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    Re: Problem shooting index error
    From: Frank Reed
    Date: 2019 Oct 13, 09:22 -0700

    Tony Oz, you wrote:
    "the horizon is too near to do checks - but VERY crude ones."

    Crude for lunars perhaps. Plenty good enough for normal celestial navigation altitudes! And the problem, normally, is not that the sea horizon is too near...

    First, let's find the distance to the horizon. Under normal weather conditions, it's given in nautical miles by approximately 1.15·sqrt[Ht(ft)]. Equivalently, take the dip in minutes of arc, add 20%, and that's the distance to the horizon in nautical miles. For height of eye over 3 meters (or 10 feet), that means that the horizon is over three nautical miles away, which is about 18000 feet. At that distance, instrument parallax in the sextant is insignificant. How do we know? The distance between the two lines of sight in the instrument (vertical distance between the mirror centers) is four inches or less in most modern sextants. The ratio of four inches to 18000 feet, which is 1/54000, gives us the sextant parallax as a pure number (angle in "radians"). We can, as always, convert a small ratio to an angle by multiplying by that "magic number" 3438: minutes of arc = 3438/54000 = 0.06' ...less than a tenth of a minute of arc. That would be excellent for index error estimation or better yet for zero-ing out index error. 

    This mathematical analysis does not mean that the sea horizon is the best choice for index error checks, but it is a very good choice. The sea horizon is a bit wobbly which reduces the expected accuracy a bit in practice. You should expect +/-0.5' when using the sea horizon to check or eliminate index error. On land, you're better off using some other sharp, linear feature. Place the sextant on its side on a table and shoot a distant radio tower, for example. You'll get excellent results as long as the tower is more than two or three miles away. As I have mentioned before, you can also subsititute an airplane contrail for your distant linear feature. Aircraft creating contrails are almost always flying near the tropopause which means that the contrail is typically 35000 feet away when straight up (and therefore about 70000 feet away if the contrail is 30° high).

    I want to emphasize again that the sea horizon is a very good choice for eliminating or testing index error, and this is especially true if you're using a plastic sextant. When using a plastic sextant, you should zero out index error immediately before every sight. At sea there's usually nothing more convenient than the sea horizon.

    You also wrote:
    "The most accurate I.E. check/measurement is measuring Sun's diameter - across - in horizontal direction; with left-right limbs. This excludes any effects of the atmospheric refraction - which are there if the upper-lower limbs are used."

    The Sun-Sun test is an excellent choice for index error estimation. But you don't need to worry much about making this observation horizontally. Atmospheric refraction is not a problem for the Sun-based index error test unless you're measuring SD as a "sanity check" on your results. Even if that's the case, you can wait until the Sun is about 30° altitude, and then the reduction in SD due to refraction is below the sextant's ability to measure. Even better, if you wait until the Sun is above 45°, you know that the horizontal and vertical diameters are both reduced by the same amount: one part in 3000 (about 0.6 seconds of arc, which is completely negligible). Wait... both? Yes, both. Most people are surprised to learn that refraction also reduces the horizontal diameter of the Sun. It's quite a small adjustment though --never measurable by sextant. You remember that rule? All angles are reduced by one in 3000 when both "objects" are above 45° altitude. In this case, the two objects are points on opposite sides of the Sun's disk. Either vertically separated or horizontally separated, refraction reduces the angular distance (in this case angular diameter of the Sun) by one part in 3000.

    So what's the best way to estimate index error? I've experimented with many different methods over the years. The best by far is to place the sextant on a table on its side and look at a very distant radio tower or other linear feature (as described above), but do this with the standard sextant scope removed and a medium power spotting scope placed in line with the sextant's collimation axis (normal line of sight for a scope).With a scope magnification of 30-40x, you can easily see changes as small as a tenth of a minute of arc reliably and repeatably. You can easily zero out index error with this technique.

    Frank Reed

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