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    About Lunars, part 3
    From: George Huxtable
    Date: 2002 Feb 13, 20:46 +0000

    3.1  CHANGE OF PLAN.
    
    In part 2, there was a prediction of what part 3 would contain. However, I
    have recently had some useful feedback from listmembers, (especially Steven
    Wepster) which shows that rather more remains to be said about measuring
    and correcting the lunar distance. So part 3 will fill in some of those
    gaps.
    
    Part 4, which should appear in the next few weeks, 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.
    
    
    
    3.2  UNDUE EMPHASIS ON ACCURACY IN MOON ALTITUDES.
    
    In part 1, under "How is a lunar distance measured?", I stated "The
    corrections are such that the Sun altitude is not needed to great accuracy,
    but the Moon altitude should be measured with precision." This overstated
    things.
    
    An error of a degree in Moon altitude will, at the most, give rise to an
    error of 1 minute in the parallax correction of the lunar distance,
    therefore the accuracy required in Moon altitude is no better than to 20
    minutes or so. This is useful in night measurements of Moon-star or
    Moon-planet lunars, when the horizon has to picked out from the ripply
    reflection of moonlight below the Moon.
    
    For the refraction of Sun, stars, or planets, when altitudes are as low as
    10 deg., altitude measurements to 20 min. of arc are needed, but from 20
    deg up, 1 degree will suffice. Again, for stars or planets, this is helpful
    when the night-horizon is hard to make out.
    
    The conclusion is that the requirements for accuracy of the altitude
    measurements are usually rather easy to meet. This leaves the observer to
    concentrate all his precision on measuring the lunar distance.
    
    3.3  UPSIDE-DOWN SEXTANT?
    
    In part 1, under "How is a lunar distance measured?", I said-
    
    "The horizon plays no part in the lunar distance observation itself, and
    the view, straight through the sextant, is instead used to observe the
    fainter of the bodies observed (usually the Moon). The brighter one
    (usually the Sun) is viewed in the index mirror. To see them both, the
    sextant may have to be tilted at what seems like a most unnatural angle,
    pointing up in the air, and over on its side, even upside-down."
    
    I should have added here that if you find the "sextant upside-down" posture
    particularly awkward (as many do), it will do no great harm to swap the two
    images over, in the two light-paths, and use the sextant in a more natural
    attitude. This presumes that your sextant offers a suitable choice of
    shades.
    
    3.4 REFRACTION CORRECTIONS. (see parts 1 and 2)
    
    In clearing a lunar distance, it's necessary to separate the refraction
    corrections from those for dip and for semidiameter. In many tables these
    will have been combined, sometimes with Moon parallax also, to minimise the
    arithmetic for a normal sextant altitude sight. Such a combined refraction
    table is not useful for lunars. You can find the refraction correction, on
    its own, for any body, from the "mean refraction" table in Norie's, or in
    the star correction table in the pull-out card in the Nautical Almanac.
    
    3.5 INVERSE LINEAR INTERPOLATION FOR TIME (see part 2).
    
    This unwieldy phrase describes the process of finding the exact GMT
    corresponding to a lunar distance, given predicted values of LD at two
    times. Although I stated that those times, T1 and T2, could be chosen at
    intervals 1, 2, or 3 hours apart, the example quoted presumed an interval
    of 3 hours. This is the maximum interval that's compatible with the
    assumption of a linear change, and is the interval on which the old
    lunar-distance tables were based. That 3-hour interval was chosen to
    minimise the size of the tables and to minimize the amount of calculation
    in their making (in the days when that mattered).
    
    If you are calculating lunar distances for yourself, it will do no harm to
    choose a shorter interval than 3 hours, if you can be sure that the GMT
    will end up within that interval.
    
    3.6  ERROR IN EXAMPLE
    
    Please note that in part 2, an error occurred in the following passage, as
    uncovered by Greg Gilbert-
    
       "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."
    
    Where this reads 105deg 50.5 min, it should have been 106deg 50.5 min.
    Sorry about that. It was a transcription error, which doesn't affect the
    following parts of the calculation.
    
    3.7  ABOUT SOME PROBLEMS IN CLEARING THE LUNAR DISTANCE
    
    In "About Lunars. part 1." it was shown how to obtain a corrected Lunar
    Distance, D, using Young's method.
    
    It included the following comment, about Young's method-
    
    "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."
    
