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    About Lunars, Part 4a
    From: George Huxtable
    Date: 2003 Jan 11, 16:15 +0000

    About Lunars, part 4a.
    
    There has been a very long gap since Part 4 appeared, for which I offer my
    apologies. I have discovered that there were gaps (indeed, errors) in some
    of my understanding of lunars, and hope that with help from Bruce Stark,
    Bill Noyce, and Herbert Prinz of the Nav-L mailing-list at least some of
    those notions have been put right.
    
    This new Part 4a is intended as a complete replacement for the earlier Part
    4, which I suggest you remove and trash.
    
    The main change is a new section 4.5a in place of 4.5, which contained some
    serious errors. Also in 4.4a, I have corrected and expanded on the
    reference to a German text which described some effects of parallax. The
    rest is nearly unaltered from Part 4.
    
    ================================
    This is part 4a of a series "About Lunars".
    
    It will deal with-
    Bruce Stark's Lunar Tables
    The serious effects of lunar parallax.
    The possibility of calculating altitudes from predictions, rather than
    measuring them.
    The possibility of calculating "lunars" from measured altitudes rather than
    observed lunar distances.
    Lunar measurements on-land.
    
    
    4.1  BRUCE STARK'S LUNAR TABLES.
    
    Thanks to the kindness of list member Bruce Stark, I am now the owner of
    his "Tables for clearing the Lunar Distance, and finding GMT by Sextant
    Observation". A long title, perhaps, but it defines exactly what these
    tables do. You can get a copy via info{at}celestaire.com at $37 + carriage.
    
    With a current Nautical Almanac, to give the coordinates (dec and GHA) of
    the Moon and other-body, the tables allow the prediction of lunar distances
    at two times, one hour apart, on-the-hour, which you hope will bracket your
    GMT. The tables then allow you to correct ("clear") a sextant observation
    of lunar distance between the Moon and other-body for the effects of
    parallax and refraction, and calculate GMT from the way that this
    observation fits between the two predicted lunar distances. No other tables
    are required; everything is supplied in Bruce's book. It uses logarithms
    throughout, but in a way that the user is hardly aware of it.
    
    I have been very impressed by the ingenuity with which Bruce has modified
    the trig and the logs to make everything as simple as possible for the
    navigator, and avoid pitfalls that might otherwise lurk in his path.
    Bruce's system avoids the need for any interpolations, yet maintains a
    precision to 0.1 minutes, as far as I can judge by limited spot-checks. He
    has allowed for small effects that are often neglected. Blank calculation
    forms (for photocopying) are supplied, and these are essential. Examples
    are given for the user to check his working.
    
    Captain Cook would have been delighted to have a copy of Bruce's lunar
    tables on board. He was provided with precomputed lunar distances in the
    Nautical Almanac of his day, which made his task a bit simpler than ours.
    
    I have one criticism, which Bruce and I have discussed, as follows. The
    user has to go through a series of steps, which are well defined, but
    nowhere are the underlying principles and equations EXPLAINED. This might
    be acceptable to an old-style navigator who is prepared to follow a rote;
    modern individuals with that mindset will nowadays all be using GPS. Anyone
    who is measuring lunars today will be doing it (to some extent, at least)
    out of intellectual curiosity, as has become clear from the correspondence
    with list members. For them, Bruce's tables would, in my own opinion, be
    rather more satisfying if background explanation were added.
    
    4.2 FINDING THE LONGITUDE
    
    The user of Bruce's tables (or of any of the other lunar techniques
    discussed so far), ends up with a measure of GMT. He is put into the same
    position as is the owner of a chronometer. If he does carry a chronometer,
    then he is enabled to check its going. But even after all that he does not
    yet have a longitude.
    
    Using his now-known GMT, the modern navigator can establish position lines
    for objects he can observe in the sky, from their measured and predicted
    altitudes. By choosing suitable objects he can cross position lines to
    provide his latitude and longitude. Those objects might be (but don't have
    to be) the same ones that were used for the lunar distance.
    
    In earlier times he would have to establish his longitude, by determining
    his Apparent Local Time from a Sun (or star) observation, and compare that
    with the known GMT to determine longitude. This process would call for a
    known latitude, readily measured at noon, and followed-up since by the
    "reckoning".
    
