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    About Lunars, part 2
    From: George Huxtable
    Date: 2002 Jan 28, 5:19 PM

    About Lunars, part 2.
    
    Part 1 explained what a Lunar Distance is, how it is measured and "cleared"
    to obtain a Corrected Lunar Distance.
    
    Part 2  explains how to obtain tabulated lunar distances, so that the GMT
    can be deduced. Some examples are given.
    
    Part 3 will come at a future date. It will discuss the possibility of
    calculating the altitudes required with a lunar distance, instead of
    measuring them: also the question of observing Moon altitudes in place of
    lunar distances. It will show how to obtain local apparent time and use it
    with GMT to obtain longitude, as early navigators had to do.
    
    TABULATED LUNAR DISTANCES.
    
    Until 1906, the Nautical Almanac carried tables of predicted lunar
    distances, at 3-hour intervals, for selected bodies. The navigator, having
    obtained a corrected lunar distance (as detailed in part 1), now had to
    enter the lunar distance table to find two predicted lunar distances for
    that body, 3 hours apart, which bracketed his observed lunar distance. At
    that point, he knew that the GMT of his observation was somewhere within
    the three-hour interval between those entries.
    
    The process of finding the exact GMT time of the observation is one of
    inverse linear interpolation, which is simpler than it sounds. Given a
    tabulated lunar distance D1, at a GMT time T1 (in whole hours), earlier
    than the observation, and a distance D2, at a time T2, (in whole hours),
    later than the observation, then assuming a linear change, the rate of
    change over the period is
     (D2-D1) / (T2-T1) degrees per hour.
    
    If the observed lunar distance at time T (as yet unknown) is the measured,
    corrected, quantity D, as determined in part 1, we know that
    
    (D-D1) / (T-T1) = (D2 - D1) / (T2-T1), by simple proportion.
    
    This assumes that the rate-of-change of lunar distance remains constant
    over the period of up to 3 hours. So we can deduce-
    
    T = T1 + (T2-T1)*(D - D1) / (D2 - D1), where T2-T1 can be chosen as 1, 2,
    or 3 whole hours, giving the GMT time of the observation in hours.
    
    This is the Inverse Linear Interpolation formula.
    
    The lunar distances are always treated as positive values, but they may be
    increasing or decreasing with time, so the sign of their difference has to
    be treated properly. Also, when the period in question crosses over
    midnight, the step of 24 hours must be allowed for in a sensible way, of
    course.
    
    Now, here's the problem. Those tables have not been published for 100
    years, nearly. Without them, how is an observer to calculate D1 and D2?
    
    CALCULATING PREDICTED LUNAR DISTANCES FOR YOURSELF.
    
    The basic data required for the calculation is the GHA (Greenwich Hour
    Angle) and dec. (declination) of the Moon, and the same for the Sun (or
    other body), at the two chosen times, T1 and T2. These numbers can come
    from the following sources-
    
    1. The Nautical Almanac provides predictions, at every exact hour of GMT,
    of the GHA (Greenwich Hour Angle) and Dec (declination) of the Sun and
    Moon, Venus, Mars, Jupiter, Saturn, and (for predicting star positions)
    Aries, all to 0.1 minutes of arc.
    
    2. Many navigators have access to a computer program which calculates these
    quantities, just for the asking, at any specified moment, GMT.
    
    3. Others have written their own program (mine runs, painfully slowly, on a
    Casio pocket calculator), using the basic astronomical data provided by
    Jean Meeus, Astronomical Algorithms, 1998.
    
    Any of these sources will provide, for the Moon, Dec.m and GHA.m
    and for Sun or other body, Dec.s and GHA.s, at any integral hour, GMT.
    
    Finding, at any time, the angular distance between the predicted positions
    of the two bodies, in degrees, is identical with familiar problems in
    spherical trig: obtaining the zenith distance of a body seen by an
    observer, or finding the great-circle distance between two points on the
    (assumed spherical) Earth's surface.
    
    The Predicted Lunar Distances, that is, the angles D1 and D2 between the
    Moon and other body, in degrees, at the two times T1 and T2, are calculated
    using-
    
    Predicted Lunar Distance =
    acs ((sin Dec.m*sin Dec.s) + (cos Dec.m * cos Dec.s * cos (GHA.m - GHA.s)))
    
    (For comments on symbols and on limitations of some computer trig
    functions, see part 1)
    
    Again, use 5-figure tables, or double-precision arithmetic, to preserve the
    required accuracy, to 0.1 minutes. Remember to treat North declinations as
    positive, and vice versa.
    