    ================ (end of quote)
    
    Bill Noyce responded by saying-
    
    >And I would like to take him up on his offer to spell out Borda's
    >method for clearing the distance longhand.  Young's formula seems
    >to require converting out of logs to do additions.
    
    ==================
    
    For modern navigators, the availability of calculators and on-board
    computers has made lunars more accessible to the rest of us, and that is
    the approach I would recommend. But readers are entitled to choose a
    technique using tables and logs, the traditional method that was employed
    right through the era of lunars.
    
    There are several problems that crop up when trying to use Young's method
    using tables, rather than a calculator. Borda offers an alternative, but
    still involves popping in and out of logs, and more than once!
    
    However, I promised Borda's method, on request, and Bill Noyce has indeed
    requested it, so here goes-
    
    3.8  BORDA'S METHOD
    
    This is more-or-less as quoted by Cotter. The quantities m, M, s, S, d, are
    given, as follows.
    
    d   (observed lunar distance between centres)
    m   (observed moon-centre altitude above true horizontal)
    M   (true Moon-centre altitude above true horizontal: it has been corrected
    for parallax and refraction)
    s   (obs Sun/body-centre altitude above true horizontal)
    S   (true Sun/body-centre altitude above true horizontal: it has been
    corrected for parallax and refraction)
    
    The result D, will be the true lunar distance, which has been corrected for
    parallax and refraction.
    
    Firstly, an angle A has to be calculated. A is just an angle that's used as
    an intermediate step in the calculation. Whether it has any physical
    reality, I rather doubt. Obtain A as follows-
    
    First work out and write down  (m+s+d)/2 and (m+s-d)/2
    
    log cos A = {log cos((m+s+d)/2) + log cos((m+s-d)/2) +log cos M + log cos S
    + log sec m + log sec s}/2
    
    This involves working out all those 6 log cos and log sec terms, adding
    them up, and dividing by two.
    
    Having calculated a value for the right-hand side of this equation, search
    for that value in the log cos table and find out what angle it corresponds
    to. This is angle A. You are now "out of logs" and back in the world of
    ordinary numbers, for a time.
    
    Now work out and write down A + (M+S)/2, and A ~ (M+S)/2.
    
    where "~" means "subtract the smaller from the larger"
    
    then go back into logs again, obtaining D from
    
    log sin (D/2) = {log sin (A + (M+S)/2) + log sin (A ~ (M+S)/2)}/2.,
    
    So having calculated a value for the right-hand side of the equation,
    search for that value in the log sin table and find out what angle it
    corresponds to. Double the result. That gives you D, the corrected lunar
    distance, all corrections made.
    
    That second term of that last equation differs from what is given in
    Cotter, who puts a minus sign, which I have changed to "~". Following his
    notation would lead you into trying to obtain the log of a negative
    quantity. I think he has got that wrong.
    
    The way I have rewritten that expression now conforms with the following
    explanation in words of how to tackle the problem, as follows (from Cotter,
    but somewhat modified)-
    
    1. Find M, S, m, s, and d, as above.
    
    D is the true lunar distance (in which the effects of parallax and
    refraction have been allowed for) that we wish to find.
    
    2. Place under one another the apparent distance d and the apparent
    altitudes m and s: and take half their sum, L. From the half-sum L,
    subtract the apparent distance d. Under this place the true altudes M and
    S.
    
    3. Take from tables log cosines of L, L-d, M, and S, log secant m, and log
    secant s. Add these six quantities and divide by 2. The result is the log
    cosine of A. So look up this quantity in the log cosine table, and find
    what angle corresponds to it. This is A.
    
    4. Take half the sum of the true altitudes M and S. Call this B. Find the
    sum of, and the difference between, A and B. Add the log sines of the sum
    and the difference. Divide by 2. The result is the log sine of half the
    true lunar difference, that is D/2. So look up that result in the log sine
    table, find the angle that corresponds to it, and double it to obtain the
    corrected lunar distance D.
    
    (Cotter also missed out an important word in his paragraph 4, which I have
    reinstated.)
    
    3.9 BORDA'S METHOD: AN EXAMPLE
    
    Let's try Borda's method for real using the numbers below, for Steven
    Wepster's Sun-lunar in the Atlantic on 2001 April 02, just as were used in
    part 2 with Young's method.
    
