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    Re: Time of meridian passage accuracy
    From: Douglas Denny
    Date: 2009 Sep 27, 16:46 -0700

    At last you have addressed the question posed; and, (surprise?) I can agree 
    with most of what you say in this posting, as it seems we are saying the same 
    thing mostly.
    My edition of Bowditch is 1995,  and states exactly the same as Andreas 
    quotes, including the paragraph number 1801, with the one minute time 
    constraint for Local Apparent Noon. (LAN).
    I note you now agree with me that Latitude from LAN within one minute 
    constraint is acceptable; and I agree with you the Longitude from a sight one 
    minute either side of LAN is dubious. 
    The LAN case is a special case with LHA at zero where Latitude can be obtained 
    quite accurately, and all that is required is 
    1) an accurate declination for the time of LAN (easily obtained from almanac or calculated)  and 
    2) accurate sight at the time of the true LAN, (which can be easily calculated 
    from Mer Passage of Sun at Greenwich with longitude correction). Actually for 
    "extraordinary accuracy" this is not good enough but for practical navigation 
    it is quite good enough.
    I have not disagreed with you that the true LAN is not the same as max 
    altitude observed. This is, or should be, common knowledge. Where I disagree 
    is that the difference for a stationary observer is small, and for practical 
    use one minute either way of a calculated LAN is acceptable -  for Latitude 
    anyway.  Using a sight at a calculated time for the actual LAN is accurate 
    enough so long as the observer uses the calculated time of sight and not the 
    max altitude per se - though there will be little difference between the two.
    It would not normally be the case however in modern practical navigation that 
    Longitude would be obtained with anything other than accutate timings for the 
    observed sight - noon or at any time.  There is no other method worth using 
    other than Marcq St Hilaire in my opinion - which is a universal method with 
    few limitations unless you are into Polar navigation.
    The whole business using the Sun at LAN or near LAN is similar to the 
    so-called 'Ex-Meridian' problem or 'Equal altitude' problem, where the main 
    difficulty is difference in declinations between sights due to it's 
    constantly changing.  These methods were only developed because of the 
    practical navigator's historical preference for the almost religious "Noon 
    Sight", and not being able to take a sight if the cloud cover did not allow 
    it at exactly local noon.
    Latitude can be obtained accurately without a knowledge of the absolute time 
    of two sights, of the same body separated in time, so long as an accurate 
    time between them is known, in other words, having a clock which you do not 
    know the actual accuracy of the absolute time but is accurate to measure the 
    interval.  I have done this myself for the fun of it using marine sextant and 
    mercury artificial horizon. It is an interesting exercise in spherical trig.
    To address some points in an earlier post:
    "If we rephrased Douglas' statement, above, to read-
    "Local Apparent Time is, by definition, 12 hours at the instant when the
    true Sun crosses the observer's meridian", then I suggest both of us might
    accept those words"
    Agreed. I was lax in fast writing. I spotted afterwards too late I should have 
    put '... when true Sun crosses the local meridian'. I was referring to the 
    Local Transit Time.
    "But that is not the instant of greatest altitude of the Sun; not even for a
    stationary observer, because of the Sun's changing declination. The
    difference may never be much more than a quarter of a minute of time, but it
    isn't zero (except at the solstices)".
    I have not disagreed with this anywhere.  It is not a large quantity however.
    "(For the Moon, because its declination changes much faster, the time
    difference between meridian passage and maximum altitude can amount to many
    Agreed. The Moon is always a nuisance - it moves so fast.  Newton declared 
    whilst trying to analyse Her position in the Heavens as the only problem that 
    gave him a headache!
    "In many situations, such as deciding on the best moment to observe for a
    noon Sun latitude, that time difference can indeed be ignored, for an
    observer who is stationary or slow-moving. But when the longitude is to be
    derived from timing the Sun's changing altitude, it must be considered;
    otherwise, it will just add a systematic error to the result, which varies
    with time-of-year. That's what Douglas has neglected to take into account.
    If he is simply claiming that such a correction is too small for him to
    bother with, in view of the limited precision of the observation, then that
    would be fair enough. However, he appears to be denying the case for such a
    correction at all, as I read his postings. Is that his position?"
    No it is not my position.  You are extrapolating and making unwarranted suppositions.  
    I am claiming Bowditch makes a reasonable satement about a one minute 
    criterion for LAN in practical navigation terms. (And that I would say, only 
    if a proper LAN is calculated).
    This is certainly OK for Latitude, but like you I am uncomfortable about any 
    such use for longitude and do not think it reaosnable for this. I wonder in 
    fact if Bowditch makes an error in suggesting this use of LAN for longitude.
    "I ask again: what happened to the claim, in [9932], that-
    "The difference is 0.000737068 degrees,  or 0.04422408 minutes,  or 2,6
    seconds of arc.
    Utterly  negligable "
    Does Douglas wish to defend that claim, or perhaps withdraw it? What does it
    It means the differenece in declination of the Sun between the three minutes 
    of change of time from Mer Passage at Greewich to Mer Passage at Bosham is 
    only 2.6 seconds of arc.
    You missed the point that the example and graph of results has a calculation 
    for latitude which uses a declination calculated for Mer passage at Greenwich 
    (three minutes earlier than at my location). The use of Dec for Greenwich is 
    negligable compared to Bosham three minutes later: it does not affect the 
    results for declination used to calculate latitude.
