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    Summary so far: Extremely poor condtions, clocks
    From: Alexandre Eremenko
    Date: 2012 Mar 23, 22:50 -0400

    I am trying to summarize (from my point of view) this long discussion so far.
    It becomes harder and harder to find a specific message in the long list.
    1. What has happened, briefly.
    On March 17 Bill B and I made 33 observations of Sun and Venus altitudes,
    from the shore of lake Michigan, over sea horizon, at St Joseph.
    Observations began at GMT 21:34
    and ended at GMT 00:15. Plus/minus 10 min.
    The results are posted under the title
    "Extremely poor conditions??".
    The results show approx 10' cosnstant systematic error;
    all observed altitudes were approx. 10' LESS THAN
    the computed altitudes. We carefully checked all possible sources
    of this error, coud not find any, and posted the question on the list.
    2. Discussion. Two explanations were proposed.
    First: some anomaly of dip/refraction.
    Second: watch was 1 minute off.
    3. I reduced again all observations assuming this 1m watch correction.
    The result is posted under "Atomic watch".
    Statistics of these results I consider as a svery strong
    evidence that the watch was 1m off.
    Indeed: after this correction, the observations show no systematic bias.
    The average error of all observations is 0. If one considers Astra observations
    only, the average error is -0'3.
    The average error of EACH individual series of 3-5 altitudes with Astra is NEVER
    more than 0'8. And sigma for each series of 5 is never more than 1'.
    I know that statistical evidence is not considered a proof in courts.
    But in experimental sciences it is sometimes considered:-)
    On my opinion, the evidence based on statistics of this 1min watch
    error looks very strong.
    4. The only alternative explanation proposed was some dip/refraction
    anomaly. Let me explain why this looks unlikely to me.
    Refraction can be considered as consisting of two parts:
    correction to the TRUE Sun altitude (over the true horizon),
    and correction to the dip of the horizon.
    a) The Sun altitude was about 24d in the beginning, the Venus altitude 44d.
    At these altitudes, refraction of more than 3' was never mentioned
    in the literature that I know.
    And it always gives NEGATIVE corrections at these altitudes.
    Our error apparently DID NOT DEPEND on the altitude!
    b) Refraction could affect the dip. Given that correction in a)
    is always negative, the horizon must have been at least 10' HIGHER
    than the normal. This amounts to 6' of POSITIVE dip correction.
    Two pieces of evidence that positive dip correction of this
    magnitude may happen in
    sea horizon observations were found. Ken provided the following reference:
      An XVIII century British astronomer, W. Wales, who compared
    meridional altutudes taken with a sea horizon with those taken with
    a plumb line in Churchil, Canada, 59N, 93 W. He describes his sea horizon
    observations in reasonable detail, but only briefly mentions the shore
    observations, giving no numbers, no dates, except that the discrepancy
    was 10', altitudes from the sea horizon beeng smaller by this amount.
    More serious evidence was found by Lars in a navigation book
    of the beginning of XX century. They cite with details two apparently reliable
    observations (one of them in Mediterranean!) which give positive dip correction.
    The author calls these cases "extraordinary".
    The article specifies conditions under which such thing might occur,
    and they exactly conicide with the conditions that we had.
    What makes me still somewhat skeptical about positive dip correction is this:
    c) There is a "theorem" in optics that says that mirages (inverted images)
    can be only seen BELOW the true horizon of the observer.
    Thanks to John H who provided a link to SEVERAL HUNDREDS of web pages
    on mirages, green flash and refraction,
    Perhaps I should retire and spend the rest of my life reading these pages:-)
    The true horizon was about 4' below, so there was simply no space
    for 10' error under the true horizon. (The historical observations
    mentioned hardly agree with this "theorem", but maybe this is not about
    inverted images. I found no theoretical explanation of positive dip
    d) There was absolutely no anomalies visible with naked eye.
    And 10' is something well visible with the naked eye.
    e) An almost constant error persisted for 4 hours before and after sunset.
    5. Of course, the hypothesis that the watch was 1min off requires
    some explanation
    how this was poissible. Here I know very little (about how atomic watches
    are corrected). And essentially there are just two pieces of evidence:
    a) Greg's statement that he has seen "atomic watches" jumping 1min.
    b) Bill's statement that at some point in our observation he checked
    his watch against the GPS and there was less than 1sec error.
    I remember him checking, but none of us remembers when exactly.
    We only agree that this was before sunset. Possibly at the beginning of
    a) and b) certainly do not contradict each other directly:-)
    If not this statement of Bill, I'd consider the watch explanation
    as proven (by statistics). Bill's statement causes me to doubt.
    Perhaps we indeed observed  something extraordinary...

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