    I will stick to the degrees-and-minutes notation rather than decimal
    degrees, as that's what is needed for looking up the tables.
    
    d=107 deg 22.9 min    (observed lunar distance between centres)
    m= 49 deg 52.5 min    (observed moon-centre altitude above true horizontal)
    M= 50 deg 29.9 min    (true Moon-centre altitude above true horizontal)
    s= 21 deg 10.4 min    (obs Sun/body-centre altitude above true horizontal)
    S= 21 deg 08.1 min    (true Sun/body-centre altitude above true horizontal)
    
    
    Following the written-out instructions in 3.8, we have now completed step 1.
    
    now for steps 2 and 3
    
    d           107deg  22.9
    m           049deg  52.5    log sec = 10.19080
    s           021deg  10.4    log sec = 10.03036
    sum (d+m+s) 178deg  25.8
    half-sum L  089deg  12.9    log cos =  8.13673
    L-d        -018deg  10.0    log cos =  9.97779 (same as log cos +18deg 10.0)
    M           050deg  29.9    log cos =  9.80353
    S           021deg  08.1    log cos =  9.96976
                                sum     = 58.10897
                                Halve   = 29.05448
                             cast off 20=  9.05448
           from log cos table,  Angle A = 083deg 29.4min
                                =====================
    
    next, step 4.
    
    M  (copied)       050deg 29.9
    S  (copied)       021deg 08.1
    sum               071deg 38.0
    half-sum, =B      035deg 49.0
    A (copied)        083deg 29.4
    sum, A and B.     119deg 18.4      log sin =  9.94052
    diff., A and B.   047deg 40.4      log sin =  9.86884
                                       sum     = 19.80936
                                       halved  =  9.90468
                 from log sin table, Angle D/2 = 053deg 24.6
                                  so Angle D   = 106deg 49.2
                                     =======================
    
    This result, of 106deg 49.2min, for the corrected lunar distance, can be
    compared with the value obtained (in part 2) for D using Young's method
    with a calculator, which was 106deg 49.3. So rather good agreement: nothing
    to complain about there!
    
    However, few navigators would find clearing the lunar distance by Borda's
    method to be a lot of fun, I admit.
    
    3.10  AN EASIER APPROACH TO CLEARING THE LUNAR DISTANCE.
    
    The two methods considered so far, Young's and Borda's, are "mathematically
    exact". Both these methods end up by multiplying the observed lunar
    distance d by a calculated factor, which is always very near 1. The
    observed distance d  can be of the order of 100 deg or so. To obtain the
    answer D to within 0.1 min, that multiplying factor of about 1 has to be
    known to 1 part in 100,000 or thereabouts. In other words the error in that
    multiplying factor should be within .00001. That is why 5-figure logs are
    required, and every step in the calculation made with scrupulous attention
    to accuracy.
    
    But what if we can correct d, to obtain D, by ADDING a correction to it,
    rather than using a multiplier? The correction is never more than about 60
    minutes, so to make that correction to provide D to 0.1 minutes, it's only
    necessary to evaluate that added correction to 0.1 in 60 minutes. An error
    of 1 part in 1,000 of that added amount now becomes acceptable. That is an
    accuracy that might even be approached by the careful wielder of a
    slide-rule that shows trig functions.
    
    Throughout the 19th century, much effort went into devising a solution to
    clearing the lunar distance which involves such an added correction, rather
    than a multiplying factor. The trouble has been that such solutions have
    involved an approximation to the geometry, not an exact solution. They rely
    on the fact that both angles are small, the parallax and the refraction. In
    practice, those angles always are acceptably small. If observations are
    made only when altitudes exceed 10 degrees, refractions are always less
    than 5.5 min. Parallax of the Moon never much exceeds 60 minutes. If we
    make an additional rule that lunar distances are measured only when they
    exceed 10 degrees the requirements are met for good accuracy of the
    "approximate solution".
    
    3.11 LETCHER'S METHOD.
    
    In "Self-contained celestial navigation using H.O.208" (1977), John S
    Letcher describes such an approximate method for clearing the lunar
    distance. The way he proposes for its use would be suitable only for lunar
    distances up to 90 degrees, whereas any lunar observer would certainly
    require its application to angles up to 120 degrees. I will therefore give
    a modified calculation method that will cover the full range of useful
    sextant angles.
    
