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    Re: Lunars (reposted from 1995)
    From: George Huxtable
    Date: 2001 Jul 10, 1:38 AM

    Testerday, 9 July 01, Dan Allen reposted an old mailing from Jeff Gottfred
    dated 18 Dec 1995, on the subject of lunar distances.
    
    It was reposted at least once before, in an earlier incarnation of this
    list in 1999.
    
    In response to that reposting, I made a number of comments, sent to the
    list on 23 March 99, and I will repost that mailing of mine below, in view
    of its relevance. Below that, once again, I will copy the original 1995
    posting that sparked it all off.
    
    But first, here's a comment on an error in that old 1999 posting of mine,
    discovered with hindsight. You will find below that somewhere it states-
    
    "However, the lunar distance predictions that were tabulated in the old
    almanacs, up to the early years of this century, were, as I understand it,
    based on apparent time (the time given by a sundial) at Greenwich, not on
    mean time."
    
    However, I have since found that the lunar distance predictions in the
    Nautical Almanac switched from apparent time to mean time somewhere in the
    early 1800s.
    
    Here goes.
    
    George Huxtable.
    
    ====================================================
    First, a copy of my response to the 1995 posting, made on 23 March 1999.
    
    ------
    
            A thread about Lunar distance measurements was active here until 16
    Feb 99, when a long posting by Mike Wescott
     appeared to finish it off. However, it
    seems to me that a few loose ends remain from this thread, and I aim here
    to tease them out and tie them off.
    
            Mike's mailing quoted an earlier submission to this list, undated,
    from Peter Smith, psmith{at}wellspring.us.dg.com, which in turn quoted a
    contribution made back on 18 Dec 95 by gottfred{at}agt.net (Jeff Gottfred). It
    seems to me that some of Jeff's statements from his 95 mailing could
    mislead readers. He is no longer a member of this mailing list, but is
    still a very active observer (land-based) of lunar distances. I have been
    in touch with him and he agrees that some of his procedures and
    recommendations, made back in 95, no longer represent his present thinking
    or practice. In addition, some of what he says refers specifically to his
    interest in land-based lunars, and would not apply to an ocean navigator.
    His present ideas and mine now largely agree; perhaps it would help readers
    if I note down where I differ from his mailing of 95. In general terms, I
    think he would now back most of what I say.
    
            Jeff Gottfred said in 95-
    
            " For my first attempt (in the absence of better information-- but
    more on that later), I simplified the problem of parallax by making my
    observation at the moment of lunar transit, and ignoring refraction by
    making sure to keep my observation above 20 degrees apparent altitude for
    both bodies. To make it easy with a reflecting horizon, I used the sun as
    the second body."
    
            My comments on this passage follow-
    
            There is no simplification (neither in observation nor in
    calculation) to be found by restricting Lunar distance observations to the
    moment of Lunar transit. An argument to justify this restriction is
    presented in Jeff's next paragraph, but I do not think it is well founded.
    The observer is free to choose his time of observation as he wishes, as
    long as the angle between the Moon and the other body (Sun or star) are
    within the range of his sextant, 120 degrees or so. Both bodies should also
    be more than 20 degrees or so in altitude, to ensure that local refraction
    anomalies near the horizon have no influence, and to ensure that the
    effects of refraction are small enough and predictable enough to be
    corrected for with precision.
            It is bad advice to suggest that refraction can be ignored if the
    altitude is over 20 degrees. For two objects on opposite sides of the
    zenith, the angle between them could easily be in error by 3 minutes of arc
    or more if refraction was neglected, which would put the longitude out by
    90 minutes; a 90-mile error in position if near the equator. Quite
    intolerable. No, all corrections to the observation, including refraction,
    have to be made to the highest possible precision, for a successful Lunar
    distance sight.
            It is very restrictive to confine Lunar distance measurements to
    those between Sun and Moon, neglecting the opportunity of using stars.
    Sextants won't cope with an angle measurement of much more than 120
    degrees, and there's a period of 10 days or so in each month, around Full
    Moon,  when the Sun-Moon angle exceeds this. (That's why the quintant was
    developed, to extend this range.) Taken together with the few days either
    side of New Moon, when Lunar distances can't be used because the Moon is
    then invisible, this would leave precious few days in each month for Lunars
    to be usable. So an ocean navigator needs to cope with star-Lunars as well
    as Sun-Lunars. Maybe it's different for an on-land navigator, if he can
    stay put for a few days and wait for the right time of the month.
            Jeff's reference to a "reflecting horizon" is because on land he
    doesn't have a natural horizon to measure altitudes from, so has to use a
    reflecting fluid pool in a dish, which gives 2 x altitude.
    