    For anyone preferring to use trig tables and logs, rather than electronic
    calculation, I would recommend the cosine-haversine method (as for  zenith
    distance), to be found in Norie's tables. If care is taken with
    interpolation, these can be worked to 0.1 minutes accuracy.
    
    I know of no precomputed altitude tables that are appropriate for
    calculating the Predicted Lunar Distance. In general, they don't work to
    sufficient accuracy, and only cover a range of 90 degrees for the result.
    
    What has just been described is the range of tools available for
    calculating the Predicted Lunar Distances, D1 and D2, at chosen times T1
    and T2. The task that Maskelyne set himself in 1766 was to precompute and
    tabulate those distances at 3-hour intervals for a useful selection of
    bodies. Now you have to use one of the methods above instead, which may
    seem complicated, but are not a large part of the overall complexity of
    taking a lunar.
    
    HOW ARE THE TIMES T1 and T2 TO BE CHOSEN?
    
    First, the navigator has to make a guess at the GMT of his lunar
    observation, to within an hour or so. This should not be difficult, but if
    it turns out to have been a bad guess, it can be readjusted later.
    
    Two times T1 and T2 should be chosen, at integral hours GMT, which should
    bracket the estimated GMT of the observation. They should not be more than
    three hours apart, or the assumption of linearity will be endangered. The
    Dec and GHA of the Moon, and of the Sun or other body, should be
    calculated, or extracted from the tables, at those times, and the two
    distance-angles calculated, D1 and D2. If these do not bracket the
    measured, corrected, lunar distance D, different hours are to be chosen for
    T1 and T2, using common-sense. Apply the Inverse Linear Interpolation
    formula, shown above.
    
    T= T1 + (T2 - T1)*(D - D1) / (D2 - D1), with times measured in hours.
    
    This will, at last, provide the GMT at the time of the observation.
    
    Once GMT is known, it can be compared with the time that was noted on the
    deckwatch to provide a correction (to within a couple of minutes) to that
    deckwatch, for as long as it continues to provide reliable time.
    
    It can be used to check our ship's chronometer (if one is carried) in the
    same way.
    
    It can be used directly to calculate the altitude and azimuth of one or
    both of the observed bodies, and then obtain one or more position lines.
    
    In Cook's time, position-line navigation had not been invented. Instead,
    navigators thought about latitude and longitude as two quite-separate
    quantities. Latitude, of course, was easy. Longitude was obtained from the
    difference between the local time by the Sun, and GMT, allowing for the
    "equation of time".
    
    Part 3 will explain how this was done, and will pick up a few more points
    about lunars.
    
    ===================
    
    At this point, it would be useful to introduce some actual lunar-distance
    measurements, taken out in the Atlantic Ocean. These were made, not by me,
    but by my e-mail friend Steven Wepster, the only man I know who actually
    takes lunar distances from a small craft (8 metres).
    
    FIRST EXAMPLE.
    
    Date, 2001 April 02. Estimated time, something between 1700 and 1800 GMT.
    DR position, 35 deg N, 015 deg 30 min W. Sun lunar taken at 17:36:43 by the
    watch.
    What is the error of that watch?
    
    Choose T1 to be 15:00, and T2 to be 18:00 GMT (these are the same times
    Maskelyne would have chosen, times at 3-hour intervals which bracket the
    estimated time of the observation.).
    
    Start off by calculating the Predicted Lunar Distance for each of those times.
    
    From the 2001 Nautical Almanac on 2 April-
    at T1, 15:00 GMT
     Sun dec (Dec.s) = N 05 deg 06.4 min, GHA.s = 044 deg 07.6 min.
    Moon dec (Dec.m) = N 21 deg 59.8 min, GHA.m = 295 deg 17.9 min.
    
    Converting these angles to decimal degrees, and using the Predicted Lunar
    Distance formula given above, I get from a pocket calculator D1 = 105.350
    deg for the lunar distance, as a decimal-degree quantity. In degrees and
    minutes it is 105 deg 21.0 min.
    
    Using the cosine-haversine formula and Norie's tables, predicts a lunar
    distance of 105 deg 20.9 min. So that shows very satisfactory agreement
    between the different methods
    
    Now do the same for T2, 3 hours later at 18:00 GMT.
     Sun dec (Dec.s) = N 05 deg 09.3 min, GHA.s = 089 deg 08.1 min.
    Moon dec (Dec.m) = N 21 deg 46.2 min, GHA.m = 338 deg 31.0 min.
    