    These matters will be considered in part 5.
    
    4.3 A SURPRISING OBSERVATION. (Well, it surprised me!)
    
    With some friends, I was examining a series of land-based Sun-lunars that
    they measured last summer, over a period of about 1 hour, at a time when
    the Moon was quite high in the sky, and drawing a plot of the observed
    (uncleared) lunar distance against time. It turned out to be a rather
    straight line, as expected. But instead of the expected slope of about 30
    arc-minutes per hour, the observed lunar distance was changing at just over
    20 arc-minutes per hour. Why so much slower? The answer lies in the way the
    Moon's correction for parallax changes so rapidly with time.
    
    4.4a EVIL EFFECTS OF MOON'S PARALLAX
    
    As long as altitudes are above 10 degrees, refractions are no more than a
    few minutes, and for the purpose of this argument we can neglect them. In
    contrast, parallax makes an enormous difference to the apparent position of
    the Moon, up to 60 arc-minutes.
    
    As explained in an earlier part, all nautical tables have been calculated
    from the point-of-view of an imaginary observer at the centre of a
    transparent Earth. A real observer on the surface of the Earth sees the
    Moon, against its background of stars, from a different perspective,
    depending on just where he is. The difference in apparent angle is the
    parallax. For the Moon, the parallax is so large because it is much closer
    to the Earth than any other object in the sky. When the Moon is overhead,
    its parallax is zero. When the Moon is on the horizon, it has its maximum
    value (the Horizontal Parallax). In between, it varies as cos(altitude).
    Parallax always makes the Moon appear lower in the sky.
    
    For simplicity, imagine a navigator near the equator, and the Moon with
    near-zero declination. This is the worst-case scenario, when parallax has
    its greatest effect, and it is also simple to visualise. The Moon will rise
    in the East, pass over his head six hours later, and set in the West.
    Parallax will depress the apparent Moon at Moonrise by about 1 deg, then as
    the Moon altitude increases, the parallax will decrease until at the moment
    it is directly overhead, the parallax is zero. After the Moon passes "over
    the top", parallax starts to increase again, but now it is in the opposite
    direction, still depressing the Moon's position, more and more, but now
    pushing it down to the Westward, until finally the Moon sets in the West,
    with parallax at its maximum value of about 1 degree again. The Sun follows
    a similar path, but a few hours apart.
    
    So it's reasonably clear, if you think about it and draw a diagram, that
    parallax increases the apparent speed of the Moon in its path across the
    sky, from East to West, because the Moon appears to travel an extra 2
    degrees all told. The rate at which the Moon is pushed by parallax becomes
    greatest when it's at it's highest, overhead, at about 15 arc-minutes per
    hour, at a time when the parallax itself is zero.
    
    But what about the movement of the Moon with respect to the other bodies,
    Sun, stars, planets? Because they are all so much further away than the
    Moon, the effect of parallax on the position of those bodies is quite
    negligible (for this argument). Parallax doesn't move them.
    
    So as the Moon and the other bodies pass across the sky, more-or-less
    together, parallax shifts the apparent Moon westwards, by up to 15
    arc-minutes per hour if it's at its zenith, with respect to everything
    else.
    
    Now remember what we are trying to measure to obtain GMT. It's the position
    of the Moon with respect to the background of Sun and stars. And with
    respect to that background, the Moon is moving, always Easterly, at about
    30 arc-minutes an hour, give or take a bit. However, we have just worked
    out above that parallax causes the APPARENT Moon, as seen by an observer on
    the Earth's equator, to be shifted Westerly at a rate of up to 15
    arc-minutes per hour. So with respect to the star background, the Moon,
    when directly above the observer, has lost half of its apparent velocity,
    because of the changing parallax. And that apparent motion of the Moon is
    what a lunar observer uses to determine GMT from the Moon's position!
    
    We might call this effect "parallactic retardation" of the apparent Moon. I
    have never seen it discussed in an English-language text. If anyone else
    has, I would be interested to learn. Steven Wepster has found a discussion
    in a German text by Carl Bremiker, writing in-
    "Zeitschrift fur Vermessungswesen", Berlin, 1875, pages 59 to 79.
    The relevant part of the text is on pages 77 to 79.
    If anyone is interested enough to ask for it, I can email a copy of the
    German text of these pages, and an amateur translation into English.
    