            Jeff's mailing said, earlier-
    
            "First, you must set your watch to local time-- I use solar time
    because its easier, and my watch is then useful for other purposes, but you
    could use any convenient body. Once you have set your watch to local
    apparent time, you apply the equation of time correction (as found at the
    bottom of each page in the Nautical Almanac) to set your watch to local
    mean time (NB, this is NOT the same as mean zone time!!) so you can relate
    it to the time data in the almanac, which uses the mean sun."
    
            My own comments follow-
    
            Personally, I would avoid all this "setting" of a watch, and so, I
    think, would most navigators. Instead, we would simply record the moment of
    local noon as read on the dial, without altering its setting. In this
    respect, Jeff and I still appear to disagree.
            As for correcting for the equation of time, to get the watch set to
    mean time, this is a consequence of the way Jeff has had to compute his
    Lunar distances. Let me quote what he says about this..."Lunar distance
    tables haven't been produced for over 80 years so I generated a little
    visual basic program that would crank out lunar distance solutions for
    every ten seconds for the hour of the observation--i.e., just enter the
    sun's GHA & dec, and moon's GHA and dec, and solve the problem..." Not
    surprisingly, his lunar distance computations were based on modern
    predictions for Sun and Moon position, which relate to mean time, as is
    always the case in modern astronomy.
            However, the lunar distance predictions that were tabulated in the
    old almanacs, up to the early years of this century, were, as I understand
    it, based on apparent time (the time given by a sundial) at Greenwich, not
    on mean time. This makes sense; mean time is only relevant to someone who
    possesses an accurate chronometer, which would generally exclude navigators
    who used Lunars. Longitude is measured as the difference between the
    apparent time according to the Sun at Greenwich and the observer's own
    local apparent time, by the Sun. The equation of time didn't come in at
    all. So, for those navigators who used Lunars in real life, with the tables
    of their day, the equation of time didn't need to be considered.
    
            Jeff ends with the following statement-
    
            "In the best case, you use three observers to measure the alitiude
    of the moon, the altitude of the sun, and the lunar distance at the same
    instant.
           If you are doing this alone, then you must take a few obs of the
    lunar and solar altitudes, then measure the distance, then a few more lunar
    and solar altitudes. You must then plot the altitudes and pick the
    altitudes at the instant of the lunar distance measurement. One of the
    benefits of Young's method is that slight errors in the altitudes do not
    have a large effect on the result."
    
            Here are my comments-
    
            The most important part of this passage is in the last sentence.
    Although it's necessary to know the altitudes of the two bodies, this is
    only for the purpose of calculating the small corrections for parallax,
    refraction, and semidiameter. The observed altitudes appear to be
    thoroughly tangled up into Young's formula, but when the calculation is
    done, the effect of altitudes nearly cancels out. As a result, there's no
    need at all for any ritual of simultaneously measuring the altitudes and
    the lunar distance, nor any real need to bracket the altitudes round the
    Lunar distance shot. The altitudes can be measured, before or after the
    lunar distance, in a leisurely manner, and will be quite accurate enough.
    However, this only applies if the Lunar site is being used to determine
    time only.
            If the observer has an almanac of the position in the sky of the
    Moon and the other object, then once the time has been determined from the
    lunar distance, it can be used to obtain predicted altitudes for the two
    objects, and these can be compared with the observed altitudes, making the
    relevant corrections, to obtain two position lines. This, then, produces a
    fix. In that case the timing of the two altitudes becomes more important.
    They don't have to be measured at the same instant as the lunar distance,
    however, as long as the time offsets are known and allowed for.
            It's difficult, sometimes impossible. to measure altitudes at
    night, when taking a star-Lunar, and the horizon can't be seen. In that
    case, it's possible to get the time from a Lunar without measuring
    altitudes at all; instead, approximate altitudes are calculated from the
    almanac for the Moon and the other body, assuming an approximate time and
    position. Because the time obtained from a Lunar is so insensitive to the
    altitudes, this allows the accurate time to be deduced.
    
            I would like to thank Jeff Gottfred for the assistance he has given
    in putting this mailing together.
    
            George Huxtable.
    
    (end of quote of my own 1999 posting)
    
    ===============================
    Copied once again below is the complete original mailing from Jeff
    Gottfred, dated 18 Dec 95.
    
    
    To: navigation{at}ronin.com
    From- gottfred{at}agt.net (Jeff Gottfred)
    Subject: Lunar Distances
    Date: Mon, 18 Dec 1995 17:09:23 -0700
    
    Why do I do lunars?
    
    The short answer is for fun. A slightly longer answer is that I am very
    interested in the navigational techniques of Alexander Mackenzie, David
    Thompson, Peter Fidler, Lewis & Clark, &c. For a hobby I do workshops
    and demonstrations of the techniques that these men used at various
    historical sites.
    