    
    Using the calculator I get D2 = 107.000 deg, or 107 deg 00.0 min, for the
    predicted LD.
    Calculated by cosine-haversine from Norie's, this is 107 deg 00.1 min.
    Again, excellent agreement between the two methods.
    
    You can see that in three hours, the Moon has moved further from the Sun by
    1.65 degrees, or 0.55 degrees (33 minutes of arc) in each hour.
    
    I have shown two alternative methods, but you simply have to choose one,
    whichever method is more appropriate for you.
    
    I have shown the Sun and Moon predicted positions extracted from the
    Nautical Almanac, but instead, if you have the gear and the program, these
    quantities could be computed from first principles instead, at the
    appropriate times. This would avoid transcription errors. My own pocket
    calculator, programmed to use all the harmonic terms in Meeus, generally
    agrees with the Nautical Almanac, sometimes showing a difference of 0.1
    minutes of arc. But it's dreadfully slow, taking 5 minutes for each Moon
    prediction! If you could do the job must faster than that, you could
    precalculate and print-out lunar distance tables, at 3-hour intervals,
    covering long periods, to rival anything Maskelyne could do.
    
    But if all you need to do is to measure the occasional lunar distance, my
    suggestion is that it's more sensible to make that measurement, then just
    compute the predictions at two times, within three hours, that bracket the
    time of the measurement, as part of the data-reduction process.
    
    To summarise what has been achieved so far, we have calculated two values
    of Predicted Lunar Distance, 3 hours apart. Each is a value for the angle
    between the centres of the Sun and Moon, as if they were viewed from an
    observer at the centre of a transparent Earth. Our observed, corrected,
    lunar distance ought to lie between those values, if we have estimated GMT
    correctly.
    
    We have substituted our own calculation in the place of the Lunar Distance
    tables (which are no longer available), just as we set out to do.
    
    THE NEXT STEP: DEALING WITH THE SEXTANT READINGS.
    
    We have not, yet, even considered the sextant observations that were made
    at this watch-time of 17:36:43 on 2 April 01..
    
    Assume that the raw sextant readings have already been corrected for any
    known sextant calibration errors and for index errors.
    
    The measurements were taken in the following sequence: Sun LL., Moon UL.,
    Lunar Distance, Moon UL., Sun LL. (LL means Lower-Limb, etc). If the mean
    times of the Sun and Moon altitudes didn't correspond to the watch-time of
    17:36:43, corresponding small adjustments were made to the altitudes.
    Assume all this has been done.
    
    First thing to do is to correct the observed Sun and Moon altitudes for dip
    and for semidiameter. The dips may differ slightly, as the bodies may have
    been observed from different parts of the vessel, with different
    height-of-eye.
    
    Observed Sun Altitude, lower limb above horizon. 20 deg 56.6 min.
    Subtract dip of 2.2 min. Add semidiam. (16.0 min from almanac).
    So centre of Sun is 21 deg 10.4 min above the true horizontal. This is s.
    
    Observed Moon Altitude, upper limb above horizon, 50 deg 10.7 min. Subtract
    dip of 1.8 min. Subtract Moon semidiameter (subtract because upper-limb was
    measured). Calculate semidiameter from Moon HP (from the Almanac, for that
    hour) * .2724 *(1+sin (alt)/55). As the HP was 59.4 min, this gives a SD
    for the Moon of 16.4 . The result is that the Moon centre is 49 deg 52.5
    min above the true horizontal. This is m.
    
    Next stage is to make refraction-and-parallax corrections, to s and m, to
    obtain the corrected altitudes S and M, as follows-.
    
    For the Sun, take the altitude s of 21 deg 10.4min. Subtract the refraction
    appropriate to the altitude, 2.4 min.
    Add the Sun parallax. The Sun horizontal parallax is only 0.15 min, and the
    parallax itself is then 0.15*cos(alt), so working to an accuracy of 0.1
    min, a sensible rule might be: take the Sun Parallax as 0.1 min for
    altitudes up to 70 deg, and above that, as zero.
    We then get the corrected Sun altitude S to be 21 deg 8.1 min.
    
    For the Moon, take the altitude m of 49 deg 52.5 min, and subtract the
    appropriate refraction for that altitude, 0.8 min.
    
    Add the Moon parallax. Compared with everything else this is an immense,
    dominant, correction. We have already observed that the Moon's HP, for that
    hour, from the almanac, was 59.4 min. As noted in part 1, this should be
    multiplied by cos(alt), and also by the Reduction (1-(sin(lat)^2)/300),
    ending up with a parallax of 38.2 min.
    So the corrected Moon altitude M is 50 deg 29.9min.
    