    I have described this effect in simplified terms, by ignoring the tilt of
    the Earth's equator to the ecliptic - quite an approximation!.
    
    This effect has some serious implications for lunar observations when the
    Moon is high in the sky. Not that one will get an erroneous answer, the
    corrections for parallax see to that. But one will get a less precise
    answer. If there's an error of 1 arc-minute in the lunar distance, then
    with a high Moon, the resulting error in GMT becomes not 2 minutes of time
    (as we presumed before), but 4. And so the resulting error in longitude
    doubles, from 30 to 60 arc-minutes.
    
    I have chosen a worst-case geometry of Moon and other-body as my example,
    and often, the adverse effects of lunar parallax will be rather less than
    described above. But the relation of the bodies in the sky is so variable,
    and so hard to assess, that it's yery difficult to predict the effects of
    parallax on the precision of lunars.
    
    One recommendation could clearly be made as follows- If lunars were always
    limited to times when the Moon altitude was less than 30 degrees, the
    adverse effects of Moon parallax would be less that half the values
    referred to above, because the rate-of-change of parallax varies with
    sin(alt). And yet, Moon altitudes should be kept above 10 deg, to minimise
    uncertainties in refraction, so it doesn't leave a big allowable range.
    
    Has any reader come across any such recommendation in a text on lunars, I
    wonder? Or any discussion at all of the evil-effect of parallax as
    described above? It seems to have been universally ignored in
    English-laguage publications.
    
    4.5a LUNARS WITHOUT ALTITUDES.
    
    ============================================
    In Part 4, section 4.5 was significantly off-beam. Here, the original
    section 4.5 has been withdrawn and replaced by 4.5a.
    
    I am aware that this section is rather hard-going, but haven't found a way
    to simplify it. Any suggestions to that end would be most welcome.
    ============================================
    
    Every method for "clearing", or correcting, the lunar distance requires a
    knowledge of the altitudes of the Moon and the other-body involved,
    obtained at the same moment (or nearly so) as the lunar distance between
    them was measured. Sometimes, however, it's impossible to measure those
    altitudes: for example, when the horizon can't be seen because the night is
    black. Because the altitudes are required only to make corrections, they
    are not required to very high accuracy. For the most demanding case, that
    of the Moon's altitude, errors up to 6 arc-minutes or so are acceptable.
    Such an error would contribute no more than 0.1 arc-minutes to the error in
    correcting the lunar distance. Even so, without a good view of the horizon,
    such accuracy in the measured altitude may be unachieveable.
    
    However, if the Local Apparent Time (LAT) is known, at the moment of the
    lunar distance observation, it's possible to deduce those altitudes from
    Almanac data, with sufficient accuracy, instead of measuring the altitude
    of the Moon and the other-body.
    
    It's likely that the Local Apparent Time will have been obtained in any
    case, at the moment of observing the lunar distance, as a necessary step in
    obtaining the observer's longitude.  Local apparent time is usually
    obtained from two altitudes of the Sun, one around noon, the other several
    hours from noon, in the morning or afternoon of the same day. Other
    techniques, and other bodies, may be used for the same purpose.
    
    Once the Local Apparent Time has been measured, it can be converted to
    Local Mean Time (LMT), and used to set, or correct, the ship's clock or
    deckwatch. Changing Apparent Time to Mean Time isn't difficult: it's only a
    matter of applying the Equation of Time as a correction, of up to 15
    minutes or so. (The difficult bit is deciding which direction to make the
    correction, which we will discuss later in this series.
    
    Fom then on, the time given by that timepiece can be used over the
    following (or preceding) hours or days, for as long as its timekeeping and
    the constancy of its rating can be trusted. In that interim. any movement
    of the vessel in the East or West direction, from the dead-reckoning,
    should be used to correct the LMT. For each degree of ship's Westing since
    the LMT was measured, the LMT should be delayed by 4 minutes of time.
    
    So, given a reliable measure of LMT, from reading the ship's clock. let's
    consider the question of calculating the altitude of a celestial body, at
    the same moment as the lunar distance observation, from data given in the
    Nautical Almanac.
    
