this would principally affect observations made with a sight tube. There's a simple rule to adopt which can keep us safe from the effects of this line of sight error: always try to overlap the Sun's image with the center of the field of view. Then the maximum distance off-center is just one quarter of a degree, the Sun's semi-diameter, and the maximum error does not even reach a tenth of a minute of arc for common altitude observations.

    Note that the center of the field of view might not be the center of the horizon mirror depending on the adjustment of the scope. Also note that this does depend on the sight tube or telescope being properly collimated. If not, then that should be arranged first. Most sextants manufactured in the past fifty years are built with fixed, near-perfect collimation so it's not an issue. Of course, most us have used older sextants and many of use sextants with adjustable telescope collimation regularly. If that's you, too, then just be aware that the distance left or right in the field of view should be left or right from the center of collimation.

    -FER


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