    What was all that work for? For the lunar itself, we aren't going to need
    those altitudes any further. We worked out those altitudes solely because
    they enter into the correction for the Lunar Distance, as will be seen.
    
    NOW TO DEAL WITH THE LUNAR DISTANCE, AT LAST!
    
    After correction for sextant errors, the Observed Lunar Distance between
    the near limbs of the Sun and Moon was 105 deg 50.5 min. There is no dip to
    consider because the horizon is not involved in the measurement.
    
    Because we want the angle between the centres, not the limbs, the
    semidiameters of Sun and Moon must be added to the observed lunar distance.
    The SD of the Sun for that day was obtained from the Almanac as 16.0 min.
    
     For the Moon (see part 1) we take the HP at the estimated GMT, of 59.4
    min. From this, as shown above, the augmented SD of the Moon for that hour
    was calculated to be 16.4 min. So we get the apparent angle between the
    centres of Moon and Sun to be 107 deg 22.9 min. This is the angle d which
    we now have to put through the "clearing" process, correcting it for
    refraction and parallax, to obtain the cleared Lunar distance D, as seen by
    a geocentric observer, to compare with predictions.
    
    The formula needed (from part 1) is Young's method-
    
    D = acs ((cos d + cos (m+s))*(cos M*cos S)/(cos m*cos s) - cos (M+S))
    
    We have all the necessary angles, summarised below, to put into that
    equation and obta1n D.-
    
    d=107 deg 22.9 min or 107.381 deg
    m= 49 deg 52.5 min or  49.875 deg
    M= 50 deg 29.9 min or  50.498 deg
    s= 21 deg 10.4 min or  21.173 deg
    S= 21 deg 08.1 min or  21.135 deg
                   (m+s) = 71.048 deg
                   (M+S) = 71.633 deg
    
    The end result is that D = 106.8209 deg, or 106 deg 49.3 min.
    
    CLEARING THE LUNAR DISTANCE, LONGHAND.
    
    The formula above is fine for electronic computation, but quite unsuitable
    for longhand calculation using logs. The trouble is that some of the
    quantities may go negative, and the log of a negative number is
    meaningless.
    
    In the era of lunar distances, the navigator relied on 5-figure logs and
    trig tables. The many different ways devised for clearing the Lunar
    distance were mostly devoted to expressing it in such a way that logs could
    be used for the solution, sometimes using auxiliary tables designed for the
    purpose.
    
    If readers find the need to do this clearance longhand using logs, my
    suggestion is to use Borda's method, to be found in Cotter. On request, I
    will spell it out for the list.
    
    THE FINAL STEP, TO GET GMT.
    
    We are nearly there, having ended up with a cleared Lunar Distance of
    106.821 deg. Look back at the predicted Lunar Distances that were
    calculated for 15:00 GMT (105.350 deg.), and for 18:00 GMT (107.000 deg.),
    and it is clear that our lunar distance fits nicely somewhere in the gap
    between them. That means that we estimated sensible times for calculating
    those predictions. If that had turned out not to be so, we would have to
    choose different times and calculate new predictions.
    
    Now the Inverse Linear Interpolation formula is needed, from the beginning
    of Part 2, as follows.
    
    T = T1 + (T2-T1)*(D - D1) / (D2 - D1), where T2-T1 can be chosen as 1, 2,
    or 3 whole hours, giving the GMT time of the observation in hours.
    
    T1 = 15 hrs., T2 = 18hrs.
    D1 = 105.350 deg., D2 = 107.000 deg., and D = 106.821 deg.
    
    This gives T = 17.6745 hrs GMT or 17:40:28 GMT. THE ANSWER!!!
    
    As the deck-watch read 17:36:43 GMT at the moment of lunar distance
    measurement, the conclusion is that it is slow by 3 min 45 sec.
    
    And you might well think: what a complex way to determine the time!
    
    ==============================
    
    I have worked through the above example step by step, in somewhat painful
    detail, to illustrate what has to be done. If any reader finds an error, or
    disagrees with any point, or simply fails to understand, please contact me,
    preferably via the NAL-L list, and I will do my best.
    
    Now, for anyone that is still with me, here is another example for you to
    work out for yourself. This is another observation made at sea by Steven
    Wepster.
    
