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    Re: Celestial Navigation without a sextant.
    From: Frank Reed
    Date: 2008 Mar 10, 21:12 -0400

    George, you wrote:
    "What does Greg base his "6 minutes of arc" figure on, when it must include
    the uncertainties in refraction for the Sun, as seen on the horizon? How
    well does he know what that refraction will be, and how much it might vary
    from the "book" value of 34 arc-minutes?"
    Strikes me as a 'not unreasonable' value. There is an awful lot of
    variability within the variability. What I mean by that is that under
    certain very common circumstances, e.g. at sea in temperate climates, the
    day-to-day variability in the refraction at the horizon is relatively small
    (a few minutes of arc), but under other circumstances, which might be common
    for some observers, e.g. near the coast in cold weather, the day-to-day
    variability in refraction could be much larger (tens of minutes of arc).
    You quoted Meeus,
    " ... According to Schaefer and Liller, the refraction at the horizon
    fluctuates by 0.3 degrees around a mean value normally, and in some cases
    apparently much more"
    And quoted Bowditch,
    "Generally, the error in tabulated refraction should not exceed two or three
    minutes, even at the horizon" ...which you called 'absurd'.
    The Bowditch statement is weak because it doesn't provide any ranges. The
    refraction tables, when properly corrected for temperature and pressure, can
    in fact be counted on as exact for all practical navigation down to THREE
    degrees altitude. Although some navigation manuals still recommend avoiding
    sights below ten degrees altitude, this is over-cautious.
    Meanwhile, claiming "two or three minutes" variability at the horizon seems
    to be in conflict with the other source which suggests 18 minutes
    variability in normal cases (incidentally, Greg's "6 minutes" is close to
    the geometric mean of these other claims, for whatever that's worth ). So
    who's right? I think they're both right under certain conditions. As I noted
    above, I think this is due to the great difference between the temperature
    profile of the atmosphere on a large landmass (or near the coast),
    especially in winter, and the temperature profile at sea. Extreme
    temperature inversions are common near a continental coastline but rare at
    sea. You can look at real data on this from daily weather balloons. Pick an
    island location like Bermuda, and you'll see much less variability than at a
    coastal or inland location.
    By the way, there are sometimes temperature inversions even at sea. Calm
    weather especially is associated with them. It's worth knowing that the same
    conditions which would lead to anomalous low altitude refraction at sea will
    also often lead to "sea fog".
    Finally then, what range in variability of the refraction should a navigator
    (or a celestial navigation enthusiast) use? I would recommend these:
    1) Trust the tables for altitudes of 3 degrees or above, but be sure to
    correct for temperature and pressure. There's no reason to worry about
    anomalous refraction above these altitudes. You DO have to be concerned
    about anomalous dip, however, as a separate issue, no matter what the
    object's altitude.
    2) Ignore tenths of a minute of arc in the refraction below 1 degree
    altitude (so you can ignore the "improved" refraction tables published in
    the Nautical Almanac starting in 2004 --they result from a relatively
    trivial change in the atmospheric temperature profile).
    3) Where temperature inversions and other low-level variability are likely
    to be small (a good distance from large landmasses, not becalmed, no sea
    fog), expect less than six minutes variability in refraction even at the
    horizon, probably less than three minutes. Also, under these same
    conditions, anomalous dip is likely to be small.
    4) If there is any reason to believe there is an unusual temperature profile
    in the lower atmosphere (particularly close to shore in cold weather,
    especially in the early morning), be aware that the refraction can easily
    vary by 20 or even 30 minutes of arc.
    Incidentally, I base these conclusions, in part, on a large number of
    refraction integrations that I ran back in 2005 (I think) when Marcel got me
    interested in the production of refraction tables. It's quite
    straight-forward to take an observed temperature profile and generate a
    refraction table just for those special conditions. Of course, navigators at
    sea do not generally have access to current atmospheric temperature
    profiles. Some naval navigators might have been able to observe the real
    temperature profile (by launching balloons or even aircraft), but the
    computing power to generate refraction tables "on the fly" probably arrived
    too late for naval celestial navigation. And again, it's totally unnecessary
    for altitudes above three degrees, so the value of such tables is limited in
    any case.
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