    To calculate the altitude of a body, the quantities required are the
    observer's latitude, the declination of the body, and the Local Hour Angle
    between the body and the observer. This is familiar ground to any
    astro-navigator: obtaining a calculated altitude, often referred to as Hc.
    
    sin alt = sin lat sin dec + col lat cos dec cos LHA
    
    Any navigator worth his salt, with clear sky around noon, will have no
    difficulty in obtaining a good value for his latitude, obtaining Sun
    declination from the Almanac. Then, over a limited period, he can keep
    track of changes in his latitude, to some extent, by dead reckoning of his
    North-South travel. At the moment of observing the lunar distance, we can
    assume that the vessel's latitude is known to within a couple of
    arc-minutes, perhaps a bit more. Maybe the latitude at the moment of the
    lunar can be deduced from the Sun's altitude at the following noon, rather
    than the preceding one.
    
    This leaves the declination and Local Hour Angle (LHA) of the body to be
    obtained, using the Almanac. A serious problem arises here. Although the
    Local Mean Time is known, the Greenwich Mean Time isn't (the first aim of a
    lunar observation is usually to determine GMT). However, GMT is the time
    that a navigator must use to look up coordinates of the two bodies in an
    almanac, at the moment of the lunar. Where does the navigator get his GMT
    from, for entering the almanac? Here's how-
    
    The difference between the Local Mean Time and Greenwich Mean Time is the
    Westerly longitude, expressed in hours at 15� per hour.
    
    I think of longitude, measured Westerly, as Westitude, increasing Westerly
    from 0 to 360 degrees. A longitude of 1� E would have a Westitude of -1�,
    or (same thing) 359�. Some authorities have chosen an East-is-positive
    convention for longitude, but, with Meeus, I prefer it the other way, in
    the same direction as Hour Angles, which are always Westerly.
    
    At any moment, then, GMT = LMT + Westitude(in hours) . At 12 hrs local mean
    time in Washington,  Greenwich mean time will be about 5 hrs (75�) later,
    or 5pm.  All this is straightforward stuff.
    
    To start with, the navigator has to make a GUESS at his longitude. It
    doesn't need to be a very precise guess: within, say, 3� (which corresponds
    to 12 minutes of time) will be perfectly adequate. If his guess is worse,
    it will show up later. Using that guess gives an estimated GMT with which
    he can enter and interpolate the almanac tables for dec. and GHA of the two
    bodies. We hope (but don't insist) that this estimated GMT will be within
    12 minutes of the truth.
    
    Looking up the declination for the two bodies is quite straightforward.
    Declinations of all bodies change slowly enough that even for the Moon (the
    worst case) an error in Greenwich time of 12 minutes results in an error of
    only 3 arc-minutes in the Moon dec. No problem there.
    
    The real difficulty arrives in assessing the GHA of the two bodies. All
    hour angles change rapidly, at roughly 15� per hour, the rate of rotation
    of the Earth. So a possible error in our estimate of GMT of 12 min of time
    would give rise to an error of 3� in the interpolated GHA of the body.
    Such an error would make the GHA prediction quite useless for calculating
    the altitude of either of the bodies involved.
    
    The difficulty is overcome by a clever trick, for which I think Maskelyne
    should get the credit: he described it in his British Mariner's Guide 0f
    1762. It goes like this-
    
    For each of the bodies in question, consider not just its own GHA, but also
    the GHA of the Sun, both taken from the almanac. Although these are both
    increasing rapidly, at nearly 15� per hour, the difference between them,
    GHA (body) - GHA (Sun), changes much more slowly: in the worst-case, for
    the Moon, at about 30' per hour.  This is the amount by which the body is
    West of the Sun. If we have estimated the GMT to within 12 minutes of time,
    then we will know the Westing of the body from the Sun to within about 6
    arc-minutes. We already have an accurate value for LHA Sun, based on
    measurement.
    
    We take: LHA (body) = LHA (Sun) + (GHA (body)- GHA (Sun))
    
    So now we have the LHA of the body to within about 6 arc-minutes, if our
    longitude guess was within our target range. Taken with our measured lat
    and the dec of the body from the almanac, this provides all the necessary
    information to predict the altitude of the body, as described above, within
    about 6 arc-minutes. And that calculated altitude then allows the lunar
    distance to be "cleared", to within about 0.1 arc-minute, and a new value
    of GMT calculated from the lunar observation. The important factor here is
    that our initial "guess" of longitude, which we hoped was within 3�, or 12
    minutes of time,  contributes an error to the resulting value of GMT, of no
    more than 12 seconds of time, if we were within our target: a perfectly
    acceptable result, which has reduced the error of our initial guess by a
    factor of 60.
    