    I am not sure where this method comes from, as Letcher does not quote any
    reference, nor does he show how it is derived from first-principles. It
    might be his own invention, perhaps. To me, it seems very clever. I will
    christen it "Letcher's method" until we find out more about its provenance.
    If anyone recognises it from another publication, I would be interested to
    learn.
    
    Let's investigate Letcher's Method.
    
    We will assume that all sextant observations have been corrected for index
    error.
    
    As before, it starts with the observed lunar distance d, after correction
    for semidiameters. This is to be corrected by adding, separately,-
    
    a) The correction P, for the combined parallaxes of the Moon and the Sun,
    which may be a positive or negative amount, never greater than about 60
    min.
    
    b) The correction R, for the refractions of Moon and Sun, always positive,
    never more than 11 min.
    
    Both P and R need to be calculated to within 0.1 min.
    
    These two corrections require knowledge of-
    
    d, the observed lunar distance between centres (i.e. corrected for
    semidiameters).
    m, the observed altitude of the Moon's centre above the true horizon (i.e.
    corrected for dip and semidiameter)
    s, the observed altitude of the centre of the Sun (or other body), above
    the true horizon (i.e. corrected for dip and, if necessary, semidiameter).
    HP, the Moon's horizontal parallax, in minutes of arc, at the hour of
    observation.
    The HP of the Sun (or other body) is ignored, thus losing a bit of precision.
    
    It is no longer necessary (as it was in the other methods) to make parallax
    and refraction corrections to m and s individually, as M and S are no
    longer needed. These corrections are taken into account automatically, as
    part of the clearance procedure. That is partly why the method is so much
    simpler.
    
    Let's go through it.
    
    First, obtain B = (cos d sin m - sin s) / sin d  (B is just an intermediate
    angle, used in the computation).
    
    Then the parallax correction in minutes is-
    
    P = HP*B + (HP)^2*[(cos m)^2 - B^2)]/ (6900*tan d)
    
    P may be a positive or negative number, in minutes, to be added or
    subtracted from d.
    
    The second term of P is usually a very small quantity, but still needs to
    be evaluated.
    
    Now for the refraction term R. This, very cleverly, includes its own
    built-in calculation of the way refraction changes with altitude.
    
    R = .95* (sin s/sin m + sin m/sin s - 2*cos d) / sin d ,in minutes of arc.
    
    So the end result is D = d + P + R , giving the Corrected Lunar Distance.
    
    3.7. LETCHER: WORKED EXAMPLE
    
    Let's compare this method with the others, using Steven Wepster's Atlantic
    observations of 2001 Apr 02 once again.
    
    The inputs we need are, as before-
    
    d = 107.382 deg    (observed lunar distance between centres)
    m = 049.875 deg   (observed moon-centre altitude above true horizontal)
    s = 021.173 deg   (obs Sun or body-centre altitude above true horizontal)
    HP = 59.4 min.
    
    By pocket calculator we get-
    
    B= -0.6178
    
    so P = -36.7 - 0.1 = -36.8 min,   and R = 3.2 min
    
    As d (in minutes) is 107deg 22.9, then for the Corrected Lunar Distance,
    
    D= d + P + R, or 107deg 22.9 - 36.8 + 3.2
    
    Therefore D = 106deg 49.3 by Letcher's method.
              ===============
    
    This should be compared with 106 deg 49.2, by Borda's method using tables,
    and 106deg 49.3 using Young's method with a calculator. Of these, I would
    choose the last, 106deg 49.3, as being the more precise. Really, what
    better agreement could anyone wish for, between three different methods?
    
    However, don't expect to always obtain that same precision using Letcher's
    method. He concedes that by ignoring some of the contributions such as the
    Sun's parallax and the ellipsoidal figure-of-the-Earth, errors may combine
    up to a total of 0.3 minutes. Even so, when compared with the likely errors
    inherent in measuring the lunar distance, due to the motion of a small
    craft, a possible error of 0.3 minutes in the clearing of it may be very
    acceptable. My vote would go to Letcher. But it's your choice...
    
    My thanks to Bill Murdoch for alerting me to this useful Letcher
    publication. If you can find a secondhand copy, I think it would be a
    worthwhile buy. It was written in the days when scientific calculators
    existed but were uncommon; hence the emphasis on using tables.
    
    =================
    
    Part 4, which should appear in the next few weeks, 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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