    [As a small aside, this summer, while giving a demonstration at the
    Lewis and Clark festival in Great Falls, I took a noon sight for the
    latitude of the Giant Springs of the Missouri River. To some
    embarrassment, I discovered that Lewis was in error by some 17 nautical
    miles. (Oops!). Even if you ignore refraction, it's hard to explain that
    kind of error. I wonder if he was just having a bad day, or if all their
    data was that bad? Perhaps some massive frontal system had just gone
    through or something...]
    
    In this message, I talk about my first successful attempts at lunars,
    and then describe an historical method by Young (1848).
    
    If anyone has fiddled around with lunars before, please let me know, I'd
    be keen to know how it worked out for you and what methods you used.
    
    If any of you are really keen to try this stuff using the methods,
    tables, &c used from about 1790-1820 let me know, I can provide you with
    them.
    
    What is a Lunar?
    
    The idea behind lunar distance observations is conceptually easy.
    First, you must set your watch to local time-- I use solar time because
    its easier, and my watch is then useful for other purposes, but you
    could use any convenient body. Once you have set your watch to local
    apparent time, you apply the equation of time correction (as found at
    the bottom of each page in the Nautical Almanac) to set your watch to
    local mean time (NB, this is NOT the same as mean zone time!!) so you
    can relate it to the time data in the almanac, which uses the mean sun.
    
    Next, you then measure the angular distance of the moon from some other
    object on the ecliptic (the sun is my fave), and then look up in a lunar
    distance table when the moon would be seen to be that distance by an
    observer in Greenwich. Voila!, you now know the relationship between
    Greenwich and local time, and therefore (at four minutes per degree)
    your longitude.
    
    Actually doing one:
    
    O.K., problem #1, just how do you correct for refraction & parallax &c
    when you make this observation?
    
    This is called the "clearing the distance" problem.
    
    For my first attempt (in the absence of better information-- but more on
    that later), I simplified the problem of parallax by making my
    observation at the moment of lunar transit, and ignoring refraction by
    making sure to keep my observation above 20 degrees apparent altitude
    for both bodies. To make it easy with a reflecting horizon, I used the
    sun as the second body.
    
    In this case, to clear the distance I had to apply corrections for the
    semi-diameter of both bodies (I used near limb to near limb, therefore
    added both corrections), and also the small correction for lunar
    augmentation. At the moment of lunar transit, there is no parallax in
    the plane perpendicular to the meridian, only the normal parallax in
    altitude (PA) which is derived from the horizontal parallax value in the
    almanac (H.P.). The PA is computed simply as:
    
          PA = HP cos A
    
    where A is the apparent altitude of the moon.
    
    I have therefore constructed a spherical triangle that looks (something)
    like this:
    
                      Z
                      .
                     /|
                    / |
                   /  |
                  /   |
                S/----+ Ma
                  \.  |
                    `\|
                      Mo
    
    
    Where Z is the zenith, Ma is the actual position of the moon. Mo is the
    observed position of the moon, and S is the sun;
    and where Z-Mo is the apparent zenith distance of the moon, and Z-S is
    the apparent zenith distance to the sun, and Ma-Mo the parallax in
    altitude of the moon (PA).
    
    
    Now for problem #2. Lunar distance tables.
    
    Lunar distance tables haven't been produced for over 80 years so I
    generated a little visual basic program that would crank out lunar
    distance solutions for every ten seconds for the hour of the
    observation--i.e., just enter the sun's GHA & dec, and moon's GHA and
    dec, and solve the problem...
    
    I then just go down the list until the find the distance that
    corresponds to what I measure, and Bingo! I know the time in Greenwich
    at the time of my obs!
    
    When I have done this using a mechanical pocket watch which gains about
    1.8 seconds per hour (and the rate varies!) I can find my longitude to
    within about 16' or 10 nm at this latitude using this technique. The
    really cool thing about this is that MOST of my errors come from trying
    to pick the exact moment of local apparent noon!
    
    The problem with this technique is that it is so restrictive, I mean,
    how many times per month do you get the moon transiting at same time
    that the sun is more than 20 degrees above the horizon, and yet far
    enough from the moon so that the moon is visible, and not more than 130
    degrees from the moon (so you can measure the distance with a sextant)?
    Well, you can count 'em on the fingers of one elbow... e.g., a couple of
    times a month...
    
    An Historical Technique:
    
    This summer I stumbled upon the most excellent source on this stuff:
    Charles H. Cotter, "A History of Nautical Astronomy", American Elsevier
    Publishing Company, Inc., 52 Vanderbilt Avenue, New York, New York,
    10017. 1968
    
    Here is a method for clearing the distance first published by Young, in
    1856.
    