    You will note that 2.6 seconds of arc in three minutes is equivalent to 0.8 
    minutes of arc in one hour - which is what the rate of change of declination 
    was at the time of observation and what you estimated somewhere in a previous 
    posting by a simple calculation.
    "Thanks for bringing that Cotter book to my attention. I hadn't come across
    it before. Cotter also treats these questions in his "History of Nautical
    Astronomy", (1968), but some aspects are confused, and several of his
    equations are wrong. Perhaps he had got things clearer by 1969."
    I have not yet looked to compare Cotter's 'History of Nautical Astronomy' with 
    his 'Complete Nautical Astronomy' and check your claim that he was inaccurate 
    in some way.   I have all of his books, some with personal annotations to me 
    as I knew him from my Cardiff student days.  It is due to Charles Cotter that 
    I have an interest in navigation and nautical astronomy today.  He was a most 
    charming and generous man and an excellent teacher.
    I would be very surprised indeed if he is in any way "Confused" or with "wrong 
    equations" as he was a recognised expert in the theroy of nautical astronomy. 
    He was a lecturer teaching at Cardiff University and wrote extensively about 
    the subject in books and in the Journal of the Institue of Navigation over 
    many years.  This subject of Meridian Passage not being the same as maximum 
    altitude was written about in the 'Journal'  by him extensively and in 
    detail.  He also wrote extensively about the Ex-Meridian and double- altitude 
    problem amongst other things.
    In another posting you suggest taking Smart's Equation of Time formula with "a pinch of salt".  (?).
    Smart is clear and says it is ..."accurate up  to the order of approximation 
    adopted"...  explained in the text.  It is quite accurate.
    Perhaps your edition is an earlier one than mine of 1977,  in which he uses 
    parameters for epoch  1975.  The values used in the final formula of 
    ecentricity and longitude of perigee are for 1975 and can be ammended going 
    back to the earlier parts of the formulae if wanted. The mean longitude of 
    the Sun I used in Smart's formula was calculated with up to date parameters 
    for obliquity and nutation included.
    Douglas Denny.
    Chichester.  England.
    P.S. I will have to start putting in a caveat to ignor obvious typos and/or 
    simple blunders as one cannot correct them after they are posted.  I always 
    seem to notice silly errors which I miss with the quick scan through after 
    writing which I pick-up on re-reading later.
    It is a pity one cannot have say fifteen minutes to go back to a posting and make corrections.
    Original Post:-
    Andres wrote-
    "BOWDITCH says this:
     1801. Equation of Time
    To calculate latitude and longitude at LAN, the navigator
    seldom requires the time of meridian passage to accuracies
    greater than one minute. Therefore, use the time listed under
    the "Mer. Pass." column to estimate LAN unless extraordinary
    accuracy is required.
    Any opinion?
    That's an interesting question. I can't find those words in my earlier
    2-volume edition of 1977 , not in para 1801 or in para 1809, which it
    devotes to equation of time.
    It's certainly true that to calculate LATITUDE from Sun altitudes at LAN,
    there's no need to know Sun-time to better than a minute. But in the
    (unusual) situation of trying to deduce LONGITUDE from
    altitudes-around-noon, knowing Sun-time only to the nearest minute adds an
    unnecessary error of +/- 7.5 arc-minutes to the result.
    Perhaps Bowditch presumes that in general, attempts to deduce longitude that
    way are going to be rough-and-ready ones, in which such an additional error
    wouldn't matter. And that any attempts to do better come into the
    "extraordinary accuracy" category. But it seems silly to me, to introduce
    such unnecessary error by taking the predicted "mer pass", given only to the
    nearest minute, when right alongside it is the prediction for equation of
    time, given to the nearest second. Why not use that?
    Or, if needing to be even more precise, why not use the Sun GHA prediction
    for noon that day at Greenwich, and convert the difference from 0-degrees
    into time?
    Of course, all those predictions are for noon that day at Greenwich, and as
    equation-of-time is continually changing, though slowly, by the time it's
    local noon where you happen to be, the EoT will be a bit different. It's
    pretty easy to allow for this by using the tabulated value in the almanac
    for that day, and also for the previous day or the next (depending on
    whether you are East or West of Greenwich), and interpolating accordingly.
    There are other ways to get Equation of Time. It can be computed from "first
    principles", in a similar way to that described by Douglas Denny, who wrote,
    in [9923]-
    "The Equation of Time is quite rigorously dealt with in W.M. Smart's book
    'Textbook on Spherical Astronomy'.
    A formula is quoted based on the mean longitude of the Sun."
    I have the 5th edition of Smart's textbook, printed in 1971, which gives the
    equation of time in equation 32 of chapter VI, "Time". And yes, the
    calculation of equation of time has been treated rather carefully. However,
    if precise results are needed, it needs to be taken with a pinch of salt.
    Smart treats the tilt of the Eart's axis, the eccentricity of the Earth's
    orbit, and the longitude of perihelion as constants, adopting their values
    as they were in 1931. In fact, all three change slowly over the years,
    altering the shape of the curve of equation of time quite dramatically when
    taken over several hundred years, as Meeus illustrates in his chapter 28.
    Over the ensuing 78 years after 1931, those changes will be less dramatic,
    but will be there. For any precise work, that Smart formula should not be
    used blindly, without taking some care to check whether it still remains
    valid (I haven't done so).
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