    If the difference between the guessed value of GMT and the final result
    happens to exceed 12 minutes of time, this indicates that the improved GMT
    value should be substituted for the earlier guess, and then a reiteration
    should shrink the resulting error by another factor of 60. There will never
    be a need for further iteration after that.
    
    The procedure I have described isn't quite as Maskelyne proposed. In his
    day, the position East-West around the sky was measured in terms of Right
    Ascenscion, not GHA, which was defined in the opposite direction, from a
    different reference-point..Also, in his day, almanacs used Apparent Time,
    and not Mean Time, as their argument, so there was no need in those days to
    correct with the Equation of Time.
    
    
    4.6 LONGITUDES WITHOUT LUNAR DISTANCES?
    
    Why is it necessary to measure lunar distabces at all? It's an unfamiliar
    and awkward process to many navigators. Can the same information be
    obtained just by the familiar process of measuring altitudes? This question
    arose recently on the Nav-L mailing list, and I will tackle it here by
    quoting (edited) versions of Chuck Griffith's question and my own reply.
    Chuck asked-
    
    >Consider an alternative approach to finding GMT. Why can't we
    >observe the altitude of the moon and one other body and, using our assumed
    >latitude, solve for  the meridian angle of both bodies. The difference between
    >the two angles should change by the rate at which the moon moves through
    >the sky
    >faster than another body. If that's true, can't we find the meridian angle
    >between the two bodies for the even hours, say on either side of what time we
    >think it is, and use the same inverse linear interpolation approach to find the
    >time of our sight?
    >
    >Of course, I can think of a couple issues with this approach worth discussion.
    >First, this only works when the altitude of the moon and the other body change
    >reasonably with time, i.e., we can't do it when either body is close to being a
    >meridian sight. Second, we need both altitudes simultaneously. I think this
    >could be solved by alternately observing one body then the other several times
    >and graphing the sights so that we could derive an averaged simultaneous
    >altitude from the graph.
    
    My response is as follows-
    
    The question is a very fair one. It has been asked before, however;
    starting in 1674. Francis Chichester, the famous single-handed
    circumnavigator, proposed such a method in 1966, and a spate of
    publications followed, on similar lines. These were answered in an
    authoritative article by David Sadler, then director of HM Nautical Almanac
    Office, in the RIN's "Journal of Navigation", 31, 2 May 1978, page 244,
    entitled "Lunar Methods for 'Longitude Without Time' ".
    
    From my point of view, the drawback of Sadler's article is that it is
    illustrated by a diagram of such devilish cunning and complexity that I am
    quite unable to make head or tail of it. If any reader manages to penetrate
    its mysteries, I would be grateful for an explanation.
    
    It's important to bear in mind that in any measurement that uses the Moon's
    motion to provide time and hence longitude, accuracy in determining the
    Moon's position is all. This follows from the fact that each minute-of-arc
    error results in an error in the final position of the vessel of 30 minutes
    of arc (which near the equator corresponds to 30 miles) or sometimes more.
    It is FAR more demanding that the normal run of astronavigation.
    
    The main virtue of a lunar is that the all-important measurement in which
    so much accuracy is required, the angle-in-the-sky between the Moon and the
    Sun (or other body) does not involve the horizon AT ALL. True, the
    altitudes of Sun and Moon do have to be measured up from the horizon as an
    auxiliary measurement, but this is only to get a correction to a
    correction, and an imprecise value for those altitudes will be perfectly
    adequate.
    
    Why is the accuracy so degraded whenever the horizon is involved?
    
    First, if there's any haziness in the air, the first thing to become
    indistinct is the line of the horizon.
    
    Second, even if the horizon is really sharp, it isn't exactly a
    well-defined straight line (except in millpond conditions), especially from
    a small craft. The horizon-line is made up from the peaks of overlapping
    waves and swell, and the vessel, too, is riding on those waves. The
    observer does what he can by timing his shots when he judges his vessel to
    be on the top of its "heave", but it is inevitably a compromise.
    