    First, construct the following spherical triangle (only better than this
    one!):
    
                        Z
                       ,/\
                     ,/   `\
                   ,/       `\
                 ,/           `\ M
              s,/--,        ,----\
             ,/     `-,----'      `\
           ,/   ,----' `-----,      `\ m
        S,/----'              `-------`\
       ,/                               `\
    H /-----------------------------------`\ O
    
    
    Z is the zenith, M is the actual position of the moon, m is the apparent
    position of the moon, S is the actual position of the sun, s is the
    apparent position of the sun.
    
    Zs is the apparent zenith distance to the sun, ZS is the actual zenith
    distance to the sun, ZM is the actual zenith distance to the moon, and
    Zm is the apparent zenith distance to the moon. HO is the observers
    horizon.
    
    Zs is less than ZS due to refraction, but Zm is greater than ZM because
    the effect of the lunar parallax in altitude (PA) is (normally) greater
    then the refraction.
    
    What we see (and measure) in the sky is the observed lunar distance ms,
    what we want to accomplish by "clearing the distance" is to compute the
    actual distance MS -- this distance can then be compared to the lunar
    distance table/output as described above.
    
    For triangle ZSM, using the law of cosines for spherical triangles we
    can write:
    
    cos Z = (cos SM - cos ZS * cos ZM) / (sin ZS * sin ZM)
    
    remembering your math teacher who insisted that:
          sin (90-x) = cos x
          cos (90-x) = sin x
    
    we can write:
    
    cos Z = (cos SM - sin HS * sin OM) / (cos HS * cos OM)
    
    Next, for triangle Zsm, we can write another equation for Z:
    
    cos Z = (cos sm - cos Zs * cos Zm) / (sin Zs * sin Zm)
    
    An again, remembering our trig:
    
    cos Z = (cos sm - sin Hs * sin Om) / (cos Hs * cos Om)
    
    This gets a bit ugly, so let's simplify the notation a bit,
    let's define some new variables:
    D = SM, the true lunar distance
    
    S = HS, the true altitude of the sun's center.
    M = OM, the true altitude of the moon's center.
    d = sm, the observed lunar distance.
    s = Hs, the sun's apparent altitude.
    m = Hm, the moon's apparent altitude.
    
    we can now re-write the above equations as:
    
            cos D - sin S * sin M
    cos Z = ---------------------
               cos S * cos M
    
    and
    
             cos d - sin s * sin m
     cos Z = ---------------------
                cos s * cos m
    
    
    
    Now, add one to both sides of each equation:
    
    
                cos S * cos M   cos D - sin S * sin M
    1 + cos Z = ------------- + ---------------------
                cos S * cos M       cos S * cos M
    
    or,
    
                cos D + cos S * cos M - sin S * sin M
    1 + cos Z = -------------------------------------
                           cos S * cos M
    
    
    and likewise to the second equation:
    
                cos d + cos s * cos m - sin s * sin m
    1 + cos Z = -------------------------------------
                           cos s * cos m
    
    
    Now, remebering that wonderful trig formula (that I just now had to go
    and look up again...)
    
    cos (x + y) = cos x * cos y - sin x * sin y
    
    we can now write:
    
                cos D + cos (M + S)
    1 + cos Z = -------------------
                   cos M * cos S
    
    and,
    
                cos d + cos (m + s)
    1 + cos Z = -------------------
                   cos m * cos s
    
    
    so, now be equating the two equations we get:
    
    cos D + cos (M + S)   cos d + cos (m + s)
    ------------------- = -------------------
       cos M * cos S         cos m * cos s
    
    
    Multiplying both sides by cos M * cos S we get:
    
                          [cos d + cos (m + s)] * cos M * cos S
    cos D + cos (M + S) = -------------------------------------
                                     cos m * cos s
    
    This can be written as:
    
    
                                                  cos M * cos S
    cos D + cos (M + S) = [cos d + cos (m + s)] * -------------
                                                  cos m * cos s
    
    
    Subtracting cos (M + S) from both sides we get (at last):
    
    
                                    cos M * cos S
    cos D = [cos d + cos (m + s)] * ------------- - cos (M + S)
                                    cos m * cos s
    
    
    This is Young's formula for clearing the distance.
    
    In the best case, you use three observers to measure the alitiude of the
    moon, the altitude of the sun, and the lunar distance at the same
    instant.
    
    If you are doing this alone, then you must take a few obs of the lunar
    and solar altitudes, then measure the distance, then a few more lunar
    and solar altitudes. You must then plot the altitudes and pick the
    altitudes at the instant of the lunar distance measurement. One of the
    benefits of Young's method is that slight errors in the altitudes do not
    have a large effect on the result.
    
    If anyone is really keen, I have more methods form Cotter's book...
    
    (end of quote from Gottfred's 1995 mailing)
    
    ==========================================
    
    ------------------------------
    
    george---.u-net.com
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.
    ------------------------------
    

       
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