    2001 April 08, DR position 44N 012W. Lunar between Moon and Mars. Deckwatch
    time 03:22:05 GMT.
    Observations corrected for index error.
    Moon observed altitude (LOWER limb)   31 deg 14.8 min, dip 2.5 min
    Mars observed altitude                17 deg 50.4 min, dip 2,2 min
    Lunar distance, Mars to Moon FAR limb 65 deg 11.9 min (i.e. measured ACROSS
    Moon).
    
    Some hints-
    
    This measurement was made at almost exactly full Moon. The Moon edge was
    therefore clear and sharp all round and any limb could have been used.
    Which limb was actually used is stated above. Think hard (draw a picture)
    about which way you need to correct, using the Moon semidiameter, to obtain
    the lunar distance between centres.
    
    Mars is treated as a point so has a semidiameter of zero.
    Consider the Mars observation as requiring a parallax correction of 0.2
    min. of arc. This can be checked from page A4 at the beginning of the
    Nautical Almanac, under "additional corrn" for Mars: this is the parallax
    correction. This correction was greatest in June 2001, when Mars was
    closest to the Earth, at opposition.
    
    See if you can work out the GMT of the observation, and therefore the error
    of the deckwatch.
    
    In case you don't possess a 2001 Nautical Almanac, here are some relevant
    quantities you might need-
    
    2001 April 08
    03:00 GMT. Moon dec S 02 deg 31.5 min  Moon GHA 042 deg 49.3 min
               Mars dec S 23 deg 08.3 min  Mars GHA 338 deg 55.7 min
    
    06:00 GMT  Moon dec S 03 deg 13,2 min  Moon GHA 086 deg 17.9 min
               Mars dec S 23 deg 08.6 min  Mars GHA 024 deg 00.5 min
    
    Moon horizontal parallax from 03:00  is 59.3 minutes, not changing over
    3-hour period.
    
    ===================================
    
    FURTHER CORRECTIONS, INDEED.
    
    You may think you have seen enough in the way of corrections, but I am
    aware that there are further additional corrections which I have neglected,
    partly because I am unsure about the details.
    
    One is due to further effects of the Earth's ellipticity on the parallax of
    the Moon, which can cause the apparent position of the Moon to shift in
    both altitude and azimuth, by amounts up to 0.2 min., depending on the
    Moon's azimuth.
    
    Another is caused by differences in the refraction at the upper and lower
    limbs, for both Sun and Moon, which makes both bodies appear to be slightly
    elliptical. When accounting for semidiameters in correcting a lunar
    distance, it is assumed that both bodies are round. There is a small
    correction to be made to put this right, which has effect mainly at low
    altitudes.
    
    There may well be even more corrections which could be made. I would be
    grateful to any listmember who can supply information about these matters.
    
    ===================================
    
    JUST THINK....
    
    By now you should have a good idea about the complexity of the
    lunar-distance procedures. Right through the lunar-distance era, navigators
    have had to go though the same calculations described here, with the
    exception of the predicted lunar distances: they could get those from an
    almanac, and you can't.
    
    If you go back in time to the early and middle 1800s, on any day there
    would  be literally thousands of ships plying the oceans (far, far more
    than today). Mostly, their navigators would be struggling with lunar
    distances, with a quill pen or a slate, under oil-lamps or candles at
    night, with only log tables to ease their arithmetic. If they made an
    arithmetical slip, there was little to check it against. It could,
    literally, put them on the rocks: and often did.
    
    They were, generally, men of little education, who had been trained to go
    through the motions of these calculations by rote. My guess is that only a
    few really understood what they were doing.
    
    They were in a dangerous trade, and most ships ended their days from the
    hazards of the sea rather than from old age. To some extent, that must have
    been true of the mariners as well.
    
    As I have wrestled with the lunar task that they had to face, I have
    appreciated, more and more, their achievements. I hope you agree.
    
    I have also come to appreciate the work of Maskelyne, who set up the whole
    procedure with Mayer's help. They stood on the shoulders of their
    predecessors: mainly French, but not forgetting Newton.
    
    =================================
    
    Part 1 explained what a Lunar Distance is, how it is measured and "cleared"
    to obtain a Corrected Lunar Distance.
    
    
    Part 3 will come at a future date. It will discuss the possibility of
    calculating the altitudes required with a lunar distance, instead of
    measuring them: also the question of observing Moon altitudes in place of
    lunar distances. It will show how to obtain local apparent time and use it
    with GMT to obtain longitude, as early navigators had to do.
    
    George Huxtable.
    
    ------------------------------
    
    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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