    Third, even if the horizon is both sharp and straight, its angle  can be
    affected by anomalous refraction, which causes the dip to vary from its
    predicted value. Air layers at different temperatures near the horizon can
    cause the sun's image to be distorted as it rises and sets, and can in
    extreme cases cause mirage effects when a distant vessel is observed as
    floating well above the horizon, sometimes even inverted. Where none of
    these objects is there to give a clue to the odd behaviour of light on its
    path from skimming the horizon to the observer's eye, anomalous dip may
    nevertheless be present, quite unsuspected and undetectable. An error in
    dip of 1 minute may be quite usual, and 2 or 3 minute errors can also occur
    occasionally. There is no way for the observer to correct for it. (Special
    instruments to measure the dip-of-the-moment have been devised but are very
    uncommon).
    
    These errors may present no real problems in normal astronavigation. After
    all, what significance has an error of 2 or 3 miles in an astro position?
    However, to the lunar observer, where any such errors are multiplied 30
    times or more in calculating his longitude, they are intolerable.
    
    When objects lie in opposite parts of the sky, the Moon to the East, say,
    and the Sun to the West, such horizon errors would actually add when
    comparing the positions of the two objects.
    
    This was well-known to eighteenth-century navigators, who accepted the
    practical and arithmetical difficulties of measuring lunar distance up in
    the sky, rather than altitudes up from the horizon, to cling on to all the
    precision that they possibly could.
    
    4.7 LUNARS IN YOUR BACKYARD.
    
    It isn't necessary to go to sea to observe lunar distances. Because the
    angle between Moon and other-body doesn't require a horizon, it can be
    measured from on-land, and that is indeed a good way to practice, when the
    observer is not being distracted and shaken-up by the motion of a vessel.
    
    However, the required auxiliary measurements, the altitudes of the Moon and
    the other-body, above the horizontal, do need to be made. If you don't live
    with a view of the sea (or a large lake) then you have to create your own
    horizon. This is easy to do.
    
    What's needed is a pool of reflecting liquid in a tray, placed on the
    ground. In the days of explorers, they would carry a few pounds weight of
    liquid mercury for this purpose. Nowadays, old black engine-oil, or a
    dollop of black treacle (molasses in American, I understand) will do the
    job. The viscosity helps, in limiting ruffling of the surface by the wind.
    If wind is a problem, surround the tray with a wind-break, or even cover it
    over with a sort-of garden 'cloche', in an inverted vee, made of glass
    which you have checked for flatness beforehand.
    A good method for making that check is to look with the sextant at the two
    views of a distant object with the index set to zero, as if checking
    index-error. Then interpose the glass plate into one view-line from the
    sextant (but not both), move the plate about, and check that there is no
    shift between the images. If shift is observed, all is not lost: just
    average two measurements between which the glass has been turned through
    180 in its plane (not overturned).
    
    Look down with your sextant (the straight-through view via the
    half-silvered mirror) at the reflected image of the object in the pool, and
    in the index mirror view the object directly, up in the sky. The angle
    shown on the sextant will be angle between the object and its reflected
    image, or twice its altitude above the true horizon. Because a normal
    sextant can measure only to 120 degrees (sometimes a few degrees more),
    then the maximum altitude that can be measured in this way is limited to 60
    degrees (or slightly over).
    
    The sextant angle, as measured, should be corrected for any sextant error
    shown in the box, and for index error, and then halved. No dip correction
    should be made.
    
    After noting the observed altitudes in this way for both bodies,
    corrections for refraction and for parallax must be made (unless Letcher's
    method is being used which includes these corrections automatically). If
    your altitude is significantly above sea level, make the appropriate
    corrections to refraction caused by the reduced pressure, and also, if
    necessary, any abnormal temperature. Note the corrected altitudes, and then
    proceed as for any other lunar distance calculation.
    
    ===========================
    
    This is the end of part 4a. Further parts will cover the final step of
    obtaining a ship's longitude from a knowledge of GMT, whether this was
    derived from a lunar or from a chronometer: and will also detail some of
    the changes in the almanac, and in the thinking about celestial positions,
    that have occurred over the many years since lunars first appeared. That
    will conclude this series, unless further questions arise that need
    answering.
    
    George Huxtable.
    
    
    

       
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